Introduction
Geometric evolution equations play a central role in geometric analysis and the calculus of variations. Among them, the harmonic map heat flow has served as a fundamental paradigm for understanding the interplay between geometry and regularity theory. Given two complete smooth Riemannian manifolds ( M , g ) (M,g) ( M , g ) ( N , h ) ↪ R ℓ (N,h) \hookrightarrow\mathbb{R}^{\ell} ( N , h ) ↪ R ℓ u : M × ( 0 , T ) → N u:M \times(0,T) \to N u : M × ( 0 , T ) → N T > 0 T>0 T > 0
∂ t u − Δ u ⊥ T u N , (1) \partial_t u-\Delta u \perp T_uN, \tag*{(1)} ∂ t u − Δ u ⊥ T u N , (1) and can be interpreted as the (negative) gradient flow of the Dirichlet energy functional
E ( u ) = 1 2 ∫ M ∣ ∇ u ∣ 2 d v g , u : M → N ↪ R ℓ . E(u)=\frac{1}{2}\int_M |\nabla u|^2\,d v_g,\quad u:M\to N\hookrightarrow\mathbb{R}^{\ell}. E ( u ) = 2 1 ∫ M ∣∇ u ∣ 2 d v g , u : M → N ↪ R ℓ . In their landmark paper, Eells and Sampson [13 ] established the existence of a unique global smooth solution of (1.1) whenever the target manifold ( N , h ) (N,h) ( N , h ) [27 ] , Chen-Struwe [7 ] , and Chen-Lin [5 ] established the global existence of partially smooth harmonic heat flows into any compact target manifold N N N ε \varepsilon ε [4 ] and Feldman [14 ] for sphere targets, and later extended by Moser [22 ] for arbitrary compact target manifolds.
In recent years, nonlocal geometric variational problems driven by fractional operators have attracted considerable interest, including fractional harmonic maps, nonlocal minimal surfaces. These developments naturally motivate the study of geometric evolution equations governed by fractional operators. For 0 < s < 1 0<s<1 0 < s < 1
∂ t u + ( − Δ ) s u ⊥ T u N , (2) \partial_tu+(-\Delta)^su\perp T_uN, \tag*{(2)} ∂ t u + ( − Δ ) s u ⊥ T u N , (2) which corresponds to the negative gradient flow of the fractional Dirichlet energy. While this formulation is natural from both physical and geometric prospectives, there seems lack of analytic tools to establish its existence and regularity, see Schikorra-Sire-Wang [30 ] , Wettstein [31 , 32 ] , Sire-Wei-Zheng [26 ] , and Struwe [29 ] for some related results in this direction.
In this paper, we will investigate an alternative, structurally advantageous formulation of heat flow of fractional harmonic maps, given by
( ∂ t − Δ ) s u ⊥ T u N (3) (\partial_t-\Delta)^su\perp T_uN \tag*{(3)} ( ∂ t − Δ ) s u ⊥ T u N (3) where ( ∂ t − Δ ) s (\partial_t-\Delta)^s ( ∂ t − Δ ) s t → ∞ t\to\infty t → ∞ R m × R \mathbb{R}^m\times\mathbb{R} R m × R A 2 A_2 A 2
Here, we focus on the fractional harmonic map flow (1.3) targeting the sphere N = S ℓ − 1 N=\mathbb{S}^{\ell-1} N = S ℓ − 1 N N N
To make this precise, we recall the definition of the fractional heat operator ( ∂ t − Δ ) s u (\partial_t-\Delta)^s u ( ∂ t − Δ ) s u R m × R \mathbb{R}^m \times\mathbb{R} R m × R [24 ] and Stinga–Torrea [25 ] . For 0 < s < 1 0<s<1 0 < s < 1 u ∈ D ( H s ) u\in D(H^s) u ∈ D ( H s )
( ∂ t − Δ ) s u ( x , t ) = ∬ R m × R + ( u ( x , t ) − u ( x − z , t − τ ) ) K s ( z , τ ) d y d τ , (4) (\partial_t-\Delta)^s u(x,t)=\iint_{\mathbb{R}^m\times\mathbb{R}_+}(u(x,t)-u(x-z,t-\tau))K_s(z,\tau)\,dyd\tau, \tag*{(4)} ( ∂ t − Δ ) s u ( x , t ) = ∬ R m × R + ( u ( x , t ) − u ( x − z , t − τ )) K s ( z , τ ) d y d τ , (4) where the kernel K s K_s K s
K s ( z , τ ) = { e − ∣ z ∣ 2 4 τ ( 4 π ) m 2 ∣ Γ ( − s ) ∣ τ m 2 + 1 + s , ∀ z ∈ R m , τ > 0 , 0 , ∀ z ∈ R m , τ ≤ 0. K_s(z,\tau)=
\begin{cases}
\dfrac{e^{-\frac{|z|^2}{4\tau}}}{(4\pi)^{\frac{m}{2}}|\Gamma(-s)|\tau^{\frac{m}{2}+1+s}}, & \forall z\in\mathbb{R}^m,\ \tau>0,\\
0, & \forall z\in\mathbb{R}^m,\ \tau\le0.
\end{cases} K s ( z , τ ) = ⎩ ⎨ ⎧ ( 4 π ) 2 m ∣Γ ( − s ) ∣ τ 2 m + 1 + s e − 4 τ ∣ z ∣ 2 , 0 , ∀ z ∈ R m , τ > 0 , ∀ z ∈ R m , τ ≤ 0. and Γ \Gamma Γ D ( H s ) D(H^s) D ( H s )
D ( H s ) = { v ∈ S ′ ( R m + 1 ) : v ^ ∈ L l o c 1 ( R m + 1 ) , ( i τ + ∣ ξ ∣ 2 ) s 2 v ^ ( ξ , τ ) ∈ L 2 ( R m + 1 ) } . (5) D(H^s)=\left\{v\in\mathcal{S}'(\mathbb{R}^{m+1}):\widehat{v}\in L^1_{\mathrm{loc}}(\mathbb{R}^{m+1}),\ (i\tau+|\xi|^2)^{\frac{s}{2}}\widehat{v}(\xi,\tau)\in L^2(\mathbb{R}^{m+1})\right\}. \tag*{(5)} D ( H s ) = { v ∈ S ′ ( R m + 1 ) : v ∈ L loc 1 ( R m + 1 ) , ( i τ + ∣ ξ ∣ 2 ) 2 s v ( ξ , τ ) ∈ L 2 ( R m + 1 ) } . (5) where S ′ ( R m + 1 ) \mathcal{S}'(\mathbb{R}^{m+1}) S ′ ( R m + 1 ) v ^ \widehat{v} v v v v D ( H s ) D(H^s) D ( H s )
H ˙ x , t s , s / 2 ( R m + 1 ) = { v ∈ L l o c 1 ( R m + 1 ) : ∥ v ∥ H ˙ x , t s , s / 2 2 ≡ ∬ R m + 1 ( ∣ τ ∣ s + ∣ ξ ∣ 2 s ) ∣ v ^ ∣ 2 ( ξ , τ ) < + ∞ } . \dot{H}^{s,s/2}_{x,t}(\mathbb{R}^{m+1})=\left\{v\in L^1_{\mathrm{loc}}(\mathbb{R}^{m+1}):\|v\|^2_{\dot{H}^{s,s/2}_{x,t}}\equiv\iint_{\mathbb{R}^{m+1}}(|\tau|^s+|\xi|^{2s})|\widehat{v}|^2(\xi,\tau)<+\infty\right\}. H ˙ x , t s , s /2 ( R m + 1 ) = { v ∈ L loc 1 ( R m + 1 ) : ∥ v ∥ H ˙ x , t s , s /2 2 ≡ ∬ R m + 1 ( ∣ τ ∣ s + ∣ ξ ∣ 2 s ) ∣ v ∣ 2 ( ξ , τ ) < + ∞ } . We define the the corresponding space for maps into the sphere S ℓ − 1 \mathbb{S}^{\ell-1} S ℓ − 1
H ˙ x , t s , s / 2 ( R m + 1 , S ℓ − 1 ) = { u = ( u 1 , ⋯ , u ℓ ) : R m + 1 → S ℓ − 1 ∣ u i ∈ H ˙ x , t s , s / 2 ( R m + 1 ) , i = 1 , ⋯ , ℓ } . \dot{H}^{s,s/2}_{x,t}(\mathbb{R}^{m+1},\mathbb{S}^{\ell-1})=\left\{u=(u^1,\cdots,u^\ell):\mathbb{R}^{m+1}\to\mathbb{S}^{\ell-1}\mid u^i\in\dot{H}^{s,s/2}_{x,t}(\mathbb{R}^{m+1}),\ i=1,\cdots,\ell\right\}. H ˙ x , t s , s /2 ( R m + 1 , S ℓ − 1 ) = { u = ( u 1 , ⋯ , u ℓ ) : R m + 1 → S ℓ − 1 ∣ u i ∈ H ˙ x , t s , s /2 ( R m + 1 ) , i = 1 , ⋯ , ℓ } . We now give the notation of weak solutions to the heat flow of s s s M = R m M=\mathbb{R}^m M = R m S ℓ − 1 ↪ R ℓ \mathbb{S}^{\ell-1}\hookrightarrow\mathbb{R}^{\ell} S ℓ − 1 ↪ R ℓ
Definition 1.1 For 0 < s < 1 0<s<1 0 < s < 1 u 0 ∈ H ˙ s ( R m , S ℓ − 1 ) u_0\in\dot{H}^s(\mathbb{R}^m,\mathbb{S}^{\ell-1}) u 0 ∈ H ˙ s ( R m , S ℓ − 1 ) u ∈ H ˙ x , t s , s / 2 ( R m × R , S ℓ − 1 ) u\in\dot{H}^{s,s/2}_{x,t}(\mathbb{R}^m\times\mathbb{R},\mathbb{S}^{\ell-1}) u ∈ H ˙ x , t s , s /2 ( R m × R , S ℓ − 1 ) s s s u 0 u_0 u 0
{ ( ∂ t − Δ ) s u = λ ( x , t ) u in R m × ( 0 , ∞ ) , u ( x , t ) = u 0 ( x ) in R m × ( − ∞ , 0 ] , (6) \begin{cases}
(\partial_t-\Delta)^s u=\lambda(x,t)u & \text{in }\mathbb{R}^m\times(0,\infty),\\
u(x,t)=u_0(x) & \text{in }\mathbb{R}^m\times(-\infty,0], \tag*{(6)}
\end{cases} { ( ∂ t − Δ ) s u = λ ( x , t ) u u ( x , t ) = u 0 ( x ) in R m × ( 0 , ∞ ) , in R m × ( − ∞ , 0 ] , (6) where λ \lambda λ
λ ( x , t ) = 1 2 ∬ R m × R + ∣ u ( x , t ) − u ( x − z , t − τ ) ∣ 2 K s ( z , τ ) d y d τ . \lambda(x,t)=\frac{1}{2}\iint_{\mathbb{R}^m\times\mathbb{R}_+}|u(x,t)-u(x-z,t-\tau)|^2K_s(z,\tau)\,dyd\tau. λ ( x , t ) = 2 1 ∬ R m × R + ∣ u ( x , t ) − u ( x − z , t − τ ) ∣ 2 K s ( z , τ ) d y d τ . Because λ ( x , t ) u ⊥ T u S ℓ − 1 \lambda(x,t)u\perp T_u\mathbb{S}^{\ell-1} λ ( x , t ) u ⊥ T u S ℓ − 1
( ∂ t − Δ ) s u ⊥ T u S ℓ − 1 . (7) (\partial_t-\Delta)^s u\perp T_u\mathbb{S}^{\ell-1}. \tag*{(7)} ( ∂ t − Δ ) s u ⊥ T u S ℓ − 1 . (7) Following the parabolic Caffarelli–Silvestre extension framework [3 , 24 , 25 ] , we reformulate (6) into a degenerate parabolic system in R + m + 1 × R \mathbb{R}^{m+1}_{+} \times\mathbb{R} R + m + 1 × R y a y^{a} y a a = 1 − 2 s ∈ ( − 1 , 1 ) a = 1 - 2s \in(-1,1) a = 1 − 2 s ∈ ( − 1 , 1 ) R m = ∂ R + m + 1 \mathbb{R}^{m} = \partial\mathbb{R}^{m+1}_{+} R m = ∂ R + m + 1 ( X , t ) = ( x , y , t ) ∈ R m × ( 0 , ∞ ) × R (X,t) = (x,y,t) \in\mathbb{R}^{m} \times(0,\infty) \times\mathbb{R} ( X , t ) = ( x , y , t ) ∈ R m × ( 0 , ∞ ) × R
{ y a ∂ t U = div X ( y a D U ) in R + m + 1 × R , U ( x , 0 , t ) = u ( x , t ) in R m × R . (8) \begin{cases}
y^{a}\partial_{t}U = \operatorname{div}_{X}(y^{a}DU) & \text{in } \mathbb{R}^{m+1}_{+} \times\mathbb{R},\\
U(x,0,t) = u(x,t) & \text{in } \mathbb{R}^{m} \times\mathbb{R}. \tag*{(8)}
\end{cases} { y a ∂ t U = div X ( y a D U ) U ( x , 0 , t ) = u ( x , t ) in R + m + 1 × R , in R m × R . (8) For any u ∈ D ( H s ) u \in\mathcal{D}(H^{s}) u ∈ D ( H s ) U U U (8) satisfying the trace condition
− 1 c s y a ∂ U ∂ y ∣ y = 0 = ( ∂ t − Δ ) s u , with c s = Γ ( 1 − s ) 2 2 s − 1 Γ ( s ) . (9) -\frac{1}{c_{s}}y^{a}\frac{\partial U}{\partial y}\bigg|_{y=0} = (\partial_{t} - \Delta)^{s}u,\quad\text{with } c_{s} = \frac{\Gamma(1-s)}{2^{2s-1}\Gamma(s)}. \tag*{(9)} − c s 1 y a ∂ y ∂ U y = 0 = ( ∂ t − Δ ) s u , with c s = 2 2 s − 1 Γ ( s ) Γ ( 1 − s ) . (9) Throughout this paper, we will denote
y a ∂ U ∂ y ∣ y = 0 = lim y → 0 + y a ∂ U ∂ y , y^{a}\frac{\partial U}{\partial y}\bigg|_{y=0} = \lim_{y\to0^{+}} y^{a}\frac{\partial U}{\partial y}, y a ∂ y ∂ U y = 0 = y → 0 + lim y a ∂ y ∂ U , where the limit is understood in the sense of distributions on R m × R \mathbb{R}^{m} \times\mathbb{R} R m × R (6) translates to the local extended system:
{ y a ∂ t U ( X , t ) = div X ( y a D U ) ( X , t ) , in R + m + 1 × R , U ( x , 0 , t ) = u 0 ( x ) , in R m × ( − ∞ , 0 ] , y a ∂ U ( X , t ) ∂ y ∣ y = 0 ⊥ T U ( x , 0 , t ) S ℓ − 1 , on R m × ( 0 , + ∞ ) . (10) \begin{cases}
y^{a}\partial_{t}U(X,t) = \operatorname{div}_{X}(y^{a}DU)(X,t), & \text{in } \mathbb{R}^{m+1}_{+} \times\mathbb{R},\\
U(x,0,t) = u_{0}(x), & \text{in } \mathbb{R}^{m} \times(-\infty,0],\\
y^{a}\dfrac{\partial U(X,t)}{\partial y}\bigg|_{y=0} \perp T_{U(x,0,t)}S^{\ell-1}, & \text{on } \mathbb{R}^{m} \times(0,+\infty). \tag*{(10)}
\end{cases} ⎩ ⎨ ⎧ y a ∂ t U ( X , t ) = div X ( y a D U ) ( X , t ) , U ( x , 0 , t ) = u 0 ( x ) , y a ∂ y ∂ U ( X , t ) y = 0 ⊥ T U ( x , 0 , t ) S ℓ − 1 , in R + m + 1 × R , in R m × ( − ∞ , 0 ] , on R m × ( 0 , + ∞ ) . (10) By the extension theorem [3 ] there exists a harmonic extension U ^ 0 ∈ H ˙ 1 ( ( R + m + 1 , y a d X ) , R ℓ ) \widehat{U}_{0} \in\dot{H}^{1}((\mathbb{R}^{m+1}_{+},y^{a}dX),\mathbb{R}^{\ell}) U 0 ∈ H ˙ 1 (( R + m + 1 , y a d X ) , R ℓ )
div X ( y a D U ^ 0 ) = 0 in R + m + 1 ; U ^ 0 ( x , 0 ) = u 0 ( x ) on R m . \operatorname{div}_{X}(y^{a}D\widehat{U}_{0}) = 0 \quad\text{in } \mathbb{R}^{m+1}_{+};\quad\widehat{U}_{0}(x,0) = u_{0}(x) \quad\text{on } \mathbb{R}^{m}. div X ( y a D U 0 ) = 0 in R + m + 1 ; U 0 ( x , 0 ) = u 0 ( x ) on R m . Using U ^ 0 \widehat{U}_{0} U 0 (10) (and hence (6) ) becomes the initial and boundary value problem as
{ y a ∂ t U ( X , t ) = div X ( y a D U ) ( X , t ) , ( X , t ) ∈ R + m + 1 × ( 0 , ∞ ) , U ( x , y , 0 ) = U ^ 0 ( x , y ) , ( x , y ) ∈ R + m + 1 , ∣ U ∣ ( x , 0 , t ) = 1 ; y a ∂ U ( x , y , t ) ∂ y ∣ y = 0 ⊥ T U ( x , 0 , t ) S ℓ − 1 , ( x , t ) ∈ R m × ( 0 , + ∞ ) . (11) \begin{cases}
y^{a}\partial_{t}U(X,t) = \operatorname{div}_{X}(y^{a}DU)(X,t), & (X,t) \in\mathbb{R}^{m+1}_{+} \times(0,\infty),\\
U(x,y,0) = \widehat{U}_{0}(x,y), & (x,y) \in\mathbb{R}^{m+1}_{+},\\
|U|(x,0,t) = 1;\quad y^{a}\dfrac{\partial U(x,y,t)}{\partial y}\bigg|_{y=0} \perp T_{U(x,0,t)}S^{\ell-1}, & (x,t) \in\mathbb{R}^{m} \times(0,+\infty). \tag*{(11)}
\end{cases} ⎩ ⎨ ⎧ y a ∂ t U ( X , t ) = div X ( y a D U ) ( X , t ) , U ( x , y , 0 ) = U 0 ( x , y ) , ∣ U ∣ ( x , 0 , t ) = 1 ; y a ∂ y ∂ U ( x , y , t ) y = 0 ⊥ T U ( x , 0 , t ) S ℓ − 1 , ( X , t ) ∈ R + m + 1 × ( 0 , ∞ ) , ( x , y ) ∈ R + m + 1 , ( x , t ) ∈ R m × ( 0 , + ∞ ) . (11) Throughout this paper, μ a = y a d X \mu_{a} = y^{a}dX μ a = y a d X μ − a = y − a d X \mu_{-a} = y^{-a}dX μ − a = y − a d X R m + 1 \mathbb{R}^{m+1} R m + 1 [17 ] established that for any u 0 ∈ H ˙ s ( R m , S ℓ − 1 ) u_{0} \in\dot{H}^{s}(\mathbb{R}^{m},S^{\ell-1}) u 0 ∈ H ˙ s ( R m , S ℓ − 1 ) (11) admits a global weak solution
U ∈ L ∞ ( R + ; H 1 ( ( R + m + 1 , μ a ) , R ℓ ) ) ∩ H 1 ( R + ; L 2 ( ( R + m + 1 , μ a ) , R ℓ ) ) . U \in L^{\infty}(\mathbb{R}_{+};H^{1}((\mathbb{R}^{m+1}_{+},\mu_{a}),\mathbb{R}^{\ell})) \cap H^{1}(\mathbb{R}_{+};L^{2}((\mathbb{R}^{m+1}_{+},\mu_{a}),\mathbb{R}^{\ell})). U ∈ L ∞ ( R + ; H 1 (( R + m + 1 , μ a ) , R ℓ )) ∩ H 1 ( R + ; L 2 (( R + m + 1 , μ a ) , R ℓ )) . Moreover, when s = 1 / 2 s = 1/2 s = 1/2 u = U ∣ R m × ( 0 , + ∞ ) u = U|_{\mathbb{R}^{m}\times(0,+\infty)} u = U ∣ R m × ( 0 , + ∞ ) (6) is smooth outside a closed set of locally finite ( m + 1 ) (m+1) ( m + 1 ) [2 ] for a related work. In a forthcoming paper [16 ] , we will extend the main theorems of [17 ] to all 0 < s < 1 0 < s < 1 0 < s < 1 s ≠ 1 / 2 s \ne1/2 s = 1/2
We would like to remark that when s = 1 / 2 s = 1/2 s = 1/2 1 / 2 1/2 1/2 (6) can be interpreted as the free boundary problem of the heat flow of harmonic map U U U R + m + 1 \mathbb{R}^{m+1}_{+} R + m + 1 R ℓ \mathbb{R}^{\ell} R ℓ U ( ∂ R + m + 1 ) ⊂ S ℓ − 1 U(\partial\mathbb{R}_{+}^{m+1}) \subset\mathbb{S}^{\ell-1} U ( ∂ R + m + 1 ) ⊂ S ℓ − 1 (11) . This view of point has played an important role in the study of steady 1 / 2 1/2 1/2 [18 ] , Millot-Pegon [20 ] , Mazowiecka-Rodiac-Schikorra [19 ] , Millot-Pegon-Schikorra [21 ] for relevant results. We would also like to mention the important work by Da Lio-Rivière [12 ] on the regularity for 1 / 2 1/2 1/2
In this paper, we address the partial regularity of stationary weak solutions of (11) , which satisfy a domain-variation stationarity condition analogously to Feldman’s framework [14 ] for the heat flow of harmonic maps.
Given admissible vector fields ξ ( X , t ) ∈ C 0 ∞ ( R ‾ + m + 1 × ( 0 , ∞ ) , R m + 1 ) \xi(X,t) \in C^\infty_0(\overline{\mathbb{R}}_{+}^{m+1} \times(0,\infty),\mathbb{R}^{m+1}) ξ ( X , t ) ∈ C 0 ∞ ( R + m + 1 × ( 0 , ∞ ) , R m + 1 ) ξ m + 1 ≥ 0 \xi^{m+1} \ge0 ξ m + 1 ≥ 0 0 ≤ θ ( X , t ) ∈ C 0 ∞ ( R ‾ + m + 1 × ( 0 , ∞ ) ) 0 \le\theta(X,t) \in C^\infty_0(\overline{\mathbb{R}}_{+}^{m+1} \times(0,\infty)) 0 ≤ θ ( X , t ) ∈ C 0 ∞ ( R + m + 1 × ( 0 , ∞ )) U τ U^\tau U τ τ > 0 \tau> 0 τ > 0
U τ ( X , t ) = U ( X + τ ξ ( X , t ) , t + τ θ ( X , t ) ) , ( X , t ) ∈ R ‾ + m + 1 × [ 0 , ∞ ) . U^\tau(X,t) = U(X+\tau\xi(X,t),t+\tau\theta(X,t)), \quad(X,t) \in\overline{\mathbb{R}}_{+}^{m+1} \times[0,\infty). U τ ( X , t ) = U ( X + τ ξ ( X , t ) , t + τ θ ( X , t )) , ( X , t ) ∈ R + m + 1 × [ 0 , ∞ ) . Definition 1.2 A map U : R ‾ + m + 1 × ( 0 , ∞ ) → R ℓ U:\overline{\mathbb{R}}_{+}^{m+1} \times(0,\infty) \to\mathbb{R}^{\ell} U : R + m + 1 × ( 0 , ∞ ) → R ℓ L ∞ ( R + ; H ˙ 1 ( R + m + 1 ; μ a , R ℓ ) ) ∩ H ˙ 1 ( R + ; L 2 ( R + m + 1 ; μ a , R ℓ ) ) L^\infty(\mathbb{R}_{+};\dot{H}^{1}(\mathbb{R}_{+}^{m+1};\mu_a,\mathbb{R}^{\ell})) \cap\dot{H}^{1}(\mathbb{R}_{+};L^{2}(\mathbb{R}_{+}^{m+1};\mu_a,\mathbb{R}^{\ell})) L ∞ ( R + ; H ˙ 1 ( R + m + 1 ; μ a , R ℓ )) ∩ H ˙ 1 ( R + ; L 2 ( R + m + 1 ; μ a , R ℓ )) s s s S ℓ − 1 \mathbb{S}^{\ell-1} S ℓ − 1 U U U (11) but also satisfies the following variational inequality:
2 ∬ R + m + 1 × R + ∂ t U ⋅ d d τ ∣ τ = 0 U τ μ a d t + d d τ ∣ τ = 0 ∬ R + m + 1 × R + ∣ D U τ ∣ 2 ( X , t ) μ a d t ≤ 0 (12) 2\iint_{\mathbb{R}_{+}^{m+1}\times\mathbb{R}_{+}} \partial_t U \cdot\left.\frac{d}{d\tau}\right|_{\tau=0} U^\tau\,\mu_a\,dt+\left.\frac{d}{d\tau}\right|_{\tau=0}\iint_{\mathbb{R}_{+}^{m+1}\times\mathbb{R}_{+}} |DU^\tau|^2(X,t)\,\mu_a\,dt \le0 \tag*{(12)} 2 ∬ R + m + 1 × R + ∂ t U ⋅ d τ d τ = 0 U τ μ a d t + d τ d τ = 0 ∬ R + m + 1 × R + ∣ D U τ ∣ 2 ( X , t ) μ a d t ≤ 0 (12) for all admissible ξ \xi ξ ξ m + 1 ≥ 0 \xi^{m+1} \ge0 ξ m + 1 ≥ 0 ξ m + 1 ∣ y = 0 = 0 \left.\xi^{m+1}\right|_{y=0}=0 ξ m + 1 y = 0 = 0 0 ≤ θ ∈ C 0 ∞ ( R ‾ + m + 1 × ( 0 , ∞ ) ) 0 \le\theta\in C^\infty_0(\overline{\mathbb{R}}_{+}^{m+1} \times(0,\infty)) 0 ≤ θ ∈ C 0 ∞ ( R + m + 1 × ( 0 , ∞ ))
Stationary solutions satisfy two pivotal structural properties:
Struwe’s type monotonicity inequality for y a y^a y a
Local y a y^a y a
These inequalities serve as the primary source for proving partial regularity. Our main theorem is as follows.
Theorem 1.3 For 0 < s < 1 0 < s < 1 0 < s < 1 U ∈ L ∞ ( R + ; H ˙ 1 ( R + m + 1 ; μ a , R ℓ ) ) ∩ H ˙ 1 ( R + ; L 2 ( R + m + 1 ; μ a , R ℓ ) ) U \in L^\infty(\mathbb{R}_{+};\dot{H}^{1}(\mathbb{R}_{+}^{m+1};\mu^a,\mathbb{R}^{\ell})) \cap\dot{H}^{1}(\mathbb{R}_{+};L^{2}(\mathbb{R}_{+}^{m+1};\mu_a,\mathbb{R}^{\ell})) U ∈ L ∞ ( R + ; H ˙ 1 ( R + m + 1 ; μ a , R ℓ )) ∩ H ˙ 1 ( R + ; L 2 ( R + m + 1 ; μ a , R ℓ )) S ℓ − 1 \mathbb{S}^{\ell-1} S ℓ − 1 Σ ⊂ R m × ( 0 , ∞ ) \Sigma\subset\mathbb{R}^{m} \times(0,\infty) Σ ⊂ R m × ( 0 , ∞ ) H m + 2 − 2 s ( Σ ) = 0 \mathcal{H}^{m+2-2s}(\Sigma)=0 H m + 2 − 2 s ( Σ ) = 0
u = U ∣ ∂ R + m + 1 × ( 0 , ∞ ) ∈ C ∞ ( R m × ( 0 , ∞ ) ∖ Σ , S ℓ − 1 ) . u = U|_{\partial\mathbb{R}_{+}^{m+1}\times(0,\infty)} \in C^\infty(\mathbb{R}^{m}\times(0,\infty)\setminus\Sigma,\mathbb{S}^{\ell-1}). u = U ∣ ∂ R + m + 1 × ( 0 , ∞ ) ∈ C ∞ ( R m × ( 0 , ∞ ) ∖ Σ , S ℓ − 1 ) . Here H m + 2 − 2 s \mathcal{H}^{m+2-2s} H m + 2 − 2 s ( m + 2 − 2 s ) (m+2-2s) ( m + 2 − 2 s )
d ( ( x , t ) , ( y , s ) ) = max { ∣ x − y ∣ , ∣ t − s ∣ } , ∀ ( x , t ) , ( y , s ) ∈ R m × ( 0 , ∞ ) . d((x,t),(y,s)) = \max\left\{|x-y|,\sqrt{|t-s|}\right\}, \quad\forall(x,t),(y,s) \in\mathbb{R}^{m}\times(0,\infty). d (( x , t ) , ( y , s )) = max { ∣ x − y ∣ , ∣ t − s ∣ } , ∀ ( x , t ) , ( y , s ) ∈ R m × ( 0 , ∞ ) . Theorem 1.3 is deduced via an ε 0 \varepsilon_0 ε 0 ∥ ∂ t U ∥ M μ a d t 2 , 4 + ∥ D U ∥ M μ a d t 2 , 2 \|\partial_t U\|_{M^{2,4}_{\mu_a\,dt}}+\|DU\|_{M^{2,2}_{\mu_a\,dt}} ∥ ∂ t U ∥ M μ a d t 2 , 4 + ∥ D U ∥ M μ a d t 2 , 2 (11) , which can be rewritten as (see Lemma 2.2 below)
y a ∂ t U k − div X ( y a D U k ) = 1 2 y a ∂ t ( ∣ U ∣ 2 − 1 ) U k + y a D U i ⋅ Ω k i − 1 2 div X ( y a D ( ∣ U ∣ 2 − 1 ) U k ) + y a D ( ∣ U ∣ 2 − 1 ) ⋅ D U k in R + m + 1 × ( 0 , ∞ ) , (13) y^a\partial_t U^k-\operatorname{div}_X(y^aDU^k)=\frac{1}{2}y^a\partial_t(|U|^2-1)U^k+y^aDU^i\cdot\Omega^{ki}-\frac{1}{2}\operatorname{div}_X\left(y^aD(|U|^2-1)U^k\right)+y^aD(|U|^2-1)\cdot DU^k \quad\text{in } \mathbb{R}_{+}^{m+1}\times(0,\infty), \tag*{(13)} y a ∂ t U k − div X ( y a D U k ) = 2 1 y a ∂ t ( ∣ U ∣ 2 − 1 ) U k + y a D U i ⋅ Ω k i − 2 1 div X ( y a D ( ∣ U ∣ 2 − 1 ) U k ) + y a D ( ∣ U ∣ 2 − 1 ) ⋅ D U k in R + m + 1 × ( 0 , ∞ ) , (13) y a ∂ U k ∂ y ∣ y = 0 = 1 2 y a ∂ ∂ y ( ( ∣ U ∣ 2 − 1 ) U k ) ∣ y = 0 on ∂ R + m + 1 × ( 0 , ∞ ) . (14) \left.y^a\frac{\partial U^k}{\partial y}\right|_{y=0}=\left.\frac{1}{2}y^a\frac{\partial}{\partial y}\left((|U|^2-1)U^k\right)\right|_{y=0}\quad\text{on } \partial\mathbb{R}_{+}^{m+1}\times(0,\infty). \tag*{(14)} y a ∂ y ∂ U k y = 0 = 2 1 y a ∂ y ∂ ( ( ∣ U ∣ 2 − 1 ) U k ) y = 0 on ∂ R + m + 1 × ( 0 , ∞ ) . (14) Here the matrix-valued connection 1-form Ω i j = U i D U j − U j D U i \Omega^{ij}=U^{i}DU^{j}-U^{j}DU^{i} Ω ij = U i D U j − U j D U i
div X ( y a Ω i j ) = y a ( U i ∂ t U j − U j ∂ t U i ) . \operatorname{div}_{X}(y^{a}\Omega^{ij})=y^{a}\left(U^{i}\partial_{t}U^{j}-U^{j}\partial_{t}U^{i}\right). div X ( y a Ω ij ) = y a ( U i ∂ t U j − U j ∂ t U i ) . This identity on divergence implies that y a D U i ⋅ Ω k i y^{a}DU^{i}\cdot\Omega^{ki} y a D U i ⋅ Ω k i R + m + 1 × ( 0 , ∞ ) \mathbb{R}_{+}^{m+1}\times(0,\infty) R + m + 1 × ( 0 , ∞ ) U U U Theorem 3.3 ), forming a key step in establishing Theorem 1.3 .
Organization of the Paper
The paper is organized as follows. In Section 2, we rewrite system (11) to uncover the hidden algebraic structure. In Section 3, we establish the improved energy estimates via Hardy–BMO duality. In Section 4, we derive the y a y^{a} y a y a y^{a} y a C α s C^{\alpha_{s}} C α s C ∞ C^{\infty} C ∞
Structure of Nonlinearities of the Heat Flow of s s s
In this section, we provide the hidden algebraic structure of system (11) governing the heat flow of s s s [19 ] in the context of free boundary problems for p p p
Lemma 2.1 Let U : R + m + 1 × R + → R ℓ U:\mathbb{R}_{+}^{m+1}\times\mathbb{R}_{+}\to\mathbb{R}^{\ell} U : R + m + 1 × R + → R ℓ (11) with boundary trace u = U ∣ ∂ R + m + 1 × R + : ∂ R + m + 1 × R + → S ℓ − 1 u=U|_{\partial\mathbb{R}_{+}^{m+1}\times\mathbb{R}_{+}}:\partial\mathbb{R}_{+}^{m+1}\times\mathbb{R}_{+}\to S^{\ell-1} u = U ∣ ∂ R + m + 1 × R + : ∂ R + m + 1 × R + → S ℓ − 1 Φ ∈ C 0 ∞ ( R + m + 1 × R + ) \Phi\in C_{0}^{\infty}(\mathbb{R}_{+}^{m+1}\times\mathbb{R}_{+}) Φ ∈ C 0 ∞ ( R + m + 1 × R + ) k = 1 , ⋯ , ℓ k=1,\cdots,\ell k = 1 , ⋯ , ℓ
∬ R + m + 1 × R + ( ∂ t U k Φ + D U k ⋅ D Φ ) μ a d t = ∬ R + m + 1 × R + ( ∂ t U i ( U k U i Φ ) + D U i ⋅ D ( U k U i Φ ) ) μ a d t . (15) \iint_{\mathbb{R}_{+}^{m+1}\times\mathbb{R}_{+}}\left(\partial_{t}U^{k}\Phi+DU^{k}\cdot D\Phi\right)\,\mu_{a}dt
=
\iint_{\mathbb{R}_{+}^{m+1}\times\mathbb{R}_{+}}\left(\partial_{t}U^{i}(U^{k}U^{i}\Phi)+DU^{i}\cdot D(U^{k}U^{i}\Phi)\right)\,\mu_{a}dt. \tag*{(15)} ∬ R + m + 1 × R + ( ∂ t U k Φ + D U k ⋅ D Φ ) μ a d t = ∬ R + m + 1 × R + ( ∂ t U i ( U k U i Φ ) + D U i ⋅ D ( U k U i Φ ) ) μ a d t . (15) Proof. Applying the equation y a ∂ t U = div X ( y a D U ) y^{a}\partial_{t}U=\operatorname{div}_{X}(y^{a}DU) y a ∂ t U = div X ( y a D U ) R + m + 1 × R + \mathbb{R}_{+}^{m+1}\times\mathbb{R}_{+} R + m + 1 × R + (15) can be written as
∬ R + m + 1 × R + ( ∂ t U k Φ + D U k ⋅ D Φ ) μ a d t − ∬ R + m + 1 × R + ( ∂ t U i ( U k U i Φ ) + D U i ⋅ D ( U k U i Φ ) ) μ a d t = ∬ R + m + 1 × R + y a ∂ t U i ( δ k i − U k U i ) Φ d X d t − ∬ R + m + 1 × R + y a D U i ⋅ D ( ( δ k i − U k U i ) Φ ) d X d t = − ∬ ∂ R + m + 1 × R + lim y → 0 + ( y a ∂ U i ∂ y ) ( δ k i − U k U i ) Φ d x d t . \begin{aligned}
&\iint_{\mathbb{R}_{+}^{m+1}\times\mathbb{R}_{+}}\left(\partial_{t}U^{k}\Phi+DU^{k}\cdot D\Phi\right)\,\mu_{a}dt-\iint_{\mathbb{R}_{+}^{m+1}\times\mathbb{R}_{+}}\left(\partial_{t}U^{i}(U^{k}U^{i}\Phi)+DU^{i}\cdot D(U^{k}U^{i}\Phi)\right)\,\mu_{a}dt\\
&=\iint_{\mathbb{R}_{+}^{m+1}\times\mathbb{R}_{+}}y^{a}\partial_{t}U^{i}\left(\delta_{ki}-U^{k}U^{i}\right)\Phi\,dXdt-\iint_{\mathbb{R}_{+}^{m+1}\times\mathbb{R}_{+}}y^{a}DU^{i}\cdot D\left((\delta_{ki}-U^{k}U^{i})\Phi\right)\,dXdt\\
&=-\iint_{\partial\mathbb{R}_{+}^{m+1}\times\mathbb{R}_{+}}\lim_{y\to0^{+}}\left(y^{a}\frac{\partial U^{i}}{\partial y}\right)\left(\delta_{ki}-U^{k}U^{i}\right)\Phi\,dxdt.
\end{aligned} ∬ R + m + 1 × R + ( ∂ t U k Φ + D U k ⋅ D Φ ) μ a d t − ∬ R + m + 1 × R + ( ∂ t U i ( U k U i Φ ) + D U i ⋅ D ( U k U i Φ ) ) μ a d t = ∬ R + m + 1 × R + y a ∂ t U i ( δ k i − U k U i ) Φ d X d t − ∬ R + m + 1 × R + y a D U i ⋅ D ( ( δ k i − U k U i ) Φ ) d X d t = − ∬ ∂ R + m + 1 × R + y → 0 + lim ( y a ∂ y ∂ U i ) ( δ k i − U k U i ) Φ d x d t . Since ∣ u ( x , t ) ∣ = ∣ U ( x , 0 , t ) ∣ = 1 |u(x,t)|=|U(x,0,t)|=1 ∣ u ( x , t ) ∣ = ∣ U ( x , 0 , t ) ∣ = 1 ∂ R + m + 1 × R + \partial\mathbb{R}_{+}^{m+1}\times\mathbb{R}_{+} ∂ R + m + 1 × R + i i i
u i ( δ k i − u k u i ) = u k − ∣ u ∣ 2 u k = 0 on ∂ R + m + 1 × R + . u^{i}\left(\delta_{ki}-u^{k}u^{i}\right)=u^{k}-|u|^{2}u^{k}=0\quad\text{on }\partial\mathbb{R}_{+}^{m+1}\times\mathbb{R}_{+}. u i ( δ k i − u k u i ) = u k − ∣ u ∣ 2 u k = 0 on ∂ R + m + 1 × R + . Consequently, the vector field ( δ k 1 − u k 1 u 1 , … , δ k ℓ − u k ℓ u ℓ ) (\delta_{k_1}-u^{k_1}u^1,\ldots,\delta_{k_\ell}-u^{k_\ell}u^\ell) ( δ k 1 − u k 1 u 1 , … , δ k ℓ − u k ℓ u ℓ ) S ℓ − 1 S^{\ell-1} S ℓ − 1 u ( x , t ) u(x,t) u ( x , t ) (11) , namely
lim y → 0 + y a ∂ U ∂ y ⊥ T u S ℓ − 1 , \lim_{y\to0^+} y^a \frac{\partial U}{\partial y} \perp T_uS^{\ell-1}, y → 0 + lim y a ∂ y ∂ U ⊥ T u S ℓ − 1 , it follows immediately that the boundary integral vanishes:
∬ ∂ R + m + 1 × R + lim y → 0 + ( y a ∂ U i ∂ y ) ( δ k i − U k U i ) Φ d x d t = 0. \iint_{\partial\mathbb{R}^{m+1}_+\times\mathbb{R}_+} \lim_{y\to0^+}\left(y^a\frac{\partial U^i}{\partial y}\right)(\delta_{k_i}-U^kU^i)\Phi\,dx\,dt=0. ∬ ∂ R + m + 1 × R + y → 0 + lim ( y a ∂ y ∂ U i ) ( δ k i − U k U i ) Φ d x d t = 0. This completes the proof of (15) . □ \square □
Recall the matrix-valued connection 1-form ( Ω i j ) 1 ≤ i , j ≤ ℓ (\Omega^{ij})_{1\leq i,j\leq\ell} ( Ω ij ) 1 ≤ i , j ≤ ℓ R + m + 1 ‾ × R + \overline{\mathbb{R}^{m+1}_+}\times\mathbb{R}_+ R + m + 1 × R + § 1 :
Ω i j = U i D U j − U j D U i . \Omega^{ij}=U^iDU^j-U^jDU^i. Ω ij = U i D U j − U j D U i . By construction, ( Ω i j ) (\Omega^{ij}) ( Ω ij ) Ω i j = − Ω j i \Omega^{ij}=-\Omega^{ji} Ω ij = − Ω j i
Lemma 2.2 Let U : R + m + 1 × R + → R ℓ U:\mathbb{R}^{m+1}_+\times\mathbb{R}_+\to\mathbb{R}^{\ell} U : R + m + 1 × R + → R ℓ (11) . Then for any Φ ∈ C 0 ∞ ( R + m + 1 ‾ × R + ) \Phi\in C^\infty_0(\overline{\mathbb{R}^{m+1}_+}\times\mathbb{R}_+) Φ ∈ C 0 ∞ ( R + m + 1 × R + ) 1 ≤ i , j ≤ ℓ 1\leq i,j\leq\ell 1 ≤ i , j ≤ ℓ
∬ R + m + 1 × R + Ω i j ⋅ D Φ μ a d t = − ∬ R + m + 1 × R + ( U i ∂ t U j − U j ∂ t U i ) Φ μ a d t . (16) \iint_{\mathbb{R}^{m+1}_+\times\mathbb{R}_+}\Omega^{ij}\cdot D\Phi\,\mu_a\,dt=-\iint_{\mathbb{R}^{m+1}_+\times\mathbb{R}_+}(U^i\partial_tU^j-U^j\partial_tU^i)\Phi\,\mu_a\,dt. \tag*{(16)} ∬ R + m + 1 × R + Ω ij ⋅ D Φ μ a d t = − ∬ R + m + 1 × R + ( U i ∂ t U j − U j ∂ t U i ) Φ μ a d t . (16) Proof. Integrating by parts on the left-hand side of (16) yields
∬ R + m + 1 × R + Ω i j ⋅ D Φ μ a d t = − ∬ R + m + 1 × R + div X ( y a Ω i j ) Φ d X d t − ∬ ∂ R + m + 1 × R + lim y → 0 + y a ( U i ∂ U j ∂ y − U j ∂ U i ∂ y ) Φ d x d t . \iint_{\mathbb{R}^{m+1}_+\times\mathbb{R}_+}\Omega^{ij}\cdot D\Phi\,\mu_a\,dt
=-\iint_{\mathbb{R}^{m+1}_+\times\mathbb{R}_+}\operatorname{div}_X(y^a\Omega^{ij})\Phi\,dX\,dt
-\iint_{\partial\mathbb{R}^{m+1}_+\times\mathbb{R}_+}\lim_{y\to0^+}y^a\left(U^i\frac{\partial U^j}{\partial y}-U^j\frac{\partial U^i}{\partial y}\right)\Phi\,dx\,dt. ∬ R + m + 1 × R + Ω ij ⋅ D Φ μ a d t = − ∬ R + m + 1 × R + div X ( y a Ω ij ) Φ d X d t − ∬ ∂ R + m + 1 × R + y → 0 + lim y a ( U i ∂ y ∂ U j − U j ∂ y ∂ U i ) Φ d x d t . Using the equation y a ∂ t U = div X ( y a D U ) y^a\partial_tU=\operatorname{div}_X(y^aDU) y a ∂ t U = div X ( y a D U ) R + m + 1 × R + \mathbb{R}^{m+1}_+\times\mathbb{R}_+ R + m + 1 × R +
div X ( y a Ω i j ) = U i div X ( y a D U j ) − U j div X ( y a D U i ) = y a ( U i ∂ t U j − U j ∂ t U i ) . \operatorname{div}_X(y^a\Omega^{ij})=U^i\operatorname{div}_X(y^aDU^j)-U^j\operatorname{div}_X(y^aDU^i)=y^a\left(U^i\partial_tU^j-U^j\partial_tU^i\right). div X ( y a Ω ij ) = U i div X ( y a D U j ) − U j div X ( y a D U i ) = y a ( U i ∂ t U j − U j ∂ t U i ) . For the boundary integral, since u = U ∣ ∂ R + m + 1 ∈ S ℓ − 1 u=U|_{\partial\mathbb{R}^{m+1}_+}\in S^{\ell-1} u = U ∣ ∂ R + m + 1 ∈ S ℓ − 1 lim y → 0 + y a ∂ U ∂ y ⊥ T u S ℓ − 1 \lim_{y\to0^+}y^a\frac{\partial U}{\partial y}\perp T_uS^{\ell-1} lim y → 0 + y a ∂ y ∂ U ⊥ T u S ℓ − 1 u ( x , t ) u(x,t) u ( x , t )
lim y → 0 + y a ∂ U ∂ y = ⟨ lim y → 0 + y a ∂ U ∂ y , u ⟩ u on ∂ R + m + 1 × R + . \lim_{y\to0^+}y^a\frac{\partial U}{\partial y}
=\left\langle\lim_{y\to0^+}y^a\frac{\partial U}{\partial y},u\right\rangle u
\quad\text{on }\partial\mathbb{R}^{m+1}_+\times\mathbb{R}_+. y → 0 + lim y a ∂ y ∂ U = ⟨ y → 0 + lim y a ∂ y ∂ U , u ⟩ u on ∂ R + m + 1 × R + . Therefore, componentwise on the boundary ∂ R + m + 1 × R + \partial\mathbb{R}^{m+1}_+\times\mathbb{R}_+ ∂ R + m + 1 × R +
u i lim y → 0 + y a ∂ U j ∂ y − u j lim y → 0 + y a ∂ U i ∂ y = u i u j ⟨ lim y → 0 + y a ∂ U ∂ y , u ⟩ − u j u i ⟨ lim y → 0 + y a ∂ U ∂ y , u ⟩ = 0. u^i\lim_{y\to0^+}y^a\frac{\partial U^j}{\partial y}-u^j\lim_{y\to0^+}y^a\frac{\partial U^i}{\partial y}
=u^iu^j\left\langle\lim_{y\to0^+}y^a\frac{\partial U}{\partial y},u\right\rangle-u^ju^i\left\langle\lim_{y\to0^+}y^a\frac{\partial U}{\partial y},u\right\rangle=0. u i y → 0 + lim y a ∂ y ∂ U j − u j y → 0 + lim y a ∂ y ∂ U i = u i u j ⟨ y → 0 + lim y a ∂ y ∂ U , u ⟩ − u j u i ⟨ y → 0 + lim y a ∂ y ∂ U , u ⟩ = 0. Hence the boundary contribution vanishes, completing the proof of (16) . □ \square □
Combining Lemmas 2.1 and 2.2, we obtain the main algebraic identity for the s s s
Lemma 2.3 Let U : R + m + 1 × R + → R ℓ U:\mathbb{R}^{m+1}_{+}\times\mathbb{R}_{+}\to\mathbb{R}^{\ell} U : R + m + 1 × R + → R ℓ Φ ∈ C ∞ ( R + m + 1 × R + ‾ ) \Phi\in C^{\infty}(\overline{\mathbb{R}^{m+1}_{+}\times\mathbb{R}_{+}}) Φ ∈ C ∞ ( R + m + 1 × R + ) 1 ≤ k ≤ ℓ 1\leq k\leq\ell 1 ≤ k ≤ ℓ
∬ R + m + 1 × R + ( ∂ t U k Φ + D U k ⋅ D Φ ) μ a d t = 1 2 ∬ R + m + 1 × R + ∂ t ( ∣ U ∣ 2 − 1 ) U k Φ μ a d t + ∬ R + m + 1 × R + D U i ⋅ Ω k i Φ μ a d t + 1 2 ∬ R + m + 1 × R + D ( ∣ U ∣ 2 − 1 ) U k ⋅ D Φ μ a d t + ∬ R + m + 1 × R + D ( ∣ U ∣ 2 − 1 ) ⋅ D U k Φ μ a d t . (17) \iint_{\mathbb{R}^{m+1}_{+}\times\mathbb{R}_{+}}\left(\partial_t U^k\Phi+DU^k\cdot D\Phi\right)\,\mu_a\,dt
=
\frac{1}{2}\iint_{\mathbb{R}^{m+1}_{+}\times\mathbb{R}_{+}}\partial_t\left(|U|^2-1\right)U^k\Phi\,\mu_a\,dt
+\iint_{\mathbb{R}^{m+1}_{+}\times\mathbb{R}_{+}}DU^i\cdot\Omega^{ki}\Phi\,\mu_a\,dt
+\frac{1}{2}\iint_{\mathbb{R}^{m+1}_{+}\times\mathbb{R}_{+}}D\left(|U|^2-1\right)U^k\cdot D\Phi\,\mu_a\,dt
+\iint_{\mathbb{R}^{m+1}_{+}\times\mathbb{R}_{+}}D\left(|U|^2-1\right)\cdot DU^k\Phi\,\mu_a\,dt. \tag*{(17)} ∬ R + m + 1 × R + ( ∂ t U k Φ + D U k ⋅ D Φ ) μ a d t = 2 1 ∬ R + m + 1 × R + ∂ t ( ∣ U ∣ 2 − 1 ) U k Φ μ a d t + ∬ R + m + 1 × R + D U i ⋅ Ω k i Φ μ a d t + 2 1 ∬ R + m + 1 × R + D ( ∣ U ∣ 2 − 1 ) U k ⋅ D Φ μ a d t + ∬ R + m + 1 × R + D ( ∣ U ∣ 2 − 1 ) ⋅ D U k Φ μ a d t . (17) Proof. We expand the integrands on the right-hand side of identity (2.1) in Lemma 2.1. First, for the temporal derivative term, we write:
∂ t U i ( U k U i Φ ) = ( U i ∂ t U i ) U k Φ = 1 2 ∂ t ( ∣ U ∣ 2 − 1 ) U k Φ . \partial_t U^i(U^kU^i\Phi)=\left(U^i\partial_tU^i\right)U^k\Phi=\frac{1}{2}\partial_t\left(|U|^2-1\right)U^k\Phi. ∂ t U i ( U k U i Φ ) = ( U i ∂ t U i ) U k Φ = 2 1 ∂ t ( ∣ U ∣ 2 − 1 ) U k Φ. Next, expanding the spatial gradient term yields:
D U i ⋅ D ( U k U i Φ ) = D U i ⋅ ( U i U k D Φ + U i Φ D U k + U k Φ D U i ) DU^i\cdot D(U^kU^i\Phi)=DU^i\cdot\left(U^iU^kD\Phi+U^i\Phi DU^k+U^k\Phi DU^i\right) D U i ⋅ D ( U k U i Φ ) = D U i ⋅ ( U i U k D Φ + U i Φ D U k + U k Φ D U i ) = ( U i D U i ) ⋅ D Φ U k + ( U i D U i ) ⋅ D U k Φ + ∣ D U ∣ 2 U k Φ . =\left(U^iDU^i\right)\cdot D\Phi\,U^k+\left(U^iDU^i\right)\cdot DU^k\Phi+|DU|^2U^k\Phi. = ( U i D U i ) ⋅ D Φ U k + ( U i D U i ) ⋅ D U k Φ + ∣ D U ∣ 2 U k Φ. Using U i D U i = 1 2 D ( ∣ U ∣ 2 − 1 ) U^iDU^i=\frac{1}{2}D(|U|^2-1) U i D U i = 2 1 D ( ∣ U ∣ 2 − 1 ) 1 2 D ( ∣ U ∣ 2 − 1 ) U k ⋅ D Φ \frac{1}{2}D(|U|^2-1)U^k\cdot D\Phi 2 1 D ( ∣ U ∣ 2 − 1 ) U k ⋅ D Φ 1 2 D ( ∣ U ∣ 2 − 1 ) ⋅ D U k Φ \frac{1}{2}D(|U|^2-1)\cdot DU^k\Phi 2 1 D ( ∣ U ∣ 2 − 1 ) ⋅ D U k Φ ∣ D U ∣ 2 U k Φ |DU|^2U^k\Phi ∣ D U ∣ 2 U k Φ Ω k i = U k D U i − U i D U k \Omega^{ki}=U^kDU^i-U^iDU^k Ω k i = U k D U i − U i D U k
D U i ⋅ Ω k i = D U i ⋅ ( U k D U i − U i D U k ) DU^i\cdot\Omega^{ki}=DU^i\cdot\left(U^kDU^i-U^iDU^k\right) D U i ⋅ Ω k i = D U i ⋅ ( U k D U i − U i D U k ) = ∣ D U ∣ 2 U k − ( U i D U i ) ⋅ D U k =|DU|^2U^k-\left(U^iDU^i\right)\cdot DU^k = ∣ D U ∣ 2 U k − ( U i D U i ) ⋅ D U k = ∣ D U ∣ 2 U k − 1 2 D ( ∣ U ∣ 2 − 1 ) ⋅ D U k . =|DU|^2U^k-\frac{1}{2}D(|U|^2-1)\cdot DU^k. = ∣ D U ∣ 2 U k − 2 1 D ( ∣ U ∣ 2 − 1 ) ⋅ D U k . Rearranging this identity gives
∣ D U ∣ 2 U k Φ = D U i ⋅ Ω k i Φ + 1 2 D ( ∣ U ∣ 2 − 1 ) ⋅ D U k Φ . |DU|^2U^k\Phi=DU^i\cdot\Omega^{ki}\Phi+\frac{1}{2}D(|U|^2-1)\cdot DU^k\Phi. ∣ D U ∣ 2 U k Φ = D U i ⋅ Ω k i Φ + 2 1 D ( ∣ U ∣ 2 − 1 ) ⋅ D U k Φ. Summing all the expanded pieces, the right-hand side of (2.1) transforms into:
∬ R + m + 1 × R + ( ∂ t U i ( U k U i Φ ) + D U i ⋅ D ( U k U i Φ ) ) μ a d t \iint_{\mathbb{R}^{m+1}_{+}\times\mathbb{R}_{+}}\left(\partial_tU^i(U^kU^i\Phi)+DU^i\cdot D(U^kU^i\Phi)\right)\,\mu_a\,dt ∬ R + m + 1 × R + ( ∂ t U i ( U k U i Φ ) + D U i ⋅ D ( U k U i Φ ) ) μ a d t = 1 2 ∬ R + m + 1 × R + ∂ t ( ∣ U ∣ 2 − 1 ) U k Φ μ a d t + ∬ R + m + 1 × R + D U i ⋅ Ω k i Φ μ a d t =\frac{1}{2}\iint_{\mathbb{R}^{m+1}_{+}\times\mathbb{R}_{+}}\partial_t\left(|U|^2-1\right)U^k\Phi\,\mu_a\,dt
+\iint_{\mathbb{R}^{m+1}_{+}\times\mathbb{R}_{+}}DU^i\cdot\Omega^{ki}\Phi\,\mu_a\,dt = 2 1 ∬ R + m + 1 × R + ∂ t ( ∣ U ∣ 2 − 1 ) U k Φ μ a d t + ∬ R + m + 1 × R + D U i ⋅ Ω k i Φ μ a d t + 1 2 ∬ R + m + 1 × R + D ( ∣ U ∣ 2 − 1 ) U k ⋅ D Φ μ a d t + ∬ R + m + 1 × R + D ( ∣ U ∣ 2 − 1 ) ⋅ D U k Φ μ a d t . +\frac{1}{2}\iint_{\mathbb{R}^{m+1}_{+}\times\mathbb{R}_{+}}D\left(|U|^2-1\right)U^k\cdot D\Phi\,\mu_a\,dt
+\iint_{\mathbb{R}^{m+1}_{+}\times\mathbb{R}_{+}}D\left(|U|^2-1\right)\cdot DU^k\Phi\,\mu_a\,dt. + 2 1 ∬ R + m + 1 × R + D ( ∣ U ∣ 2 − 1 ) U k ⋅ D Φ μ a d t + ∬ R + m + 1 × R + D ( ∣ U ∣ 2 − 1 ) ⋅ D U k Φ μ a d t . Substituting this back into (2.1) establishes (2.3), completing the proof. ✷
Hybrid inequality under small y a y^{a} y a
In this section, we will derive a hybrid energy inequality for the heat flow of s s s y a y^{a} y a y a y^{a} y a
Definition 3.1 Let O ⊂ R m + 1 × ( 0 , ∞ ) O \subset\mathbb{R}^{m+1} \times(0,\infty) O ⊂ R m + 1 × ( 0 , ∞ ) O + = O ∩ ( R + m + 1 × ( 0 , ∞ ) ) O_{+}=O\cap\left(\mathbb{R}^{m+1}_{+}\times(0,\infty)\right) O + = O ∩ ( R + m + 1 × ( 0 , ∞ ) ) 0 < λ < m + 3 + a 0<\lambda<m+3+a 0 < λ < m + 3 + a M 2 , λ ( O + , μ a d t ) M^{2,\lambda}(O_{+},\mu_{a}dt) M 2 , λ ( O + , μ a d t ) f ∈ L l o c 2 ( O + , μ a d t ) f\in L^{2}_{\mathrm{loc}}(O_{+},\mu_{a}dt) f ∈ L loc 2 ( O + , μ a d t )
∥ f ∥ M 2 , λ ( O + , μ a d t ) 2 = sup ( x , t ) ∈ O + , r > 0 { r λ − ( m + 3 + a ) ∬ P r ( x , t ) ∩ O + ∣ f ∣ 2 μ a d t } < ∞ . (18) \|f\|^{2}_{M^{2,\lambda}(O_{+},\mu_{a}dt)}
=\sup_{(x,t)\in O_{+},r>0}\left\{r^{\lambda-(m+3+a)}\iint_{P_{r}(x,t)\cap O_{+}}|f|^{2}\mu_{a}dt\right\}<\infty. \tag*{(18)} ∥ f ∥ M 2 , λ ( O + , μ a d t ) 2 = ( x , t ) ∈ O + , r > 0 sup { r λ − ( m + 3 + a ) ∬ P r ( x , t ) ∩ O + ∣ f ∣ 2 μ a d t } < ∞. (18) Throughout this paper, denote by P r ( X , t ) = B r ( X ) × ( t − r 2 , t + r 2 ) P_{r}(X,t)=B_{r}(X)\times(t-r^{2},t+r^{2}) P r ( X , t ) = B r ( X ) × ( t − r 2 , t + r 2 ) ( X , t ) (X,t) ( X , t ) r r r P r + ( X , t ) = P r ( X , t ) ∩ ( R + m + 1 × R ) P_{r}^{+}(X,t)=P_{r}(X,t)\cap(\mathbb{R}^{m+1}_{+}\times\mathbb{R}) P r + ( X , t ) = P r ( X , t ) ∩ ( R + m + 1 × R ) U U U P r + ( z 0 ) P_{r}^{+}(z_{0}) P r + ( z 0 ) μ a d t \mu_{a}dt μ a d t
( U ) z 0 , r , + a = 1 ∣ P r + ( z 0 ) ∣ a ∬ P r + ( z 0 ) U μ a d t , where ∣ P r + ( z 0 ) ∣ a = ∬ P r + ( z 0 ) μ a d t . (U)^{a}_{z_{0},r,+}=\frac{1}{|P_{r}^{+}(z_{0})|_{a}}\iint_{P_{r}^{+}(z_{0})}U\mu_{a}dt,\qquad\text{where }|P_{r}^{+}(z_{0})|_{a}=\iint_{P_{r}^{+}(z_{0})}\mu_{a}dt. ( U ) z 0 , r , + a = ∣ P r + ( z 0 ) ∣ a 1 ∬ P r + ( z 0 ) U μ a d t , where ∣ P r + ( z 0 ) ∣ a = ∬ P r + ( z 0 ) μ a d t . When a = 0 a=0 a = 0 ∣ P r + ( z 0 ) ∣ 0 |P_{r}^{+}(z_{0})|_{0} ∣ P r + ( z 0 ) ∣ 0 ( U ) z 0 , r , + 0 (U)^{0}_{z_{0},r,+} ( U ) z 0 , r , + 0 ∣ P r + ( z 0 ) ∣ |P_{r}^{+}(z_{0})| ∣ P r + ( z 0 ) ∣ U z 0 , r , + U_{z_{0},r,+} U z 0 , r , +
Before proving the main hybrid inequality, we establish a crucial point-wise distance estimate showing that small Morrey bounds force U U U S ℓ − 1 S^{\ell-1} S ℓ − 1
Lemma 3.2 For any z 0 = ( x 0 , 0 , t 0 ) ∈ ∂ R + m + 1 × ( 0 , ∞ ) z_{0}=(x_{0},0,t_{0})\in\partial\mathbb{R}^{m+1}_{+}\times(0,\infty) z 0 = ( x 0 , 0 , t 0 ) ∈ ∂ R + m + 1 × ( 0 , ∞ ) 0 < r 0 < t 0 0<r_{0}<\sqrt{t_{0}} 0 < r 0 < t 0
sup z ∈ P r 0 / 4 + ( z 0 ) dist ( U ( z ) , S ℓ − 1 ) ≤ C [ ∥ D U ∥ M 2 , 2 ( P r 0 + ( z 0 ) , μ a d t ) + ∥ ∂ t U ∥ M 2 , 4 ( P r 0 + ( z 0 ) , μ a d t ) ] . \sup_{z\in P^{+}_{r_{0}/4}(z_{0})}\operatorname{dist}(U(z),S^{\ell-1})
\le C\left[\|DU\|_{M^{2,2}(P^{+}_{r_{0}}(z_{0}),\mu_{a}dt)}
+\|\partial_{t}U\|_{M^{2,4}(P^{+}_{r_{0}}(z_{0}),\mu_{a}dt)}\right]. z ∈ P r 0 /4 + ( z 0 ) sup dist ( U ( z ) , S ℓ − 1 ) ≤ C [ ∥ D U ∥ M 2 , 2 ( P r 0 + ( z 0 ) , μ a d t ) + ∥ ∂ t U ∥ M 2 , 4 ( P r 0 + ( z 0 ) , μ a d t ) ] . Proof. Pick any z = ( x , y , t ) ∈ P r 0 / 4 + ( z 0 ) z=(x,y,t)\in P^{+}_{r_{0}/4}(z_{0}) z = ( x , y , t ) ∈ P r 0 /4 + ( z 0 ) U ( x , 0 , t ) ∈ S ℓ − 1 U(x,0,t)\in S^{\ell-1} U ( x , 0 , t ) ∈ S ℓ − 1 y > 0 y>0 y > 0 U U U R + m + 1 \mathbb{R}^{m+1}_{+} R + m + 1 P y / 2 ( z ) ⊂ P y ( z ) ⊂ P r 0 / 2 + ( z 0 ) P_{y/2}(z)\subset P_{y}(z)\subset P^{+}_{r_{0}/2}(z_{0}) P y /2 ( z ) ⊂ P y ( z ) ⊂ P r 0 /2 + ( z 0 )
∣ U ( z ) − ( U ) z , y / 2 ∣ ≤ C ( ∣ y ∣ − ( m + 3 + a ) ∬ P ∣ y ∣ ( z ) ( ∣ D U ∣ 2 + ∣ y ∣ 2 ∣ ∂ t U ∣ 2 ) μ a d t ) 1 2 ≤ C ( ∥ D U ∥ M 2 , 2 ( P r 0 + ( z 0 ) , μ a d t ) + ∥ ∂ t U ∥ M 2 , 4 ( P r 0 + ( z 0 ) , μ a d t ) ) . (19) \begin{aligned}
\left|U(z)-(U)_{z,y/2}\right|
\le C\left(|y|^{-(m+3+a)}\iint_{P_{|y|}(z)}\left(|DU|^{2}+|y|^{2}|\partial_{t}U|^{2}\right)\mu_{a}dt\right)^{\frac12} \\
&\le C\left(\|DU\|_{M^{2,2}(P^{+}_{r_{0}}(z_{0}),\mu_{a}dt)}
+\|\partial_{t}U\|_{M^{2,4}(P^{+}_{r_{0}}(z_{0}),\mu_{a}dt)}\right). \tag*{(19)}
\end{aligned} U ( z ) − ( U ) z , y /2 ≤ C ( ∣ y ∣ − ( m + 3 + a ) ∬ P ∣ y ∣ ( z ) ( ∣ D U ∣ 2 + ∣ y ∣ 2 ∣ ∂ t U ∣ 2 ) μ a d t ) 2 1 ≤ C ( ∥ D U ∥ M 2 , 2 ( P r 0 + ( z 0 ) , μ a d t ) + ∥ ∂ t U ∥ M 2 , 4 ( P r 0 + ( z 0 ) , μ a d t ) ) . (19) Set z ˉ = ( x , 0 , t ) \bar z=(x,0,t) z ˉ = ( x , 0 , t ) P y / 2 ( z ) ⊂ P 3 y / 2 + ( z ˉ ) ⊂ P r 0 + ( z 0 ) P_{y/2}(z)\subset P^{+}_{3y/2}(\bar z)\subset P^{+}_{r_{0}}(z_{0}) P y /2 ( z ) ⊂ P 3 y /2 + ( z ˉ ) ⊂ P r 0 + ( z 0 ) y a y^{a} y a A 2 A_{2} A 2 y a y^{a} y a
∣ ( U ) z , y / 2 − ( U ) z ˉ , 3 y / 2 , + a ∣ ≤ 3 m + 2 ∣ P 3 y / 2 + ( z ˉ ) ∣ ∬ P 3 y / 2 + ( z ˉ ) ∣ U − ( U ) z ˉ , 3 y / 2 , + a ∣ d X d t ≤ C ( 1 ∣ P 3 y / 2 + ( z ˉ ) ∣ a ∬ P 3 y / 2 + ( z ˉ ) ∣ U − ( U ) z ˉ , 3 y / 2 , + a ∣ 2 μ a d t ) 1 2 ≤ C ( ∥ D U ∥ M 2 , 2 ( P r 0 + ( z 0 ) , μ a d t ) + ∥ ∂ t U ∥ M 2 , 4 ( P r 0 + ( z 0 ) , μ a d t ) ) . (20) \begin{aligned}
\left|(U)_{z,y/2}-(U)_{\bar z,3y/2,+}^{a}\right|
\le\frac{3^{m+2}}{|P^{+}_{3y/2}(\bar z)|}\iint_{P^{+}_{3y/2}(\bar z)}|U-(U)_{\bar z,3y/2,+}^{a}|\,dXdt \\
&\le C\left(\frac{1}{|P^{+}_{3y/2}(\bar z)|_{a}}\iint_{P^{+}_{3y/2}(\bar z)}
|U-(U)_{\bar z,3y/2,+}^{a}|^{2}\mu_{a}dt\right)^{\frac12} \\
&\le C\left(\|DU\|_{M^{2,2}(P^{+}_{r_{0}}(z_{0}),\mu_{a}dt)}
+\|\partial_{t}U\|_{M^{2,4}(P^{+}_{r_{0}}(z_{0}),\mu_{a}dt)}\right). \tag*{(20)}
\end{aligned} ( U ) z , y /2 − ( U ) z ˉ , 3 y /2 , + a ≤ ∣ P 3 y /2 + ( z ˉ ) ∣ 3 m + 2 ∬ P 3 y /2 + ( z ˉ ) ∣ U − ( U ) z ˉ , 3 y /2 , + a ∣ d X d t ≤ C ( ∣ P 3 y /2 + ( z ˉ ) ∣ a 1 ∬ P 3 y /2 + ( z ˉ ) ∣ U − ( U ) z ˉ , 3 y /2 , + a ∣ 2 μ a d t ) 2 1 ≤ C ( ∥ D U ∥ M 2 , 2 ( P r 0 + ( z 0 ) , μ a d t ) + ∥ ∂ t U ∥ M 2 , 4 ( P r 0 + ( z 0 ) , μ a d t ) ) . (20) Since u ( x , t ) = U ( x , 0 , t ) ∈ S ℓ − 1 u(x,t)=U(x,0,t)\in\mathbb{S}^{\ell-1} u ( x , t ) = U ( x , 0 , t ) ∈ S ℓ − 1 w ∈ ∂ R + m + 1 × R + w\in\partial\mathbb{R}^{m+1}_{+}\times\mathbb{R}_{+} w ∈ ∂ R + m + 1 × R +
dist ( ( U ) z ˉ , 3 y 2 , + , S ℓ − 1 ) ≤ ∣ ( U ) z ˉ , 3 y 2 , + − u ( w ) ∣ ≤ C y − ( m + 3 ) ∬ P 3 y 2 + ( z ˉ ) ∣ U − u ( w ) ∣ . \operatorname{dist}\left((U)_{\bar z,\frac{3y}{2},+},\mathbb{S}^{\ell-1}\right)\leq\left|(U)_{\bar z,\frac{3y}{2},+}-u(w)\right|\leq Cy^{-(m+3)}\iint_{P^{+}_{\frac{3y}{2}}(\bar z)}|U-u(w)|. dist ( ( U ) z ˉ , 2 3 y , + , S ℓ − 1 ) ≤ ( U ) z ˉ , 2 3 y , + − u ( w ) ≤ C y − ( m + 3 ) ∬ P 2 3 y + ( z ˉ ) ∣ U − u ( w ) ∣. Averaging w w w P ^ 3 y 2 ( z ˉ ) : = D 3 y 2 ( x ) × [ t − 9 y 2 4 , t + 9 y 2 4 ] \widehat{P}_{\frac{3y}{2}}(\bar z):=D_{\frac{3y}{2}}(x)\times[t-\frac{9y^{2}}{4},t+\frac{9y^{2}}{4}] P 2 3 y ( z ˉ ) := D 2 3 y ( x ) × [ t − 4 9 y 2 , t + 4 9 y 2 ] R m × R \mathbb{R}^{m}\times\mathbb{R} R m × R
dist ( ( U ) z ˉ , 3 y 2 , + , S ℓ − 1 ) ≤ C y − ( m + 3 ) ∬ P 3 y 2 + ( z ˉ ) ∣ U − ( u ) P ^ 3 y 2 ( z ˉ ) ∣ + C y − ( m + 2 ) ∬ P ^ 3 y 2 ( z ˉ ) ∣ u − ( u ) P ^ 3 y 2 ( z ˉ ) ∣ ≤ C ( ∥ D U ∥ M 2 , 2 ( P r 0 + ( z 0 ) , μ a d t ) + ∥ ∂ t U ∥ M 2 , 4 ( P r 0 + ( z 0 ) , μ a d t ) ) . (21) \begin{aligned}
\operatorname{dist}\left((U)_{\bar z,\frac{3y}{2},+},\mathbb{S}^{\ell-1}\right)
&\leq Cy^{-(m+3)}\iint_{P^{+}_{\frac{3y}{2}}(\bar z)}\left|U-(u)_{\widehat{P}_{\frac{3y}{2}}(\bar z)}\right| \\
&\quad+Cy^{-(m+2)}\iint_{\widehat{P}_{\frac{3y}{2}}(\bar z)}\left|u-(u)_{\widehat{P}_{\frac{3y}{2}}(\bar z)}\right| \\
&\leq C\left(\|DU\|_{M^{2,2}(P^{+}_{r_{0}}(z_{0}),\mu_{a}dt)}+\|\partial_{t}U\|_{M^{2,4}(P^{+}_{r_{0}}(z_{0}),\mu_{a}dt)}\right). \tag*{(21)}
\end{aligned} dist ( ( U ) z ˉ , 2 3 y , + , S ℓ − 1 ) ≤ C y − ( m + 3 ) ∬ P 2 3 y + ( z ˉ ) U − ( u ) P 2 3 y ( z ˉ ) + C y − ( m + 2 ) ∬ P 2 3 y ( z ˉ ) u − ( u ) P 2 3 y ( z ˉ ) ≤ C ( ∥ D U ∥ M 2 , 2 ( P r 0 + ( z 0 ) , μ a d t ) + ∥ ∂ t U ∥ M 2 , 4 ( P r 0 + ( z 0 ) , μ a d t ) ) . (21) Combining (3.3), (3.4), and (3.5) completes the proof of (3.2) □ \square □
Equipped with Lemma 3.2, we establish the key hybrid inequality under a smallness condition.
Theorem 3.3 There exist ε 0 > 0 \varepsilon_{0}>0 ε 0 > 0 θ 0 > 0 \theta_{0}>0 θ 0 > 0 z 0 = ( x 0 , t 0 ) ∈ ∂ R + m + 1 × ( 0 , ∞ ) z_{0}=(x_{0},t_{0})\in\partial\mathbb{R}^{m+1}_{+}\times(0,\infty) z 0 = ( x 0 , t 0 ) ∈ ∂ R + m + 1 × ( 0 , ∞ ) U ∈ L t ∞ H X 1 ( P r 0 + ( z 0 ) , μ a d t ) U\in L_{t}^{\infty}H_{X}^{1}(P^{+}_{r_{0}}(z_{0}),\mu_{a}dt) U ∈ L t ∞ H X 1 ( P r 0 + ( z 0 ) , μ a d t )
r 0 − ( m + 1 + a ) ∬ P r 0 + ( z 0 ) ∣ D U ∣ 2 μ a d t ≤ ε 0 2 , (22) r_{0}^{-(m+1+a)}\iint_{P^{+}_{r_{0}}(z_{0})}|DU|^{2}\mu_{a}dt\leq\varepsilon_{0}^{2}, \tag*{(22)} r 0 − ( m + 1 + a ) ∬ P r 0 + ( z 0 ) ∣ D U ∣ 2 μ a d t ≤ ε 0 2 , (22) then for all 0 < r ≤ θ 0 r 0 0<r\leq\theta_{0}r_{0} 0 < r ≤ θ 0 r 0
r − ( m + 1 + a ) ∬ P r + ( z 0 ) ∣ D U ∣ 2 μ a d t ≤ C ε 0 r − ( m + 1 + a ) ∬ P 2 r + ( z 0 ) ( r 2 ∣ ∂ t U ∣ 2 + ∣ D U ∣ 2 ) μ a d t + C ε 0 r − ( m − 1 + a ) sup t ∈ [ t 0 − 4 r 2 , t 0 + 4 r 2 ] ∫ B 2 r + ( x 0 ) ∣ D U ∣ 2 μ a + C ε 0 − 1 r − ( m + 3 + a ) ∬ P 2 r + ( z 0 ) ( ∣ U − ( U ) z 0 , 2 r , + a ∣ 2 + ∣ U − ( U ) z 0 , 2 r , + 4 ∣ 2 ) μ a d t + C ε 0 − 1 r − ( m + 3 + a ) ∬ P 2 r + ( z 0 ) ( ∣ U ∣ 2 − 1 ) 2 μ a d t , (23) \begin{aligned}
r^{-(m+1+a)}\iint_{P^{+}_{r}(z_{0})}|DU|^{2}\mu_{a}dt
&\leq C\varepsilon_{0}r^{-(m+1+a)}\iint_{P^{+}_{2r}(z_{0})}\left(r^{2}|\partial_{t}U|^{2}+|DU|^{2}\right)\mu_{a}dt \\
&\quad+C\varepsilon_{0}r^{-(m-1+a)}
\sup_{t\in[t_{0}-4r^{2},t_{0}+4r^{2}]}\int_{B^{+}_{2r}(x_{0})}|DU|^{2}\mu_{a} \\
&\quad+C\varepsilon_{0}^{-1}r^{-(m+3+a)}\iint_{P^{+}_{2r}(z_{0})}
\left(|U-(U)^{a}_{z_{0},2r,+}|^{2}+|U-(U)^{4}_{z_{0},2r,+}|^{2}\right)\mu_{a}dt \\
&\quad+C\varepsilon_{0}^{-1}r^{-(m+3+a)}\iint_{P^{+}_{2r}(z_{0})}(|U|^{2}-1)^{2}\mu_{a}dt, \tag*{(23)}
\end{aligned} r − ( m + 1 + a ) ∬ P r + ( z 0 ) ∣ D U ∣ 2 μ a d t ≤ C ε 0 r − ( m + 1 + a ) ∬ P 2 r + ( z 0 ) ( r 2 ∣ ∂ t U ∣ 2 + ∣ D U ∣ 2 ) μ a d t + C ε 0 r − ( m − 1 + a ) t ∈ [ t 0 − 4 r 2 , t 0 + 4 r 2 ] sup ∫ B 2 r + ( x 0 ) ∣ D U ∣ 2 μ a + C ε 0 − 1 r − ( m + 3 + a ) ∬ P 2 r + ( z 0 ) ( ∣ U − ( U ) z 0 , 2 r , + a ∣ 2 + ∣ U − ( U ) z 0 , 2 r , + 4 ∣ 2 ) μ a d t + C ε 0 − 1 r − ( m + 3 + a ) ∬ P 2 r + ( z 0 ) ( ∣ U ∣ 2 − 1 ) 2 μ a d t , (23) holds for some $C=C(m)>0.
Proof. Without loss of generality, assume z 0 = ( 0 , 1 ) z_{0}=(0,1) z 0 = ( 0 , 1 ) r 0 = 1 2 r_{0}=\frac{1}{2} r 0 = 2 1 0 < r < 1 8 0<r<\frac{1}{8} 0 < r < 8 1 η ∈ C 0 ∞ ( P 1 ( 0 , 1 ) ) \eta\in C_{0}^{\infty}(P_{1}(0,1)) η ∈ C 0 ∞ ( P 1 ( 0 , 1 )) P r ( 0 , 1 ) P_{r}(0,1) P r ( 0 , 1 ) 0 ≤ η ≤ 1 0\leq\eta\leq1 0 ≤ η ≤ 1 η ≡ 1 \eta\equiv1 η ≡ 1 P r ( 0 , 1 ) P_{r}(0,1) P r ( 0 , 1 ) η ≡ 0 \eta\equiv0 η ≡ 0 P 2 r ( 0 , 1 ) P_{2r}(0,1) P 2 r ( 0 , 1 ) ∣ D 2 η ∣ + ∣ D η ∣ 2 + ∣ ∂ t η ∣ ≤ 16 r − 2 |D^{2}\eta|+|D\eta|^{2}+|\partial_{t}\eta|\leq16r^{-2} ∣ D 2 η ∣ + ∣ D η ∣ 2 + ∣ ∂ t η ∣ ≤ 16 r − 2 Φ = ( U k − ( U k ) ( 0 , 1 ) , 2 r , + a ) η 2 \Phi=(U^k-(U^k)^a_{(0,1),2r,+})\eta^2 Φ = ( U k − ( U k ) ( 0 , 1 ) , 2 r , + a ) η 2
∬ P 1 + ( 0 , 1 ) ∣ D U ∣ 2 η 2 μ a d t = − ∬ P 1 + ( 0 , 1 ) ∂ t U k ( U k − ( U k ) ( 0 , 1 ) , 2 r , + a ) η 2 μ a d t − ∬ P 1 + ( 0 , 1 ) D U k ( U k − ( U k ) ( 0 , 1 ) , 2 r , + a ) ⋅ D η 2 μ a d t + ∬ P 1 + ( 0 , 1 ) 1 2 ∂ t ( ∣ U ∣ 2 − 1 ) U k ( U k − ( U k ) ( 0 , 1 ) , 2 r , + a ) η 2 μ a d t + ∬ P 1 + ( 0 , 1 ) 1 2 D ( ∣ U ∣ 2 − 1 ) U k ⋅ D ( ( U k − ( U k ) ( 0 , 1 ) , 2 r , + a ) η 2 ) μ a d t + ∬ P 1 + ( 0 , 1 ) D ( ∣ U ∣ 2 − 1 ) ⋅ D U k ( U k − ( U k ) ( 0 , 1 ) , 2 r , + a ) η 2 μ a d t + ∬ P 1 + ( 0 , 1 ) D U i ⋅ Ω k i ( U k − ( U k ) ( 0 , 1 ) , 2 r , + a ) η 2 μ a d t = I + I I + I I I + I V + V + V I . \begin{aligned}
\iint_{P^+_1(0,1)} |DU|^2\eta^2\mu_a\,dt
={}&-\iint_{P^+_1(0,1)}\partial_tU^k(U^k-(U^k)^a_{(0,1),2r,+})\eta^2\mu_a\,dt\\
&-\iint_{P^+_1(0,1)}DU^k(U^k-(U^k)^a_{(0,1),2r,+})\cdot D\eta^2\mu_a\,dt\\
&+\iint_{P^+_1(0,1)}\frac{1}{2}\partial_t(|U|^2-1)U^k(U^k-(U^k)^a_{(0,1),2r,+})\eta^2\mu_a\,dt\\
&+\iint_{P^+_1(0,1)}\frac{1}{2}D(|U|^2-1)U^k\cdot D((U^k-(U^k)^a_{(0,1),2r,+})\eta^2)\mu_a\,dt\\
&+\iint_{P^+_1(0,1)}D(|U|^2-1)\cdot DU^k(U^k-(U^k)^a_{(0,1),2r,+})\eta^2\mu_a\,dt\\
&+\iint_{P^+_1(0,1)}DU^i\cdot\Omega^{ki}(U^k-(U^k)^a_{(0,1),2r,+})\eta^2\mu_a\,dt\\
={}&I+II+III+IV+V+VI.
\end{aligned} ∬ P 1 + ( 0 , 1 ) ∣ D U ∣ 2 η 2 μ a d t = = − ∬ P 1 + ( 0 , 1 ) ∂ t U k ( U k − ( U k ) ( 0 , 1 ) , 2 r , + a ) η 2 μ a d t − ∬ P 1 + ( 0 , 1 ) D U k ( U k − ( U k ) ( 0 , 1 ) , 2 r , + a ) ⋅ D η 2 μ a d t + ∬ P 1 + ( 0 , 1 ) 2 1 ∂ t ( ∣ U ∣ 2 − 1 ) U k ( U k − ( U k ) ( 0 , 1 ) , 2 r , + a ) η 2 μ a d t + ∬ P 1 + ( 0 , 1 ) 2 1 D ( ∣ U ∣ 2 − 1 ) U k ⋅ D (( U k − ( U k ) ( 0 , 1 ) , 2 r , + a ) η 2 ) μ a d t + ∬ P 1 + ( 0 , 1 ) D ( ∣ U ∣ 2 − 1 ) ⋅ D U k ( U k − ( U k ) ( 0 , 1 ) , 2 r , + a ) η 2 μ a d t + ∬ P 1 + ( 0 , 1 ) D U i ⋅ Ω k i ( U k − ( U k ) ( 0 , 1 ) , 2 r , + a ) η 2 μ a d t I + I I + I I I + I V + V + V I . Before estimating I , I I , I I I , I V , V I,II,III,IV,V I , I I , I I I , I V , V θ 0 ∈ ( 0 , 1 4 ) \theta_0\in(0,\frac14) θ 0 ∈ ( 0 , 4 1 )
∣ 1 − ∣ U ( z ) ∣ ∣ = dist ( U ( z ) , S ℓ − 1 ) ≤ C ε 0 , ∀ z ∈ P 2 θ 0 + ( 0 , 1 ) . (24) |1-|U(z)||=\operatorname{dist}(U(z),\mathbb{S}^{\ell-1})\le C\varepsilon_0,\quad\forall z\in P^+_{2\theta_0}(0,1). \tag*{(24)} ∣1 − ∣ U ( z ) ∣∣ = dist ( U ( z ) , S ℓ − 1 ) ≤ C ε 0 , ∀ z ∈ P 2 θ 0 + ( 0 , 1 ) . (24) In particular, ∣ U ∣ ≤ 2 |U|\le2 ∣ U ∣ ≤ 2 P 2 θ 0 + ( 0 , 1 ) P^+_{2\theta_0}(0,1) P 2 θ 0 + ( 0 , 1 ) ε 0 > 0 \varepsilon_0>0 ε 0 > 0
For 0 < r < min { 1 8 , θ 0 2 } 0<r<\min\{\frac18,\frac{\theta_0}{2}\} 0 < r < min { 8 1 , 2 θ 0 } I , I I , I I I I,II,III I , I I , I I I
∣ I ∣ ≤ ε 0 ( 2 r ) 2 ∬ P 2 r + ( 0 , 1 ) ∣ ∂ t U ∣ 2 μ a d t + C ε 0 − 1 r − 2 ∬ P 2 r + ( 0 , 1 ) ∣ U − ( U ) ( 0 , 1 ) , 2 r , + a ∣ 2 μ a d t , |I|\le\varepsilon_0(2r)^2\iint_{P^+_{2r}(0,1)}|\partial_tU|^2\mu_a\,dt+C\varepsilon_0^{-1}r^{-2}\iint_{P^+_{2r}(0,1)}|U-(U)^a_{(0,1),2r,+}|^2\mu_a\,dt, ∣ I ∣ ≤ ε 0 ( 2 r ) 2 ∬ P 2 r + ( 0 , 1 ) ∣ ∂ t U ∣ 2 μ a d t + C ε 0 − 1 r − 2 ∬ P 2 r + ( 0 , 1 ) ∣ U − ( U ) ( 0 , 1 ) , 2 r , + a ∣ 2 μ a d t , ∣ I I ∣ ≤ ε 0 ∬ P 2 r + ( 0 , 1 ) ∣ D U ∣ 2 μ a d t + C ε 0 r − 2 ∬ P 2 r + ( 0 , 1 ) ∣ U − ( U ) ( 0 , 1 ) , 2 r , + a ∣ 2 μ a d t , |II|\le\varepsilon_0\iint_{P^+_{2r}(0,1)}|DU|^2\mu_a\,dt+C\varepsilon_0r^{-2}\iint_{P^+_{2r}(0,1)}|U-(U)^a_{(0,1),2r,+}|^2\mu_a\,dt, ∣ I I ∣ ≤ ε 0 ∬ P 2 r + ( 0 , 1 ) ∣ D U ∣ 2 μ a d t + C ε 0 r − 2 ∬ P 2 r + ( 0 , 1 ) ∣ U − ( U ) ( 0 , 1 ) , 2 r , + a ∣ 2 μ a d t , ∣ I I I ∣ ≤ 1 4 ε 0 ( 2 r ) 2 ∥ U ∥ L ∞ ( P 1 4 + ( 0 , 1 ) ) 2 ∬ P 2 r + ( 0 , 1 ) ∣ ∂ t U ∣ 2 μ a d t + C ε 0 r − 2 ∬ P 2 r + ( 0 , 1 ) ∣ U − ( U ) ( 0 , 1 ) , 2 r , + a ∣ 2 μ a d t ≤ ε 0 ( 2 r ) 2 ∬ P 2 r + ( 0 , 1 ) ∣ ∂ t U ∣ 2 μ a d t + C ε 0 r − 2 ∬ P 2 r + ( 0 , 1 ) ∣ U − ( U ) ( 0 , 1 ) , 2 r , + a ∣ 2 μ a d t . \begin{aligned}
|III|
&\le\frac14\varepsilon_0(2r)^2\|U\|^2_{L^\infty(P^+_{\frac14}(0,1))}\iint_{P^+_{2r}(0,1)}|\partial_tU|^2\mu_a\,dt\\
&\quad+C\varepsilon_0r^{-2}\iint_{P^+_{2r}(0,1)}|U-(U)^a_{(0,1),2r,+}|^2\mu_a\,dt\\
&\le\varepsilon_0(2r)^2\iint_{P^+_{2r}(0,1)}|\partial_tU|^2\mu_a\,dt+C\varepsilon_0r^{-2}\iint_{P^+_{2r}(0,1)}|U-(U)^a_{(0,1),2r,+}|^2\mu_a\,dt.
\end{aligned} ∣ I I I ∣ ≤ 4 1 ε 0 ( 2 r ) 2 ∥ U ∥ L ∞ ( P 4 1 + ( 0 , 1 )) 2 ∬ P 2 r + ( 0 , 1 ) ∣ ∂ t U ∣ 2 μ a d t + C ε 0 r − 2 ∬ P 2 r + ( 0 , 1 ) ∣ U − ( U ) ( 0 , 1 ) , 2 r , + a ∣ 2 μ a d t ≤ ε 0 ( 2 r ) 2 ∬ P 2 r + ( 0 , 1 ) ∣ ∂ t U ∣ 2 μ a d t + C ε 0 r − 2 ∬ P 2 r + ( 0 , 1 ) ∣ U − ( U ) ( 0 , 1 ) , 2 r , + a ∣ 2 μ a d t . To estimate I V IV I V ( ∣ U ∣ 2 − 1 ) η 2 = 0 (|U|^2-1)\eta^2=0 ( ∣ U ∣ 2 − 1 ) η 2 = 0 ∂ P 1 + ( 0 , 1 ) \partial P^+_1(0,1) ∂ P 1 + ( 0 , 1 )
I V = ∬ P 1 + ( 0 , 1 ) [ D ( ∣ U ∣ 2 − 1 ) ⟨ U , D U ⟩ η 2 + D ( ∣ U ∣ 2 − 1 ) ⟨ U , U − ( U ) ( 0 , 1 ) , 2 r , + a ⟩ D η 2 ] μ a d t = − ∬ P 1 + ( 0 , 1 ) ( ∣ U ∣ 2 − 1 ) div X ( y a ⟨ U , D U ⟩ η 2 ) μ a d t + ∬ P 1 + ( 0 , 1 ) D ( ∣ U ∣ 2 − 1 ) ⟨ U , U − ( U ) ( 0 , 1 ) , 2 r , + a ⟩ D η 2 μ a d t = − ∬ P 1 + ( 0 , 1 ) ( ∣ U ∣ 2 − 1 ) ( ⟨ U , ∂ t U ⟩ η 2 + ∣ D U ∣ 2 η 2 + ⟨ U , D U ⟩ D η 2 ) μ a d t + ∬ P 1 + ( 0 , 1 ) D ( ∣ U ∣ 2 − 1 ) ⟨ U , U − ( U ) ( 0 , 1 ) , 2 r , + a ⟩ D η 2 μ a d t = I V 1 + I V 2 . \begin{aligned}
IV
={}&\iint_{P^+_1(0,1)}
\left[D(|U|^2-1)\langle U,DU\rangle\eta^2+D(|U|^2-1)\langle U,U-(U)^a_{(0,1),2r,+}\rangle D\eta^2\right]\mu_a\,dt\\
={}&-\iint_{P^+_1(0,1)}(|U|^2-1)\operatorname{div}_X(y^a\langle U,DU\rangle\eta^2)\mu_a\,dt\\
&+\iint_{P^+_1(0,1)}D(|U|^2-1)\langle U,U-(U)^a_{(0,1),2r,+}\rangle D\eta^2\mu_a\,dt\\
={}&-\iint_{P^+_1(0,1)}(|U|^2-1)
\left(\langle U,\partial_tU\rangle\eta^2+|DU|^2\eta^2+\langle U,DU\rangle D\eta^2\right)\mu_a\,dt\\
&+\iint_{P^+_1(0,1)}D(|U|^2-1)\langle U,U-(U)^a_{(0,1),2r,+}\rangle D\eta^2\mu_a\,dt\\
={}&IV_1+IV_2.
\end{aligned} I V = = = = ∬ P 1 + ( 0 , 1 ) [ D ( ∣ U ∣ 2 − 1 ) ⟨ U , D U ⟩ η 2 + D ( ∣ U ∣ 2 − 1 ) ⟨ U , U − ( U ) ( 0 , 1 ) , 2 r , + a ⟩ D η 2 ] μ a d t − ∬ P 1 + ( 0 , 1 ) ( ∣ U ∣ 2 − 1 ) div X ( y a ⟨ U , D U ⟩ η 2 ) μ a d t + ∬ P 1 + ( 0 , 1 ) D ( ∣ U ∣ 2 − 1 ) ⟨ U , U − ( U ) ( 0 , 1 ) , 2 r , + a ⟩ D η 2 μ a d t − ∬ P 1 + ( 0 , 1 ) ( ∣ U ∣ 2 − 1 ) ( ⟨ U , ∂ t U ⟩ η 2 + ∣ D U ∣ 2 η 2 + ⟨ U , D U ⟩ D η 2 ) μ a d t + ∬ P 1 + ( 0 , 1 ) D ( ∣ U ∣ 2 − 1 ) ⟨ U , U − ( U ) ( 0 , 1 ) , 2 r , + a ⟩ D η 2 μ a d t I V 1 + I V 2 . Applying (3.8) and Hölder’s inequality, we can bound I V 1 IV_1 I V 1
∣ I V 1 ∣ ≤ ∥ ∣ U ∣ 2 − 1 ∥ L ∞ ( P 1 4 + ( 0 , 1 ) ) ∬ P 1 + ( 0 , 1 ) ∣ D U ∣ 2 η 2 μ a d t + 2 ∬ P 1 4 + ( 0 , 1 ) ∣ U ∣ 2 − 1 ∣ ( ∣ ∂ t U ∣ η 2 + ∣ D U ∣ ∣ D η 2 ∣ ) μ a d t ≤ ε 0 2 ∬ P 2 r + ( 0 , 1 ) ( r 2 ∣ ∂ t U ∣ 2 + ∣ D U ∣ 2 ) μ a d t + C ε 0 − 1 r − 2 ∬ P 2 r + ( 0 , 1 ) ( ∣ U ∣ 2 − 1 ) 2 μ a d t . \begin{aligned}
|IV_1| &\le\| |U|^2-1\|_{L^\infty(P_{\frac{1}{4}}^+(0,1))}\iint_{P_1^+(0,1)} |DU|^2\eta^2\,\mu_a\,dt \\
&\quad+2\iint_{P^+_{\frac{1}{4}}(0,1)} |U|^2-1|(|\partial_t U|\eta^2+|DU||D\eta^2|)\,\mu_a\,dt \\
&\le\frac{\varepsilon_0}{2}\iint_{P^+_{2r}(0,1)}(r^2|\partial_t U|^2+|DU|^2)\,\mu_a\,dt+C\varepsilon_0^{-1}r^{-2}\iint_{P^+_{2r}(0,1)}(|U|^2-1)^2\,\mu_a\,dt.
\end{aligned} ∣ I V 1 ∣ ≤ ∥∣ U ∣ 2 − 1 ∥ L ∞ ( P 4 1 + ( 0 , 1 )) ∬ P 1 + ( 0 , 1 ) ∣ D U ∣ 2 η 2 μ a d t + 2 ∬ P 4 1 + ( 0 , 1 ) ∣ U ∣ 2 − 1∣ ( ∣ ∂ t U ∣ η 2 + ∣ D U ∣∣ D η 2 ∣ ) μ a d t ≤ 2 ε 0 ∬ P 2 r + ( 0 , 1 ) ( r 2 ∣ ∂ t U ∣ 2 + ∣ D U ∣ 2 ) μ a d t + C ε 0 − 1 r − 2 ∬ P 2 r + ( 0 , 1 ) ( ∣ U ∣ 2 − 1 ) 2 μ a d t . while I V 2 IV_2 I V 2
∣ I V 2 ∣ ≤ ε 0 2 ∬ P 2 r + ( 0 , 1 ) ∣ D U ∣ 2 μ a d t + C ε 0 − 1 r − 2 ∬ P 2 r + ( 0 , 1 ) ∣ U − ( U ) ( 0 , 1 ) , 2 r , + a ∣ 2 μ a d t . |IV_2|\le\frac{\varepsilon_0}{2}\iint_{P^+_{2r}(0,1)}|DU|^2\,\mu_a\,dt+C\varepsilon_0^{-1}r^{-2}\iint_{P^+_{2r}(0,1)}|U-(U)^a_{(0,1),2r,+}|^2\,\mu_a\,dt. ∣ I V 2 ∣ ≤ 2 ε 0 ∬ P 2 r + ( 0 , 1 ) ∣ D U ∣ 2 μ a d t + C ε 0 − 1 r − 2 ∬ P 2 r + ( 0 , 1 ) ∣ U − ( U ) ( 0 , 1 ) , 2 r , + a ∣ 2 μ a d t . Substituting the estimates of I V 1 IV_1 I V 1 I V 2 IV_2 I V 2 I V IV I V
∣ I V ∣ ≤ ε 0 ∬ P 2 r + ( 0 , 1 ) ( r 2 ∣ ∂ t U ∣ 2 + ∣ D U ∣ 2 ) μ a d t + C ε 0 − 1 r − 2 ∬ P 2 r + ( 0 , 1 ) ∣ U − ( U ) ( 0 , 1 ) , 2 r , + a ∣ 2 μ a d t + C ε 0 − 1 r − 2 ∬ P 2 r + ( 0 , 1 ) ( ∣ U ∣ 2 − 1 ) 2 μ a d t . \begin{aligned}
|IV|&\le\varepsilon_0\iint_{P^+_{2r}(0,1)}(r^2|\partial_t U|^2+|DU|^2)\,\mu_a\,dt+C\varepsilon_0^{-1}r^{-2}\iint_{P^+_{2r}(0,1)}|U-(U)^a_{(0,1),2r,+}|^2\,\mu_a\,dt\\
&\quad+C\varepsilon_0^{-1}r^{-2}\iint_{P^+_{2r}(0,1)}(|U|^2-1)^2\,\mu_a\,dt.
\end{aligned} ∣ I V ∣ ≤ ε 0 ∬ P 2 r + ( 0 , 1 ) ( r 2 ∣ ∂ t U ∣ 2 + ∣ D U ∣ 2 ) μ a d t + C ε 0 − 1 r − 2 ∬ P 2 r + ( 0 , 1 ) ∣ U − ( U ) ( 0 , 1 ) , 2 r , + a ∣ 2 μ a d t + C ε 0 − 1 r − 2 ∬ P 2 r + ( 0 , 1 ) ( ∣ U ∣ 2 − 1 ) 2 μ a d t . The term V V V I V IV I V
∣ V ∣ = ∣ ∬ P r + ( 0 , 1 ) ( ∣ U ∣ 2 − 1 ) ( ∂ t U ⋅ ( U − ( U ) ( 0 , 1 ) , 2 r , + a ) η 2 + ∣ D U ∣ 2 η 2 + D U ( U − ( U ) ( 0 , 1 ) , 2 r , + a ) ⋅ D η 2 ) μ a d t ∣ ≤ ∥ ∣ U ∣ 2 − 1 ∥ L ∞ ( P 1 4 + ( 0 , 1 ) ) ∬ P 2 r + ( 0 , 1 ) ( ( r ∣ ∂ t U ∣ + ∣ D U ∣ ) r − 1 ∣ U − ( U ) ( 0 , 1 ) , 2 r , + a ∣ + ∣ D U ∣ 2 ) μ a d t ≤ ε 0 ∬ P 2 r + ( 0 , 1 ) ( r 2 ∣ ∂ t U ∣ 2 + ∣ D U ∣ 2 ) μ a d t + C ε 0 r − 2 ∬ P 2 r + ( 0 , 1 ) ∣ U − ( U ) ( 0 , 1 ) , 2 r , + a ∣ 2 μ a d t . \begin{aligned}
|V|=\left|\iint_{P^+_r(0,1)}(|U|^2-1)\left(\partial_tU\cdot(U-(U)^a_{(0,1),2r,+})\eta^2+|DU|^2\eta^2+DU(U-(U)^a_{(0,1),2r,+})\cdot D\eta^2\right)\,\mu_a\,dt\right| \\
&\le\| |U|^2-1\|_{L^\infty(P^+_{\frac{1}{4}}(0,1))}\iint_{P^+_{2r}(0,1)}\left((r|\partial_tU|+|DU|)r^{-1}|U-(U)^a_{(0,1),2r,+}|+|DU|^2\right)\,\mu_a\,dt \\
&\le\varepsilon_0\iint_{P^+_{2r}(0,1)}(r^2|\partial_tU|^2+|DU|^2)\,\mu_a\,dt+C\varepsilon_0r^{-2}\iint_{P^+_{2r}(0,1)}|U-(U)^a_{(0,1),2r,+}|^2\,\mu_a\,dt.
\end{aligned} ∣ V ∣ = ∬ P r + ( 0 , 1 ) ( ∣ U ∣ 2 − 1 ) ( ∂ t U ⋅ ( U − ( U ) ( 0 , 1 ) , 2 r , + a ) η 2 + ∣ D U ∣ 2 η 2 + D U ( U − ( U ) ( 0 , 1 ) , 2 r , + a ) ⋅ D η 2 ) μ a d t ≤ ∥∣ U ∣ 2 − 1 ∥ L ∞ ( P 4 1 + ( 0 , 1 )) ∬ P 2 r + ( 0 , 1 ) ( ( r ∣ ∂ t U ∣ + ∣ D U ∣ ) r − 1 ∣ U − ( U ) ( 0 , 1 ) , 2 r , + a ∣ + ∣ D U ∣ 2 ) μ a d t ≤ ε 0 ∬ P 2 r + ( 0 , 1 ) ( r 2 ∣ ∂ t U ∣ 2 + ∣ D U ∣ 2 ) μ a d t + C ε 0 r − 2 ∬ P 2 r + ( 0 , 1 ) ∣ U − ( U ) ( 0 , 1 ) , 2 r , + a ∣ 2 μ a d t . In order to estimate V I VI V I
V I = ∫ 1 − 4 r 2 1 + 4 r 2 ∫ B 2 r + ( 0 ) D U i ⋅ Ω k i ( U k − ( U ) ( 0 , 1 ) , 2 r , + a ) η 2 μ a d t = − ∫ 1 − 4 r 2 1 + 4 r 2 ∫ B 2 r + ( 0 ) ( U i − ( U i ) ( 0 , 1 ) , 2 r , + a ) ( U i ∂ t U k − U k ∂ t U i ) ( U k − ( U k ) ( 0 , 1 ) , 2 r , + a ) η 2 μ a d t − ∫ 1 − 4 r 2 1 + 4 r 2 ∫ B 2 r + ( 0 ) ( U i − ( U i ) ( 0 , 1 ) , 2 r , + a ) Ω k i D ( ( U k − ( U k ) ( 0 , 1 ) , 2 r , + a ) η 2 ) μ a d t = V I 1 + V I 2 . \begin{aligned}
VI={}&\int_{1-4r^2}^{1+4r^2}\int_{B^+_{2r}(0)}DU^i\cdot\Omega^{ki}(U^k-(U)^a_{(0,1),2r,+})\eta^2\,\mu_a\,dt\\
={}&-\int_{1-4r^2}^{1+4r^2}\int_{B^+_{2r}(0)}(U^i-(U^i)^a_{(0,1),2r,+})(U^i\partial_tU^k-U^k\partial_tU^i)(U^k-(U^k)^a_{(0,1),2r,+})\eta^2\,\mu_a\,dt\\
&-\int_{1-4r^2}^{1+4r^2}\int_{B^+_{2r}(0)}(U^i-(U^i)^a_{(0,1),2r,+})\Omega^{ki}D((U^k-(U^k)^a_{(0,1),2r,+})\eta^2)\,\mu_a\,dt\\
={}&VI_1+VI_2.
\end{aligned} V I = = = ∫ 1 − 4 r 2 1 + 4 r 2 ∫ B 2 r + ( 0 ) D U i ⋅ Ω k i ( U k − ( U ) ( 0 , 1 ) , 2 r , + a ) η 2 μ a d t − ∫ 1 − 4 r 2 1 + 4 r 2 ∫ B 2 r + ( 0 ) ( U i − ( U i ) ( 0 , 1 ) , 2 r , + a ) ( U i ∂ t U k − U k ∂ t U i ) ( U k − ( U k ) ( 0 , 1 ) , 2 r , + a ) η 2 μ a d t − ∫ 1 − 4 r 2 1 + 4 r 2 ∫ B 2 r + ( 0 ) ( U i − ( U i ) ( 0 , 1 ) , 2 r , + a ) Ω k i D (( U k − ( U k ) ( 0 , 1 ) , 2 r , + a ) η 2 ) μ a d t V I 1 + V I 2 . By Hölder’s inequality, V I 1 VI_1 V I 1
∣ V I 1 ∣ ≤ ε 0 2 ∬ P 2 r + ( 0 , 1 ) r 2 ∣ ∂ t U ∣ 2 μ a d t + C ε 0 − 1 r − 2 ∬ P 2 r + ( 0 , 1 ) ∣ U − ( U ) ( 0 , 1 ) , 2 r , + a ∣ 4 μ a d t . |VI_1|\le\frac{\varepsilon_0}{2}\iint_{P^+_{2r}(0,1)}r^2|\partial_tU|^2\,\mu_a\,dt+C\varepsilon_0^{-1}r^{-2}\iint_{P^+_{2r}(0,1)}|U-(U)^a_{(0,1),2r,+}|^4\,\mu_a\,dt. ∣ V I 1 ∣ ≤ 2 ε 0 ∬ P 2 r + ( 0 , 1 ) r 2 ∣ ∂ t U ∣ 2 μ a d t + C ε 0 − 1 r − 2 ∬ P 2 r + ( 0 , 1 ) ∣ U − ( U ) ( 0 , 1 ) , 2 r , + a ∣ 4 μ a d t . The estimate of main term V I 2 VI_2 V I 2 t ∈ ( 1 − 4 r 2 , 1 + 4 r 2 ) t\in(1-4r^2,1+4r^2) t ∈ ( 1 − 4 r 2 , 1 + 4 r 2 ) y a Ω k i ( ⋅ , t ) ∈ L 2 ( B 2 r + ( 0 ) , μ − a ) y^a\Omega^{ki}(\cdot,t)\in L^2(B^+_{2r}(0),\mu_{-a}) y a Ω k i ( ⋅ , t ) ∈ L 2 ( B 2 r + ( 0 ) , μ − a ) μ − a = y − a d X \mu_{-a}=y^{-a}dX μ − a = y − a d X y − a y^{-a} y − a A 2 A_2 A 2
G k i ( t ) ∈ H 1 ( B 2 r + ( 0 ) , μ − a ) and H k i ( t ) ∈ H 0 1 ( B 2 r + ( 0 ) , μ − a ; ∧ 2 R m + 1 ) G^{ki}(t)\in H^1(B^+_{2r}(0),\mu_{-a})\quad\text{and}\quad H^{ki}(t)\in H^1_0(B^+_{2r}(0),\mu_{-a};\wedge^2\mathbb{R}^{m+1}) G k i ( t ) ∈ H 1 ( B 2 r + ( 0 ) , μ − a ) and H k i ( t ) ∈ H 0 1 ( B 2 r + ( 0 ) , μ − a ; ∧ 2 R m + 1 ) such that
y a Ω k i ( ⋅ , t ) = D G k i ( t ) + curl H k i ( t ) in B 2 r + ( 0 ) , (25) y^{a}\Omega^{ki}(\cdot,t)=DG^{ki}(t)+\operatorname{curl}H^{ki}(t)\quad\text{in }B^+_{2r}(0), \tag*{(25)} y a Ω k i ( ⋅ , t ) = D G k i ( t ) + curl H k i ( t ) in B 2 r + ( 0 ) , (25) and
∫ B 2 r + ( 0 ) ∣ D G k i ( t ) ∣ 2 μ − a + ∫ B 2 r + ( 0 ) ∣ D H k i ( t ) ∣ 2 μ − a ≤ C ∫ B 2 r + ( 0 ) ( y a ∣ Ω k i ( ⋅ , t ) ∣ ) 2 ( X , t ) μ − a = C ∫ B 2 r + ( 0 ) ∣ Ω k i ( ⋅ , t ) ∣ 2 ( X , t ) μ a . (26) \int_{B^+_{2r}(0)}|DG^{ki}(t)|^2\mu_{-a}+\int_{B^+_{2r}(0)}|DH^{ki}(t)|^2\mu_{-a}
\le C\int_{B^+_{2r}(0)}(y^a|\Omega^{ki}(\cdot,t)|)^2(X,t)\mu_{-a}
=C\int_{B^+_{2r}(0)}|\Omega^{ki}(\cdot,t)|^2(X,t)\mu_a. \tag*{(26)} ∫ B 2 r + ( 0 ) ∣ D G k i ( t ) ∣ 2 μ − a + ∫ B 2 r + ( 0 ) ∣ D H k i ( t ) ∣ 2 μ − a ≤ C ∫ B 2 r + ( 0 ) ( y a ∣ Ω k i ( ⋅ , t ) ∣ ) 2 ( X , t ) μ − a = C ∫ B 2 r + ( 0 ) ∣ Ω k i ( ⋅ , t ) ∣ 2 ( X , t ) μ a . (26) In view of (2.2), G k i ( t ) G^{ki}(t) G k i ( t )
{ − Δ X G k i ( t ) = y a ( U i ∂ t U k − U k ∂ t U i ) ( ⋅ , t ) in B 2 r + ( 0 ) , ∂ G k i ( t ) ∂ y = 0 on ∂ B 2 r + ( 0 ) ∩ ∂ R + m + 1 . (27) \begin{cases}
-\Delta_XG^{ki}(t)=y^a(U^i\partial_tU^k-U^k\partial_tU^i)(\cdot,t)&\text{in }B^+_{2r}(0),\\
\dfrac{\partial G^{ki}(t)}{\partial y}=0&\text{on }\partial B^+_{2r}(0)\cap\partial\mathbb{R}^{m+1}_+. \tag*{(27)}
\end{cases} ⎩ ⎨ ⎧ − Δ X G k i ( t ) = y a ( U i ∂ t U k − U k ∂ t U i ) ( ⋅ , t ) ∂ y ∂ G k i ( t ) = 0 in B 2 r + ( 0 ) , on ∂ B 2 r + ( 0 ) ∩ ∂ R + m + 1 . (27) By the y − a y^{-a} y − a W 2 , 2 W^{2,2} W 2 , 2 G k i ∈ H 2 ( B 3 r 2 + ( 0 ) , μ − a ) G^{ki}\in H^2(B^+_{\frac{3r}{2}}(0),\mu_{-a}) G k i ∈ H 2 ( B 2 3 r + ( 0 ) , μ − a )
∫ B 3 r 2 + ( 0 ) ( ∣ D G k i ∣ 2 ( X , t ) + r 2 ∣ D 2 G k i ∣ 2 ( X , t ) ) μ − a ≤ C ∫ B 2 r + ( 0 ) ( ∣ D G k i ∣ 2 ( X , t ) + r 2 y 2 a ∣ ∂ t U ∣ 2 ( X , t ) ) μ − a ≤ C ∫ B 2 r + ( 0 ) ( ∣ y ∣ 2 a ∣ Ω k i ∣ 2 ( X , t ) + r 2 y 2 a ∣ ∂ t U ∣ 2 ( X , t ) ) μ − a ≤ C ∫ B 2 r + ( 0 ) ( ∣ D U ∣ 2 ( X , t ) + r 2 ∣ ∂ t U ∣ 2 ( X , t ) ) μ a . (28) \begin{aligned}
\int_{B^+_{\frac{3r}{2}}(0)}\left(|DG^{ki}|^2(X,t)+r^2|D^2G^{ki}|^2(X,t)\right)\mu_{-a}
\le C\int_{B^+_{2r}(0)}\left(|DG^{ki}|^2(X,t)+r^2y^{2a}|\partial_tU|^2(X,t)\right)\mu_{-a} \\
&\le C\int_{B^+_{2r}(0)}\left(|y|^{2a}|\Omega^{ki}|^2(X,t)+r^2y^{2a}|\partial_tU|^2(X,t)\right)\mu_{-a}
\le C\int_{B^+_{2r}(0)}\left(|DU|^2(X,t)+r^2|\partial_tU|^2(X,t)\right)\mu_a. \tag*{(28)}
\end{aligned} ∫ B 2 3 r + ( 0 ) ( ∣ D G k i ∣ 2 ( X , t ) + r 2 ∣ D 2 G k i ∣ 2 ( X , t ) ) μ − a ≤ C ∫ B 2 r + ( 0 ) ( ∣ D G k i ∣ 2 ( X , t ) + r 2 y 2 a ∣ ∂ t U ∣ 2 ( X , t ) ) μ − a ≤ C ∫ B 2 r + ( 0 ) ( ∣ y ∣ 2 a ∣ Ω k i ∣ 2 ( X , t ) + r 2 y 2 a ∣ ∂ t U ∣ 2 ( X , t ) ) μ − a ≤ C ∫ B 2 r + ( 0 ) ( ∣ D U ∣ 2 ( X , t ) + r 2 ∣ ∂ t U ∣ 2 ( X , t ) ) μ a . (28) By the y a y^a y a
∥ D G k i ( t ) ∥ L 2 a ∗ ( B 3 r 2 + ( 0 ) , μ − a ) ≤ C ( ∥ D 2 G k i ( t ) ∥ L 2 ( B 3 r 2 + ( 0 ) , μ − a ) + r − 1 ∥ D G k i ( t ) ∥ L 2 ( B 3 r 2 + ( 0 ) , μ − a ) ) ≤ C r − 1 ( ∥ D U ( ⋅ , t ) ∥ L 2 ( B 2 r + ( 0 ) , μ a ) + r ∥ ∂ t U ( ⋅ , t ) ∥ L 2 ( B 2 r + ( 0 ) , μ a ) ) , \begin{aligned}
\|DG^{ki}(t)\|_{L^{2_a^*}(B^+_{\frac{3r}{2}}(0),\mu_{-a})}
\le C\left(\|D^2G^{ki}(t)\|_{L^2(B^+_{\frac{3r}{2}}(0),\mu_{-a})}+r^{-1}\|DG^{ki}(t)\|_{L^2(B^+_{\frac{3r}{2}}(0),\mu_{-a})}\right) \\
&\le Cr^{-1}\left(\|DU(\cdot,t)\|_{L^2(B^+_{2r}(0),\mu_a)}+r\|\partial_tU(\cdot,t)\|_{L^2(B^+_{2r}(0),\mu_a)}\right),
\end{aligned} ∥ D G k i ( t ) ∥ L 2 a ∗ ( B 2 3 r + ( 0 ) , μ − a ) ≤ C ( ∥ D 2 G k i ( t ) ∥ L 2 ( B 2 3 r + ( 0 ) , μ − a ) + r − 1 ∥ D G k i ( t ) ∥ L 2 ( B 2 3 r + ( 0 ) , μ − a ) ) ≤ C r − 1 ( ∥ D U ( ⋅ , t ) ∥ L 2 ( B 2 r + ( 0 ) , μ a ) + r ∥ ∂ t U ( ⋅ , t ) ∥ L 2 ( B 2 r + ( 0 ) , μ a ) ) , where
2 a ∗ = { 2 ( m + 1 − a ) m − 1 − a , m ≥ 2 or m = 1 and a < 0 , ∞ , m = 1 and a > 0. 2_a^*=
\begin{cases}
\dfrac{2(m+1-a)}{m-1-a},&m\ge2\text{ or }m=1\text{ and }a<0,\\
\infty,&m=1\text{ and }a>0.
\end{cases} 2 a ∗ = ⎩ ⎨ ⎧ m − 1 − a 2 ( m + 1 − a ) , ∞ , m ≥ 2 or m = 1 and a < 0 , m = 1 and a > 0. Combining condition (3.6), the interior Hölder continuity of U U U y a y^a y a θ 0 ∈ ( 0 , 1 ) \theta_0\in(0,1) θ 0 ∈ ( 0 , 1 ) ε 0 , t 0 , E 0 \varepsilon_0,t_0,E_0 ε 0 , t 0 , E 0
1 ∣ P r + ( z ) ∣ a ∬ P r + ( z ) ∣ U − ( U ) z , r , + a ∣ 2 μ a d t ≤ C r 2 ∣ P r + ( z ) ∣ a ∬ P r + ( z ) ( r 2 ∣ ∂ t U ∣ 2 + ∣ D U ∣ 2 ) μ a d t ≤ C ∬ P 1 2 + ( 0 , 1 ) ( ∣ ∂ t U ∣ 2 + ∣ D U ∣ 2 ) μ a d t ≤ C ε 0 2 . (29) \begin{aligned}
\frac{1}{|P_r^+(z)|_a}\iint_{P_r^+(z)}|U-(U)^a_{z,r,+}|^2\mu_a\,dt
\le\frac{Cr^2}{|P_r^+(z)|_a}\iint_{P_r^+(z)}(r^2|\partial_tU|^2+|DU|^2)\mu_a\,dt \\
&\le C\iint_{P^+_{\frac12}(0,1)}(|\partial_tU|^2+|DU|^2)\mu_a\,dt
\le C\varepsilon_0^2. \tag*{(29)}
\end{aligned} ∣ P r + ( z ) ∣ a 1 ∬ P r + ( z ) ∣ U − ( U ) z , r , + a ∣ 2 μ a d t ≤ ∣ P r + ( z ) ∣ a C r 2 ∬ P r + ( z ) ( r 2 ∣ ∂ t U ∣ 2 + ∣ D U ∣ 2 ) μ a d t ≤ C ∬ P 2 1 + ( 0 , 1 ) ( ∣ ∂ t U ∣ 2 + ∣ D U ∣ 2 ) μ a d t ≤ C ε 0 2 . (29) holds for all z ∈ P 1 2 + ( 0 , 1 ) z \in P_{\frac{1}{2}}^+(0,1) z ∈ P 2 1 + ( 0 , 1 ) 0 < r < θ 0 0<r<\theta_0 0 < r < θ 0 U ∈ BMO ( P θ 0 + ( 0 , 1 ) , μ a d t ) U\in\operatorname{BMO}(P_{\theta_0}^+(0,1),\mu_a dt) U ∈ BMO ( P θ 0 + ( 0 , 1 ) , μ a d t )
[ u ] BMO ( P θ 0 + ( 0 , 1 ) , μ a d t ) ≤ C ε 0 . (30) [u]_{\operatorname{BMO}(P_{\theta_0}^+(0,1),\mu_a dt)}\le C\varepsilon_0. \tag*{(30)} [ u ] BMO ( P θ 0 + ( 0 , 1 ) , μ a d t ) ≤ C ε 0 . (30) Since y a y^a y a A 2 A_2 A 2 (99) to obtain that for any z ∈ P θ 0 + ( 0 , 1 ) z\in P_{\theta_0}^+(0,1) z ∈ P θ 0 + ( 0 , 1 ) 0 < r < θ 0 0<r<\theta_0 0 < r < θ 0 ( U − ( U ) z , r , + a ) ∈ L p ( P r + ( z ) , μ a d t ) (U-(U)^a_{z,r,+})\in L^p(P_r^+(z),\mu_a dt) ( U − ( U ) z , r , + a ) ∈ L p ( P r + ( z ) , μ a d t ) p ∈ ( 1 , ∞ ) p\in(1,\infty) p ∈ ( 1 , ∞ )
1 ∣ P r + ( z ) ∣ a ∬ P r + ( z ) ∣ U − ( U ) z , r , + a ∣ p μ a d t ≤ C ( m , p ) [ u ] BMO ( P θ 0 + ( 0 , 1 ) , μ a d t ) p ≤ C ( m , p ) ε 0 p . (31) \frac{1}{|P_r^+(z)|_a}\iint_{P_r^+(z)}|U-(U)^a_{z,r,+}|^p\mu_a dt\le C(m,p)[u]_{\operatorname{BMO}(P_{\theta_0}^+(0,1),\mu_a dt)}^p\le C(m,p)\varepsilon_0^p. \tag*{(31)} ∣ P r + ( z ) ∣ a 1 ∬ P r + ( z ) ∣ U − ( U ) z , r , + a ∣ p μ a d t ≤ C ( m , p ) [ u ] BMO ( P θ 0 + ( 0 , 1 ) , μ a d t ) p ≤ C ( m , p ) ε 0 p . (31) Decomposing I V 2 IV_2 I V 2 (25) , we have
− V I 2 = ∫ 1 − 4 r 2 1 + 4 r 2 ∫ B 2 r + ( 0 ) ( U i − ( U i ) ( 0 , 1 ) , 2 r , + a ) Ω k i D ( ( U k − ( U k ) ( 0 , 1 ) , 2 r , + a ) η 2 ) μ a d t = ∫ 1 − 4 r 2 1 + 4 r 2 ∫ B 2 r + ( 0 ) ( U i − ( U i ) ( 0 , 1 ) , 2 r , + a ) D G k i D ( ( U k − ( U k ) ( 0 , 1 ) , 2 r , + a ) η 2 ) + ∫ 1 − 4 r 2 1 + 4 r 2 ∫ B 2 r + ( 0 ) ( U i − ( U i ) ( 0 , 1 ) , 2 r , + a ) curl H k i D ( ( U k − ( U k ) ( 0 , 1 ) , 2 r , + a ) η 2 ) = V I 21 + V I 22 . \begin{aligned}
-VI_2={}&\int_{1-4r^2}^{1+4r^2}\int_{B_{2r}^+(0)}(U^i-(U^i)^a_{(0,1),2r,+})\Omega^{ki}D((U^k-(U^k)^a_{(0,1),2r,+})\eta^2)\mu_a dt\\
={}&\int_{1-4r^2}^{1+4r^2}\int_{B_{2r}^+(0)}(U^i-(U^i)^a_{(0,1),2r,+})DG^{ki}D((U^k-(U^k)^a_{(0,1),2r,+})\eta^2)\\
&+\int_{1-4r^2}^{1+4r^2}\int_{B_{2r}^+(0)}(U^i-(U^i)^a_{(0,1),2r,+})\operatorname{curl}H^{ki}D((U^k-(U^k)^a_{(0,1),2r,+})\eta^2)\\
={}&VI_{21}+VI_{22}.
\end{aligned} − V I 2 = = = ∫ 1 − 4 r 2 1 + 4 r 2 ∫ B 2 r + ( 0 ) ( U i − ( U i ) ( 0 , 1 ) , 2 r , + a ) Ω k i D (( U k − ( U k ) ( 0 , 1 ) , 2 r , + a ) η 2 ) μ a d t ∫ 1 − 4 r 2 1 + 4 r 2 ∫ B 2 r + ( 0 ) ( U i − ( U i ) ( 0 , 1 ) , 2 r , + a ) D G k i D (( U k − ( U k ) ( 0 , 1 ) , 2 r , + a ) η 2 ) + ∫ 1 − 4 r 2 1 + 4 r 2 ∫ B 2 r + ( 0 ) ( U i − ( U i ) ( 0 , 1 ) , 2 r , + a ) curl H k i D (( U k − ( U k ) ( 0 , 1 ) , 2 r , + a ) η 2 ) V I 21 + V I 22 . Using Hölder’s inequality with conjugate exponent ( 2 a ∗ ) ′ = 2 a ∗ 2 a ∗ − 1 (2_a^*)'=\frac{2_a^*}{2_a^*-1} ( 2 a ∗ ) ′ = 2 a ∗ − 1 2 a ∗ V I 21 VI_{21} V I 21
∣ V I 21 ∣ ≤ ( ∫ 1 − 4 r 2 1 + 4 r 2 ∥ D G ∥ L a ∗ 2 ( B 2 r + ( 0 ) , μ − a ) 2 d t ) 1 2 ⋅ ( ∫ 1 − 4 r 2 1 + 4 r 2 ∥ D ( ( U − ( U ) ( 0 , 1 ) , 2 r , + a ) η 2 ) ( U − ( U ) ( 0 , 1 ) , 2 r , + a ) ∥ L ( 2 a ∗ ) ′ ( B 2 r + ( 0 ) , μ a ) 2 d t ) 1 2 ≤ C r − 1 ( ∬ P 2 r + ( 0 , 1 ) ( r 2 ∣ ∂ t U ∣ 2 + ∣ D U ∣ 2 ) μ a d t ) 1 2 ⋅ ( ∫ 1 − 4 r 2 1 + 4 r 2 ∥ r − 1 ∣ U − ( U ) ( 0 , 1 ) , 2 r , + a ∣ 2 + ∣ D U ∣ ∣ U − ( U ) ( 0 , 1 ) , 2 r , + a ∣ ∥ L ( 2 a ∗ ) ′ ( B 2 r + ( 0 ) , μ a ) 2 d t ) 1 2 ≤ C r − 1 ( ∬ P 2 r + ( 0 , 1 ) ( r 2 ∣ ∂ t U ∣ 2 + ∣ D U ∣ 2 ) μ a d t ) 1 2 ⋅ ( ∫ 1 − 4 r 2 1 + 4 r 2 ∥ r − 1 ∣ U − ( U ) ( 0 , 1 ) , 2 r , + a ∣ 2 + ∣ D U ∣ ∣ U − ( U ) ( 0 , 1 ) , 2 r , + a ∣ ∥ L ( 2 a ∗ ) ′ ( B 2 r + ( 0 ) , μ a ) 2 d t ) 1 2 ≤ C ε 0 ∬ P 2 r + ( 0 , 1 ) ( r 2 ∣ ∂ t U ∣ 2 + ∣ D U ∣ 2 ) μ a d t + C ε 0 − 1 r − 2 ∬ P 2 r + ( 0 , 1 ) ∣ U − ( U ) ( 0 , 1 ) , 2 r , + a ∣ 4 μ a d t . \begin{aligned}
|VI_{21}|\le{}&
\left(\int_{1-4r^2}^{1+4r^2}\|DG\|_{L^2_{a^*}(B_{2r}^+(0),\mu_{-a})}^2dt\right)^{\frac12}\\
&\cdot\left(\int_{1-4r^2}^{1+4r^2}\|D((U-(U)^a_{(0,1),2r,+})\eta^2)(U-(U)^a_{(0,1),2r,+})\|_{L^{(2_a^*)'}(B_{2r}^+(0),\mu_a)}^2dt\right)^{\frac12}\\
\le{}&Cr^{-1}\left(\iint_{P_{2r}^+(0,1)}(r^2|\partial_tU|^2+|DU|^2)\mu_a dt\right)^{\frac12}\\
&\cdot\left(\int_{1-4r^2}^{1+4r^2}\left\|r^{-1}|U-(U)^a_{(0,1),2r,+}|^2+|DU||U-(U)^a_{(0,1),2r,+}|\right\|_{L^{(2_a^*)'}(B_{2r}^+(0),\mu_a)}^2dt\right)^{\frac12}\\
\le{}&Cr^{-1}\left(\iint_{P_{2r}^+(0,1)}(r^2|\partial_tU|^2+|DU|^2)\mu_a dt\right)^{\frac12}\\
&\cdot\left(\int_{1-4r^2}^{1+4r^2}\left\|r^{-1}|U-(U)^a_{(0,1),2r,+}|^2+|DU||U-(U)^a_{(0,1),2r,+}|\right\|_{L^{(2_a^*)'}(B_{2r}^+(0),\mu_a)}^2dt\right)^{\frac12}\\
\le{}&C\varepsilon_0\iint_{P_{2r}^+(0,1)}(r^2|\partial_tU|^2+|DU|^2)\mu_a dt\\
&+C\varepsilon_0^{-1}r^{-2}\iint_{P_{2r}^+(0,1)}|U-(U)^a_{(0,1),2r,+}|^4\mu_a dt.
\end{aligned} ∣ V I 21 ∣ ≤ ≤ ≤ ≤ ( ∫ 1 − 4 r 2 1 + 4 r 2 ∥ D G ∥ L a ∗ 2 ( B 2 r + ( 0 ) , μ − a ) 2 d t ) 2 1 ⋅ ( ∫ 1 − 4 r 2 1 + 4 r 2 ∥ D (( U − ( U ) ( 0 , 1 ) , 2 r , + a ) η 2 ) ( U − ( U ) ( 0 , 1 ) , 2 r , + a ) ∥ L ( 2 a ∗ ) ′ ( B 2 r + ( 0 ) , μ a ) 2 d t ) 2 1 C r − 1 ( ∬ P 2 r + ( 0 , 1 ) ( r 2 ∣ ∂ t U ∣ 2 + ∣ D U ∣ 2 ) μ a d t ) 2 1 ⋅ ( ∫ 1 − 4 r 2 1 + 4 r 2 r − 1 ∣ U − ( U ) ( 0 , 1 ) , 2 r , + a ∣ 2 + ∣ D U ∣∣ U − ( U ) ( 0 , 1 ) , 2 r , + a ∣ L ( 2 a ∗ ) ′ ( B 2 r + ( 0 ) , μ a ) 2 d t ) 2 1 C r − 1 ( ∬ P 2 r + ( 0 , 1 ) ( r 2 ∣ ∂ t U ∣ 2 + ∣ D U ∣ 2 ) μ a d t ) 2 1 ⋅ ( ∫ 1 − 4 r 2 1 + 4 r 2 r − 1 ∣ U − ( U ) ( 0 , 1 ) , 2 r , + a ∣ 2 + ∣ D U ∣∣ U − ( U ) ( 0 , 1 ) , 2 r , + a ∣ L ( 2 a ∗ ) ′ ( B 2 r + ( 0 ) , μ a ) 2 d t ) 2 1 C ε 0 ∬ P 2 r + ( 0 , 1 ) ( r 2 ∣ ∂ t U ∣ 2 + ∣ D U ∣ 2 ) μ a d t + C ε 0 − 1 r − 2 ∬ P 2 r + ( 0 , 1 ) ∣ U − ( U ) ( 0 , 1 ) , 2 r , + a ∣ 4 μ a d t . Finally, we estimate V I 22 VI_{22} V I 22 H k i ( t ) H^{ki}(t) H k i ( t ) 0 0 0 B 2 r + ( 0 ) B_{2r}^+(0) B 2 r + ( 0 ) U U U ∂ R + m + 1 \partial\mathbb{R}_+^{m+1} ∂ R + m + 1 y a y^a y a
∣ V I 22 ∣ = ∣ ∫ 1 − 4 r 2 1 + 4 r 2 ∫ R + m + 1 ( U i − ( U i ) ( 0 , 1 ) , 2 r , + a ) ⋅ ( curl H k i ) ⋅ D ( ( U k − ( U k ) ( 0 , 1 ) , 2 r a ) η 2 ) ∣ ≤ C ∥ D ( ( U k − ( U k ) ( 0 , 1 ) , 2 r a ) η 2 ) ⋅ curl H k i ∥ H 1 ( R m + 1 × R ) [ U i ] BMO ( P 2 r + ( 0 , 1 ) ) ≤ C ∥ D U i ∥ L 2 ( P 2 r + ( 0 , 1 ) , μ a d t ) ∥ D H k i ∥ L 2 ( P 2 r + ( 0 , 1 ) , μ − a d t ) [ U i ] BMO ( P 2 r + ( 0 , 1 ) ) ≤ C ε 0 ∬ P 2 r + ( 0 , 1 ) ( r 2 ∣ ∂ t U ∣ 2 + ∣ D U ∣ 2 ) μ a d t . \begin{aligned}
|VI_{22}|&=\left|\int_{1-4r^2}^{1+4r^2}\int_{\mathbb{R}^{m+1}_+}\left(U^i-(U^i)^{a}_{(0,1),2r,+}\right)\cdot(\operatorname{curl}H^{ki})\cdot D\left((U^k-(U^k)^{a}_{(0,1),2r})\eta^2\right)\right|\\
&\le C\left\|D\left((U^k-(U^k)^{a}_{(0,1),2r})\eta^2\right)\cdot\operatorname{curl}H^{ki}\right\|_{\mathcal{H}^1(\mathbb{R}^{m+1}\times\mathbb{R})}\left[U^i\right]_{\operatorname{BMO}(P^+_{2r}(0,1))}\\
&\le C\left\|D U^i\right\|_{L^2(P^+_{2r}(0,1),\mu_a dt)}\left\|D H^{ki}\right\|_{L^2(P^+_{2r}(0,1),\mu_{-a}dt)}\left[U^i\right]_{\operatorname{BMO}(P^+_{2r}(0,1))}\\
&\le C\varepsilon_0\iint_{P^+_{2r}(0,1)}\left(r^2|\partial_tU|^2+|DU|^2\right)\mu_a dt.
\end{aligned} ∣ V I 22 ∣ = ∫ 1 − 4 r 2 1 + 4 r 2 ∫ R + m + 1 ( U i − ( U i ) ( 0 , 1 ) , 2 r , + a ) ⋅ ( curl H k i ) ⋅ D ( ( U k − ( U k ) ( 0 , 1 ) , 2 r a ) η 2 ) ≤ C D ( ( U k − ( U k ) ( 0 , 1 ) , 2 r a ) η 2 ) ⋅ curl H k i H 1 ( R m + 1 × R ) [ U i ] BMO ( P 2 r + ( 0 , 1 )) ≤ C D U i L 2 ( P 2 r + ( 0 , 1 ) , μ a d t ) D H k i L 2 ( P 2 r + ( 0 , 1 ) , μ − a d t ) [ U i ] BMO ( P 2 r + ( 0 , 1 )) ≤ C ε 0 ∬ P 2 r + ( 0 , 1 ) ( r 2 ∣ ∂ t U ∣ 2 + ∣ D U ∣ 2 ) μ a d t . where we have used (3.10) and the BMO estimate (see (7.3))
[ U ] BMO ( P 2 r + ( 0 , 1 ) ) ≤ [ U ] BMO ( P θ 0 + ( 0 , 1 ) ) ≤ C [ U ] BMO ( P θ 0 + ( 0 , 1 ) , μ a d t ) ≤ C ( ∥ D U ∥ M 2 , 2 ( P 1 2 + ( 0 , 1 ) , μ a d t ) + ∥ ∂ t U ∥ M 2 , 4 ( P 1 2 + ( 0 , 1 ) , μ a d t ) ) ≤ C ε 0 . \begin{aligned}
\left[U\right]_{\operatorname{BMO}(P^+_{2r}(0,1))}\le\left[U\right]_{\operatorname{BMO}(P^+_{\theta_0}(0,1))}\le C\left[U\right]_{\operatorname{BMO}(P^+_{\theta_0}(0,1),\mu_a dt)} \\
&\le C\left(\|DU\|_{\mathcal{M}^{2,2}(P^+_{\frac12}(0,1),\mu_a dt)}+\|\partial_tU\|_{\mathcal{M}^{2,4}(P^+_{\frac12}(0,1),\mu_a dt)}\right) \\
&\le C\varepsilon_0.
\end{aligned} [ U ] BMO ( P 2 r + ( 0 , 1 )) ≤ [ U ] BMO ( P θ 0 + ( 0 , 1 )) ≤ C [ U ] BMO ( P θ 0 + ( 0 , 1 ) , μ a d t ) ≤ C ( ∥ D U ∥ M 2 , 2 ( P 2 1 + ( 0 , 1 ) , μ a d t ) + ∥ ∂ t U ∥ M 2 , 4 ( P 2 1 + ( 0 , 1 ) , μ a d t ) ) ≤ C ε 0 . Combining the bounds for I I I V I VI V I □ \square □
Energy monotonicity inequality and local energy inequality
In this section, motivated by the definition of stationarity of heat flow of harmonic maps, we introduce the notation of stationary heat flow of s s s
For convenience, we first recall the Definition 1.2. Let ξ ( X , t ) ∈ C 0 ∞ ( R + m + 1 ‾ × ( 0 , ∞ ) , R m + 1 ) \xi(X,t)\in C^\infty_0(\overline{\mathbb{R}^{m+1}_+}\times(0,\infty),\mathbb{R}^{m+1}) ξ ( X , t ) ∈ C 0 ∞ ( R + m + 1 × ( 0 , ∞ ) , R m + 1 ) ξ m + 1 ≥ 0 \xi^{m+1}\ge0 ξ m + 1 ≥ 0 θ ( X , t ) ∈ C 0 ∞ ( R + m + 1 × ( 0 , ∞ ) ) \theta(X,t)\in C^\infty_0(\mathbb{R}^{m+1}_+\times(0,\infty)) θ ( X , t ) ∈ C 0 ∞ ( R + m + 1 × ( 0 , ∞ )) θ ≥ 0 \theta\ge0 θ ≥ 0 U ∈ L ∞ ( R + ; H 1 ( ( R + m + 1 , μ a ) , R ℓ ) ) ∩ H 1 ( R + ; L 2 ( ( R + m + 1 , μ a ) , R ℓ ) ) U\in L^\infty(\mathbb{R}_+;H^1((\mathbb{R}^{m+1}_+,\mu_a),\mathbb{R}^{\ell}))\cap H^1(\mathbb{R}_+;L^2((\mathbb{R}^{m+1}_+,\mu_a),\mathbb{R}^{\ell})) U ∈ L ∞ ( R + ; H 1 (( R + m + 1 , μ a ) , R ℓ )) ∩ H 1 ( R + ; L 2 (( R + m + 1 , μ a ) , R ℓ )) τ > 0 \tau>0 τ > 0 U τ U^\tau U τ U U U
U τ ( X , t ) = U ( X + τ ξ ( X , t ) , t + τ θ ( X , t ) ) , ( X , t ) ∈ R + m + 1 × [ 0 , ∞ ) . U^\tau(X,t)=U(X+\tau\xi(X,t),t+\tau\theta(X,t)),\qquad(X,t)\in\mathbb{R}^{m+1}_+\times[0,\infty). U τ ( X , t ) = U ( X + τ ξ ( X , t ) , t + τ θ ( X , t )) , ( X , t ) ∈ R + m + 1 × [ 0 , ∞ ) . Definition 4.1 A weak solution U : R + m + 1 × ( 0 , ∞ ) → R ℓ U:\mathbb{R}^{m+1}_+\times(0,\infty)\to\mathbb{R}^{\ell} U : R + m + 1 × ( 0 , ∞ ) → R ℓ s s s
2 ∬ R + m + 1 × R + ∂ t U ⋅ d d τ ∣ τ = 0 U τ μ a d t + d d τ ∣ τ = 0 ∬ R + m + 1 × R + ∣ D U τ ∣ 2 μ a d t ≤ 0. (32) 2\iint_{\mathbb{R}^{m+1}_+\times\mathbb{R}_+}\partial_tU\cdot\left.\frac{d}{d\tau}\right|_{\tau=0}U^\tau\mu_a dt+\left.\frac{d}{d\tau}\right|_{\tau=0}\iint_{\mathbb{R}^{m+1}_+\times\mathbb{R}_+}|DU^\tau|^2\mu_a dt\le0. \tag*{(32)} 2 ∬ R + m + 1 × R + ∂ t U ⋅ d τ d τ = 0 U τ μ a d t + d τ d τ = 0 ∬ R + m + 1 × R + ∣ D U τ ∣ 2 μ a d t ≤ 0. (32) holds for all ξ ( X , t ) ∈ C 0 ∞ ( R + m + 1 ‾ × ( 0 , ∞ ) , R m + 1 ) \xi(X,t)\in C^\infty_0(\overline{\mathbb{R}^{m+1}_+}\times(0,\infty),\mathbb{R}^{m+1}) ξ ( X , t ) ∈ C 0 ∞ ( R + m + 1 × ( 0 , ∞ ) , R m + 1 ) ξ m + 1 ≥ 0 \xi^{m+1}\ge0 ξ m + 1 ≥ 0 ξ m + 1 ( ( x , 0 ) , t ) = 0 \xi^{m+1}((x,0),t)=0 ξ m + 1 (( x , 0 ) , t ) = 0 ( x , t ) ∈ R m × ( 0 , ∞ ) (x,t)\in\mathbb{R}^m\times(0,\infty) ( x , t ) ∈ R m × ( 0 , ∞ ) 0 ≤ θ ( X , t ) ∈ C 0 ∞ ( R + m + 1 × ( 0 , ∞ ) ) 0\le\theta(X,t)\in C^\infty_0(\mathbb{R}^{m+1}_+\times(0,\infty)) 0 ≤ θ ( X , t ) ∈ C 0 ∞ ( R + m + 1 × ( 0 , ∞ ))
Since ξ ( ⋅ , t ) : ∂ R + m + 1 → ∂ R + m + 1 \xi(\cdot,t):\partial\mathbb{R}^{m+1}_+\to\partial\mathbb{R}^{m+1}_+ ξ ( ⋅ , t ) : ∂ R + m + 1 → ∂ R + m + 1 t > 0 t>0 t > 0 U τ ( x , 0 , t ) ∈ S ℓ − 1 U^\tau(x,0,t)\in S^{\ell-1} U τ ( x , 0 , t ) ∈ S ℓ − 1 ( x , t ) ∈ R m × ( 0 , ∞ ) (x,t)\in\mathbb{R}^m\times(0,\infty) ( x , t ) ∈ R m × ( 0 , ∞ ) U τ U^\tau U τ U U U
d d τ ∣ τ = 0 U τ = ξ ⋅ D U + θ ∂ t U , \left.\frac{d}{d\tau}\right|_{\tau=0}U^\tau=\xi\cdot DU+\theta\partial_tU, d τ d τ = 0 U τ = ξ ⋅ D U + θ ∂ t U , so that condition (4.1) takes the form
∬ R + m + 1 × R + ∂ t U ⋅ ( ξ ⋅ D U + θ ∂ t U ) μ a d t + ∬ R + m + 1 × R + D U ⋅ D ( ξ ⋅ D U + θ ∂ t U ) μ a d t ≤ 0. (33) \iint_{\mathbb{R}^{m+1}_+\times\mathbb{R}_+}\partial_tU\cdot(\xi\cdot DU+\theta\partial_tU)\mu_a dt+\iint_{\mathbb{R}^{m+1}_+\times\mathbb{R}_+}DU\cdot D(\xi\cdot DU+\theta\partial_tU)\mu_a dt\le0. \tag*{(33)} ∬ R + m + 1 × R + ∂ t U ⋅ ( ξ ⋅ D U + θ ∂ t U ) μ a d t + ∬ R + m + 1 × R + D U ⋅ D ( ξ ⋅ D U + θ ∂ t U ) μ a d t ≤ 0. (33) For any 0 < t 1 < t 2 < ∞ 0<t_1<t_2<\infty 0 < t 1 < t 2 < ∞ θ ≡ 0 \theta\equiv0 θ ≡ 0 ξ ∈ C ∞ ( R + m + 1 ‾ × ( t 1 , t 2 ) ) \xi\in C^\infty(\overline{\mathbb{R}^{m+1}_+}\times(t_1,t_2)) ξ ∈ C ∞ ( R + m + 1 × ( t 1 , t 2 )) ξ m + 1 = 0 \xi^{m+1}=0 ξ m + 1 = 0 ∂ R + m + 1 × ( 0 , ∞ ) \partial\mathbb{R}^{m+1}_+\times(0,\infty) ∂ R + m + 1 × ( 0 , ∞ ) ξ \xi ξ − ξ -\xi − ξ U U U ∂ R + m + 1 × ( 0 , ∞ ) \partial\mathbb{R}^{m+1}_+\times(0,\infty) ∂ R + m + 1 × ( 0 , ∞ )
∬ R + m + 1 × [ t 1 , t 2 ] y a ∂ t U ⋅ ( ξ ⋅ D U ) − 1 2 div X ( y a ξ ) ∣ D U ∣ 2 + y a D U ⊗ D U : D ξ = 0. (34) \iint_{\mathbb{R}^{m+1}_+\times[t_1,t_2]} y^a\partial_tU\cdot(\xi\cdot DU)-\frac{1}{2}\operatorname{div}_X(y^a\xi)|DU|^2+y^aDU\otimes DU:D\xi=0. \tag*{(34)} ∬ R + m + 1 × [ t 1 , t 2 ] y a ∂ t U ⋅ ( ξ ⋅ D U ) − 2 1 div X ( y a ξ ) ∣ D U ∣ 2 + y a D U ⊗ D U : D ξ = 0. (34) On the other hand, setting ξ ≡ 0 \xi\equiv0 ξ ≡ 0 0 ≤ θ ∈ C 0 ∞ ( R + m + 1 × [ t 1 , t 2 ] ) 0\leq\theta\in C^\infty_0(\mathbb{R}^{m+1}_+\times[t_1,t_2]) 0 ≤ θ ∈ C 0 ∞ ( R + m + 1 × [ t 1 , t 2 ])
∬ R + m + 1 × { t 2 } ∣ D U ∣ 2 θ μ a − ∬ R + m + 1 × { t 1 } ∣ D U ∣ 2 θ μ a ≤ ∬ R + m + 1 × [ t 1 , t 2 ] ( ∣ D U ∣ 2 ∂ t θ − 2 D θ ⋅ D U ⋅ ∂ t U − 2 ∣ ∂ t U ∣ 2 θ ) μ a d t . (35) \iint_{\mathbb{R}^{m+1}_+\times\{t_2\}}|DU|^2\theta\mu_a-\iint_{\mathbb{R}^{m+1}_+\times\{t_1\}}|DU|^2\theta\mu_a
\leq\iint_{\mathbb{R}^{m+1}_+\times[t_1,t_2]}\left(|DU|^2\partial_t\theta-2D\theta\cdot DU\cdot\partial_tU-2|\partial_tU|^2\theta\right)\mu_a\,dt. \tag*{(35)} ∬ R + m + 1 × { t 2 } ∣ D U ∣ 2 θ μ a − ∬ R + m + 1 × { t 1 } ∣ D U ∣ 2 θ μ a ≤ ∬ R + m + 1 × [ t 1 , t 2 ] ( ∣ D U ∣ 2 ∂ t θ − 2 D θ ⋅ D U ⋅ ∂ t U − 2∣ ∂ t U ∣ 2 θ ) μ a d t . (35) For X 0 = ( X 0 , 0 ) ∈ ∂ R + m + 1 X_0=(X_0,0)\in\partial\mathbb{R}^{m+1}_+ X 0 = ( X 0 , 0 ) ∈ ∂ R + m + 1 t 0 > 0 t_0>0 t 0 > 0 G ( X 0 , t 0 ) a G^a_{(X_0,t_0)} G ( X 0 , t 0 ) a L a V : = ∣ y ∣ a ∂ t V − div ( ∣ y ∣ a D V ) \mathcal{L}_aV:=|y|^a\partial_tV-\operatorname{div}(|y|^aDV) L a V := ∣ y ∣ a ∂ t V − div ( ∣ y ∣ a D V ) R + m + 1 \mathbb{R}^{m+1}_+ R + m + 1
G ( X 0 , t 0 ) a ( X , t ) = 1 Γ ( 1 − a 2 ) ( 4 π ) m 2 ∣ t − t 0 ∣ m + 1 + a 2 exp ( − ∣ X − X 0 ∣ 2 4 ∣ t − t 0 ∣ ) , t < t 0 . (36) G^a_{(X_0,t_0)}(X,t)=\frac{1}{\Gamma\left(\frac{1-a}{2}\right)(4\pi)^{\frac{m}{2}}|t-t_0|^{\frac{m+1+a}{2}}}\exp\left(-\frac{|X-X_0|^2}{4|t-t_0|}\right),\qquad t<t_0. \tag*{(36)} G ( X 0 , t 0 ) a ( X , t ) = Γ ( 2 1 − a ) ( 4 π ) 2 m ∣ t − t 0 ∣ 2 m + 1 + a 1 exp ( − 4∣ t − t 0 ∣ ∣ X − X 0 ∣ 2 ) , t < t 0 . (36) In particular, when s = 1 / 2 s=1/2 s = 1/2 a = 0 a=0 a = 0 G ( X 0 , t 0 ) 0 G^0_{(X_0,t_0)} G ( X 0 , t 0 ) 0 G ( X 0 , t 0 ) G_{(X_0,t_0)} G ( X 0 , t 0 ) R m + 1 \mathbb{R}^{m+1} R m + 1 0 < R ≤ t 0 2 0<R\leq\frac{\sqrt{t_0}}{2} 0 < R ≤ 2 t 0
T R + ( t 0 ) = { ( X , t ) ∈ R + m + 1 × R + : t 0 − 4 R 2 < t < t 0 − R 2 } , T_R^+(t_0)=\{(X,t)\in\mathbb{R}^{m+1}_+\times\mathbb{R}_+:t_0-4R^2<t<t_0-R^2\}, T R + ( t 0 ) = {( X , t ) ∈ R + m + 1 × R + : t 0 − 4 R 2 < t < t 0 − R 2 } , ∂ + T R + ( t 0 ) = { ( x , t ) ∈ R m × R + : t 0 − 4 R 2 < t < t 0 − R 2 } , \partial^+T_R^+(t_0)=\{(x,t)\in\mathbb{R}^{m}\times\mathbb{R}_+:t_0-4R^2<t<t_0-R^2\}, ∂ + T R + ( t 0 ) = {( x , t ) ∈ R m × R + : t 0 − 4 R 2 < t < t 0 − R 2 } , and abbreviate T 1 + = T 1 + ( 0 ) T_1^+=T_1^+(0) T 1 + = T 1 + ( 0 ) ∂ + T 1 + = ∂ T 1 + ( 0 ) \partial^+T_1^+=\partial T_1^+(0) ∂ + T 1 + = ∂ T 1 + ( 0 )
Lemma 4.2 If U U U s s s a = 1 − 2 s a=1-2s a = 1 − 2 s Z 0 = ( X 0 , t 0 ) Z_0=(X_0,t_0) Z 0 = ( X 0 , t 0 ) X 0 ∈ ∂ R + m + 1 X_0\in\partial\mathbb{R}^{m+1}_+ X 0 ∈ ∂ R + m + 1 t 0 > 0 t_0>0 t 0 > 0
D a ( U , Z 0 , R ) : = R 2 2 ∬ R + m + 1 × { t 0 − R 2 } ∣ D U ∣ 2 G ( X 0 , t 0 ) a ( X , t ) μ a \mathcal{D}^a(U,Z_0,R):=\frac{R^2}{2}\iint_{\mathbb{R}^{m+1}_+\times\{t_0-R^2\}}|DU|^2G^a_{(X_0,t_0)}(X,t)\mu_a D a ( U , Z 0 , R ) := 2 R 2 ∬ R + m + 1 × { t 0 − R 2 } ∣ D U ∣ 2 G ( X 0 , t 0 ) a ( X , t ) μ a and
E a ( U , Z 0 , R ) : = 1 2 ∬ T R + ( t 0 ) ∣ D U ∣ 2 G ( X 0 , t 0 ) a ( X , t ) μ a d t \mathcal{E}^a(U,Z_0,R):=\frac{1}{2}\iint_{T_R^+(t_0)}|DU|^2G^a_{(X_0,t_0)}(X,t)\mu_a\,dt E a ( U , Z 0 , R ) := 2 1 ∬ T R + ( t 0 ) ∣ D U ∣ 2 G ( X 0 , t 0 ) a ( X , t ) μ a d t are monotone nondecreasing with respect to R R R
D a ( U , Z 0 , r ) ≤ D a ( U , Z 0 , R ) , 0 < r ≤ R < t 0 , (37) \mathcal{D}^a(U,Z_0,r)\leq\mathcal{D}^a(U,Z_0,R),\qquad0<r\leq R<\sqrt{t_0}, \tag*{(37)} D a ( U , Z 0 , r ) ≤ D a ( U , Z 0 , R ) , 0 < r ≤ R < t 0 , (37) and
E a ( U , Z 0 , r ) ≤ E a ( U , Z 0 , R ) , 0 < r ≤ R < 1 2 t 0 . (38) \mathcal{E}^a(U,Z_0,r)\leq\mathcal{E}^a(U,Z_0,R),\qquad0<r\leq R<\frac{1}{2}\sqrt{t_0}. \tag*{(38)} E a ( U , Z 0 , r ) ≤ E a ( U , Z 0 , R ) , 0 < r ≤ R < 2 1 t 0 . (38) Proof. Without loss of generality, assume X 0 = ( 0 , 0 ) X_0=(0,0) X 0 = ( 0 , 0 ) t 0 = 1 t_0=1 t 0 = 1 G a ( X , t ) = G ( X 0 , t 0 ) a ( X , t ) G^a(X,t)=G^a_{(X_0,t_0)}(X,t) G a ( X , t ) = G ( X 0 , t 0 ) a ( X , t ) − ξ -\xi − ξ ξ = X G a ( X , t ) \xi=XG^a(X,t) ξ = X G a ( X , t ) ξ m + 1 \xi^{m+1} ξ m + 1 ξ m + 1 = 0 \xi^{m+1}=0 ξ m + 1 = 0 ∂ R + m + 1 × ( 0 , ∞ ) \partial\mathbb{R}^{m+1}_+\times(0,\infty) ∂ R + m + 1 × ( 0 , ∞ ) 1
div X ( y a ξ ) = y a ( m + 1 + a − ∣ X ∣ 2 2 ( 1 − t ) ) G a , \operatorname{div}_{X}(y^a\xi)=y^a\left(m+1+a-\frac{|X|^2}{2(1-t)}\right)G^a, div X ( y a ξ ) = y a ( m + 1 + a − 2 ( 1 − t ) ∣ X ∣ 2 ) G a , ∇ X i ξ j = { ( δ i j − X i X j 2 ( 1 − t ) ) G a , 1 ≤ i , j ≤ m , − y X i 2 ( 1 − t ) G a , 1 ≤ i ≤ m , j = m + 1 , − y X j 2 ( 1 − t ) G a , i = m + 1 , 1 ≤ j ≤ m , ( 1 − y 2 2 ( 1 − t ) ) G a , i = j = m + 1. \nabla_{X_i}\xi^j=
\begin{cases}
\left(\delta_{ij}-\frac{X_iX_j}{2(1-t)}\right)G^a, & 1\le i,j\le m,\\
-\frac{yX_i}{2(1-t)}G^a, & 1\le i\le m,\ j=m+1,\\
-\frac{yX_j}{2(1-t)}G^a, & i=m+1,\ 1\le j\le m,\\
\left(1-\frac{y^2}{2(1-t)}\right)G^a, & i=j=m+1.
\end{cases} ∇ X i ξ j = ⎩ ⎨ ⎧ ( δ ij − 2 ( 1 − t ) X i X j ) G a , − 2 ( 1 − t ) y X i G a , − 2 ( 1 − t ) y X j G a , ( 1 − 2 ( 1 − t ) y 2 ) G a , 1 ≤ i , j ≤ m , 1 ≤ i ≤ m , j = m + 1 , i = m + 1 , 1 ≤ j ≤ m , i = j = m + 1. For 0 < r < R < 1 0<r<R<1 0 < r < R < 1 t 1 = 1 − R 2 t_1=1-R^2 t 1 = 1 − R 2 t 2 = 1 − r 2 t_2=1-r^2 t 2 = 1 − r 2 ξ \xi ξ
0 = ∬ R + m + 1 × [ 1 − R 2 , 1 − r 2 ] ∂ t U ⋅ ( X ⋅ D U ) G a μ a d t + ∬ R + m + 1 × [ 1 − R 2 , 1 − r 2 ] [ ( − m − 1 + a 2 + ∣ X ∣ 2 4 ( 1 − t ) ) ∣ D U ∣ 2 − ∣ X ⋅ D U ∣ 2 2 ( 1 − t ) ] G a μ a d t . (39) \begin{aligned}
0={}&\iint_{\mathbb{R}^{m+1}_+\times[1-R^2,1-r^2]}\partial_tU\cdot(X\cdot DU)G^a\mu_a\,dt\\
&+\iint_{\mathbb{R}^{m+1}_+\times[1-R^2,1-r^2]}
\left[\left(-\frac{m-1+a}{2}+\frac{|X|^2}{4(1-t)}\right)|DU|^2-\frac{|X\cdot DU|^2}{2(1-t)}\right]G^a\mu_a\,dt. \tag*{(39)}
\end{aligned} 0 = ∬ R + m + 1 × [ 1 − R 2 , 1 − r 2 ] ∂ t U ⋅ ( X ⋅ D U ) G a μ a d t + ∬ R + m + 1 × [ 1 − R 2 , 1 − r 2 ] [ ( − 2 m − 1 + a + 4 ( 1 − t ) ∣ X ∣ 2 ) ∣ D U ∣ 2 − 2 ( 1 − t ) ∣ X ⋅ D U ∣ 2 ] G a μ a d t . (39) Next, choosing θ = ( 1 − t ) G a \theta=(1-t)G^a θ = ( 1 − t ) G a θ t = m − 1 + a 2 G a − ∣ X ∣ 2 4 ( 1 − t ) G a \theta_t=\frac{m-1+a}{2}G^a-\frac{|X|^2}{4(1-t)}G^a θ t = 2 m − 1 + a G a − 4 ( 1 − t ) ∣ X ∣ 2 G a θ \theta θ
∬ R + m + 1 × [ 1 − R 2 , 1 − r 2 ] ( − 2 ( 1 − t ) ∣ ∂ t U ∣ 2 + X ⋅ D U ⋅ ∂ t U + ( m − 1 + a 2 − ∣ X ∣ 2 4 ( 1 − t ) ) ∣ D U ∣ 2 ) G a μ a d t ≥ ∫ R + m + 1 × { 1 − r 2 } ∣ D U ∣ 2 r 2 G a μ a − ∫ R + m + 1 × { 1 − R 2 } ∣ D U ∣ 2 R 2 G a μ a . (40) \begin{aligned}
&\iint_{\mathbb{R}^{m+1}_+\times[1-R^2,1-r^2]}
\left(-2(1-t)|\partial_tU|^2+X\cdot DU\cdot\partial_tU+\left(\frac{m-1+a}{2}-\frac{|X|^2}{4(1-t)}\right)|DU|^2\right)G^a\mu_a\,dt\\
&\qquad\ge\int_{\mathbb{R}^{m+1}_+\times\{1-r^2\}}|DU|^2r^2G^a\mu_a-\int_{\mathbb{R}^{m+1}_+\times\{1-R^2\}}|DU|^2R^2G^a\mu_a. \tag*{(40)}
\end{aligned} ∬ R + m + 1 × [ 1 − R 2 , 1 − r 2 ] ( − 2 ( 1 − t ) ∣ ∂ t U ∣ 2 + X ⋅ D U ⋅ ∂ t U + ( 2 m − 1 + a − 4 ( 1 − t ) ∣ X ∣ 2 ) ∣ D U ∣ 2 ) G a μ a d t ≥ ∫ R + m + 1 × { 1 − r 2 } ∣ D U ∣ 2 r 2 G a μ a − ∫ R + m + 1 × { 1 − R 2 } ∣ D U ∣ 2 R 2 G a μ a . (40) Adding (4.8) and (4.9), we obtain
R 2 ∫ R + m + 1 × { 1 − R 2 } ∣ D U ∣ 2 G a μ a − r 2 ∫ R + m + 1 × { 1 − r 2 } ∣ D U ∣ 2 G a μ a ≥ ∬ R + m + 1 × [ 1 − R 2 , 1 − r 2 ] ( 2 ( 1 − t ) ∣ ∂ t U ∣ 2 − 2 X ⋅ D U ⋅ ∂ t U + 1 2 ( 1 − t ) ∣ X ⋅ D U ∣ 2 ) G a μ a d t = ∬ R + m + 1 × [ 1 − R 2 , 1 − r 2 ] ∣ X ⋅ D U + 2 ( t − 1 ) ∂ t U ∣ 2 2 ( 1 − t ) G a μ a d t . (41) \begin{aligned}
&R^2\int_{\mathbb{R}^{m+1}_+\times\{1-R^2\}}|DU|^2G^a\mu_a-r^2\int_{\mathbb{R}^{m+1}_+\times\{1-r^2\}}|DU|^2G^a\mu_a\\
&\qquad\ge\iint_{\mathbb{R}^{m+1}_+\times[1-R^2,1-r^2]}
\left(2(1-t)|\partial_tU|^2-2X\cdot DU\cdot\partial_tU+\frac{1}{2(1-t)}|X\cdot DU|^2\right)G^a\mu_a\,dt\\
&\qquad=\iint_{\mathbb{R}^{m+1}_+\times[1-R^2,1-r^2]}
\frac{|X\cdot DU+2(t-1)\partial_tU|^2}{2(1-t)}G^a\mu_a\,dt. \tag*{(41)}
\end{aligned} R 2 ∫ R + m + 1 × { 1 − R 2 } ∣ D U ∣ 2 G a μ a − r 2 ∫ R + m + 1 × { 1 − r 2 } ∣ D U ∣ 2 G a μ a ≥ ∬ R + m + 1 × [ 1 − R 2 , 1 − r 2 ] ( 2 ( 1 − t ) ∣ ∂ t U ∣ 2 − 2 X ⋅ D U ⋅ ∂ t U + 2 ( 1 − t ) 1 ∣ X ⋅ D U ∣ 2 ) G a μ a d t = ∬ R + m + 1 × [ 1 − R 2 , 1 − r 2 ] 2 ( 1 − t ) ∣ X ⋅ D U + 2 ( t − 1 ) ∂ t U ∣ 2 G a μ a d t . (41) This proves (4.6). Moreover, observing that
E a ( U , Z 0 , R ) = 2 ∫ R 2 R D a ( U , Z 0 , τ ) τ d τ , \mathcal{E}^a(U,Z_0,R)=2\int_R^{2R}\frac{\mathcal{D}^a(U,Z_0,\tau)}{\tau}\,d\tau, E a ( U , Z 0 , R ) = 2 ∫ R 2 R τ D a ( U , Z 0 , τ ) d τ , so that differentiating with respect to R R R
d d R E a ( U , Z 0 , R ) = 2 R ( D a ( U , Z 0 , 2 R ) − D a ( U , Z 0 , R ) ) ≥ 0. \frac{d}{dR}\mathcal{E}^a(U,Z_0,R)=\frac{2}{R}\left(\mathcal{D}^a(U,Z_0,2R)-\mathcal{D}^a(U,Z_0,R)\right)\ge0. d R d E a ( U , Z 0 , R ) = R 2 ( D a ( U , Z 0 , 2 R ) − D a ( U , Z 0 , R ) ) ≥ 0. This implies (4.7). □
Next, we obtain a local version of y a y^a y a
Lemma 4.3 Let U U U s s s R + m + 1 \mathbb{R}^{m+1}_{+} R + m + 1 a = 1 − 2 s a=1-2s a = 1 − 2 s ε > 0 \varepsilon>0 ε > 0 K ( ε ) ∼ ∣ ln ε ∣ K(\varepsilon)\sim\sqrt{|\ln\varepsilon|} K ( ε ) ∼ ∣ ln ε ∣ Z 0 = ( X 0 , t 0 ) ∈ ∂ R + m + 1 × ( 0 , ∞ ) Z_{0}=(X_{0},t_{0})\in\partial\mathbb{R}^{m+1}_{+}\times(0,\infty) Z 0 = ( X 0 , t 0 ) ∈ ∂ R + m + 1 × ( 0 , ∞ ) 0 < R < t 0 2 K ( ε ) 0<R<\frac{\sqrt{t_{0}}}{2K(\varepsilon)} 0 < R < 2 K ( ε ) t 0
ρ − ( m + 1 + a ) ∬ P ρ + ( Z 0 ) ∣ D U ∣ 2 μ a d t ≤ 2 ln ( 5 2 ) t 0 1 − m − a 2 E 0 + R − ( m + 1 + a ) ∬ P K ( ε ) R + ( Z 0 ) ∣ D U ∣ 2 μ a d t . (42) \rho^{-(m+1+a)}\iint_{P_{\rho}^{+}(Z_{0})}|DU|^{2}\mu_{a}\,dt\leq2\ln\left(\frac{5}{2}\right)t_{0}^{\frac{1-m-a}{2}}E_{0}+R^{-(m+1+a)}\iint_{P_{K(\varepsilon)R}^{+}(Z_{0})}|DU|^{2}\mu_{a}\,dt. \tag*{(42)} ρ − ( m + 1 + a ) ∬ P ρ + ( Z 0 ) ∣ D U ∣ 2 μ a d t ≤ 2 ln ( 2 5 ) t 0 2 1 − m − a E 0 + R − ( m + 1 + a ) ∬ P K ( ε ) R + ( Z 0 ) ∣ D U ∣ 2 μ a d t . (42) holds for 0 < ρ < R 2 0<\rho<\frac{R}{2} 0 < ρ < 2 R E 0 = ∫ R + m + 1 y a ∣ D U ∣ 2 ( ⋅ , 0 ) E_{0}=\int_{\mathbb{R}^{m+1}_{+}}y^{a}|DU|^{2}(\cdot,0) E 0 = ∫ R + m + 1 y a ∣ D U ∣ 2 ( ⋅ , 0 )
Proof. By the definition of G a G^{a} G a c = c ( m , a ) > 0 c=c(m,a)>0 c = c ( m , a ) > 0
ρ − ( m + 1 + a ) ≤ c G ( X 0 , t 0 + 2 ρ 2 ) a ( X , t ) , ∀ ( X , t ) ∈ P ρ + ( Z 0 ) . \rho^{-(m+1+a)}\leq cG^{a}_{(X_{0},t_{0}+2\rho^{2})}(X,t),\quad\forall(X,t)\in P_{\rho}^{+}(Z_{0}). ρ − ( m + 1 + a ) ≤ c G ( X 0 , t 0 + 2 ρ 2 ) a ( X , t ) , ∀ ( X , t ) ∈ P ρ + ( Z 0 ) . Since P ρ + ( Z 0 ) ⊂ T ρ + ( t 0 + 2 ρ 2 ) P_{\rho}^{+}(Z_{0})\subset T_{\rho}^{+}(t_{0}+2\rho^{2}) P ρ + ( Z 0 ) ⊂ T ρ + ( t 0 + 2 ρ 2 ) (38) we have that
ρ − ( m + 1 + a ) ∬ P ρ + ( Z 0 ) ∣ D U ∣ 2 μ a d t ≤ c ∬ P ρ + ( Z 0 ) ∣ D U ∣ 2 G ( X 0 , t 0 + 2 ρ 2 ) a μ a d t ≤ c ∬ T ρ + ( t 0 + 2 ρ 2 ) ∣ D U ∣ 2 G ( X 0 , t 0 + 2 ρ 2 ) a μ a d t ≤ c ∬ T R + ( t 0 + 2 ρ 2 ) ∣ D U ∣ 2 G ( X 0 , t 0 + 2 ρ 2 ) a μ a d t . (43) \begin{aligned}
\rho^{-(m+1+a)}\iint_{P_{\rho}^{+}(Z_{0})}|DU|^{2}\mu_{a}\,dt\leq c\iint_{P_{\rho}^{+}(Z_{0})}|DU|^{2}G^{a}_{(X_{0},t_{0}+2\rho^{2})}\mu_{a}\,dt \\
&\leq c\iint_{T_{\rho}^{+}(t_{0}+2\rho^{2})}|DU|^{2}G^{a}_{(X_{0},t_{0}+2\rho^{2})}\mu_{a}\,dt \\
&\leq c\iint_{T_{R}^{+}(t_{0}+2\rho^{2})}|DU|^{2}G^{a}_{(X_{0},t_{0}+2\rho^{2})}\mu_{a}\,dt. \tag*{(43)}
\end{aligned} ρ − ( m + 1 + a ) ∬ P ρ + ( Z 0 ) ∣ D U ∣ 2 μ a d t ≤ c ∬ P ρ + ( Z 0 ) ∣ D U ∣ 2 G ( X 0 , t 0 + 2 ρ 2 ) a μ a d t ≤ c ∬ T ρ + ( t 0 + 2 ρ 2 ) ∣ D U ∣ 2 G ( X 0 , t 0 + 2 ρ 2 ) a μ a d t ≤ c ∬ T R + ( t 0 + 2 ρ 2 ) ∣ D U ∣ 2 G ( X 0 , t 0 + 2 ρ 2 ) a μ a d t . (43) Now, we claim that
∬ T R + ( t 0 + 2 ρ 2 ) ∣ D U ∣ 2 G ( X 0 , t 0 + 2 ρ 2 ) a μ a d t ≤ 2 ln ( 5 2 ) t 0 1 − m − a 2 E 0 + R − ( m + 1 + a ) ∬ P K ( ε ) R + ( Z 0 ) ∣ D U ∣ 2 μ a d t . (44) \iint_{T_{R}^{+}(t_{0}+2\rho^{2})}|DU|^{2}G^{a}_{(X_{0},t_{0}+2\rho^{2})}\mu_{a}\,dt\leq2\ln\left(\frac{5}{2}\right)t_{0}^{\frac{1-m-a}{2}}E_{0}
+R^{-(m+1+a)}\iint_{P_{K(\varepsilon)R}^{+}(Z_{0})}|DU|^{2}\mu_{a}\,dt. \tag*{(44)} ∬ T R + ( t 0 + 2 ρ 2 ) ∣ D U ∣ 2 G ( X 0 , t 0 + 2 ρ 2 ) a μ a d t ≤ 2 ln ( 2 5 ) t 0 2 1 − m − a E 0 + R − ( m + 1 + a ) ∬ P K ( ε ) R + ( Z 0 ) ∣ D U ∣ 2 μ a d t . (44) The proof of (44) is based on suitable modifications of [8 Theorem 3] (see also [14 Lemma 3.1] ). Here we sketch it. First, as in the proof of [8 Lemma 1] , if we choose K ( ε ) ∼ ∣ log ε ∣ K(\varepsilon)\sim\sqrt{|\log\varepsilon|} K ( ε ) ∼ ∣ log ε ∣
G ( X 0 , t 0 + 2 ρ 2 ) a ( X , t ) ≤ { R − ( m + 1 + a ) ( X , t ) ∈ T R + ( t 0 + 2 ρ 2 ) , ε G ( X 0 , t 0 + 2 ρ 2 + R 2 ) a ( X , t ) ( X , t ) ∈ T R + ( t 0 + 2 ρ 2 ) with ∣ X − X 0 ∣ ≥ K ( ε ) R . G^{a}_{(X_{0},t_{0}+2\rho^{2})}(X,t)\leq
\begin{cases}
R^{-(m+1+a)} & (X,t)\in T_{R}^{+}(t_{0}+2\rho^{2}),\\
\varepsilon G^{a}_{(X_{0},t_{0}+2\rho^{2}+R^{2})}(X,t) & (X,t)\in T_{R}^{+}(t_{0}+2\rho^{2})\text{ with }|X-X_{0}|\geq K(\varepsilon)R.
\end{cases} G ( X 0 , t 0 + 2 ρ 2 ) a ( X , t ) ≤ { R − ( m + 1 + a ) ε G ( X 0 , t 0 + 2 ρ 2 + R 2 ) a ( X , t ) ( X , t ) ∈ T R + ( t 0 + 2 ρ 2 ) , ( X , t ) ∈ T R + ( t 0 + 2 ρ 2 ) with ∣ X − X 0 ∣ ≥ K ( ε ) R . Indeed, for ( X , t ) ∈ T R + ( t 0 + 2 ρ 2 ) (X,t)\in T_{R}^{+}(t_{0}+2\rho^{2}) ( X , t ) ∈ T R + ( t 0 + 2 ρ 2 ) ∣ X − X 0 ∣ ≥ K ( ε ) R |X-X_{0}|\geq K(\varepsilon)R ∣ X − X 0 ∣ ≥ K ( ε ) R
G ( X 0 , t 0 + 2 ρ 2 ) a ( X , t ) G ( X 0 , t 0 + 2 ρ 2 + R 2 ) a ( X , t ) = ( t 0 + 2 ρ 2 + R 2 − t t 0 + 2 ρ 2 − t ) m + 1 + a 2 exp ( − ∣ X − X 0 ∣ 2 4 ( t 0 + 2 ρ 2 − t ) + ∣ X − X 0 ∣ 2 4 ( t 0 + 2 ρ 2 + R 2 − t ) ) \frac{G^{a}_{(X_{0},t_{0}+2\rho^{2})}(X,t)}{G^{a}_{(X_{0},t_{0}+2\rho^{2}+R^{2})}(X,t)}
=
\left(\frac{t_{0}+2\rho^{2}+R^{2}-t}{t_{0}+2\rho^{2}-t}\right)^{\frac{m+1+a}{2}}
\exp\left(-\frac{|X-X_{0}|^{2}}{4(t_{0}+2\rho^{2}-t)}+\frac{|X-X_{0}|^{2}}{4(t_{0}+2\rho^{2}+R^{2}-t)}\right) G ( X 0 , t 0 + 2 ρ 2 + R 2 ) a ( X , t ) G ( X 0 , t 0 + 2 ρ 2 ) a ( X , t ) = ( t 0 + 2 ρ 2 − t t 0 + 2 ρ 2 + R 2 − t ) 2 m + 1 + a exp ( − 4 ( t 0 + 2 ρ 2 − t ) ∣ X − X 0 ∣ 2 + 4 ( t 0 + 2 ρ 2 + R 2 − t ) ∣ X − X 0 ∣ 2 ) ≤ 5 m + 1 + a 2 exp ( − R 2 ∣ X − X 0 ∣ 2 4 ( t 0 + 2 ρ 2 − t ) ( t 0 + 2 ρ 2 + R 2 − t ) ) \leq5^{\frac{m+1+a}{2}}\exp\left(-\frac{R^{2}|X-X_{0}|^{2}}{4(t_{0}+2\rho^{2}-t)(t_{0}+2\rho^{2}+R^{2}-t)}\right) ≤ 5 2 m + 1 + a exp ( − 4 ( t 0 + 2 ρ 2 − t ) ( t 0 + 2 ρ 2 + R 2 − t ) R 2 ∣ X − X 0 ∣ 2 ) ≤ 5 m + 1 + a 2 exp ( − K 2 ( ε ) 8 ) ≤ ε . \leq5^{\frac{m+1+a}{2}}\exp\left(-\frac{K^{2}(\varepsilon)}{8}\right)\leq\varepsilon. ≤ 5 2 m + 1 + a exp ( − 8 K 2 ( ε ) ) ≤ ε . Decomposing the integration domain into B K ( ε ) R + ( X 0 ) B^{+}_{K(\varepsilon)R}(X_{0}) B K ( ε ) R + ( X 0 ) T R + ( t 0 + 2 ρ 2 ) ∖ B K ( ε ) R + ( X 0 ) T^{+}_{R}(t_{0}+2\rho^{2})\setminus B^{+}_{K(\varepsilon)R}(X_{0}) T R + ( t 0 + 2 ρ 2 ) ∖ B K ( ε ) R + ( X 0 )
∬ T R + ( t 0 + 2 ρ 2 ) ∣ D U ∣ 2 G ( X 0 , t 0 + 2 ρ 2 ) a μ a d t ≤ R − ( m + 1 + a ) ∫ t 0 + 2 ρ 2 − R 2 t 0 + 2 ρ 2 − 4 R 2 ∫ B K ( ε ) R + ( X 0 ) ∣ D U ∣ 2 μ a d t + ε ∬ T R + ( t 0 + 2 ρ 2 ) ∣ D U ∣ 2 G ( X 0 , t 0 + 2 ρ 2 + R 2 ) a μ a d t ≤ R − ( m + 1 + 1 ) ∬ P K ( ε ) R + ( Z 0 ) ∣ D U ∣ 2 μ a d t + ε ∬ T R + ( t 0 + 2 ρ 2 ) ∣ D U ∣ 2 G ( X 0 , t 0 + 2 ρ 2 + R 2 ) a μ a d t . (45) \begin{aligned}
\iint_{T^{+}_{R}(t_{0}+2\rho^{2})}|DU|^{2}G^{a}_{(X_{0},t_{0}+2\rho^{2})}\mu_{a}\,dt
&\leq R^{-(m+1+a)}\int_{t_{0}+2\rho^{2}-R^{2}}^{t_{0}+2\rho^{2}-4R^{2}}\int_{B^{+}_{K(\varepsilon)R}(X_{0})}|DU|^{2}\mu_{a}\,dt \\
&\quad+\varepsilon\iint_{T^{+}_{R}(t_{0}+2\rho^{2})}|DU|^{2}G^{a}_{(X_{0},t_{0}+2\rho^{2}+R^{2})}\mu_{a}\,dt \\
&\leq R^{-(m+1+1)}\iint_{P^{+}_{K(\varepsilon)R}(Z_{0})}|DU|^{2}\mu_{a}\,dt \\
&\quad+\varepsilon\iint_{T^{+}_{R}(t_{0}+2\rho^{2})}|DU|^{2}G^{a}_{(X_{0},t_{0}+2\rho^{2}+R^{2})}\mu_{a}\,dt. \tag*{(45)}
\end{aligned} ∬ T R + ( t 0 + 2 ρ 2 ) ∣ D U ∣ 2 G ( X 0 , t 0 + 2 ρ 2 ) a μ a d t ≤ R − ( m + 1 + a ) ∫ t 0 + 2 ρ 2 − R 2 t 0 + 2 ρ 2 − 4 R 2 ∫ B K ( ε ) R + ( X 0 ) ∣ D U ∣ 2 μ a d t + ε ∬ T R + ( t 0 + 2 ρ 2 ) ∣ D U ∣ 2 G ( X 0 , t 0 + 2 ρ 2 + R 2 ) a μ a d t ≤ R − ( m + 1 + 1 ) ∬ P K ( ε ) R + ( Z 0 ) ∣ D U ∣ 2 μ a d t + ε ∬ T R + ( t 0 + 2 ρ 2 ) ∣ D U ∣ 2 G ( X 0 , t 0 + 2 ρ 2 + R 2 ) a μ a d t . (45) As in [8 page 172] , we can apply Lemma 4.2 to estimate
∬ T R + ( t 0 + 2 ρ 2 ) ∣ D U ∣ 2 G ( X 0 , t 0 + 2 ρ 2 + R 2 ) a μ a d t = 2 ∫ t 0 + 2 ρ 2 − 4 R 2 t 0 + 2 ρ 2 − R 2 ( t 0 + 2 ρ 2 + R 2 − t ) − 1 ⋅ D a ( U , ( X 0 , t 0 + 2 ρ 2 + R 2 ) , t 0 + 2 ρ 2 + R 2 − t ) d t ≤ 2 D a ( U , ( X 0 , t 0 + 2 ρ 2 + R 2 ) , t 0 + 2 ρ 2 ) ⋅ ∫ t 0 + 2 ρ 2 − 4 R 2 t 0 + 2 ρ 2 − R 2 ( t 0 + 2 ρ 2 + R 2 − t ) − 1 d t ≤ 2 ln ( 5 2 ) ( t 0 + 2 ρ 2 + R 2 ) 1 − m − a 2 E 0 ≤ 2 ln ( 5 2 ) t 0 1 − m − a 2 E 0 . (46) \begin{aligned}
\iint_{T^{+}_{R}(t_{0}+2\rho^{2})}|DU|^{2}G^{a}_{(X_{0},t_{0}+2\rho^{2}+R^{2})}\mu_{a}\,dt
&=2\int_{t_{0}+2\rho^{2}-4R^{2}}^{t_{0}+2\rho^{2}-R^{2}}(t_{0}+2\rho^{2}+R^{2}-t)^{-1} \\
&\quad\cdot D^{a}\left(U,(X_{0},t_{0}+2\rho^{2}+R^{2}),\sqrt{t_{0}+2\rho^{2}+R^{2}-t}\right)\,dt \\
&\leq2D^{a}\left(U,(X_{0},t_{0}+2\rho^{2}+R^{2}),\sqrt{t_{0}+2\rho^{2}}\right) \\
&\quad\cdot\int_{t_{0}+2\rho^{2}-4R^{2}}^{t_{0}+2\rho^{2}-R^{2}}(t_{0}+2\rho^{2}+R^{2}-t)^{-1}\,dt \\
&\leq2\ln\left(\frac{5}{2}\right)(t_{0}+2\rho^{2}+R^{2})^{\frac{1-m-a}{2}}E_{0} \\
&\leq2\ln\left(\frac{5}{2}\right)t_{0}^{\frac{1-m-a}{2}}E_{0}. \tag*{(46)}
\end{aligned} ∬ T R + ( t 0 + 2 ρ 2 ) ∣ D U ∣ 2 G ( X 0 , t 0 + 2 ρ 2 + R 2 ) a μ a d t = 2 ∫ t 0 + 2 ρ 2 − 4 R 2 t 0 + 2 ρ 2 − R 2 ( t 0 + 2 ρ 2 + R 2 − t ) − 1 ⋅ D a ( U , ( X 0 , t 0 + 2 ρ 2 + R 2 ) , t 0 + 2 ρ 2 + R 2 − t ) d t ≤ 2 D a ( U , ( X 0 , t 0 + 2 ρ 2 + R 2 ) , t 0 + 2 ρ 2 ) ⋅ ∫ t 0 + 2 ρ 2 − 4 R 2 t 0 + 2 ρ 2 − R 2 ( t 0 + 2 ρ 2 + R 2 − t ) − 1 d t ≤ 2 ln ( 2 5 ) ( t 0 + 2 ρ 2 + R 2 ) 2 1 − m − a E 0 ≤ 2 ln ( 2 5 ) t 0 2 1 − m − a E 0 . (46) Combining (4.14) with (4.15) yields (4.13). Substituting (4.13) into (4.12) completes the proof of (4.11). □
The stationarity condition (4.1) also leads to the local energy inequality. More precisely, let 0 ≤ η ∈ C 0 ∞ ( R m + 1 × ( 0 , ∞ ) ) 0\leq\eta\in C^{\infty}_{0}(\mathbb{R}^{m+1}\times(0,\infty)) 0 ≤ η ∈ C 0 ∞ ( R m + 1 × ( 0 , ∞ )) y y y η ( ( x , − y ) , t ) = η ( x , y , t ) \eta((x,-y),t)=\eta(x,y,t) η (( x , − y ) , t ) = η ( x , y , t ) X = ( x , y ) ∈ R m + 1 X=(x,y)\in\mathbb{R}^{m+1} X = ( x , y ) ∈ R m + 1 t > 0 t>0 t > 0 ξ ( X , t ) = D η ∈ C 0 ∞ ( R + m + 1 ) \xi(X,t)=D\eta\in C^{\infty}_{0}(\mathbb{R}^{m+1}_{+}) ξ ( X , t ) = D η ∈ C 0 ∞ ( R + m + 1 ) ξ ( ( x , 0 ) , t ) = 0 \xi((x,0),t)=0 ξ (( x , 0 ) , t ) = 0 ( x , t ) ∈ R m × ( 0 , ∞ ) (x,t)\in\mathbb{R}^{m}\times(0,\infty) ( x , t ) ∈ R m × ( 0 , ∞ ) ξ \xi ξ θ \theta θ η \eta η
∫ R + m + 1 × { t 2 } ∣ D U ∣ 2 η μ a − ∫ R + m + 1 × { t 1 } ∣ D U ∣ 2 η μ a + 2 ∫ R + m + 1 × [ t 1 , t 2 ] ∣ ∂ t U ∣ 2 η μ a d t ≤ ∬ R + m + 1 × [ t 1 , t 2 ] [ ∣ D U ∣ 2 ( y a ∂ t η + div X ( y a D η ) ) − 2 y a D U ⊗ D U : D 2 η ] d X d t . (47) \begin{aligned}
\int_{\mathbb{R}^{m+1}_{+}\times\{t_{2}\}}|DU|^{2}\eta\mu_{a}
-\int_{\mathbb{R}^{m+1}_{+}\times\{t_{1}\}}|DU|^{2}\eta\mu_{a}
+2\int_{\mathbb{R}^{m+1}_{+}\times[t_{1},t_{2}]}|\partial_{t}U|^{2}\eta\mu_{a}\,dt \\
\leq\iint_{\mathbb{R}^{m+1}_{+}\times[t_{1},t_{2}]}
\left[|DU|^{2}\left(y^{a}\partial_{t}\eta+\operatorname{div}_{X}(y^{a}D\eta)\right)
-2y^{a}DU\otimes DU:D^{2}\eta\right]\,dX\,dt. \tag*{(47)}
\end{aligned} ∫ R + m + 1 × { t 2 } ∣ D U ∣ 2 η μ a − ∫ R + m + 1 × { t 1 } ∣ D U ∣ 2 η μ a + 2 ∫ R + m + 1 × [ t 1 , t 2 ] ∣ ∂ t U ∣ 2 η μ a d t ≤ ∬ R + m + 1 × [ t 1 , t 2 ] [ ∣ D U ∣ 2 ( y a ∂ t η + div X ( y a D η ) ) − 2 y a D U ⊗ D U : D 2 η ] d X d t . (47) As an immediate consequence of (4.16), we establish the local y a y^{a} y a
Lemma 4.4 If U U U s s s R + m + 1 \mathbb{R}^{m+1}_{+} R + m + 1 a = 1 − 2 s a=1-2s a = 1 − 2 s
∬ P R + ( Z 0 ) ∣ ∂ t U ∣ 2 μ a d t ≤ C R − 2 ∬ P 2 R + ( Z 0 ) ∣ D U ∣ 2 μ a d t . (48) \iint_{P^{+}_{R}(Z_{0})}|\partial_{t}U|^{2}\mu_{a}\,dt\leq CR^{-2}\iint_{P^{+}_{2R}(Z_{0})}|DU|^{2}\mu_{a}\,dt. \tag*{(48)} ∬ P R + ( Z 0 ) ∣ ∂ t U ∣ 2 μ a d t ≤ C R − 2 ∬ P 2 R + ( Z 0 ) ∣ D U ∣ 2 μ a d t . (48) holds for every Z 0 = ( X 0 , t 0 ) ∈ ∂ R + m + 1 × ( 0 , ∞ ) Z_{0}=(X_{0},t_{0})\in\partial\mathbb{R}^{m+1}_{+}\times(0,\infty) Z 0 = ( X 0 , t 0 ) ∈ ∂ R + m + 1 × ( 0 , ∞ ) 0 < R < t 0 4 0<R<\frac{\sqrt{t_{0}}}{4} 0 < R < 4 t 0
Proof. By Fubini’s theorem there exists t ∗ ∈ ( t 0 − R 2 , t 0 − 4 R 2 ) t_{*}\in(t_{0}-R^{2},t_{0}-4R^{2}) t ∗ ∈ ( t 0 − R 2 , t 0 − 4 R 2 )
∫ B 2 R + ( X 0 ) ∣ D U ∣ 2 ( X , t ∗ ) μ a ≤ 1 R 2 ∬ P 2 R + ( Z 0 ) ∣ D U ∣ 2 μ a d t . \int_{B^{+}_{2R}(X_{0})}|DU|^{2}(X,t_{*})\mu_{a}
\leq\frac{1}{R^{2}}\iint_{P^{+}_{2R}(Z_{0})}|DU|^{2}\mu_{a}\,dt. ∫ B 2 R + ( X 0 ) ∣ D U ∣ 2 ( X , t ∗ ) μ a ≤ R 2 1 ∬ P 2 R + ( Z 0 ) ∣ D U ∣ 2 μ a d t . Choose a parabolic cut-off function η ∈ C ∞ ( P 2 R + ( Z 0 ) ) \eta\in C^\infty(P_{2R}^{+}(Z_0)) η ∈ C ∞ ( P 2 R + ( Z 0 )) η ( x , − y , t ) = η ( x , y , t ) \eta(x,-y,t)=\eta(x,y,t) η ( x , − y , t ) = η ( x , y , t ) 0 ≤ η ≤ 1 0\leq\eta\leq1 0 ≤ η ≤ 1 η ≡ 1 \eta\equiv1 η ≡ 1 P R + ( Z 0 ) P_R^{+}(Z_0) P R + ( Z 0 ) ∣ D 2 η ∣ + ∣ D η ∣ 2 + ∣ ∂ t η ∣ ≤ 24 R 2 |D^2\eta|+|D\eta|^2+|\partial_t\eta|\leq\frac{24}{R^2} ∣ D 2 η ∣ + ∣ D η ∣ 2 + ∣ ∂ t η ∣ ≤ R 2 24 ∂ η ∂ y ∣ y = 0 = 0 \left.\frac{\partial\eta}{\partial y}\right|_{y=0}=0 ∂ y ∂ η y = 0 = 0
∣ ∂ η ∂ y ∣ ≤ ∣ y ∣ ∥ ∂ 2 η ∂ y 2 ∥ L ∞ ≤ C R − 2 ∣ y ∣ . \left|\frac{\partial\eta}{\partial y}\right|\leq|y|\left\|\frac{\partial^2\eta}{\partial y^2}\right\|_{L^\infty}\leq CR^{-2}|y|. ∂ y ∂ η ≤ ∣ y ∣ ∂ y 2 ∂ 2 η L ∞ ≤ C R − 2 ∣ y ∣. With help of this, we can apply (48) with t = t ∗ t=t_* t = t ∗ t 2 = t 0 + R 2 t_2=t_0+R^2 t 2 = t 0 + R 2
∬ P R + ( Z 0 ) ∣ ∂ t U ∣ 2 μ a d t ≤ ∬ R + m + 1 × [ t ∗ , t 0 + R 2 ] ∣ ∂ t U ∣ 2 η μ a d t ≤ ∬ R + m + 1 × { t ∗ } ∣ D U ∣ 2 η μ a + ∬ R + m + 1 × [ t ∗ , t 0 + R 2 ] [ ∣ D U ∣ 2 ( y a ∂ t η + div X ( y a D η ) ) − 2 y a D U ⊗ D U : D 2 η ] d X d t ≤ ∬ B 2 R + ( X 0 ) × { t ∗ } ∣ D U ∣ 2 μ a + C ∬ P 2 R + ( Z 0 ) ( R − 2 ∣ D U ∣ 2 + y − 1 ∣ ∂ η ∂ y ∣ ∣ D U ∣ 2 ) μ a d t ≤ C R − 2 ∬ P 2 R + ( Z 0 ) ∣ D U ∣ 2 μ a d t . \begin{aligned}
\iint_{P_R^+(Z_0)}|\partial_tU|^2\mu_a\,dt\leq\iint_{\mathbb{R}^{m+1}_+\times[t_*,t_0+R^2]}|\partial_tU|^2\eta\mu_a\,dt \\
&\leq\iint_{\mathbb{R}^{m+1}_+\times\{t_*\}}|DU|^2\eta\mu_a
+\iint_{\mathbb{R}^{m+1}_+\times[t_*,t_0+R^2]}\left[|DU|^2\left(y^a\partial_t\eta+\operatorname{div}_X(y^aD\eta)\right)-2y^aDU\otimes DU:D^2\eta\right]\,dX\,dt \\
&\leq\iint_{B_{2R}^+(X_0)\times\{t_*\}}|DU|^2\mu_a+C\iint_{P_{2R}^+(Z_0)}\left(R^{-2}|DU|^2+y^{-1}\left|\frac{\partial\eta}{\partial y}\right||DU|^2\right)\mu_a\,dt \\
&\leq CR^{-2}\iint_{P_{2R}^+(Z_0)}|DU|^2\mu_a\,dt.
\end{aligned} ∬ P R + ( Z 0 ) ∣ ∂ t U ∣ 2 μ a d t ≤ ∬ R + m + 1 × [ t ∗ , t 0 + R 2 ] ∣ ∂ t U ∣ 2 η μ a d t ≤ ∬ R + m + 1 × { t ∗ } ∣ D U ∣ 2 η μ a + ∬ R + m + 1 × [ t ∗ , t 0 + R 2 ] [ ∣ D U ∣ 2 ( y a ∂ t η + div X ( y a D η ) ) − 2 y a D U ⊗ D U : D 2 η ] d X d t ≤ ∬ B 2 R + ( X 0 ) × { t ∗ } ∣ D U ∣ 2 μ a + C ∬ P 2 R + ( Z 0 ) ( R − 2 ∣ D U ∣ 2 + y − 1 ∂ y ∂ η ∣ D U ∣ 2 ) μ a d t ≤ C R − 2 ∬ P 2 R + ( Z 0 ) ∣ D U ∣ 2 μ a d t . This yields (48) . The proof is complete. □
Proof of Theorem 1.3: Hölder continuity
In this section, we will prove the following partial regularity theorem for stationary solutions of the heat flow of s s s ϵ 0 \epsilon_0 ϵ 0
Lemma 5.1 There exist ϵ 0 > 0 \epsilon_0>0 ϵ 0 > 0 θ 0 ∈ ( 0 , 1 ) \theta_0\in(0,1) θ 0 ∈ ( 0 , 1 ) U U U s s s R + m + 1 \mathbb{R}^{m+1}_+ R + m + 1 ( X 0 , t 0 ) ∈ ∂ R + m + 1 × ( 0 , ∞ ) (X_0,t_0)\in\partial\mathbb{R}^{m+1}_+\times(0,\infty) ( X 0 , t 0 ) ∈ ∂ R + m + 1 × ( 0 , ∞ ) 0 < r < t 0 4 0<r<\frac{\sqrt{t_0}}{4} 0 < r < 4 t 0
( 2 r ) − ( m + 1 + a ) ∬ P 2 r + ( X 0 , t 0 ) ∣ D U ∣ 2 μ a d t ≤ ϵ 0 2 , (49) (2r)^{-(m+1+a)}\iint_{P_{2r}^+(X_0,t_0)}|DU|^2\mu_a\,dt\leq\epsilon_0^2, \tag*{(49)} ( 2 r ) − ( m + 1 + a ) ∬ P 2 r + ( X 0 , t 0 ) ∣ D U ∣ 2 μ a d t ≤ ϵ 0 2 , (49) then
( θ 0 r ) − ( m + 1 + a ) ∬ P θ 0 r + ( X 0 , t 0 ) ∣ D U ∣ 2 μ a d t ≤ 1 2 ( 2 r ) − ( m + 1 + a ) ∬ P 2 r + ( X 0 , t 0 ) ∣ D U ∣ 2 μ a d t . (50) (\theta_0r)^{-(m+1+a)}\iint_{P_{\theta_0r}^+(X_0,t_0)}|DU|^2\mu_a\,dt\leq\frac{1}{2}(2r)^{-(m+1+a)}\iint_{P_{2r}^+(X_0,t_0)}|DU|^2\mu_a\,dt. \tag*{(50)} ( θ 0 r ) − ( m + 1 + a ) ∬ P θ 0 r + ( X 0 , t 0 ) ∣ D U ∣ 2 μ a d t ≤ 2 1 ( 2 r ) − ( m + 1 + a ) ∬ P 2 r + ( X 0 , t 0 ) ∣ D U ∣ 2 μ a d t . (50) Proof. We may assume X 0 = 0 X_0=0 X 0 = 0 t 0 = 1 t_0=1 t 0 = 1 θ ∈ ( 0 , 1 ) \theta\in(0,1) θ ∈ ( 0 , 1 ) r i > 0 r_i>0 r i > 0 ϵ i → 0 \epsilon_i\to0 ϵ i → 0
( 2 r i ) − ( m + 1 + a ) ∬ P 2 r i + ( 0 , 1 ) ∣ D U ∣ 2 μ a d t = ϵ i 2 , (51) (2r_i)^{-(m+1+a)}\iint_{P_{2r_i}^+(0,1)}|DU|^2\mu_a\,dt=\epsilon_i^2, \tag*{(51)} ( 2 r i ) − ( m + 1 + a ) ∬ P 2 r i + ( 0 , 1 ) ∣ D U ∣ 2 μ a d t = ϵ i 2 , (51) then
( θ r i ) − ( m + 1 + a ) ∬ P θ r i + ( 0 , 1 ) ∣ D U ∣ 2 μ a d t > 1 2 ( 2 r i ) − ( m + 1 + a ) ∬ P 2 r i + ( 0 , 1 ) ∣ D U ∣ 2 μ a d t . (52) (\theta r_i)^{-(m+1+a)}\iint_{P_{\theta r_i}^+(0,1)}|DU|^2\mu_a\,dt>\frac{1}{2}(2r_i)^{-(m+1+a)}\iint_{P_{2r_i}^+(0,1)}|DU|^2\mu_a\,dt. \tag*{(52)} ( θ r i ) − ( m + 1 + a ) ∬ P θ r i + ( 0 , 1 ) ∣ D U ∣ 2 μ a d t > 2 1 ( 2 r i ) − ( m + 1 + a ) ∬ P 2 r i + ( 0 , 1 ) ∣ D U ∣ 2 μ a d t . (52) Define the blow-up sequence U i ( X , t ) = U ( r i X , 1 + r i 2 t ) − U ( 0 , 1 ) ϵ i : R + m + 1 × ( − 4 , 4 ) → R ℓ U_i(X,t)=\frac{U(r_iX,1+r_i^2t)-U(0,1)}{\epsilon_i}:\mathbb{R}^{m+1}_+\times(-4,4)\to\mathbb{R}^{\ell} U i ( X , t ) = ϵ i U ( r i X , 1 + r i 2 t ) − U ( 0 , 1 ) : R + m + 1 × ( − 4 , 4 ) → R ℓ
2 − ( m + 1 + a ) ∬ P 2 + ( 0 , 0 ) ∣ D U i ∣ 2 μ a d t = 1 , (53) 2^{-(m+1+a)}\iint_{P_2^+(0,0)}|DU_i|^2\mu_a\,dt=1, \tag*{(53)} 2 − ( m + 1 + a ) ∬ P 2 + ( 0 , 0 ) ∣ D U i ∣ 2 μ a d t = 1 , (53) but
θ − ( m + 1 + a ) ∬ P 2 + ( 0 , 0 ) ∣ D U i ∣ 2 μ a d t > 1 2 . (54) \theta^{-(m+1+a)}\iint_{P_2^+(0,0)} |DU_i|^2\,\mu_a\,dt > \frac{1}{2}. \tag*{(54)} θ − ( m + 1 + a ) ∬ P 2 + ( 0 , 0 ) ∣ D U i ∣ 2 μ a d t > 2 1 . (54) Since U U U U i U_i U i
{ y a ∂ t U i = div X ( y a D U i ) in P 2 + ( 0 , 0 ) , U i ( ∂ R + m × ( − 4 , 4 ) ) ⊂ ϵ i − 1 ( S ℓ − 1 − U ( 0 , 1 ) ) , y a ∂ U i ∂ y ∣ y = 0 ⊥ T U ( ( r i x , 0 ) , r 2 t ) S ℓ − 1 , ( x , t ) ∈ B 2 m ( 0 ) × ( − 4 , 4 ) . (55) \begin{cases}
y^a\partial_t U_i=\operatorname{div}_X(y^aDU_i) & \text{in } P_2^+(0,0),\\
U_i(\partial\mathbb{R}_+^m\times(-4,4))\subset\epsilon_i^{-1}(S^{\ell-1}-U(0,1)),\\
y^a\left.\dfrac{\partial U_i}{\partial y}\right|_{y=0}\perp T_{U((ri x,0),r_2t)}S^{\ell-1}, & (x,t)\in B_2^m(0)\times(-4,4). \tag*{(55)}
\end{cases} ⎩ ⎨ ⎧ y a ∂ t U i = div X ( y a D U i ) U i ( ∂ R + m × ( − 4 , 4 )) ⊂ ϵ i − 1 ( S ℓ − 1 − U ( 0 , 1 )) , y a ∂ y ∂ U i y = 0 ⊥ T U (( r i x , 0 ) , r 2 t ) S ℓ − 1 , in P 2 + ( 0 , 0 ) , ( x , t ) ∈ B 2 m ( 0 ) × ( − 4 , 4 ) . (55) From (4.16) we can deduce that
∬ P 1 + ( 0 , 0 ) ∣ ∂ t U i ∣ 2 μ a d t ≤ C ∬ P 2 + ( 0 , 0 ) ∣ D U i ∣ 2 μ a d t ≤ C . (56) \iint_{P_1^+(0,0)}|\partial_tU_i|^2\mu_a\,dt\le C\iint_{P_2^+(0,0)}|DU_i|^2\mu_a\,dt\le C. \tag*{(56)} ∬ P 1 + ( 0 , 0 ) ∣ ∂ t U i ∣ 2 μ a d t ≤ C ∬ P 2 + ( 0 , 0 ) ∣ D U i ∣ 2 μ a d t ≤ C . (56) By the y a y^a y a
∬ P 1 + ( 0 , 0 ) ∣ U i ∣ 2 μ a d t ≤ C ∬ P 1 + ( 0 , 0 ) ( ∣ ∂ t U i ∣ 2 + ∣ D U i ∣ 2 ) μ a d t ≤ C . (57) \iint_{P_1^+(0,0)}|U_i|^2\mu_a\,dt\le C\iint_{P_1^+(0,0)}(|\partial_tU_i|^2+|DU_i|^2)\mu_a\,dt\le C. \tag*{(57)} ∬ P 1 + ( 0 , 0 ) ∣ U i ∣ 2 μ a d t ≤ C ∬ P 1 + ( 0 , 0 ) ( ∣ ∂ t U i ∣ 2 + ∣ D U i ∣ 2 ) μ a d t ≤ C . (57) It follows from (5.5), (5.8), and (5.9) that we may assume that there exists U ∞ ∈ H 1 ( P 1 + ( 0 , 0 ) , R ℓ ) U_\infty\in H^1(P_1^+(0,0),\mathbb{R}^{\ell}) U ∞ ∈ H 1 ( P 1 + ( 0 , 0 ) , R ℓ )
( ∂ t U i , D U i ) ⇀ ( ∂ t U ∞ , D U ∞ ) ; U i → U ∞ in L 2 ( P 1 + ( 0 , 0 ) , μ a d t ) . (\partial_tU_i,DU_i)\rightharpoonup(\partial_tU_\infty,DU_\infty); \quad U_i\to U_\infty\quad\text{in }L^2(P_1^+(0,0),\mu_a\,dt). ( ∂ t U i , D U i ) ⇀ ( ∂ t U ∞ , D U ∞ ) ; U i → U ∞ in L 2 ( P 1 + ( 0 , 0 ) , μ a d t ) . Moreover, by using the y a y^a y a
U i → U ∞ in L 4 ( P 1 + ( 0 , 0 ) , μ a d t ) . U_i\to U_\infty\quad\text{in }L^4(P_1^+(0,0),\mu_a\,dt). U i → U ∞ in L 4 ( P 1 + ( 0 , 0 ) , μ a d t ) . It is not hard to check that
∬ P 1 + ( 0 , 0 ) ( ∣ ∂ t U ∞ ∣ 2 + ∣ D U ∞ ∣ 2 ) μ a d t ≤ 1 , (58) \iint_{P_1^+(0,0)}(|\partial_tU_\infty|^2+|DU_\infty|^2)\mu_a\,dt\le1, \tag*{(58)} ∬ P 1 + ( 0 , 0 ) ( ∣ ∂ t U ∞ ∣ 2 + ∣ D U ∞ ∣ 2 ) μ a d t ≤ 1 , (58) and
{ y a ∂ t U ∞ = div X ( y a D U ∞ ) in P 1 + ( 0 , 0 ) , U ∞ ( B 1 m ( 0 ) × ( − 1 , 1 ) ) ⊂ T U ( 0 , 1 ) S ℓ − 1 , y a ∂ U ∞ ∂ y ∣ y = 0 ⊥ T U ( 0 , 1 ) S ℓ − 1 , ( x , t ) ∈ B 1 m ( 0 ) × ( − 1 , 1 ) . (59) \begin{cases}
y^a\partial_tU_\infty=\operatorname{div}_X(y^aDU_\infty) & \text{in }P_1^+(0,0),\\
U_\infty(B_1^m(0)\times(-1,1))\subset T_{U(0,1)}S^{\ell-1},\\
y^a\left.\dfrac{\partial U_\infty}{\partial y}\right|_{y=0}\perp T_{U(0,1)}S^{\ell-1}, & (x,t)\in B_1^m(0)\times(-1,1). \tag*{(59)}
\end{cases} ⎩ ⎨ ⎧ y a ∂ t U ∞ = div X ( y a D U ∞ ) U ∞ ( B 1 m ( 0 ) × ( − 1 , 1 )) ⊂ T U ( 0 , 1 ) S ℓ − 1 , y a ∂ y ∂ U ∞ y = 0 ⊥ T U ( 0 , 1 ) S ℓ − 1 , in P 1 + ( 0 , 0 ) , ( x , t ) ∈ B 1 m ( 0 ) × ( − 1 , 1 ) . (59) By choosing a suitable coordinate system in R ℓ \mathbb{R}^{\ell} R ℓ U ( 0 , 1 ) = ( 0 ′ , 1 ) U(0,1)=(0',1) U ( 0 , 1 ) = ( 0 ′ , 1 ) T U ( 0 , 1 ) S ℓ − 1 = R ℓ − 1 × { 0 } T_{U(0,1)}S^{\ell-1}=\mathbb{R}^{\ell-1}\times\{0\} T U ( 0 , 1 ) S ℓ − 1 = R ℓ − 1 × { 0 }
U ∞ ℓ = 0 , y a ∂ U ∞ i ∂ y ∣ y = 0 = 0 ( 1 ≤ i ≤ ℓ − 1 ) on B 1 m ( 0 ) × ( − 1 , 1 ) . (60) U_\infty^\ell=0,\quad y^a\left.\dfrac{\partial U_\infty^i}{\partial y}\right|_{y=0}=0\ (1\le i\le\ell-1)\quad\text{on }B_1^m(0)\times(-1,1). \tag*{(60)} U ∞ ℓ = 0 , y a ∂ y ∂ U ∞ i y = 0 = 0 ( 1 ≤ i ≤ ℓ − 1 ) on B 1 m ( 0 ) × ( − 1 , 1 ) . (60) Let us extend U ∞ U_\infty U ∞ P 1 ( 0 , 0 ) P_1(0,0) P 1 ( 0 , 0 ) U ∞ ℓ U_\infty^\ell U ∞ ℓ U ∞ i U_\infty^i U ∞ i 1 ≤ i ≤ ℓ − 1 1\le i\le\ell-1 1 ≤ i ≤ ℓ − 1 y y y U ∞ U_\infty U ∞
∣ y ∣ a ∂ t U ∞ = div X ( ∣ y ∣ a D U ∞ ) in P 1 ( 0 , 0 ) . (61) |y|^a\partial_tU_\infty=\operatorname{div}_X(|y|^aDU_\infty)\quad\text{in }P_1(0,0). \tag*{(61)} ∣ y ∣ a ∂ t U ∞ = div X ( ∣ y ∣ a D U ∞ ) in P 1 ( 0 , 0 ) . (61) Hence by the regularity theorem that U ∞ ∈ C α s ( P 1 ( 0 , 0 ) ) U_{\infty} \in C^{\alpha_s}(P_1(0,0)) U ∞ ∈ C α s ( P 1 ( 0 , 0 )) θ ∈ ( 0 , 1 4 ) \theta\in(0,\frac{1}{4}) θ ∈ ( 0 , 4 1 )
θ − ( m + 3 + a ) ∬ P θ + ( 0 , 0 ) ∣ U ∞ − ( U ∞ ) ( 0 , 0 ) , θ , + a ∣ 2 μ a d t ≤ C θ 2 α s , (62) \theta^{-(m+3+a)} \iint_{P_{\theta}^{+}(0,0)} \lvert U_{\infty}-(U_{\infty})_{(0,0),\theta,+}^{a}\rvert^{2}\mu_a\,dt \leq C\theta^{2\alpha_s}, \tag*{(62)} θ − ( m + 3 + a ) ∬ P θ + ( 0 , 0 ) ∣ U ∞ − ( U ∞ ) ( 0 , 0 ) , θ , + a ∣ 2 μ a d t ≤ C θ 2 α s , (62) where
α s = { 2 s 0 < s < 1 2 , 1 1 2 ≤ s < 1. \alpha_s =
\begin{cases}
2s & 0<s<\frac{1}{2},\\
1 & \frac{1}{2}\leq s<1.
\end{cases} α s = { 2 s 1 0 < s < 2 1 , 2 1 ≤ s < 1. With (5.14) at the hand, we can apply (3.7) to reach a contradiction to (5.6) as follows. First of all, by the energy monotonicity inequality (4.11) and the local energy inequality (4.17) for U U U z ∈ P r i + ( 0 , 1 ) z\in P_{r_i}^{+}(0,1) z ∈ P r i + ( 0 , 1 ) 0 < r < θ 0 r i 0<r<\theta_0r_i 0 < r < θ 0 r i
r − ( m + 1 + a ) ∬ P r + ( z ) ∣ D U ∣ 2 μ a d t ≤ C r i − ( m + 1 + a ) ∬ P 2 r i + ( 0 , 1 ) ∣ D U ∣ 2 μ a d t , (63) r^{-(m+1+a)}\iint_{P_r^{+}(z)}\lvert DU\rvert^2\mu_a\,dt\leq Cr_i^{-(m+1+a)}\iint_{P_{2r_i}^{+}(0,1)}\lvert DU\rvert^2\mu_a\,dt, \tag*{(63)} r − ( m + 1 + a ) ∬ P r + ( z ) ∣ D U ∣ 2 μ a d t ≤ C r i − ( m + 1 + a ) ∬ P 2 r i + ( 0 , 1 ) ∣ D U ∣ 2 μ a d t , (63) and
r − ( m + 1 + a ) ∬ P r + ( z ) r 2 ∣ ∂ t U ∣ 2 μ a d t ≤ C r i − ( m + 1 + a ) ∬ P 2 r i + ( 0 , 1 ) ∣ D U ∣ 2 μ a d t . (64) r^{-(m+1+a)}\iint_{P_r^{+}(z)}r^2\lvert\partial_tU\rvert^2\mu_a\,dt\leq Cr_i^{-(m+1+a)}\iint_{P_{2r_i}^{+}(0,1)}\lvert DU\rvert^2\mu_a\,dt. \tag*{(64)} r − ( m + 1 + a ) ∬ P r + ( z ) r 2 ∣ ∂ t U ∣ 2 μ a d t ≤ C r i − ( m + 1 + a ) ∬ P 2 r i + ( 0 , 1 ) ∣ D U ∣ 2 μ a d t . (64) It follows from (5.15) and (5.16) that
∥ D U ∥ M 2 , 2 ( P r i + ( 0 , 1 ) , μ a d t ) + ∥ ∂ t U ∥ M 2 , 4 ( P r i + ( 0 , 1 ) , μ a d t ) ≤ C ( r i − ( m + 1 + a ) ∬ P 2 r i + ( 0 , 1 ) ∣ D U ∣ 2 μ a d t ) 1 2 ≤ C ε i , (65) \lVert DU\rVert_{M^{2,2}(P_{r_i}^{+}(0,1),\mu_a\,dt)}+\lVert\partial_tU\rVert_{M^{2,4}(P_{r_i}^{+}(0,1),\mu_a\,dt)}
\leq C\left(r_i^{-(m+1+a)}\iint_{P_{2r_i}^{+}(0,1)}\lvert DU\rvert^2\mu_a\,dt\right)^{\frac{1}{2}}
\leq C\varepsilon_i, \tag*{(65)} ∥ D U ∥ M 2 , 2 ( P r i + ( 0 , 1 ) , μ a d t ) + ∥ ∂ t U ∥ M 2 , 4 ( P r i + ( 0 , 1 ) , μ a d t ) ≤ C ( r i − ( m + 1 + a ) ∬ P 2 r i + ( 0 , 1 ) ∣ D U ∣ 2 μ a d t ) 2 1 ≤ C ε i , (65) Let ε 0 > 0 \varepsilon_0>0 ε 0 > 0 C ε i ≤ ε 0 C\varepsilon_i\leq\varepsilon_0 C ε i ≤ ε 0 i i i ε i → 0 \varepsilon_i\to0 ε i → 0 0 < θ < θ 0 0<\theta<\theta_0 0 < θ < θ 0
θ − ( m + 1 + a ) ∬ P θ + ∣ D U i ∣ 2 μ a d t ≤ ε 0 ( 2 θ ) − ( m + 1 + a ) ∬ P 2 θ + ( ( 2 θ ) 2 ∣ ∂ t U i ∣ 2 + ∣ D U i ∣ 2 ) μ a d t + ε 0 ( 2 θ ) − ( m − 1 + a ) sup t ∈ [ − 4 θ 2 , 4 θ 2 ] ∫ B 2 θ + ∣ D U i ∣ 2 μ a + C ε 0 − 1 ( 2 θ ) − ( m + 3 + a ) ∬ P 2 θ + ∣ U i − ( U i ) 0 , 2 θ , + a ∣ 2 μ a d t + C ε 0 − 1 ( 2 θ ) − ( m + 3 + a ) ∬ P 2 θ + ∣ U i − ( U i ) 0 , 2 θ , + a ∣ 4 μ a d t + C ε 0 − 1 ε i − 2 ( 2 θ ) − ( m + 3 + a ) ∬ P 2 θ + ( ∣ U ‾ i ∣ 2 − 1 ) 2 μ a d t , (66) \begin{aligned}
\theta^{-(m+1+a)}\iint_{P_{\theta}^{+}}\lvert DU_i\rvert^2\mu_a\,dt
&\leq\varepsilon_0(2\theta)^{-(m+1+a)}\iint_{P_{2\theta}^{+}}\left((2\theta)^2\lvert\partial_tU_i\rvert^2+\lvert DU_i\rvert^2\right)\mu_a\,dt\\
&\quad+\varepsilon_0(2\theta)^{-(m-1+a)}
\sup_{t\in[-4\theta^2,4\theta^2]}\int_{B_{2\theta}^{+}}\lvert DU_i\rvert^2\mu_a\\
&\quad+C\varepsilon_0^{-1}(2\theta)^{-(m+3+a)}
\iint_{P_{2\theta}^{+}}\lvert U_i-(U_i)_{0,2\theta,+}^{a}\rvert^2\mu_a\,dt\\
&\quad+C\varepsilon_0^{-1}(2\theta)^{-(m+3+a)}
\iint_{P_{2\theta}^{+}}\lvert U_i-(U_i)_{0,2\theta,+}^{a}\rvert^4\mu_a\,dt\\
&\quad+C\varepsilon_0^{-1}\varepsilon_i^{-2}(2\theta)^{-(m+3+a)}
\iint_{P_{2\theta}^{+}}(\lvert\overline{U}_i\rvert^2-1)^2\mu_a\,dt, \tag*{(66)}
\end{aligned} θ − ( m + 1 + a ) ∬ P θ + ∣ D U i ∣ 2 μ a d t ≤ ε 0 ( 2 θ ) − ( m + 1 + a ) ∬ P 2 θ + ( ( 2 θ ) 2 ∣ ∂ t U i ∣ 2 + ∣ D U i ∣ 2 ) μ a d t + ε 0 ( 2 θ ) − ( m − 1 + a ) t ∈ [ − 4 θ 2 , 4 θ 2 ] sup ∫ B 2 θ + ∣ D U i ∣ 2 μ a + C ε 0 − 1 ( 2 θ ) − ( m + 3 + a ) ∬ P 2 θ + ∣ U i − ( U i ) 0 , 2 θ , + a ∣ 2 μ a d t + C ε 0 − 1 ( 2 θ ) − ( m + 3 + a ) ∬ P 2 θ + ∣ U i − ( U i ) 0 , 2 θ , + a ∣ 4 μ a d t + C ε 0 − 1 ε i − 2 ( 2 θ ) − ( m + 3 + a ) ∬ P 2 θ + (∣ U i ∣ 2 − 1 ) 2 μ a d t , (66) where U ‾ i ( x , t ) = U ( r i x , 1 + r i 2 t ) \overline{U}_i(x,t)=U(r_i x,1+r_i^2t) U i ( x , t ) = U ( r i x , 1 + r i 2 t )
Since ∣ U ‾ i ∣ = 1 \lvert\overline{U}_i\rvert=1 ∣ U i ∣ = 1 P ^ 1 ( 0 , 0 ) \widehat{P}_1(0,0) P 1 ( 0 , 0 ) P ^ r ( 0 , 0 ) = B r m ( 0 ) × ( − r 2 , r 2 ) \widehat{P}_r(0,0)=B_r^m(0)\times(-r^2,r^2) P r ( 0 , 0 ) = B r m ( 0 ) × ( − r 2 , r 2 )
( 2 θ ) − ( m + 3 + a ) ∬ P 2 θ + ( ∣ U ‾ i ∣ 2 − 1 ) 2 μ a d t ≤ ( 1 + ∥ U ‾ i ∥ L ∞ ( P 1 2 + ) ) 2 ( 2 θ ) − ( m + 3 + a ) ∬ P 2 θ + ∣ U ‾ i − c ‾ i ( θ ) ∣ 2 μ a d t ≤ C ( 1 + ∥ U ‾ i ∥ L ∞ ( P 1 2 + ) ) 2 ( 2 θ ) − ( m + 3 + a ) ( ∬ P 2 θ + ∣ U ‾ i − ( U ‾ i ) 0 , 2 θ , + a ∣ 2 μ a d t + ∬ P 2 θ + ∣ U ‾ i − c ‾ i ( θ ) ∣ 2 μ a d t ) \begin{aligned}
(2\theta)^{-(m+3+a)}\iint_{P_{2\theta}^{+}}(\lvert\overline{U}_i\rvert^2-1)^2\mu_a\,dt
\leq\left(1+\lVert\overline{U}_i\rVert_{L^\infty(P_{\frac12}^{+})}\right)^2(2\theta)^{-(m+3+a)}
\iint_{P_{2\theta}^{+}}\lvert\overline{U}_i-\overline{c}_i(\theta)\rvert^2\mu_a\,dt \\
&\leq C\left(1+\lVert\overline{U}_i\rVert_{L^\infty(P_{\frac12}^{+})}\right)^2(2\theta)^{-(m+3+a)}
\left(\iint_{P_{2\theta}^{+}}\lvert\overline{U}_i-(\overline{U}_i)_{0,2\theta,+}^{a}\rvert^2\mu_a\,dt+\iint_{P_{2\theta}^{+}}\lvert\overline{U}_i-\overline{c}_i(\theta)\rvert^2\mu_a\,dt\right)
\end{aligned} ( 2 θ ) − ( m + 3 + a ) ∬ P 2 θ + (∣ U i ∣ 2 − 1 ) 2 μ a d t ≤ ( 1 + ∥ U i ∥ L ∞ ( P 2 1 + ) ) 2 ( 2 θ ) − ( m + 3 + a ) ∬ P 2 θ + ∣ U i − c i ( θ ) ∣ 2 μ a d t ≤ C ( 1 + ∥ U i ∥ L ∞ ( P 2 1 + ) ) 2 ( 2 θ ) − ( m + 3 + a ) ( ∬ P 2 θ + ∣ U i − ( U i ) 0 , 2 θ , + a ∣ 2 μ a d t + ∬ P 2 θ + ∣ U i − c i ( θ ) ∣ 2 μ a d t ) where c ˉ i ( θ ) \bar{c}_i(\theta) c ˉ i ( θ ) U ‾ i \overline{U}_i U i P ^ 2 θ ( 0 , 0 ) \widehat{P}_{2\theta}(0,0) P 2 θ ( 0 , 0 ) d x d t dxdt d x d t
From (5.17), and (3.2) of Lemma 3.2, we have
∥ U ‾ i ∥ L ∞ ( P 1 2 + ) ≤ 1 + C ε i ≤ 2. \left\|\overline{U}_i\right\|_{L^\infty(P^+_{\frac{1}{2}})} \leq1+C\varepsilon_i\leq2. U i L ∞ ( P 2 1 + ) ≤ 1 + C ε i ≤ 2. Hence, after rescaling, we obtain
ε i − 2 ( 2 θ ) − ( m + 3 + a ) ∬ P 2 θ + ( ∣ U ‾ i ∣ 2 − 1 ) 2 μ a d t ≤ C ( 2 θ ) − ( m + 3 + a ) ( ∬ P 2 θ + ∣ U i − ( U i ) 0 , 2 θ , + a ∣ 2 μ a d t + ∬ P 2 θ + ∣ U i − c i ( θ ) ∣ 2 μ a d t ) . (67) \varepsilon_i^{-2}(2\theta)^{-(m+3+a)}\iint_{P^+_{2\theta}} \left(\left|\overline{U}_i\right|^2-1\right)^2\mu_a\,dt
\leq C(2\theta)^{-(m+3+a)}\left(\iint_{P^+_{2\theta}}\left|U_i-(U_i)^a_{0,2\theta,+}\right|^2\mu_a\,dt+\iint_{P^+_{2\theta}}\left|U_i-c_i(\theta)\right|^2\mu_a\,dt\right). \tag*{(67)} ε i − 2 ( 2 θ ) − ( m + 3 + a ) ∬ P 2 θ + ( U i 2 − 1 ) 2 μ a d t ≤ C ( 2 θ ) − ( m + 3 + a ) ( ∬ P 2 θ + U i − ( U i ) 0 , 2 θ , + a 2 μ a d t + ∬ P 2 θ + ∣ U i − c i ( θ ) ∣ 2 μ a d t ) . (67) with c i ( θ ) c_i(\theta) c i ( θ ) U i U_i U i P ^ 2 θ ( 0 , 0 ) \widehat{P}_{2\theta}(0,0) P 2 θ ( 0 , 0 )
Since U i → U ∞ U_i\to U_\infty U i → U ∞ L 4 ( P θ 0 + ( 0 , 0 ) , μ a d t ) L^4(P^+_{\theta_0}(0,0),\mu_a\,dt) L 4 ( P θ 0 + ( 0 , 0 ) , μ a d t ) U ∞ U_\infty U ∞
( 2 θ ) − ( m + 3 + a ) ( ∬ P 2 θ + ∣ U i − ( U i ) 0 , 2 θ , + a ∣ 2 μ a d t + ∬ P 2 θ + ∣ U i − c i ( θ ) ∣ 2 μ a d t ) ≤ 2 ( 2 θ ) − ( m + 3 + a ) ( ∬ P 2 θ + ∣ U ∞ − ( U ∞ ) 0 , 2 θ , + a ∣ 2 μ a d t + ∬ P 2 θ + ∣ U ∞ − c ∞ ( θ ) ∣ 2 μ a d t ) + 4 ( 2 θ ) − ( m + 3 + a ) ∬ P 2 θ + ∣ U i − U ∞ ∣ 2 μ a d t ≤ C θ 2 + C θ − ( m + 3 + a ) o i ( 1 ) . \begin{aligned}
&(2\theta)^{-(m+3+a)}\left(\iint_{P^+_{2\theta}}\left|U_i-(U_i)^a_{0,2\theta,+}\right|^2\mu_a\,dt+\iint_{P^+_{2\theta}}\left|U_i-c_i(\theta)\right|^2\mu_a\,dt\right)\\
&\leq2(2\theta)^{-(m+3+a)}\left(\iint_{P^+_{2\theta}}\left|U_\infty-(U_\infty)^a_{0,2\theta,+}\right|^2\mu_a\,dt+\iint_{P^+_{2\theta}}\left|U_\infty-c_\infty(\theta)\right|^2\mu_a\,dt\right)\\
&\quad+4(2\theta)^{-(m+3+a)}\iint_{P^+_{2\theta}}\left|U_i-U_\infty\right|^2\mu_a\,dt\\
&\leq C\theta^2+C\theta^{-(m+3+a)}o_i(1).
\end{aligned} ( 2 θ ) − ( m + 3 + a ) ( ∬ P 2 θ + U i − ( U i ) 0 , 2 θ , + a 2 μ a d t + ∬ P 2 θ + ∣ U i − c i ( θ ) ∣ 2 μ a d t ) ≤ 2 ( 2 θ ) − ( m + 3 + a ) ( ∬ P 2 θ + U ∞ − ( U ∞ ) 0 , 2 θ , + a 2 μ a d t + ∬ P 2 θ + ∣ U ∞ − c ∞ ( θ ) ∣ 2 μ a d t ) + 4 ( 2 θ ) − ( m + 3 + a ) ∬ P 2 θ + ∣ U i − U ∞ ∣ 2 μ a d t ≤ C θ 2 + C θ − ( m + 3 + a ) o i ( 1 ) . where o i ( 1 ) → 0 o_i(1)\to0 o i ( 1 ) → 0 i → ∞ i\to\infty i → ∞
ε i − 2 ( 2 θ ) − ( m + 3 + a ) ∬ P 2 θ + ( ∣ U ‾ i ∣ 2 − 1 ) 2 μ a d t ≤ C θ 2 + C θ − ( m + 3 + a ) o i ( 1 ) . (68) \varepsilon_i^{-2}(2\theta)^{-(m+3+a)}\iint_{P^+_{2\theta}}\left(\left|\overline{U}_i\right|^2-1\right)^2\mu_a\,dt\leq C\theta^2+C\theta^{-(m+3+a)}o_i(1). \tag*{(68)} ε i − 2 ( 2 θ ) − ( m + 3 + a ) ∬ P 2 θ + ( U i 2 − 1 ) 2 μ a d t ≤ C θ 2 + C θ − ( m + 3 + a ) o i ( 1 ) . (68) Substituting (5.20) and the scaled version of estimate (5.17) into (5.18), we arrive at
θ − ( m + 1 + a ) ∬ P θ + ∣ D U i ∣ 2 μ a d t ≤ C ε 0 + C θ 2 + C θ − m − 3 − a o i ( 1 ) . (69) \theta^{-(m+1+a)}\iint_{P^+_{\theta}}\left|DU_i\right|^2\mu_a\,dt\leq C\varepsilon_0+C\theta^2+C\theta^{-m-3-a}o_i(1). \tag*{(69)} θ − ( m + 1 + a ) ∬ P θ + ∣ D U i ∣ 2 μ a d t ≤ C ε 0 + C θ 2 + C θ − m − 3 − a o i ( 1 ) . (69) We now choose the parameters orderly: (i) First, choose ε 0 > 0 \varepsilon_0>0 ε 0 > 0 C ε 0 < 1 8 C\varepsilon_0<\frac{1}{8} C ε 0 < 8 1 θ > 0 \theta>0 θ > 0 C θ 2 < 1 8 C\theta^2<\frac{1}{8} C θ 2 < 8 1 o i ( 1 ) → 0 o_i(1)\to0 o i ( 1 ) → 0 i → ∞ i\to\infty i → ∞ i i i C θ − m − 3 − a o i ( 1 ) < 1 8 C\theta^{-m-3-a}o_i(1)<\frac{1}{8} C θ − m − 3 − a o i ( 1 ) < 8 1
θ − ( m + 1 + a ) ∬ P θ + ∣ D U i ∣ 2 < 3 8 . \theta^{-(m+1+a)}\iint_{P^+_{\theta}}\left|DU_i\right|^2<\frac{3}{8}. θ − ( m + 1 + a ) ∬ P θ + ∣ D U i ∣ 2 < 8 3 . which contradicts to (5.6). The proof is complete. □
Iterations of (5.2) give rise to Hölder continuity of U U U
Lemma 5.2 There exist ϵ 0 > 0 \epsilon_0>0 ϵ 0 > 0 α 0 ∈ ( 0 , 1 ) \alpha_0\in(0,1) α 0 ∈ ( 0 , 1 ) U U U s s s R + m + 1 \mathbb{R}^{m+1}_+ R + m + 1 ( X 0 , t 0 ) ∈ ∂ R + m + 1 × ( 0 , ∞ ) (X_0,t_0)\in\partial\mathbb{R}^{m+1}_+\times(0,\infty) ( X 0 , t 0 ) ∈ ∂ R + m + 1 × ( 0 , ∞ ) 0 < r < t 0 4 0<r<\frac{\sqrt{t_0}}{4} 0 < r < 4 t 0
( 2 r ) − ( m + 1 + a ) ∬ P 2 r + ( X 0 , t 0 ) ∣ D U ∣ 2 μ a d t ≤ ϵ 0 2 , (70) (2r)^{-(m+1+a)}\iint_{P^+_{2r}(X_0,t_0)}|DU|^2\mu_a\,dt\leq\epsilon_0^2, \tag*{(70)} ( 2 r ) − ( m + 1 + a ) ∬ P 2 r + ( X 0 , t 0 ) ∣ D U ∣ 2 μ a d t ≤ ϵ 0 2 , (70) then U ∈ C X , t α 0 , α 0 2 ( P r 2 + ( X 0 , t 0 ) ) U\in C^{\alpha_0,\frac{\alpha_0}{2}}_{X,t}(P^+_{\frac{r}{2}}(X_0,t_0)) U ∈ C X , t α 0 , 2 α 0 ( P 2 r + ( X 0 , t 0 )) Z ∈ P r + ( Z 0 ) Z\in P^+_r(Z_0) Z ∈ P r + ( Z 0 ) 0 < τ < r 0<\tau<r 0 < τ < r
τ − ( m + 1 + a ) ∬ P τ + ( Z 1 ) ∣ D U ∣ 2 μ a d t ≤ ( τ r ) 2 α 0 r − ( m + 1 + a ) ∬ P r + ( Z 1 ) ∣ D U ∣ 2 μ a d t . (71) \tau^{-(m+1+a)}\iint_{P^+_{\tau}(Z_1)}|DU|^2\mu_a\,dt\leq\left(\frac{\tau}{r}\right)^{2\alpha_0}r^{-(m+1+a)}\iint_{P^+_r(Z_1)}|DU|^2\mu_a\,dt. \tag*{(71)} τ − ( m + 1 + a ) ∬ P τ + ( Z 1 ) ∣ D U ∣ 2 μ a d t ≤ ( r τ ) 2 α 0 r − ( m + 1 + a ) ∬ P r + ( Z 1 ) ∣ D U ∣ 2 μ a d t . (71) Proof. Let Z 0 = ( X 0 , t 0 ) Z_0=(X_0,t_0) Z 0 = ( X 0 , t 0 ) (70) that
r − ( m + 1 + a ) ∬ P r + ( X ∗ , t ∗ ) ∣ D U ∣ 2 μ a d t ≤ 2 m + 1 + a ϵ 0 2 (72) r^{-(m+1+a)}\iint_{P_r^+(X_*,t_*)}|DU|^2\mu_a\,dt\leq2^{m+1+a}\epsilon_0^2 \tag*{(72)} r − ( m + 1 + a ) ∬ P r + ( X ∗ , t ∗ ) ∣ D U ∣ 2 μ a d t ≤ 2 m + 1 + a ϵ 0 2 (72) holds for all ( X ∗ , t ∗ ) ∈ P ^ r ( Z 0 ) (X_*,t_*)\in\widehat{P}_r(Z_0) ( X ∗ , t ∗ ) ∈ P r ( Z 0 ) θ 0 ∈ ( 0 , 1 ) \theta_0\in(0,1) θ 0 ∈ ( 0 , 1 ) k ≥ 1 k\geq1 k ≥ 1
( θ 0 k r ) − ( m + 1 + a ) ∬ P θ 0 k r + ( X ∗ , t ∗ ) ∣ D U ∣ 2 μ a d t ≤ 2 − k r − ( m + 1 + a ) ∬ P r + ( X ∗ , t ∗ ) ∣ D U ∣ 2 μ a d t (73) (\theta_0^k r)^{-(m+1+a)}\iint_{P_{\theta_0^k r}^+(X_*,t_*)}|DU|^2\mu_a\,dt
\leq2^{-k}r^{-(m+1+a)}\iint_{P_r^+(X_*,t_*)}|DU|^2\mu_a\,dt \tag*{(73)} ( θ 0 k r ) − ( m + 1 + a ) ∬ P θ 0 k r + ( X ∗ , t ∗ ) ∣ D U ∣ 2 μ a d t ≤ 2 − k r − ( m + 1 + a ) ∬ P r + ( X ∗ , t ∗ ) ∣ D U ∣ 2 μ a d t (73) holds for all ( X ∗ , t ∗ ) ∈ P ^ r ( Z 0 ) (X_*,t_*)\in\widehat{P}_r(Z_0) ( X ∗ , t ∗ ) ∈ P r ( Z 0 ) (47) , yields
( θ 0 k r ) − ( m + 1 + a ) ∬ P θ 0 k r + ( X ∗ , t ∗ ) ( ∣ D U ∣ 2 + ( θ 0 k r ) 2 ∣ ∂ t U ∣ 2 ) μ a d t ≤ C 2 − k r − ( m + 1 + a ) ∬ P r + ( X ∗ , t ∗ ) ∣ D U ∣ 2 μ a d t (74) \begin{aligned}
(\theta_0^k r)^{-(m+1+a)}\iint_{P_{\theta_0^k r}^+(X_*,t_*)}\left(|DU|^2+(\theta_0^k r)^2|\partial_tU|^2\right)\mu_a\,dt\\
\leq C2^{-k}r^{-(m+1+a)}\iint_{P_r^+(X_*,t_*)}|DU|^2\mu_a\,dt \tag*{(74)}
\end{aligned} ( θ 0 k r ) − ( m + 1 + a ) ∬ P θ 0 k r + ( X ∗ , t ∗ ) ( ∣ D U ∣ 2 + ( θ 0 k r ) 2 ∣ ∂ t U ∣ 2 ) μ a d t ≤ C 2 − k r − ( m + 1 + a ) ∬ P r + ( X ∗ , t ∗ ) ∣ D U ∣ 2 μ a d t (74) holds for all ( X ∗ , t ∗ ) ∈ P ^ r ( Z 0 ) (X_*,t_*)\in\widehat{P}_r(Z_0) ( X ∗ , t ∗ ) ∈ P r ( Z 0 ) α 1 ∈ ( 0 , 1 ) \alpha_1\in(0,1) α 1 ∈ ( 0 , 1 )
τ − ( m + 1 + a ) ∬ P τ + ( X ∗ , t ∗ ) ( ∣ D U ∣ 2 + τ 2 ∣ ∂ t U ∣ 2 ) μ a d t ≤ C ( τ r ) 2 α 1 r − ( m + 1 + a ) ∬ P r + ( X ∗ , t ∗ ) ∣ D U ∣ 2 μ a d t (75) \begin{aligned}
\tau^{-(m+1+a)}\iint_{P_\tau^+(X_*,t_*)}\left(|DU|^2+\tau^2|\partial_tU|^2\right)\mu_a\,dt\\
\leq C\left(\frac{\tau}{r}\right)^{2\alpha_1}r^{-(m+1+a)}\iint_{P_r^+(X_*,t_*)}|DU|^2\mu_a\,dt \tag*{(75)}
\end{aligned} τ − ( m + 1 + a ) ∬ P τ + ( X ∗ , t ∗ ) ( ∣ D U ∣ 2 + τ 2 ∣ ∂ t U ∣ 2 ) μ a d t ≤ C ( r τ ) 2 α 1 r − ( m + 1 + a ) ∬ P r + ( X ∗ , t ∗ ) ∣ D U ∣ 2 μ a d t (75) holds for all Z ∗ = ( X ∗ , t ∗ ) ∈ P ^ r ( Z 0 ) Z_*=(X_*,t_*)\in\widehat{P}_r(Z_0) Z ∗ = ( X ∗ , t ∗ ) ∈ P r ( Z 0 ) 0 < τ < r 0<\tau<r 0 < τ < r
On the other hand, by the interior Hölder continuity of U U U α 2 ∈ ( 0 , 1 ) \alpha_2\in(0,1) α 2 ∈ ( 0 , 1 ) Z ∗ = ( x ∗ , y ∗ , t ∗ ) ∈ P r + ( Z 0 ) Z_*=(x_*,y_*,t_*)\in P_r^+(Z_0) Z ∗ = ( x ∗ , y ∗ , t ∗ ) ∈ P r + ( Z 0 ) y ∗ > 0 y_*>0 y ∗ > 0
τ − ( m + 1 + a ) ∬ P τ ( Z ∗ ) ( ∣ D U ∣ 2 + τ 2 ∣ ∂ t U ∣ 2 ) μ a d t ≤ C ( τ r ) 2 α 2 y ∗ − ( m + 1 + a ) ∬ P y ∗ + ( Z ∗ ) ∣ D U ∣ 2 μ a d t ≤ C ( τ r ) 2 α 2 ( 2 y ∗ ) − ( m + 1 + a ) ∬ P 2 y ∗ + ( ( X ∗ , 0 , t ∗ ) ) ∣ D U ∣ 2 μ a d t (76) \begin{aligned}
\tau^{-(m+1+a)}\iint_{P_\tau(Z_*)}\left(|DU|^2+\tau^2|\partial_tU|^2\right)\mu_a\,dt\\
\leq C\left(\frac{\tau}{r}\right)^{2\alpha_2}y_*^{-(m+1+a)}\iint_{P_{y_*}^+(Z_*)}|DU|^2\mu_a\,dt\\
\leq C\left(\frac{\tau}{r}\right)^{2\alpha_2}(2y_*)^{-(m+1+a)}\iint_{P_{2y_*}^+((X_*,0,t_*))}|DU|^2\mu_a\,dt \tag*{(76)}
\end{aligned} τ − ( m + 1 + a ) ∬ P τ ( Z ∗ ) ( ∣ D U ∣ 2 + τ 2 ∣ ∂ t U ∣ 2 ) μ a d t ≤ C ( r τ ) 2 α 2 y ∗ − ( m + 1 + a ) ∬ P y ∗ + ( Z ∗ ) ∣ D U ∣ 2 μ a d t ≤ C ( r τ ) 2 α 2 ( 2 y ∗ ) − ( m + 1 + a ) ∬ P 2 y ∗ + (( X ∗ , 0 , t ∗ )) ∣ D U ∣ 2 μ a d t (76) holds for all 0 < τ < y ∗ 0<\tau<y_* 0 < τ < y ∗
Combining (75) and (5.28, it is not hard to show that
τ − ( m + 1 + a ) ∬ P τ + ( Z ∗ ) ( ∣ D U ∣ 2 + τ 2 ∣ ∂ t U ∣ 2 ) μ a d t ≤ C ( τ r ) 2 α 0 r − ( m + 1 + a ) ∬ P r + ( Z 0 ) ∣ D U ∣ 2 μ a d t (77) \tau^{-(m+1+a)}\iint_{P_\tau^+(Z_*)}\left(|DU|^2+\tau^2|\partial_tU|^2\right)\mu_a\,dt
\leq C\left(\frac{\tau}{r}\right)^{2\alpha_0}r^{-(m+1+a)}\iint_{P_r^+(Z_0)}|DU|^2\mu_a\,dt \tag*{(77)} τ − ( m + 1 + a ) ∬ P τ + ( Z ∗ ) ( ∣ D U ∣ 2 + τ 2 ∣ ∂ t U ∣ 2 ) μ a d t ≤ C ( r τ ) 2 α 0 r − ( m + 1 + a ) ∬ P r + ( Z 0 ) ∣ D U ∣ 2 μ a d t (77) holds for all Z ∗ ∈ P r + ( Z 0 ) Z_*\in P_r^+(Z_0) Z ∗ ∈ P r + ( Z 0 ) 0 < τ < r 0<\tau<r 0 < τ < r α 0 = min { α 1 , α 2 } \alpha_0=\min\{\alpha_1,\alpha_2\} α 0 = min { α 1 , α 2 } [15 ] ) and Lemma 7.1 and Lemma 7.2 below, we conclude that U ∈ C X , t α 0 , α 0 2 ( P r + ( Z 0 ) ) U\in C_{X,t}^{\alpha_0,\frac{\alpha_0}{2}}(P_r^+(Z_0)) U ∈ C X , t α 0 , 2 α 0 ( P r + ( Z 0 )) (71) holds. □ \square □
Proof of Theorem 1.3: Higher order regularity
This section is devoted to the proof of C 1 , α C^{1,\alpha} C 1 , α s s s u = U ∣ ∂ R + m + 1 × ( 0 , ∞ ) u = \left.U\right|_{\partial\mathbb{R}^{m+1}_{+}\times(0,\infty)} u = U ∣ ∂ R + m + 1 × ( 0 , ∞ ) Z 0 = ( X 0 , t 0 ) ∈ ∂ R + m + 1 × ( 0 , ∞ ) Z_{0}=(X_{0},t_{0})\in\partial\mathbb{R}^{m+1}_{+}\times(0,\infty) Z 0 = ( X 0 , t 0 ) ∈ ∂ R + m + 1 × ( 0 , ∞ ) U U U ( ∂ t k u , D x k u ) (\partial_{t}^{k}u,D_{x}^{k}u) ( ∂ t k u , D x k u ) Z 0 Z_{0} Z 0 k ≥ 1 k\geq1 k ≥ 1
To achieve this, we first choose a small R 0 > 0 R_{0}>0 R 0 > 0 U ‾ \overline{U} U P R 0 + ( Z 0 ) P^{+}_{R_{0}}(Z_{0}) P R 0 + ( Z 0 ) P R 0 ( Z 0 ) P_{R_{0}}(Z_{0}) P R 0 ( Z 0 ) U ‾ \overline{U} U ∣ y ∣ a |y|^{a} ∣ y ∣ a
Since U U U Z 0 = ( X 0 , t 0 ) ∈ ∂ R + m + 1 × ( 0 , ∞ ) Z_{0}=(X_{0},t_{0})\in\partial\mathbb{R}^{m+1}_{+}\times(0,\infty) Z 0 = ( X 0 , t 0 ) ∈ ∂ R + m + 1 × ( 0 , ∞ ) R 0 > 0 R_{0}>0 R 0 > 0
1 2 ≤ ∣ U ( Z ) ∣ ≤ 1 , ∀ Z ∈ P R 0 + ( Z 0 ) . (78) \frac{1}{2}\leq|U(Z)|\leq1,\quad\forall Z\in P^{+}_{R_{0}}(Z_{0}). \tag*{(78)} 2 1 ≤ ∣ U ( Z ) ∣ ≤ 1 , ∀ Z ∈ P R 0 + ( Z 0 ) . (78) Now we define the extension map U ‾ : P R 0 ( Z 0 ) → R ℓ \overline{U}:P_{R_{0}}(Z_{0})\to\mathbb{R}^{\ell} U : P R 0 ( Z 0 ) → R ℓ
U ‾ ( X , t ) = { U ( X , t ) , ( X , t ) = ( ( x , y ) , t ) ∈ P R 0 + ( Z 0 ) , ( 2 U ∣ U ∣ − U ) ( x , − y , t ) , ( X , t ) = ( ( x , y ) , t ) ∈ P R 0 − ( Z 0 ) . \overline{U}(X,t)=
\begin{cases}
U(X,t), & (X,t)=((x,y),t)\in P^{+}_{R_{0}}(Z_{0}),\\
\left(2\frac{U}{|U|}-U\right)(x,-y,t), & (X,t)=((x,y),t)\in P^{-}_{R_{0}}(Z_{0}).
\end{cases} U ( X , t ) = { U ( X , t ) , ( 2 ∣ U ∣ U − U ) ( x , − y , t ) , ( X , t ) = (( x , y ) , t ) ∈ P R 0 + ( Z 0 ) , ( X , t ) = (( x , y ) , t ) ∈ P R 0 − ( Z 0 ) . We remark that this type of extension has played crucial roles in the study of boundary regularity for heat flow of harmonic maps subject to free boundary conditions (see for example [28 , 6 ] ).
By direct calculations and application of the boundary condition of U U U
Lemma 6.1 The extension map U ‾ : P R 0 ( Z 0 ) → R ℓ \overline{U}:P_{R_{0}}(Z_{0})\to\mathbb{R}^{\ell} U : P R 0 ( Z 0 ) → R ℓ
∣ y ∣ a ∂ t U ‾ − div X ( ∣ y ∣ a D U ‾ ) = ∣ y ∣ a Γ , (79) |y|^{a}\partial_{t}\overline{U}-\operatorname{div}_{X}\left(|y|^{a}D\overline{U}\right)=|y|^{a}\Gamma, \tag*{(79)} ∣ y ∣ a ∂ t U − div X ( ∣ y ∣ a D U ) = ∣ y ∣ a Γ , (79) where
Γ ( X , t ) = { 0 , ( X , t ) ∈ P R 0 + ( Z 0 ) , 2 ( ∣ D U ∣ 2 U ∣ U ∣ 3 + 2 D U ⟨ D U , U ⟩ ∣ U ∣ 3 − 3 ∣ ⟨ D U , U ⟩ ∣ 2 U ∣ U ∣ 5 ) ( x , − y , t ) , ( X , t ) ∈ P R 0 − ( Z 0 ) . (80) \Gamma(X,t)=
\begin{cases}
0, & (X,t)\in P^{+}_{R_{0}}(Z_{0}),\\
2\left(\frac{|DU|^{2}U}{|U|^{3}}+2\frac{DU\langle DU,U\rangle}{|U|^{3}}-3\frac{|\langle DU,U\rangle|^{2}U}{|U|^{5}}\right)(x,-y,t), & (X,t)\in P^{-}_{R_{0}}(Z_{0}).
\end{cases} \tag*{(80)} Γ ( X , t ) = { 0 , 2 ( ∣ U ∣ 3 ∣ D U ∣ 2 U + 2 ∣ U ∣ 3 D U ⟨ D U , U ⟩ − 3 ∣ U ∣ 5 ∣ ⟨ D U , U ⟩ ∣ 2 U ) ( x , − y , t ) , ( X , t ) ∈ P R 0 + ( Z 0 ) , ( X , t ) ∈ P R 0 − ( Z 0 ) . (80) Observe that (6.2) and (6.3) can be viewed as a ∣ y ∣ a |y|^{a} ∣ y ∣ a
∣ Γ ( X , t ) ∣ ≤ { 0 , ( X , t ) ∈ P R 0 + ( Z 0 ) , C ∣ D U ∣ 2 ( x , − y , t ) , ( X , t ) ∈ P R 0 − ( Z 0 ) . |\Gamma(X,t)|\leq
\begin{cases}
0, & (X,t)\in P^{+}_{R_{0}}(Z_{0}),\\
C|DU|^{2}(x,-y,t), & (X,t)\in P^{-}_{R_{0}}(Z_{0}).
\end{cases} ∣Γ ( X , t ) ∣ ≤ { 0 , C ∣ D U ∣ 2 ( x , − y , t ) , ( X , t ) ∈ P R 0 + ( Z 0 ) , ( X , t ) ∈ P R 0 − ( Z 0 ) . It turns out that some of the techniques to prove higher order boundary regularity for the heat flow of harmonic maps can be adapted to establish the smoothness of u u u Z 0 Z_{0} Z 0
Step 1. Improving the Hölder continuity of U U U U ‾ ∈ C X , t γ a , γ a 2 ( P R 0 2 ( Z 0 ) ) \overline{U}\in C^{\gamma_{a},\frac{\gamma_{a}}{2}}_{X,t}\left(P_{\frac{R_{0}}{2}}(Z_{0})\right) U ∈ C X , t γ a , 2 γ a ( P 2 R 0 ( Z 0 ) )
γ a = { 2 s , 0 < s < 1 2 , 1 , 1 2 ≤ s < 1. \gamma_{a}=
\begin{cases}
2s, & 0<s<\frac{1}{2},\\
1, & \frac{1}{2}\leq s<1.
\end{cases} γ a = { 2 s , 1 , 0 < s < 2 1 , 2 1 ≤ s < 1. To show this, let R ∈ ( R 0 2 , R 0 ) R\in\left(\frac{R_{0}}{2},R_{0}\right) R ∈ ( 2 R 0 , R 0 ) V : P R ( Z 0 ) → R ℓ V:P_{R}(Z_{0})\to\mathbb{R}^{\ell} V : P R ( Z 0 ) → R ℓ
{ ∣ y ∣ a ∂ t V − div X ( ∣ y ∣ a D V ) = 0 in P R ( Z 0 ) , V = U ‾ on ∂ p P R ( Z 0 ) . (81) \begin{cases}
|y|^{a}\partial_{t}V-\operatorname{div}_{X}\left(|y|^{a}DV\right)=0 & \text{in }P_{R}(Z_{0}),\\
V=\overline{U} & \text{on }\partial_{p}P_{R}(Z_{0}). \tag*{(81)}
\end{cases} { ∣ y ∣ a ∂ t V − div X ( ∣ y ∣ a D V ) = 0 V = U in P R ( Z 0 ) , on ∂ p P R ( Z 0 ) . (81) Here ∂ p P R ( Z 0 ) \partial_pP_R(Z_0) ∂ p P R ( Z 0 ) P R ( Z 0 ) P_R(Z_0) P R ( Z 0 ) U ‾ \overline{U} U α 0 ∈ ( 0 , 1 ) \alpha_0\in(0,1) α 0 ∈ ( 0 , 1 )
osc P R ( Z 0 ) V ≤ osc P R ( Z 0 ) U ‾ ≤ C R α 0 . \operatorname{osc}_{P_R(Z_0)}V\leq\operatorname{osc}_{P_R(Z_0)}\overline{U}\leq CR^{\alpha_0}. osc P R ( Z 0 ) V ≤ osc P R ( Z 0 ) U ≤ C R α 0 . Testing (79) and (81) with U ‾ − V \overline{U}-V U − V P R ( Z 0 ) P_R(Z_0) P R ( Z 0 )
∬ P R ( Z 0 ) ∣ D ( U ‾ − V ) ∣ 2 ∣ y ∣ a d X d t ≤ C ∥ U ‾ − V ∥ L ∞ ( P R ( Z 0 ) ) ∬ P R + ( Z 0 ) ∣ D U ‾ ∣ 2 ∣ y ∣ a d X d t ≤ C ( osc P R ( Z 0 ) U ‾ ) R m + 1 + a + 2 α 0 ≤ C R m + 1 + a + 3 α 0 . (82) \begin{aligned}
\iint_{P_R(Z_0)}|D(\overline{U}-V)|^2|y|^a\,dXdt\leq C\|\overline{U}-V\|_{L^\infty(P_R(Z_0))}\iint_{P_R^+(Z_0)}|D\overline{U}|^2|y|^a\,dXdt \\
&\leq C\left(\operatorname{osc}_{P_R(Z_0)}\overline{U}\right)R^{m+1+a+2\alpha_0} \\
&\leq CR^{m+1+a+3\alpha_0}. \tag*{(82)}
\end{aligned} ∬ P R ( Z 0 ) ∣ D ( U − V ) ∣ 2 ∣ y ∣ a d X d t ≤ C ∥ U − V ∥ L ∞ ( P R ( Z 0 )) ∬ P R + ( Z 0 ) ∣ D U ∣ 2 ∣ y ∣ a d X d t ≤ C ( osc P R ( Z 0 ) U ) R m + 1 + a + 2 α 0 ≤ C R m + 1 + a + 3 α 0 . (82) On the other hand, it is well-known (see [11 ] ) that V ∈ C γ a , γ a 2 ( P 3 R 4 ( Z 0 ) ) V\in C^{\gamma_a,\frac{\gamma_a}{2}}(P_{\frac{3R}{4}}(Z_0)) V ∈ C γ a , 2 γ a ( P 4 3 R ( Z 0 ))
( θ R ) − ( m + 1 + a ) ∬ P θ R ( Z 0 ) ∣ D V ∣ 2 ∣ y ∣ a d X d t ≤ C θ 2 γ a R − ( m + 1 + a ) ∬ P R ( Z 0 ) ∣ D V ∣ 2 ∣ y ∣ a d X d t ≤ C θ 2 γ a R − ( m + 1 + a ) ∬ P R ( Z 0 ) ∣ D U ‾ ∣ 2 ∣ y ∣ a d X d t ≤ C θ 2 γ a R − ( m + 1 + a ) ∬ P R + ( Z 0 ) ∣ D U ‾ ∣ 2 μ a d t d x . (83) \begin{aligned}
(\theta R)^{-(m+1+a)}\iint_{P_{\theta R}(Z_0)}|DV|^2|y|^a\,dXdt\leq C\theta^{2\gamma_a}R^{-(m+1+a)}\iint_{P_R(Z_0)}|DV|^2|y|^a\,dXdt \\
&\leq C\theta^{2\gamma_a}R^{-(m+1+a)}\iint_{P_R(Z_0)}|D\overline{U}|^2|y|^a\,dXdt \\
&\leq C\theta^{2\gamma_a}R^{-(m+1+a)}\iint_{P_R^+(Z_0)}|D\overline{U}|^2\mu_a\,dtdx. \tag*{(83)}
\end{aligned} ( θ R ) − ( m + 1 + a ) ∬ P θ R ( Z 0 ) ∣ D V ∣ 2 ∣ y ∣ a d X d t ≤ C θ 2 γ a R − ( m + 1 + a ) ∬ P R ( Z 0 ) ∣ D V ∣ 2 ∣ y ∣ a d X d t ≤ C θ 2 γ a R − ( m + 1 + a ) ∬ P R ( Z 0 ) ∣ D U ∣ 2 ∣ y ∣ a d X d t ≤ C θ 2 γ a R − ( m + 1 + a ) ∬ P R + ( Z 0 ) ∣ D U ∣ 2 μ a d t d x . (83) holds for all 0 < θ < 3 4 0<\theta<\frac{3}{4} 0 < θ < 4 3 (82) with (83) , we obtain that for any θ ∈ ( 0 , 3 4 ) \theta\in(0,\frac{3}{4}) θ ∈ ( 0 , 4 3 )
( θ R ) − ( m + 1 + a ) ∬ P θ R + ( Z 0 ) ∣ D U ‾ ∣ 2 μ a d t (\theta R)^{-(m+1+a)}\iint_{P_{\theta R}^+(Z_0)}|D\overline{U}|^2\mu_a\,dt ( θ R ) − ( m + 1 + a ) ∬ P θ R + ( Z 0 ) ∣ D U ∣ 2 μ a d t ≤ 2 ( θ R ) − ( m + 1 + a ) ∬ P θ R ( Z 0 ) ( ∣ D V ∣ 2 + ∣ D ( U ‾ − V ) ∣ 2 ) ∣ y ∣ a d X d t ≤ C θ 2 γ a R − ( m + 1 + a ) ∬ P R + ( Z 0 ) ∣ D U ‾ ∣ 2 μ a d t + C θ − ( m + 1 + a ) R 3 α 0 . \begin{aligned}
\leq2(\theta R)^{-(m+1+a)}\iint_{P_{\theta R}(Z_0)}\left(|DV|^2+|D(\overline{U}-V)|^2\right)|y|^a\,dXdt \\
&\leq C\theta^{2\gamma_a}R^{-(m+1+a)}\iint_{P_R^+(Z_0)}|D\overline{U}|^2\mu_a\,dt+C\theta^{-(m+1+a)}R^{3\alpha_0}.
\end{aligned} ≤ 2 ( θ R ) − ( m + 1 + a ) ∬ P θ R ( Z 0 ) ( ∣ D V ∣ 2 + ∣ D ( U − V ) ∣ 2 ) ∣ y ∣ a d X d t ≤ C θ 2 γ a R − ( m + 1 + a ) ∬ P R + ( Z 0 ) ∣ D U ∣ 2 μ a d t + C θ − ( m + 1 + a ) R 3 α 0 . Without loss of generality, we start the iterations by assuming 3 α 0 ≤ 2 γ a 3\alpha_0\leq2\gamma_a 3 α 0 ≤ 2 γ a [15 ] , we conclude that for 0 < ρ < R ≤ R 0 2 0<\rho<R\leq\frac{R_0}{2} 0 < ρ < R ≤ 2 R 0
ρ − ( m + 1 + a ) ∬ P ρ + ( Z 0 ) ∣ D U ‾ ∣ 2 μ a d t ≤ C [ ( ρ R ) 3 α 0 R − ( m + 1 + a ) ∬ P R + ( Z 0 ) ∣ D U ‾ ∣ 2 μ a d t + ρ 3 α 0 ] . (84) \rho^{-(m+1+a)}\iint_{P_\rho^+(Z_0)}|D\overline{U}|^2\mu_a\,dt\leq C\left[\left(\frac{\rho}{R}\right)^{3\alpha_0}R^{-(m+1+a)}\iint_{P_R^+(Z_0)}|D\overline{U}|^2\mu_a\,dt+\rho^{3\alpha_0}\right]. \tag*{(84)} ρ − ( m + 1 + a ) ∬ P ρ + ( Z 0 ) ∣ D U ∣ 2 μ a d t ≤ C [ ( R ρ ) 3 α 0 R − ( m + 1 + a ) ∬ P R + ( Z 0 ) ∣ D U ∣ 2 μ a d t + ρ 3 α 0 ] . (84) Together with the local energy inequality (48) , this leads to
ρ − ( m + 1 + a ) ∬ P ρ + ( Z 0 ) ( ρ 2 ∣ ∂ t U ‾ ∣ 2 + ∣ D U ‾ ∣ 2 ) μ a d t ≤ C [ ( ρ R ) 3 α 0 R − ( m + 1 + a ) ∬ P R + ( Z 0 ) ∣ D U ‾ ∣ 2 μ a d t + ρ 3 α 0 ] . (85) \begin{aligned}
\rho^{-(m+1+a)}\iint_{P_\rho^+(Z_0)}\left(\rho^2|\partial_t\overline{U}|^2+|D\overline{U}|^2\right)\mu_a\,dt \\
&\leq C\left[\left(\frac{\rho}{R}\right)^{3\alpha_0}R^{-(m+1+a)}\iint_{P_R^+(Z_0)}|D\overline{U}|^2\mu_a\,dt+\rho^{3\alpha_0}\right]. \tag*{(85)}
\end{aligned} ρ − ( m + 1 + a ) ∬ P ρ + ( Z 0 ) ( ρ 2 ∣ ∂ t U ∣ 2 + ∣ D U ∣ 2 ) μ a d t ≤ C [ ( R ρ ) 3 α 0 R − ( m + 1 + a ) ∬ P R + ( Z 0 ) ∣ D U ∣ 2 μ a d t + ρ 3 α 0 ] . (85) On the other hand, if Z ∗ = ( x ∗ , y ∗ , t ∗ ) ∈ R + m + 1 × ( 0 , ∞ ) Z_*=(x_*,y_*,t_*)\in\mathbb{R}^{m+1}_+\times(0,\infty) Z ∗ = ( x ∗ , y ∗ , t ∗ ) ∈ R + m + 1 × ( 0 , ∞ ) U U U 0 < ρ < y ∗ 0<\rho<y_* 0 < ρ < y ∗
ρ − ( m + 1 + a ) ∬ P ρ ( Z ∗ ) ( ρ 2 ∣ ∂ t U ∣ 2 + ∣ D U ∣ 2 ) μ a d t ≤ C ( ρ y ∗ ) 2 γ a y ∗ − ( m + 1 + a ) ∬ P y 0 ( Z ∗ ) ∣ D U ∣ 2 μ a d t ≤ C ( ρ y ∗ ) 2 γ a y ∗ − ( m + 1 + a ) ∬ P 2 y ∗ + ( x ∗ , 0 , t ∗ ) ∣ D U ∣ 2 μ a d t . (86) \begin{aligned}
\rho^{-(m+1+a)}\iint_{P_\rho(Z_*)}\left(\rho^2|\partial_tU|^2+|DU|^2\right)\mu_a\,dt
&\le C\left(\frac{\rho}{y_*}\right)^{2\gamma_a}y_*^{-(m+1+a)}\iint_{P_{y_0}(Z_*)}|DU|^2\mu_a\,dt\\
&\le C\left(\frac{\rho}{y_*}\right)^{2\gamma_a}y_*^{-(m+1+a)}\iint_{P^+_{2y_*}(x_*,0,t_*)}|DU|^2\mu_a\,dt. \tag*{(86)}
\end{aligned} ρ − ( m + 1 + a ) ∬ P ρ ( Z ∗ ) ( ρ 2 ∣ ∂ t U ∣ 2 + ∣ D U ∣ 2 ) μ a d t ≤ C ( y ∗ ρ ) 2 γ a y ∗ − ( m + 1 + a ) ∬ P y 0 ( Z ∗ ) ∣ D U ∣ 2 μ a d t ≤ C ( y ∗ ρ ) 2 γ a y ∗ − ( m + 1 + a ) ∬ P 2 y ∗ + ( x ∗ , 0 , t ∗ ) ∣ D U ∣ 2 μ a d t . (86) Based on (84) and (86) , we can show that the following inequality
ρ − ( m + 1 + a ) ∬ P ρ + ( Z 1 ) ( ρ 2 ∣ ∂ t U ∣ 2 + ∣ D U ∣ 2 ) μ a d t ≤ C [ ( ρ R 0 ) 2 α 1 R 0 − ( m + 1 + a ) ∬ P R 0 + ( Z 0 ) ∣ D U ∣ 2 μ a d t + ρ 2 α 1 ] (87) \rho^{-(m+1+a)}\iint_{P^+_\rho(Z_1)}\left(\rho^2|\partial_tU|^2+|DU|^2\right)\mu_a\,dt
\le C\left[\left(\frac{\rho}{R_0}\right)^{2\alpha_1}R_0^{-(m+1+a)}\iint_{P^+_{R_0}(Z_0)}|DU|^2\mu_a\,dt+\rho^{2\alpha_1}\right] \tag*{(87)} ρ − ( m + 1 + a ) ∬ P ρ + ( Z 1 ) ( ρ 2 ∣ ∂ t U ∣ 2 + ∣ D U ∣ 2 ) μ a d t ≤ C [ ( R 0 ρ ) 2 α 1 R 0 − ( m + 1 + a ) ∬ P R 0 + ( Z 0 ) ∣ D U ∣ 2 μ a d t + ρ 2 α 1 ] (87) holds for all Z 1 ∈ P R 0 2 + ( Z 0 ) Z_1\in P^+_{\frac{R_0}{2}}(Z_0) Z 1 ∈ P 2 R 0 + ( Z 0 ) 0 < ρ ≤ R 0 4 0<\rho\le\frac{R_0}{4} 0 < ρ ≤ 4 R 0 α 1 = min { γ a , 3 α 0 4 } > α 0 \alpha_1=\min\{\gamma_a,\frac{3\alpha_0}{4}\}>\alpha_0 α 1 = min { γ a , 4 3 α 0 } > α 0
Applying the y a y^a y a [15 ] to (87) , we have U ∈ C X , t α 1 , α 1 2 ( P R 0 2 + ( Z 0 ) ‾ ) U\in C^{\alpha_1,\frac{\alpha_1}{2}}_{X,t}\left(\overline{P^+_{\frac{R_0}{2}}(Z_0)}\right) U ∈ C X , t α 1 , 2 α 1 ( P 2 R 0 + ( Z 0 ) )
If α 1 < γ a \alpha_1<\gamma_a α 1 < γ a U ∈ C X , t α 2 , α 2 2 ( P R 0 4 + ( Z 0 ) ‾ ) U\in C^{\alpha_2,\frac{\alpha_2}{2}}_{X,t}\left(\overline{P^+_{\frac{R_0}{4}}(Z_0)}\right) U ∈ C X , t α 2 , 2 α 2 ( P 4 R 0 + ( Z 0 ) ) α 2 = min { γ a , 9 α 0 4 } \alpha_2=\min\{\gamma_a,\frac{9\alpha_0}{4}\} α 2 = min { γ a , 4 9 α 0 } k k k k ≥ 1 k\ge1 k ≥ 1 α k = γ a \alpha_k=\gamma_a α k = γ a U ∈ C X , t γ a , γ a 2 ( P R 0 2 k + ( Z 0 ) ‾ ) U\in C^{\gamma_a,\frac{\gamma_a}{2}}_{X,t}\left(\overline{P^+_{\frac{R_0}{2^k}}(Z_0)}\right) U ∈ C X , t γ a , 2 γ a ( P 2 k R 0 + ( Z 0 ) )
Step 2. Regularity of λ u \lambda_u λ u 0 < s < 1 2 0<s<\frac{1}{2} 0 < s < 2 1 1 2 ≤ s < 1 \frac{1}{2}\le s<1 2 1 ≤ s < 1 0 < s < 1 2 0<s<\frac{1}{2} 0 < s < 2 1 Z 0 = ( 0 , 1 ) ∈ ∂ R + m + 1 × ( 0 , ∞ ) Z_0=(0,1)\in\partial\mathbb{R}^{m+1}_+\times(0,\infty) Z 0 = ( 0 , 1 ) ∈ ∂ R + m + 1 × ( 0 , ∞ ) u ∈ C X , t 2 s , s ( P 1 2 + ( 0 , 1 ) ^ ) u\in C^{2s,s}_{X,t}\left(\widehat{P^+_{\frac{1}{2}}(0,1)}\right) u ∈ C X , t 2 s , s ( P 2 1 + ( 0 , 1 ) )
Claim. For any small 0 < ε < s 0<\varepsilon<s 0 < ε < s λ u ∈ C X , t 2 ( s − ε ) , s − ε ( P 1 4 + ( 0 , 1 ) ^ ) \lambda_u\in C^{2(s-\varepsilon),s-\varepsilon}_{X,t}\left(\widehat{P^+_{\frac{1}{4}}(0,1)}\right) λ u ∈ C X , t 2 ( s − ε ) , s − ε ( P 4 1 + ( 0 , 1 ) ) R m × ( 0 , ∞ ) \mathbb{R}^m\times(0,\infty) R m × ( 0 , ∞ )
d ( ( x , t ) , ( y , τ ) ) = ∣ x − y ∣ + ∣ t − τ ∣ , ( x , t ) , ( y , τ ) ∈ R m × ( 0 , ∞ ) . d((x,t),(y,\tau))=|x-y|+\sqrt{|t-\tau|},\qquad(x,t),(y,\tau)\in\mathbb{R}^m\times(0,\infty). d (( x , t ) , ( y , τ )) = ∣ x − y ∣ + ∣ t − τ ∣ , ( x , t ) , ( y , τ ) ∈ R m × ( 0 , ∞ ) . Given two different points z 1 = ( x 1 , t 1 ) , z 2 = ( x 2 , t 2 ) ∈ P 1 4 + ( 0 , 1 ) ^ z_1=(x_1,t_1),z_2=(x_2,t_2)\in\widehat{P^+_{\frac{1}{4}}(0,1)} z 1 = ( x 1 , t 1 ) , z 2 = ( x 2 , t 2 ) ∈ P 4 1 + ( 0 , 1 ) 0 < δ = d ( z 1 , z 2 ) ≤ 1 2 0<\delta=d(z_1,z_2)\le\frac{1}{2} 0 < δ = d ( z 1 , z 2 ) ≤ 2 1
∣ λ u ( z 1 ) − λ u ( z 2 ) ∣ ≤ ∣ ∬ { d ( ( y , τ ) , z 1 ) < 2 δ } [ ∣ u ( z 1 ) − u ( y , τ ) ∣ 2 K s ( x 1 − y , t 1 − τ ) − ∣ u ( z 2 ) − u ( y , τ ) ∣ 2 K s ( x 2 − y , t 2 − τ ) ] d y d τ ∣ + ∣ ∬ { d ( ( y , τ ) , z 1 ) ≥ 2 δ } [ ∣ u ( z 1 ) − u ( y , τ ) ∣ 2 K s ( x 1 − y , t 1 − τ ) − ∣ u ( z 2 ) − u ( y , τ ) ∣ 2 K s ( x 2 − y , t 2 − τ ) ] d y d τ ∣ = I + I I . \begin{aligned}
|\lambda_u(z_1)-\lambda_u(z_2)|
&\le\left|\iint_{\{d((y,\tau),z_1)<2\delta\}}\left[|u(z_1)-u(y,\tau)|^2K_s(x_1-y,t_1-\tau)\right.\right.\\
&\qquad\left.\left.-|u(z_2)-u(y,\tau)|^2K_s(x_2-y,t_2-\tau)\right]\,dy\,d\tau\right|\\
&\quad+\left|\iint_{\{d((y,\tau),z_1)\ge2\delta\}}\left[|u(z_1)-u(y,\tau)|^2K_s(x_1-y,t_1-\tau)\right.\right.\\
&\qquad\left.\left.-|u(z_2)-u(y,\tau)|^2K_s(x_2-y,t_2-\tau)\right]\,dy\,d\tau\right|\\
&=I+II.
\end{aligned} ∣ λ u ( z 1 ) − λ u ( z 2 ) ∣ ≤ ∬ { d (( y , τ ) , z 1 ) < 2 δ } [ ∣ u ( z 1 ) − u ( y , τ ) ∣ 2 K s ( x 1 − y , t 1 − τ ) − ∣ u ( z 2 ) − u ( y , τ ) ∣ 2 K s ( x 2 − y , t 2 − τ ) ] d y d τ + ∬ { d (( y , τ ) , z 1 ) ≥ 2 δ } [ ∣ u ( z 1 ) − u ( y , τ ) ∣ 2 K s ( x 1 − y , t 1 − τ ) − ∣ u ( z 2 ) − u ( y , τ ) ∣ 2 K s ( x 2 − y , t 2 − τ ) ] d y d τ = I + I I . Since
∣ u ( z ) − u ( w ) ∣ ≤ C d ( z , w ) 2 s , ∀ z , w ∈ P 1 2 + ( 0 , 1 ) ^ , |u(z)-u(w)|\le Cd(z,w)^{2s},\qquad\forall z,w\in\widehat{P^+_{\frac{1}{2}}(0,1)}, ∣ u ( z ) − u ( w ) ∣ ≤ C d ( z , w ) 2 s , ∀ z , w ∈ P 2 1 + ( 0 , 1 ) , and
∣ K s ( x − y , t − τ ) ∣ ≤ C d ( ( x , t ) , ( y , τ ) ) n + 2 + 2 s , ( x , t ) , ( y , τ ) ∈ R m × ( 0 , ∞ ) , |K_s(x-y,t-\tau)|\le\frac{C}{d((x,t),(y,\tau))^{n+2+2s}},\qquad(x,t),(y,\tau)\in\mathbb{R}^m\times(0,\infty), ∣ K s ( x − y , t − τ ) ∣ ≤ d (( x , t ) , ( y , τ ) ) n + 2 + 2 s C , ( x , t ) , ( y , τ ) ∈ R m × ( 0 , ∞ ) , and d ( z , z 1 ) ≤ 2 δ d(z,z_1) \le2\delta d ( z , z 1 ) ≤ 2 δ d ( z , z 2 ) ≤ 3 δ d(z,z_2) \le3\delta d ( z , z 2 ) ≤ 3 δ I I I
∣ I ∣ ≤ ∬ { d ( ( y , τ ) , z 1 ) < 2 δ } ∣ u ( z 1 ) − u ( y , τ ) ∣ 2 ∣ K s ( x 1 − y , t 1 − τ ) ∣ + ∬ { d ( ( y , τ ) , z 2 ) < 3 δ } ∣ u ( z 2 ) − u ( y , τ ) ∣ 2 ∣ K s ( x 2 − y , t 2 − τ ) ∣ ≤ C ∬ { d ( ( y , τ ) , z 1 ) < 2 δ } 1 d ( ( y , τ ) , z 1 ) n + 2 − 2 s + ∬ { d ( ( y , τ ) , z 2 ) < 3 δ } 1 d ( ( y , τ ) , z 2 ) n + 2 − 2 s ≤ C ∫ 0 2 δ r − 1 + 2 s d r + C ∫ 0 3 δ r − 1 + 2 s d r ≤ C δ 2 s . \begin{aligned}
|I| &\le\iint_{\{d((y,\tau),z_1)<2\delta\}} |u(z_1)-u(y,\tau)|^2 |K_s(x_1-y,t_1-\tau)| \\
&\quad+ \iint_{\{d((y,\tau),z_2)<3\delta\}} |u(z_2)-u(y,\tau)|^2 |K_s(x_2-y,t_2-\tau)| \\
&\le C\iint_{\{d((y,\tau),z_1)<2\delta\}} \frac{1}{d((y,\tau),z_1)^{n+2-2s}} + \iint_{\{d((y,\tau),z_2)<3\delta\}} \frac{1}{d((y,\tau),z_2)^{n+2-2s}} \\
&\le C\int_0^{2\delta} r^{-1+2s}\,dr+C\int_0^{3\delta}r^{-1+2s}\,dr\le C\delta^{2s}.
\end{aligned} ∣ I ∣ ≤ ∬ { d (( y , τ ) , z 1 ) < 2 δ } ∣ u ( z 1 ) − u ( y , τ ) ∣ 2 ∣ K s ( x 1 − y , t 1 − τ ) ∣ + ∬ { d (( y , τ ) , z 2 ) < 3 δ } ∣ u ( z 2 ) − u ( y , τ ) ∣ 2 ∣ K s ( x 2 − y , t 2 − τ ) ∣ ≤ C ∬ { d (( y , τ ) , z 1 ) < 2 δ } d (( y , τ ) , z 1 ) n + 2 − 2 s 1 + ∬ { d (( y , τ ) , z 2 ) < 3 δ } d (( y , τ ) , z 2 ) n + 2 − 2 s 1 ≤ C ∫ 0 2 δ r − 1 + 2 s d r + C ∫ 0 3 δ r − 1 + 2 s d r ≤ C δ 2 s . For I I II I I
∣ I I ∣ ≤ ∬ { d ( ( y , τ ) , z 1 ) ≥ 2 δ } ∣ u ( z 1 ) − u ( y , τ ) ∣ 2 ∣ K s ( x 1 − y , t 1 − τ ) − K s ( x 2 − y , t 2 − τ ) ∣ + ∬ { d ( ( y , τ ) , z 1 ) ≥ 2 δ } ∣ ∣ u ( z 1 ) − u ( y , τ ) ∣ 2 − ∣ u ( z 2 ) − u ( y , τ ) ∣ 2 ∣ ∣ K s ( x 2 − y , t 2 − τ ) ∣ = I I I + I V . \begin{aligned}
|II| &\le\iint_{\{d((y,\tau),z_1)\ge2\delta\}} |u(z_1)-u(y,\tau)|^2\left|K_s(x_1-y,t_1-\tau)-K_s(x_2-y,t_2-\tau)\right| \\
&\quad+\iint_{\{d((y,\tau),z_1)\ge2\delta\}}\left||u(z_1)-u(y,\tau)|^2-|u(z_2)-u(y,\tau)|^2\right||K_s(x_2-y,t_2-\tau)| \\
&= III+IV.
\end{aligned} ∣ I I ∣ ≤ ∬ { d (( y , τ ) , z 1 ) ≥ 2 δ } ∣ u ( z 1 ) − u ( y , τ ) ∣ 2 ∣ K s ( x 1 − y , t 1 − τ ) − K s ( x 2 − y , t 2 − τ ) ∣ + ∬ { d (( y , τ ) , z 1 ) ≥ 2 δ } ∣ u ( z 1 ) − u ( y , τ ) ∣ 2 − ∣ u ( z 2 ) − u ( y , τ ) ∣ 2 ∣ K s ( x 2 − y , t 2 − τ ) ∣ = I I I + I V . Since
∣ K s ( x 1 − y , t 1 − τ ) − K s ( x 2 − y , t 2 − τ ) ∣ ≤ C d ( z 1 , z 2 ) d ( ( x 1 , t 1 ) , ( y , τ ) ) n + 3 + 2 s , for d ( ( y , τ ) , z 1 ) ≥ 2 δ , \left|K_s(x_1-y,t_1-\tau)-K_s(x_2-y,t_2-\tau)\right|\le\frac{Cd(z_1,z_2)}{d((x_1,t_1),(y,\tau))^{n+3+2s}},\qquad\text{for }d((y,\tau),z_1)\ge2\delta, ∣ K s ( x 1 − y , t 1 − τ ) − K s ( x 2 − y , t 2 − τ ) ∣ ≤ d (( x 1 , t 1 ) , ( y , τ ) ) n + 3 + 2 s C d ( z 1 , z 2 ) , for d (( y , τ ) , z 1 ) ≥ 2 δ , and ∣ u ( z ) − u ( w ) ∣ 2 ≤ 4 |u(z)-u(w)|^2\le4 ∣ u ( z ) − u ( w ) ∣ 2 ≤ 4 z , w ∈ R m × ( 0 , ∞ ) z,w\in\mathbb{R}^m\times(0,\infty) z , w ∈ R m × ( 0 , ∞ ) I I I III I I I
∣ I I I ∣ ≤ ∬ { 2 δ ≤ d ( ( y , τ ) , z 1 ) < 2 } ∣ u ( z 1 ) − u ( y , τ ) ∣ 2 ∣ K s ( x 1 − y , t 1 − τ ) − K s ( x 2 − y , t 2 − τ ) ∣ + ∬ { d ( ( y , τ ) , z 1 ) ≥ 2 } ∣ u ( z 1 ) − u ( y , τ ) ∣ 2 ∣ K s ( x 1 − y , t 1 − τ ) − K s ( x 2 − y , t 2 − τ ) ∣ ≤ C δ ( ∬ { 2 δ ≤ d ( ( y , τ ) , z 1 ) < 2 } d ( ( y , τ ) , z 1 ) 4 s d ( ( y , τ ) , z 1 ) n + 3 + 2 s + ∬ { d ( ( y , τ ) , z 1 ) ≥ 2 } 4 d ( ( y , τ ) , z 1 ) n + 3 + 2 s ) ≤ C δ ∫ 2 δ 2 r − ( n + 3 − 2 s ) r n + 1 d r + C δ ∫ 2 ∞ r − ( n + 3 + 2 s ) r n + 1 d r ≤ C δ 2 s + C δ ≤ C δ 2 s , \begin{aligned}
|III| &\le\iint_{\{2\delta\le d((y,\tau),z_1)<2\}} |u(z_1)-u(y,\tau)|^2\left|K_s(x_1-y,t_1-\tau)-K_s(x_2-y,t_2-\tau)\right| \\
&\quad+\iint_{\{d((y,\tau),z_1)\ge2\}} |u(z_1)-u(y,\tau)|^2\left|K_s(x_1-y,t_1-\tau)-K_s(x_2-y,t_2-\tau)\right| \\
&\le C\delta\left(\iint_{\{2\delta\le d((y,\tau),z_1)<2\}}\frac{d((y,\tau),z_1)^{4s}}{d((y,\tau),z_1)^{n+3+2s}}+\iint_{\{d((y,\tau),z_1)\ge2\}}\frac{4}{d((y,\tau),z_1)^{n+3+2s}}\right) \\
&\le C\delta\int_{2\delta}^{2}r^{-(n+3-2s)}r^{n+1}\,dr+C\delta\int_2^\infty r^{-(n+3+2s)}r^{n+1}\,dr \\
&\le C\delta^{2s}+C\delta\le C\delta^{2s},
\end{aligned} ∣ I I I ∣ ≤ ∬ { 2 δ ≤ d (( y , τ ) , z 1 ) < 2 } ∣ u ( z 1 ) − u ( y , τ ) ∣ 2 ∣ K s ( x 1 − y , t 1 − τ ) − K s ( x 2 − y , t 2 − τ ) ∣ + ∬ { d (( y , τ ) , z 1 ) ≥ 2 } ∣ u ( z 1 ) − u ( y , τ ) ∣ 2 ∣ K s ( x 1 − y , t 1 − τ ) − K s ( x 2 − y , t 2 − τ ) ∣ ≤ C δ ( ∬ { 2 δ ≤ d (( y , τ ) , z 1 ) < 2 } d (( y , τ ) , z 1 ) n + 3 + 2 s d (( y , τ ) , z 1 ) 4 s + ∬ { d (( y , τ ) , z 1 ) ≥ 2 } d (( y , τ ) , z 1 ) n + 3 + 2 s 4 ) ≤ C δ ∫ 2 δ 2 r − ( n + 3 − 2 s ) r n + 1 d r + C δ ∫ 2 ∞ r − ( n + 3 + 2 s ) r n + 1 d r ≤ C δ 2 s + C δ ≤ C δ 2 s , as 0 < 2 s < 1 0<2s<1 0 < 2 s < 1
For I V IV I V d ( w , z 1 ) ≥ 2 δ d(w,z_1) \ge2\delta d ( w , z 1 ) ≥ 2 δ d ( w , z 2 ) ≥ δ d(w,z_2) \ge\delta d ( w , z 2 ) ≥ δ
∣ I V ∣ ≤ C ∬ { d ( ( y , τ ) , z 2 ) > δ } ∣ ⟨ u ( z 1 ) − u ( z 2 ) , u ( y , τ ) ⟩ ∣ d ( ( y , τ ) , z 2 ) − ( n + 2 + 2 s ) ≤ C ∬ { d ( ( y , τ ) , z 2 ) > δ } ∣ ⟨ u ( z 1 ) − u ( z 2 ) , u ( y , τ ) − u ( z 2 ) ⟩ ∣ + ∣ u ( z 1 ) − u ( z 2 ) ∣ 2 d ( ( y , τ ) , z 2 ) n + 2 + 2 s ≤ C ∬ { d ( ( y , τ ) , z 2 ) > δ } ∣ u ( z 1 ) − u ( z 2 ) ∣ ∣ u ( y , τ ) − u ( z 2 ) ∣ + ∣ u ( z 1 ) − u ( z 2 ) ∣ 2 d ( ( y , τ ) , z 2 ) n + 2 + 2 s ≤ C δ 2 s ∬ { δ ≤ d ( ( y , τ ) , z 2 ) ≤ 1 } d ( ( y , τ ) , z 2 ) 2 s d ( ( y , τ ) , z 2 ) n + 2 + 2 s + C δ 2 s ∬ { d ( ( y , τ ) , z 2 ) ≥ 1 } 4 d ( ( y , τ ) , z 2 ) n + 2 + 2 s + C δ 4 s ∬ { d ( ( y , τ ) , z 2 ) ≥ δ } 1 d ( ( y , τ ) , z 2 ) n + 2 + 2 s ≤ C δ 2 s ( ∫ δ 1 r − ( n + 2 + 2 s ) r 2 s r n + 1 d r + ∫ 1 ∞ r − ( n + 2 + 2 s ) r n + 1 d r + δ 2 s ∫ δ ∞ r − ( n + 2 + 2 s ) r n + 1 d r ) ≤ C δ 2 s ( 1 + ∣ ln δ ∣ ) . \begin{aligned}
|IV| &\le C \iint_{\{d((y,\tau),z_2)>\delta\}} \left|\langle u(z_1)-u(z_2),u(y,\tau)\rangle\right|d((y,\tau),z_2)^{-(n+2+2s)} \\
&\le C \iint_{\{d((y,\tau),z_2)>\delta\}} \frac{\left|\langle u(z_1)-u(z_2),u(y,\tau)-u(z_2)\rangle\right|+|u(z_1)-u(z_2)|^2}{d((y,\tau),z_2)^{n+2+2s}} \\
&\le C \iint_{\{d((y,\tau),z_2)>\delta\}} \frac{|u(z_1)-u(z_2)||u(y,\tau)-u(z_2)|+|u(z_1)-u(z_2)|^2}{d((y,\tau),z_2)^{n+2+2s}} \\
&\le C\delta^{2s}\iint_{\{\delta\le d((y,\tau),z_2)\le1\}}\frac{d((y,\tau),z_2)^{2s}}{d((y,\tau),z_2)^{n+2+2s}}+C\delta^{2s}\iint_{\{d((y,\tau),z_2)\ge1\}}\frac{4}{d((y,\tau),z_2)^{n+2+2s}} \\
&\quad+C\delta^{4s}\iint_{\{d((y,\tau),z_2)\ge\delta\}}\frac{1}{d((y,\tau),z_2)^{n+2+2s}} \\
&\le C\delta^{2s}\left(\int_\delta^1r^{-(n+2+2s)}r^{2s}r^{n+1}\,dr+\int_1^\infty r^{-(n+2+2s)}r^{n+1}\,dr+\delta^{2s}\int_\delta^\infty r^{-(n+2+2s)}r^{n+1}\,dr\right) \\
&\le C\delta^{2s}(1+|\ln\delta|).
\end{aligned} ∣ I V ∣ ≤ C ∬ { d (( y , τ ) , z 2 ) > δ } ∣ ⟨ u ( z 1 ) − u ( z 2 ) , u ( y , τ )⟩ ∣ d (( y , τ ) , z 2 ) − ( n + 2 + 2 s ) ≤ C ∬ { d (( y , τ ) , z 2 ) > δ } d (( y , τ ) , z 2 ) n + 2 + 2 s ∣ ⟨ u ( z 1 ) − u ( z 2 ) , u ( y , τ ) − u ( z 2 )⟩ ∣ + ∣ u ( z 1 ) − u ( z 2 ) ∣ 2 ≤ C ∬ { d (( y , τ ) , z 2 ) > δ } d (( y , τ ) , z 2 ) n + 2 + 2 s ∣ u ( z 1 ) − u ( z 2 ) ∣∣ u ( y , τ ) − u ( z 2 ) ∣ + ∣ u ( z 1 ) − u ( z 2 ) ∣ 2 ≤ C δ 2 s ∬ { δ ≤ d (( y , τ ) , z 2 ) ≤ 1 } d (( y , τ ) , z 2 ) n + 2 + 2 s d (( y , τ ) , z 2 ) 2 s + C δ 2 s ∬ { d (( y , τ ) , z 2 ) ≥ 1 } d (( y , τ ) , z 2 ) n + 2 + 2 s 4 + C δ 4 s ∬ { d (( y , τ ) , z 2 ) ≥ δ } d (( y , τ ) , z 2 ) n + 2 + 2 s 1 ≤ C δ 2 s ( ∫ δ 1 r − ( n + 2 + 2 s ) r 2 s r n + 1 d r + ∫ 1 ∞ r − ( n + 2 + 2 s ) r n + 1 d r + δ 2 s ∫ δ ∞ r − ( n + 2 + 2 s ) r n + 1 d r ) ≤ C δ 2 s ( 1 + ∣ ln δ ∣ ) . Putting all these estimates together, we obtain that
∣ λ u ( z 1 ) − λ u ( z 2 ) ∣ ≤ C d ( z 1 , z 2 ) 2 s ( 1 + ∣ ln d ( z 1 , z 2 ) ∣ ) , ∀ z 1 , z 2 ∈ P ^ 1 4 + ( Z 0 ) . |\lambda_u(z_1)-\lambda_u(z_2)|\le Cd(z_1,z_2)^{2s}(1+|\ln d(z_1,z_2)|),\qquad\forall z_1,z_2\in\widehat{P}^{+}_{\frac14}(Z_0). ∣ λ u ( z 1 ) − λ u ( z 2 ) ∣ ≤ C d ( z 1 , z 2 ) 2 s ( 1 + ∣ ln d ( z 1 , z 2 ) ∣ ) , ∀ z 1 , z 2 ∈ P 4 1 + ( Z 0 ) . This clearly yields the claim.
Smoothness of u = U ∣ ∂ R + m + 1 × ( 0 , ∞ ) u=U|_{\partial\mathbb{R}^{m+1}_+\times(0,\infty)} u = U ∣ ∂ R + m + 1 × ( 0 , ∞ )
For Z 0 = ( x 0 , 0 , t 0 ) ∈ ∂ R + m + 1 × ( 0 , ∞ ) Z_0=(x_0,0,t_0)\in\partial\mathbb{R}^{m+1}_+\times(0,\infty) Z 0 = ( x 0 , 0 , t 0 ) ∈ ∂ R + m + 1 × ( 0 , ∞ ) 0 < R 0 < t 0 2 0<R_0<\frac{\sqrt{t_0}}{2} 0 < R 0 < 2 t 0 u ∈ C x , t γ , γ 2 ( D R 0 ( x 0 ) × ( t 0 − R 0 2 , t 0 + R 0 2 ) ) u\in C^{\gamma,\frac{\gamma}{2}}_{x,t}(D_{R_0}(x_0)\times(t_0-R_0^2,t_0+R_0^2)) u ∈ C x , t γ , 2 γ ( D R 0 ( x 0 ) × ( t 0 − R 0 2 , t 0 + R 0 2 )) u ∈ C ∞ ( P ^ R 0 ( x 0 , t 0 ) ) u\in C^\infty(\widehat{P}_{R_0}(x_0,t_0)) u ∈ C ∞ ( P R 0 ( x 0 , t 0 )) 0 < s < 1 2 0<s<\frac12 0 < s < 2 1 ( x 0 , t 0 ) = ( 0 , 1 ) (x_0,t_0)=(0,1) ( x 0 , t 0 ) = ( 0 , 1 ) R 0 = 2 R_0=2 R 0 = 2 λ u ( x , t ) u ( x , t ) ∈ C x , t γ a , γ a 2 ( P ^ 1 4 + ( 0 , 1 ) ) \lambda_u(x,t)u(x,t)\in C^{\gamma_a,\frac{\gamma_a}{2}}_{x,t}(\widehat{P}^{+}_{\frac14}(0,1)) λ u ( x , t ) u ( x , t ) ∈ C x , t γ a , 2 γ a ( P 4 1 + ( 0 , 1 )) η ∈ C 0 ∞ ( R m × ( 0 , ∞ ) ) \eta\in C^\infty_0(\mathbb{R}^m\times(0,\infty)) η ∈ C 0 ∞ ( R m × ( 0 , ∞ )) P ^ 1 5 + ( 0 , 1 ) \widehat{P}^{+}_{\frac15}(0,1) P 5 1 + ( 0 , 1 ) v : R m × R → R ℓ v:\mathbb{R}^m\times\mathbb{R}\to\mathbb{R}^{\ell} v : R m × R → R ℓ
( ∂ t − Δ ) s v = η 2 λ u u in R m × R . (\partial_t-\Delta)^s v=\eta^2\lambda_u u\quad\text{in }\mathbb{R}^m\times\mathbb{R}. ( ∂ t − Δ ) s v = η 2 λ u u in R m × R . Then, by the Schauder estimate for fractional parabolic equations (see [25 Theorem 1.19] ), we deduce that v ∈ C x , t 4 s − 2 ε , 2 s − ε ( R m × ( 0 , ∞ ) ) v\in C^{4s-2\varepsilon,2s-\varepsilon}_{x,t}(\mathbb{R}^m\times(0,\infty)) v ∈ C x , t 4 s − 2 ε , 2 s − ε ( R m × ( 0 , ∞ )) ∀ 0 < ε < s \forall0<\varepsilon<s ∀0 < ε < s
Since u u u
( ∂ t − Δ ) s u = λ u u , in R m × ( 0 , ∞ ) , (\partial_t-\Delta)^s u=\lambda_u u,\quad\text{in }\mathbb{R}^m\times(0,\infty), ( ∂ t − Δ ) s u = λ u u , in R m × ( 0 , ∞ ) , If we denote w = u − v w=u-v w = u − v
( ∂ t − Δ ) s w = 0 in P ^ 1 5 + ( 0 , 1 ) . (\partial_t-\Delta)^s w=0\quad\text{in }\widehat{P}^{+}_{\frac15}(0,1). ( ∂ t − Δ ) s w = 0 in P 5 1 + ( 0 , 1 ) . Applying the Schauder estimate for fractional parabolic equations once again, we deduce that w ∈ C ∞ ( P ^ 1 5 + ( 0 , 1 ) ) w\in C^\infty(\widehat{P}^{+}_{\frac15}(0,1)) w ∈ C ∞ ( P 5 1 + ( 0 , 1 )) u ∈ C x , t 4 s − 2 ε , 2 s − ε ( P ^ 1 5 + ( 0 , 1 ) ) u\in C^{4s-2\varepsilon,2s-\varepsilon}_{x,t}(\widehat{P}^{+}_{\frac15}(0,1)) u ∈ C x , t 4 s − 2 ε , 2 s − ε ( P 5 1 + ( 0 , 1 )) ∀ 0 < ε < s \forall0<\varepsilon<s ∀0 < ε < s
By the bootstrap argument, we can conclude that u ∈ C x , t 2 k s − 2 ε , k s − ε ( P ^ 1 8 + ( 0 , 1 ) ) u\in C^{2ks-2\varepsilon,ks-\varepsilon}_{x,t}(\widehat{P}^{+}_{\frac18}(0,1)) u ∈ C x , t 2 k s − 2 ε , k s − ε ( P 8 1 + ( 0 , 1 )) k ≥ 1 k\ge1 k ≥ 1 0 < ε < s 0<\varepsilon<s 0 < ε < s u u u P ^ 1 8 + ( 0 , 1 ) \widehat{P}^{+}_{\frac18}(0,1) P 8 1 + ( 0 , 1 ) u u u Z 0 Z_0 Z 0
Appendix
In this section, we will provide several important Lemmas that are used in the previous sections. We begin this section by pointing out that for − 1 < a < 1 -1<a<1 − 1 < a < 1
w a ( x , y ) = ∣ y ∣ a , ( x , y ) ∈ R m + 1 , w_a(x,y)=|y|^a,\qquad(x,y)\in\mathbb{R}^{m+1}, w a ( x , y ) = ∣ y ∣ a , ( x , y ) ∈ R m + 1 , is a Muckenhoupt A 2 A_2 A 2 R m + 1 \mathbb{R}^{m+1} R m + 1
[ w a ] A 2 = sup Q ( 1 ∣ Q ∣ ∬ Q w a d X d t ) ( 1 ∣ Q ∣ ∬ Q w − a d X d t ) < + ∞ , (88) [w_a]_{A_2}=\sup_Q\left(\frac{1}{|Q|}\iint_Q w_a\,dXdt\right)\left(\frac{1}{|Q|}\iint_Q w_{-a}\,dXdt\right)<+\infty, \tag*{(88)} [ w a ] A 2 = Q sup ( ∣ Q ∣ 1 ∬ Q w a d X d t ) ( ∣ Q ∣ 1 ∬ Q w − a d X d t ) < + ∞ , (88) where the supremum is taken among all cubes Q ⊂ R m + 1 Q\subset\mathbb{R}^{m+1} Q ⊂ R m + 1
Now, we recall the definition of y a y^a y a R + m + 1 × ( 0 , ∞ ) \mathbb{R}^{m+1}_+\times(0,\infty) R + m + 1 × ( 0 , ∞ ) O ⊂ R m + 1 × ( 0 , ∞ ) O\subset\mathbb{R}^{m+1}\times(0,\infty) O ⊂ R m + 1 × ( 0 , ∞ ) O + = O ∩ R + m + 1 × ( 0 , ∞ ) O_+=O\cap\mathbb{R}^{m+1}_+\times(0,\infty) O + = O ∩ R + m + 1 × ( 0 , ∞ ) BMO ( O + , μ a d t ) \operatorname{BMO}(O_+,\mu_a\,dt) BMO ( O + , μ a d t ) U ∈ L l o c 1 ( O + , μ a d t ) U\in L^1_{\mathrm{loc}}(O_+,\mu_a\,dt) U ∈ L loc 1 ( O + , μ a d t )
[ U ] BMO ( O + , μ a d t ) 2 = sup { 1 ∣ P r + ( z ) ∣ a ∬ P r + ( z ) ∣ U − ( U ) P r + ( z ) a ∣ 2 μ a d t : P r ( z ) ⊂ O + } < ∞ , (89) [U]^2_{\operatorname{BMO}(O_+,\mu_a\,dt)}
=\sup\left\{\frac{1}{|P_r^+(z)|_a}\iint_{P_r^+(z)}|U-(U)^a_{P_r^+(z)}|^2\mu_a\,dt:P_r(z)\subset O_+\right\}<\infty, \tag*{(89)} [ U ] BMO ( O + , μ a d t ) 2 = sup { ∣ P r + ( z ) ∣ a 1 ∬ P r + ( z ) ∣ U − ( U ) P r + ( z ) a ∣ 2 μ a d t : P r ( z ) ⊂ O + } < ∞ , (89) where
∣ P r + ( z ) ∣ a = ∬ P r + ( z ) μ a d t and ( U ) P r + ( z ) a = 1 ∣ P r + ( z ) ∣ a ∬ P r + ( z ) U μ a d t . |P_r^+(z)|_a=\iint_{P_r^+(z)}\mu_a\,dt
\qquad\text{and}\qquad
(U)^a_{P_r^+(z)}=\frac{1}{|P_r^+(z)|_a}\iint_{P_r^+(z)}U\mu_a\,dt. ∣ P r + ( z ) ∣ a = ∬ P r + ( z ) μ a d t and ( U ) P r + ( z ) a = ∣ P r + ( z ) ∣ a 1 ∬ P r + ( z ) U μ a d t . Notice that when a = 0 a=0 a = 0 μ 0 d t = d X d t \mu_0dt=dXdt μ 0 d t = d X d t BMO ( O + , μ a d t ) \operatorname{BMO}(O_+,\mu_a\,dt) BMO ( O + , μ a d t ) BMO ( O + ) \operatorname{BMO}(O_+) BMO ( O + )
The following simple fact on the relationship between the BMO space and the y a y^a y a
Lemma 7.1 If U ∈ BMO ( O + , μ a d t ) U\in\operatorname{BMO}(O_+,\mu_a\,dt) U ∈ BMO ( O + , μ a d t ) U ∈ BMO ( O + ) U\in\operatorname{BMO}(O_+) U ∈ BMO ( O + )
[ U ] BMO ( O + ) ≤ C [ U ] BMO ( O + , μ a d t ) . (90) [U]_{\operatorname{BMO}(O_+)}\leq C[U]_{\operatorname{BMO}(O_+,\mu_a\,dt)}. \tag*{(90)} [ U ] BMO ( O + ) ≤ C [ U ] BMO ( O + , μ a d t ) . (90) Proof. For any f ∈ L l o c 1 ( O + , R ℓ ) f\in L^1_{\mathrm{loc}}(O_+,\mathbb{R}^{\ell}) f ∈ L loc 1 ( O + , R ℓ ) P r ( z ) ⊂ O + P_r(z)\subset O_+ P r ( z ) ⊂ O +
1 ∣ P r ( z ) ∣ ∬ P r ( z ) ∣ f ∣ d X d t ≤ 1 ∣ P r ( z ) ∣ ( ∬ P r ( z ) ∣ f ∣ 2 μ a d t ) 1 2 ( ∬ P r ( z ) y − a d X d t ) 1 2 ≤ C ( ∬ P r ( z ) ∣ f ∣ 2 μ a d t ) 1 2 ( ∬ P r ( z ) y a d X d t ) − 1 2 ≤ C ( 1 ∣ P r ( z ) ∣ a ∬ P r ( z ) ∣ f ∣ 2 μ a d t ) 1 2 . (91) \begin{aligned}
\frac{1}{|P_r(z)|}\iint_{P_r(z)}|f|\,dXdt
\leq\frac{1}{|P_r(z)|}\left(\iint_{P_r(z)}|f|^2\mu_a\,dt\right)^{\frac12}\left(\iint_{P_r(z)}y^{-a}\,dXdt\right)^{\frac12} \\
&\leq C\left(\iint_{P_r(z)}|f|^2\mu_a\,dt\right)^{\frac12}\left(\iint_{P_r(z)}y^a\,dXdt\right)^{-\frac12} \\
&\leq C\left(\frac{1}{|P_r(z)|_a}\iint_{P_r(z)}|f|^2\mu_a\,dt\right)^{\frac12}. \tag*{(91)}
\end{aligned} ∣ P r ( z ) ∣ 1 ∬ P r ( z ) ∣ f ∣ d X d t ≤ ∣ P r ( z ) ∣ 1 ( ∬ P r ( z ) ∣ f ∣ 2 μ a d t ) 2 1 ( ∬ P r ( z ) y − a d X d t ) 2 1 ≤ C ( ∬ P r ( z ) ∣ f ∣ 2 μ a d t ) 2 1 ( ∬ P r ( z ) y a d X d t ) − 2 1 ≤ C ( ∣ P r ( z ) ∣ a 1 ∬ P r ( z ) ∣ f ∣ 2 μ a d t ) 2 1 . (91) Applying (7.4) with f = U − ( U ) P r ( z ) a f=U-(U)^a_{P_r(z)} f = U − ( U ) P r ( z ) a
inf c ∈ R ℓ 1 ∣ P r ( z ) ∣ ∬ P r ( z ) ∣ U − c ∣ d X d t ≤ 1 ∣ P r ( z ) ∣ ∬ P r ( z ) ∣ U − ( U ) P r ( z ) a ∣ d X d t ≤ C ( 1 ∣ P r ( z ) ∣ a ∬ P r ( z ) ∣ U − ( U ) P r ( z ) a ∣ 2 μ a d t ) 1 2 ≤ C [ U ] BMO ( O + , μ a d t ) . \begin{aligned}
\inf_{c\in\mathbb{R}^{\ell}}\frac{1}{|P_r(z)|}\iint_{P_r(z)}|U-c|\,dXdt
\leq\frac{1}{|P_r(z)|}\iint_{P_r(z)}|U-(U)^a_{P_r(z)}|\,dXdt \\
&\leq C\left(\frac{1}{|P_r(z)|_a}\iint_{P_r(z)}|U-(U)^a_{P_r(z)}|^2\mu_a\,dt\right)^{\frac12} \\
&\leq C[U]_{\operatorname{BMO}(O_+,\mu_a\,dt)}.
\end{aligned} c ∈ R ℓ inf ∣ P r ( z ) ∣ 1 ∬ P r ( z ) ∣ U − c ∣ d X d t ≤ ∣ P r ( z ) ∣ 1 ∬ P r ( z ) ∣ U − ( U ) P r ( z ) a ∣ d X d t ≤ C ( ∣ P r ( z ) ∣ a 1 ∬ P r ( z ) ∣ U − ( U ) P r ( z ) a ∣ 2 μ a d t ) 2 1 ≤ C [ U ] BMO ( O + , μ a d t ) . this, after taking supremum over all P r ( z ) ⊂ O + P_r(z)\subset O_+ P r ( z ) ⊂ O + □ \square □
It is well-known that the following y a y^a y a y a y^a y a A 2 A_2 A 2
Lemma 7.2 There exists C > 0 C>0 C > 0 f ∈ H l o c 1 ( R + m + 1 × ( 0 , ∞ ) ; μ a d t ) f\in H^1_{\mathrm{loc}}(\mathbb{R}^{m+1}_+\times(0,\infty);\mu_a dt) f ∈ H loc 1 ( R + m + 1 × ( 0 , ∞ ) ; μ a d t )
∬ P R + ( Z ) ∣ f − ( f ) P R + ( Z ) a ∣ 2 μ a d t ≤ C R 2 ∬ P R + ( Z ) ( ∣ D f ∣ 2 + R 2 ∣ ∂ t f ∣ 2 ) μ a d t . (92) \iint_{P_R^+(Z)} |f-(f)^a_{P_R^+(Z)}|^2\mu_a dt\le CR^2\iint_{P_R^+(Z)}\left(|Df|^2+R^2|\partial_t f|^2\right)\mu_a dt. \tag*{(92)} ∬ P R + ( Z ) ∣ f − ( f ) P R + ( Z ) a ∣ 2 μ a d t ≤ C R 2 ∬ P R + ( Z ) ( ∣ D f ∣ 2 + R 2 ∣ ∂ t f ∣ 2 ) μ a d t . (92) and
( 1 ∣ P R + ( Z ) ∣ a ∬ P R + ( Z ) ∣ f − ( f ) P R + ( Z ) a ∣ 2 a ∗ μ a d t ) 1 2 a ∗ ≤ C R ( 1 ∣ P R + ( Z ) ∣ a ∬ P R + ( Z ) ( ∣ D f ∣ 2 + R 2 ∣ ∂ t f ∣ 2 ) μ a d t ) 1 2 , (93) \left(\frac{1}{|P_R^+(Z)|_a}\iint_{P_R^+(Z)}|f-(f)^a_{P_R^+(Z)}|^{2_a^*}\mu_a dt\right)^{\frac{1}{2_a^*}}
\le CR\left(\frac{1}{|P_R^+(Z)|_a}\iint_{P_R^+(Z)}\left(|Df|^2+R^2|\partial_t f|^2\right)\mu_a dt\right)^{\frac{1}{2}}, \tag*{(93)} ( ∣ P R + ( Z ) ∣ a 1 ∬ P R + ( Z ) ∣ f − ( f ) P R + ( Z ) a ∣ 2 a ∗ μ a d t ) 2 a ∗ 1 ≤ C R ( ∣ P R + ( Z ) ∣ a 1 ∬ P R + ( Z ) ( ∣ D f ∣ 2 + R 2 ∣ ∂ t f ∣ 2 ) μ a d t ) 2 1 , (93) for any P R ( Z ) ⊂ R m + 1 × ( 0 , ∞ ) P_R(Z)\subset\mathbb{R}^{m+1}\times(0,\infty) P R ( Z ) ⊂ R m + 1 × ( 0 , ∞ ) 2 a ∗ 2_a^* 2 a ∗ y a y^a y a
2 a ∗ = { 2 ( m + 1 − a ) m − 1 − a , m ≥ 2 , or m = 1 and a < 0 , ∞ , m = 1 and a > 0. (94) 2_a^*=
\begin{cases}
\frac{2(m+1-a)}{m-1-a}, & m\ge2,\ \text{or }m=1\text{ and }a<0,\\
\infty, & m=1\text{ and }a>0.
\end{cases} \tag*{(94)} 2 a ∗ = { m − 1 − a 2 ( m + 1 − a ) , ∞ , m ≥ 2 , or m = 1 and a < 0 , m = 1 and a > 0. (94) In the proof of Lemma 3.2, we need the following modified Poincaré inequality. Let D R ( x 0 ) ⊂ R m D_R(x_0)\subset\mathbb{R}^m D R ( x 0 ) ⊂ R m x 0 ∈ R m x_0\in\mathbb{R}^m x 0 ∈ R m R > 0 R>0 R > 0 D R ( x 0 ) = { x ∈ R m : ∣ x − x 0 ∣ ≤ R } D_R(x_0)=\{x\in\mathbb{R}^m:|x-x_0|\le R\} D R ( x 0 ) = { x ∈ R m : ∣ x − x 0 ∣ ≤ R } P ^ R ( x 0 , t 0 ) = D R ( x 0 ) × [ t 0 − R 2 , t 0 + R 2 ] \widehat{P}_R(x_0,t_0)=D_R(x_0)\times[t_0-R^2,t_0+R^2] P R ( x 0 , t 0 ) = D R ( x 0 ) × [ t 0 − R 2 , t 0 + R 2 ]
Lemma 7.3 There exists a constant C > 0 C>0 C > 0 x 0 ∈ R m x_0\in\mathbb{R}^m x 0 ∈ R m t 0 > 0 t_0>0 t 0 > 0 0 < R < t 0 0<R<\sqrt{t_0} 0 < R < t 0
R − 1 ∬ P R + ( x 0 , t 0 ) ∣ U − [ u ] ( x 0 , t 0 ) , R ∣ d X d t + ∬ P ^ R ( x 0 , t 0 ) ∣ u − [ u ] ( x 0 , t 0 ) , R ∣ d x d t ≤ C ∬ P R + ( x 0 , t 0 ) ( ∣ D U ∣ + R ∣ ∂ t U ∣ ) d X d t . (95) R^{-1}\iint_{P_R^+(x_0,t_0)}|U-[u]_{(x_0,t_0),R}|\,dXdt+\iint_{\widehat{P}_R(x_0,t_0)}|u-[u]_{(x_0,t_0),R}|\,dxdt
\le C\iint_{P_R^+(x_0,t_0)}\left(|DU|+R|\partial_tU|\right)\,dXdt. \tag*{(95)} R − 1 ∬ P R + ( x 0 , t 0 ) ∣ U − [ u ] ( x 0 , t 0 ) , R ∣ d X d t + ∬ P R ( x 0 , t 0 ) ∣ u − [ u ] ( x 0 , t 0 ) , R ∣ d x d t ≤ C ∬ P R + ( x 0 , t 0 ) ( ∣ D U ∣ + R ∣ ∂ t U ∣ ) d X d t . (95) where u = U ∣ ∂ R + m + 1 × ( 0 , ∞ ) u=U|_{\partial\mathbb{R}^{m+1}_+\times(0,\infty)} u = U ∣ ∂ R + m + 1 × ( 0 , ∞ ) [ u ] ( x 0 , t 0 ) , R [u]_{(x_0,t_0),R} [ u ] ( x 0 , t 0 ) , R u u u P ^ R ( x 0 , t 0 ) \widehat{P}_R(x_0,t_0) P R ( x 0 , t 0 ) d x d t dxdt d x d t
Proof. Here we give a sketch of proof. By scaling, it is sufficient to show (7.8) for R = 1 R=1 R = 1 U i U^i U i ( ∂ t U i , D U i ) ∈ L 2 ( P 1 + ( x 0 , t 0 ) ) (\partial_tU^i,DU^i)\in L^2(P_1^+(x_0,t_0)) ( ∂ t U i , D U i ) ∈ L 2 ( P 1 + ( x 0 , t 0 ))
α i : = ∥ U i − [ u i ] ( x 0 , t 0 ) , 1 ∥ L 1 ( P 1 + ( x 0 , t 0 ) ) + ∥ u i − [ u i ] ( x 0 , t 0 ) , 1 ∥ L 1 ( P ^ 1 ( x 0 , t 0 ) ) > i ∬ P 1 + ( x 0 , t 0 ) ( ∣ D U i ∣ + ∣ ∂ t U i ∣ ) d X d t . (96) \alpha_i:=\left\|U^i-[u^i]_{(x_0,t_0),1}\right\|_{L^1(P_1^+(x_0,t_0))}+\left\|u^i-[u^i]_{(x_0,t_0),1}\right\|_{L^1(\widehat{P}_1(x_0,t_0))}
>i\iint_{P_1^+(x_0,t_0)}\left(|DU^i|+|\partial_tU^i|\right)\,dXdt. \tag*{(96)} α i := U i − [ u i ] ( x 0 , t 0 ) , 1 L 1 ( P 1 + ( x 0 , t 0 )) + u i − [ u i ] ( x 0 , t 0 ) , 1 L 1 ( P 1 ( x 0 , t 0 )) > i ∬ P 1 + ( x 0 , t 0 ) ( ∣ D U i ∣ + ∣ ∂ t U i ∣ ) d X d t . (96) Replacing U i U^i U i
V i = U i − [ u i ] ( x 0 , t 0 ) , 1 α i , V^i=\frac{U^i-[u^i]_{(x_0,t_0),1}}{\alpha_i}, V i = α i U i − [ u i ] ( x 0 , t 0 ) , 1 , we may further assume that
[ u i ] ( x 0 , t 0 ) , 1 = 0 and ∥ U i − [ u i ] ( x 0 , t 0 ) , 1 ∥ L 1 ( P 1 + ( x 0 , t 0 ) ) + ∥ u i ∥ L 1 ( P ^ 1 ( x 0 , t 0 ) ) = 1. [u^i]_{(x_0,t_0),1}=0\quad\text{and}\quad\left\|U^i-[u^i]_{(x_0,t_0),1}\right\|_{L^1(P_1^+(x_0,t_0))}+\left\|u^i\right\|_{L^1(\widehat{P}_1(x_0,t_0))}=1. [ u i ] ( x 0 , t 0 ) , 1 = 0 and U i − [ u i ] ( x 0 , t 0 ) , 1 L 1 ( P 1 + ( x 0 , t 0 )) + u i L 1 ( P 1 ( x 0 , t 0 )) = 1. Hence we have
∬ P 1 + ( x 0 , t 0 ) ( ∣ D U i ∣ + ∣ ∂ t U i ∣ ) d X d t ≤ 1 i . (97) \iint_{P_1^+(x_0,t_0)}\left(|DU^i|+|\partial_tU^i|\right)\,dXdt\le\frac{1}{i}. \tag*{(97)} ∬ P 1 + ( x 0 , t 0 ) ( ∣ D U i ∣ + ∣ ∂ t U i ∣ ) d X d t ≤ i 1 . (97) Recall by the standard Poincaré inequality, we have that
∬ P 1 + ( x 0 , t 0 ) ∣ U i ∣ d X d t ≤ C ( ∬ P 1 + ( x 0 , t 0 ) ( ∣ D U i ∣ + ∣ ∂ t U i ∣ ) d X d t + ∬ P ^ 1 ( x 0 , t 0 ) ∣ u i ∣ d x d t ) ≤ C ( 1 + i − 1 ) ≤ 2 C . (98) \iint_{P_1^+(x_0,t_0)} |U^i|\,dXdt \le C\left(\iint_{P_1^+(x_0,t_0)} (|DU^i|+|\partial_tU^i|)\,dXdt+\iint_{\widehat{P}_1(x_0,t_0)} |u^i|\,dxdt\right)
\le C(1+i^{-1})\le2C. \tag*{(98)} ∬ P 1 + ( x 0 , t 0 ) ∣ U i ∣ d X d t ≤ C ( ∬ P 1 + ( x 0 , t 0 ) ( ∣ D U i ∣ + ∣ ∂ t U i ∣ ) d X d t + ∬ P 1 ( x 0 , t 0 ) ∣ u i ∣ d x d t ) ≤ C ( 1 + i − 1 ) ≤ 2 C . (98) Thus { U i } ⊂ W 1 , 1 ( P 1 + ( x 0 , t 0 ) ) \{U^i\}\subset W^{1,1}(P_1^+(x_0,t_0)) { U i } ⊂ W 1 , 1 ( P 1 + ( x 0 , t 0 ))
W 1 , 1 ( P 1 + ( x 0 , t 0 ) ) ⊂ L 1 ( P ^ 1 ( x 0 , t 0 ) ) W^{1,1}(P_1^+(x_0,t_0))\subset L^1(\widehat{P}_1(x_0,t_0)) W 1 , 1 ( P 1 + ( x 0 , t 0 )) ⊂ L 1 ( P 1 ( x 0 , t 0 )) is compact, we can conclude that there exists a U ∈ B V ( P 1 + ( x 0 , t 0 ) ) U\in BV(P_1^+(x_0,t_0)) U ∈ B V ( P 1 + ( x 0 , t 0 ))
U i ⇀ U in B V ( P 1 + ( x 0 , t 0 ) ) , U i → U in L 1 ( P 1 + ( x 0 , t 0 ) ) , U^i\rightharpoonup U\quad\text{in }BV(P_1^+(x_0,t_0)),\qquad U_i\to U\quad\text{in }L^1(P_1^+(x_0,t_0)), U i ⇀ U in B V ( P 1 + ( x 0 , t 0 )) , U i → U in L 1 ( P 1 + ( x 0 , t 0 )) , and
u i → u = U ∣ P ^ 1 ( x 0 , t 0 ) in L 1 ( P ^ 1 ( x 0 , t 0 ) ) . u^i\to u=U|_{\widehat{P}_1(x_0,t_0)}\quad\text{in }L^1(\widehat{P}_1(x_0,t_0)). u i → u = U ∣ P 1 ( x 0 , t 0 ) in L 1 ( P 1 ( x 0 , t 0 )) . Hence
[ u ] ( x 0 , t 0 ) , 1 = 0 , ∥ U ∥ L 1 ( P 1 + ( x 0 , t 0 ) ) + ∥ u ∥ L 1 ( P ^ 1 ( x 0 , t 0 ) ) = 1. [u]_{(x_0,t_0),1}=0,\qquad\|U\|_{L^1(P_1^+(x_0,t_0))}+\|u\|_{L^1(\widehat{P}_1(x_0,t_0))}=1. [ u ] ( x 0 , t 0 ) , 1 = 0 , ∥ U ∥ L 1 ( P 1 + ( x 0 , t 0 )) + ∥ u ∥ L 1 ( P 1 ( x 0 , t 0 )) = 1. On the other hand, it follows from (97) and the lower semicontinuity that
( ∂ t U , D U ) = ( 0 , 0 ) in P 1 + ( x 0 , t 0 ) . (\partial_tU,DU)=(0,0)\quad\text{in }P_1^+(x_0,t_0). ( ∂ t U , D U ) = ( 0 , 0 ) in P 1 + ( x 0 , t 0 ) . Hence U U U [ u ] ( x 0 , t 0 ) , 1 = 0 [u]_{(x_0,t_0),1}=0 [ u ] ( x 0 , t 0 ) , 1 = 0 U ≡ 0 U\equiv0 U ≡ 0 ∥ U ∥ L 1 ( P 1 + ( x 0 , t 0 ) ) + ∥ u ∥ L 1 ( P ^ 1 ( x 0 , t 0 ) ) = 1 \|U\|_{L^1(P_1^+(x_0,t_0))}+\|u\|_{L^1(\widehat{P}_1(x_0,t_0))}=1 ∥ U ∥ L 1 ( P 1 + ( x 0 , t 0 )) + ∥ u ∥ L 1 ( P 1 ( x 0 , t 0 )) = 1
For the proof of Theorem 3.3, we need the John-Nirenberg inequality in BMO ( O + , μ a d t ) \operatorname{BMO}(\mathcal{O}_+,\mu_a dt) BMO ( O + , μ a d t ) [10 ] .
Lemma 7.4 Let f ∈ BMO ( P 1 2 + ( 0 , 1 ) , μ a d t ) f\in\operatorname{BMO}(P_{\frac12}^+(0,1),\mu_a dt) f ∈ BMO ( P 2 1 + ( 0 , 1 ) , μ a d t ) 1 ≤ p < ∞ 1\le p<\infty 1 ≤ p < ∞ C p > 0 C_p>0 C p > 0
1 ∣ P r + ( z ) ∣ a ∬ P r + ( z ) ∣ f − ( f ) P r + ( z ) a ∣ p μ a d t ≤ C p [ f ] BMO ( P 1 2 + ( 0 , 1 ) , μ a d t ) p , ∀ P r ( z ) ⊂ P 1 2 + ( 0 , 1 ) . (99) \frac{1}{|P_r^+(z)|_a}\iint_{P_r^+(z)} |f-(f)_{P_r^+(z)}^a|^p\mu_a dt\le C_p[f]_{\operatorname{BMO}(P_{\frac12}^+(0,1),\mu_a dt)}^p,\qquad\forall P_r(z)\subset P_{\frac12}^+(0,1). \tag*{(99)} ∣ P r + ( z ) ∣ a 1 ∬ P r + ( z ) ∣ f − ( f ) P r + ( z ) a ∣ p μ a d t ≤ C p [ f ] BMO ( P 2 1 + ( 0 , 1 ) , μ a d t ) p , ∀ P r ( z ) ⊂ P 2 1 + ( 0 , 1 ) . (99) For the proof of Theorem 3.3, we also need the following ∣ y ∣ a |y|^a ∣ y ∣ a [9 ] , see Alvarez-Bagby-Kurtz-Perez [1 ] for the proof.
Lemma 7.5 If f ∈ L 2 ( R m + 1 × R ; μ a d t ) , R m + 1 ) f\in L^2(\mathbb{R}^{m+1}\times\mathbb{R};\mu_a dt),\mathbb{R}^{m+1}) f ∈ L 2 ( R m + 1 × R ; μ a d t ) , R m + 1 ) curl X f ( X , t ) = 0 \operatorname{curl}_X f(X,t)=0 curl X f ( X , t ) = 0 t t t g ∈ L 2 ( ( R m + 1 × R ; μ − a d t ) , R m + 1 ) g\in L^2((\mathbb{R}^{m+1}\times\mathbb{R};\mu_{-a}dt),\mathbb{R}^{m+1}) g ∈ L 2 (( R m + 1 × R ; μ − a d t ) , R m + 1 ) div X g ( X , t ) = 0 \operatorname{div}_Xg(X,t)=0 div X g ( X , t ) = 0 t t t f ⋅ g ∈ H 1 ( R m + 1 × R ) f\cdot g\in H^1(\mathbb{R}^{m+1}\times\mathbb{R}) f ⋅ g ∈ H 1 ( R m + 1 × R )
∥ f ⋅ g ∥ H 1 ( R m + 1 × R ) ≤ C ∥ f ∥ L 2 ( R m + 1 × R , μ a d t ) ∥ g ∥ L 2 ( R m + 1 × R , μ − a d t ) . (100) \|f\cdot g\|_{H^1(\mathbb{R}^{m+1}\times\mathbb{R})}\le C\|f\|_{L^2(\mathbb{R}^{m+1}\times\mathbb{R},\mu_a dt)}\|g\|_{L^2(\mathbb{R}^{m+1}\times\mathbb{R},\mu_{-a}dt)}. \tag*{(100)} ∥ f ⋅ g ∥ H 1 ( R m + 1 × R ) ≤ C ∥ f ∥ L 2 ( R m + 1 × R , μ a d t ) ∥ g ∥ L 2 ( R m + 1 × R , μ − a d t ) . (100) Finally, we would like to remark on the function spaces to which the stationary heat flow of s s s u ∈ H x s , s 2 ( R m , S ℓ − 1 ) u\in H_x^{s,\frac{s}{2}}(\mathbb{R}^m,S^{\ell-1}) u ∈ H x s , 2 s ( R m , S ℓ − 1 ) u u u U : R + m + 1 × ( 0 , ∞ ) → R ℓ U : \mathbb{R}^{m+1}_{+} \times(0,\infty) \to\mathbb{R}^{\ell} U : R + m + 1 × ( 0 , ∞ ) → R ℓ y a y^{a} y a (11) and the Definition 1.2 of stationarity condition. The validity naturally implies
U ∈ L ∞ ( R + ; H ˙ 1 ( R + m + 1 , μ a ) , R ℓ ) ∩ H ˙ 1 ( R + ; L 2 ( R + m + 1 , μ a ) , R ℓ ) , U \in L^{\infty}\left(\mathbb{R}_{+}; \dot{H}^{1}\left(\mathbb{R}^{m+1}_{+},\mu_{a}\right),\mathbb{R}^{\ell}\right) \cap\dot{H}^{1}\left(\mathbb{R}_{+};L^{2}\left(\mathbb{R}^{m+1}_{+},\mu_{a}\right),\mathbb{R}^{\ell}\right), U ∈ L ∞ ( R + ; H ˙ 1 ( R + m + 1 , μ a ) , R ℓ ) ∩ H ˙ 1 ( R + ; L 2 ( R + m + 1 , μ a ) , R ℓ ) , which, with the help of the trace theorem by [23 ] on y a y^{a} y a u u u t t t u ∈ H ( x , t ) , l o c s , s ( R m × ( 0 , ∞ ) , S ℓ − 1 ) u \in H^{s,s}_{(x,t),\mathrm{loc}}\left(\mathbb{R}^{m} \times(0,\infty),\mathbb{S}^{\ell-1}\right) u ∈ H ( x , t ) , loc s , s ( R m × ( 0 , ∞ ) , S ℓ − 1 ) ( − Δ ) s 2 u ∈ L ∞ ( R + , L 2 ( R m ) ) (-\Delta)^{\frac{s}{2}}u \in L^{\infty}(\mathbb{R}_{+},L^{2}(\mathbb{R}^{m})) ( − Δ ) 2 s u ∈ L ∞ ( R + , L 2 ( R m )) ( ∣ D t ∣ ) s u ∈ L 2 ( R + , L 2 ( R m ) ) (|D_{t}|)^{s}u \in L^{2}(\mathbb{R}_{+},L^{2}(\mathbb{R}^{m})) ( ∣ D t ∣ ) s u ∈ L 2 ( R + , L 2 ( R m ))
Acknowledgments. Wang is partially supported by NSF DMS 2453789 and Simons Travel Grant TSM-00007723. Schikorra is partially supported by NSF Career DMS-2044898.
References 