Introduction
Let Ω ⊂ R 2 \Omega\subset\mathbb{R}^2 Ω ⊂ R 2 ∂ Ω \partial\Omega ∂ Ω
{ ∂ t u ν + ( u ν ⋅ ∇ ) u ν + ∇ p ν = ν Δ u ν in ( 0 , T ) × Ω , div u ν = 0 in ( 0 , T ) × Ω , u ν ⋅ n = 0 on ( 0 , T ) × ∂ Ω , 2 ( D u ν ) S n ⋅ τ + α ν u ν ⋅ τ = 0 on ( 0 , T ) × ∂ Ω , u ν ( 0 , ⋅ ) = u 0 in Ω . (1) \begin{cases}
\partial_t u^\nu+ (u^\nu\cdot\nabla)u^\nu+ \nabla p^\nu= \nu\Delta u^\nu& \text{in } (0,T) \times\Omega,\\
\operatorname{div} u^\nu= 0 & \text{in } (0,T) \times\Omega,\\
u^\nu\cdot n = 0 & \text{on } (0,T) \times\partial\Omega,\\
2(Du^\nu)_S n \cdot\tau+ \alpha^\nu u^\nu\cdot\tau= 0 & \text{on } (0,T) \times\partial\Omega,\\
u^\nu(0,\cdot) = u_0 & \text{in } \Omega. \tag*{(1)}
\end{cases} ⎩ ⎨ ⎧ ∂ t u ν + ( u ν ⋅ ∇ ) u ν + ∇ p ν = ν Δ u ν div u ν = 0 u ν ⋅ n = 0 2 ( D u ν ) S n ⋅ τ + α ν u ν ⋅ τ = 0 u ν ( 0 , ⋅ ) = u 0 in ( 0 , T ) × Ω , in ( 0 , T ) × Ω , on ( 0 , T ) × ∂ Ω , on ( 0 , T ) × ∂ Ω , in Ω. (1) The unknowns are the fluid velocity u ν u^\nu u ν p ν p^\nu p ν ν > 0 \nu> 0 ν > 0 n n n τ \tau τ ( D v ) S = 1 2 ( ∇ v + ∇ v T ) (Dv)_S = \frac{1}{2}(\nabla v + \nabla v^T) ( D v ) S = 2 1 ( ∇ v + ∇ v T ) α ν ∈ C 2 ( ∂ Ω ) \alpha^\nu\in C^2(\partial\Omega) α ν ∈ C 2 ( ∂ Ω )
A key feature of the Navier boundary condition is that it implies a boundary condition for the vorticity [8 ] , Lemma 2.1 due to the formal equality
curl u ν = 2 ( D u ν ) s n ⋅ τ + 2 κ u ν ⋅ τ on ∂ Ω , \operatorname{curl} u^\nu= 2(Du^\nu)sn \cdot\tau+ 2\kappa u^\nu\cdot\tau\text{ on } \partial\Omega, curl u ν = 2 ( D u ν ) s n ⋅ τ + 2 κ u ν ⋅ τ on ∂ Ω , where κ \kappa κ ∂ Ω \partial\Omega ∂ Ω
{ ∂ t ω ν + u ν ⋅ ∇ ω ν = ν Δ ω ν in ( 0 , T ) × Ω , ω ν = curl u ν in ( 0 , T ) × Ω , u ν ⋅ n = 0 on ( 0 , T ) × ∂ Ω , ω ν = ( 2 κ − α ν ) u ν ⋅ τ on ( 0 , T ) × ∂ Ω , ω ν ( 0 , ⋅ ) = ω 0 in Ω . (2) \begin{cases}
\partial_t\omega^\nu+ u^\nu\cdot\nabla\omega^\nu= \nu\Delta\omega^\nu\text{ in } (0,T)\times\Omega,\\
\omega^\nu= \operatorname{curl}u^\nu\text{ in } (0,T)\times\Omega,\\
u^\nu\cdot n = 0 \text{ on } (0,T)\times\partial\Omega,\\
\omega^\nu= (2\kappa-\alpha^\nu)u^\nu\cdot\tau\text{ on } (0,T)\times\partial\Omega,\\
\omega^\nu(0,\cdot)=\omega_0 \text{ in }\Omega. \tag*{(2)}
\end{cases} ⎩ ⎨ ⎧ ∂ t ω ν + u ν ⋅ ∇ ω ν = ν Δ ω ν in ( 0 , T ) × Ω , ω ν = curl u ν in ( 0 , T ) × Ω , u ν ⋅ n = 0 on ( 0 , T ) × ∂ Ω , ω ν = ( 2 κ − α ν ) u ν ⋅ τ on ( 0 , T ) × ∂ Ω , ω ν ( 0 , ⋅ ) = ω 0 in Ω. (2) The vorticity formulation is formally equivalent, as the velocity u ν u^\nu u ν ω ν \omega^\nu ω ν Lemma A.2 ). Explicitly, we write
u ν ( t ) = K Ω ( ω ν ( t ) ) , with K Ω ( f ) ( y ) : = ∫ Ω ∇ y G Ω ( x , y ) f ( x ) d x , u^\nu(t)=K_\Omega(\omega^\nu(t)),\text{ with }K_\Omega(f)(y):=\int_\Omega\nabla_yG_\Omega(x,y)f(x)\,dx, u ν ( t ) = K Ω ( ω ν ( t )) , with K Ω ( f ) ( y ) := ∫ Ω ∇ y G Ω ( x , y ) f ( x ) d x , where G Ω G_\Omega G Ω ω 0 ∈ L p ( Ω ) \omega_0\in L^p(\Omega) ω 0 ∈ L p ( Ω ) p > 1 p>1 p > 1 u 0 = K Ω ( ω 0 ) ∈ W 1 , p ( Ω ) ⊂ L 2 ( Ω ) u_0=K_\Omega(\omega_0)\in W^{1,p}(\Omega)\subset L^2(\Omega) u 0 = K Ω ( ω 0 ) ∈ W 1 , p ( Ω ) ⊂ L 2 ( Ω )
Our goal is to study the vanishing viscosity limit, i.e. ν → 0 \nu\to0 ν → 0
{ ∂ t u + u ⋅ ∇ u + ∇ p = 0 in ( 0 , T ) × Ω , div u = 0 in ( 0 , T ) × Ω , u ⋅ n = 0 on ( 0 , T ) × ∂ Ω , u ( 0 , ⋅ ) = u 0 in Ω . (3) \begin{cases}
\partial_tu+u\cdot\nabla u+\nabla p=0 \text{ in } (0,T)\times\Omega,\\
\operatorname{div}u=0 \text{ in } (0,T)\times\Omega,\\
u\cdot n=0 \text{ on } (0,T)\times\partial\Omega,\\
u(0,\cdot)=u_0 \text{ in }\Omega. \tag*{(3)}
\end{cases} ⎩ ⎨ ⎧ ∂ t u + u ⋅ ∇ u + ∇ p = 0 in ( 0 , T ) × Ω , div u = 0 in ( 0 , T ) × Ω , u ⋅ n = 0 on ( 0 , T ) × ∂ Ω , u ( 0 , ⋅ ) = u 0 in Ω. (3) The transition from Navier–Stokes to Euler as viscosity vanishes has been a classical problem with extensive research for decades (see [22 ] for an overview). Without physical boundaries, convergence in two dimensions is known for regular Euler solutions [24 ] , and also for weak Euler solution classes with some control on the initial vorticity [6 , 7 , 28 ] . With physical boundaries, the Navier–Stokes and Euler boundary conditions generally do not match, because the viscous term disappears in the limit. This mismatch leads to boundary layer effects and makes the inviscid limit more difficult. For the no-slip condition u ν = 0 u^\nu=0 u ν = 0
One difficulty here is that the no-slip condition does not prescribe boundary values for the vorticity, which blocks the usual compactness approach based on viscosity-independent a priori estimates for the vorticity.
As already mentioned, the Navier boundary condition, originally introduced by C. Navier in 1827 [26 ] , admits a formulation in terms of vorticity that makes the analysis more manageable. It can also be seen as a relaxation of the no-slip condition, as u ν ⋅ τ = 0 u^\nu\cdot\tau= 0 u ν ⋅ τ = 0 α ν = ∞ \alpha^\nu= \infty α ν = ∞ [8 ] , showing strong convergence of velocity for bounded initial vorticity, and Lopes Filho, Nussenzveig Lopes, and Planas [12 ] , extending it to initial vorticity in L p ( Ω ) L^p(\Omega) L p ( Ω ) p > 2 p > 2 p > 2 [12 ] is based on a uniform vorticity bound
∥ ω ν ∥ L ∞ ( 0 , T ; L p ( Ω ) ) ≤ C ( ∥ ω 0 ∥ L p ( Ω ) ) for p > 2 , \|\omega^\nu\|_{L^\infty(0,T;L^p(\Omega))} \le C\left(\|\omega_0\|_{L^p(\Omega)}\right) \quad\text{for } p > 2, ∥ ω ν ∥ L ∞ ( 0 , T ; L p ( Ω )) ≤ C ( ∥ ω 0 ∥ L p ( Ω ) ) for p > 2 , where the constant C > 0 C > 0 C > 0 ν \nu ν p > 2 p > 2 p > 2 L ∞ L^\infty L ∞ L p L^p L p p ∈ ( 1 , 2 ] p \in(1,2] p ∈ ( 1 , 2 ] p ∈ ( 1 , 2 ] p \in(1,2] p ∈ ( 1 , 2 ] [12 Section 4, Remark 1] and was the starting point for this work, and we can remove this restriction (Corollary 5.3). Moreover, we can strengthen the velocity convergence in [12 ] to strong convergence of vorticity: Let ω 0 ∈ L p ( Ω ) \omega_0 \in L^p(\Omega) ω 0 ∈ L p ( Ω ) p ∈ ( 1 , ∞ ) p \in(1,\infty) p ∈ ( 1 , ∞ ) ν k → 0 \nu_k \to0 ν k → 0
ω ν k → curl u strongly in C ( [ 0 , T ] ; L p ( Ω ) ) . \omega^{\nu_k} \to\operatorname{curl} u \text{ strongly in } C([0,T];L^p(\Omega)). ω ν k → curl u strongly in C ([ 0 , T ] ; L p ( Ω )) . For p > 2 p > 2 p > 2 [3 , 9 ] .
When the friction coefficient is positive, 1 / α ν 1/\alpha^\nu 1/ α ν [14 , 30 ] . This motivates the study of coefficients α ν \alpha^\nu α ν ν \nu ν
∥ α ν ∥ L ∞ ( ∂ Ω ) ≤ C ν − q , q ≥ 0 , (4) \|\alpha^\nu\|_{L^\infty(\partial\Omega)} \le C\nu^{-q}, \qquad q \ge0, \tag*{(4)} ∥ α ν ∥ L ∞ ( ∂ Ω ) ≤ C ν − q , q ≥ 0 , (4) where the constant C > 0 C > 0 C > 0 ν \nu ν (4) , we include for q > 0 q > 0 q > 0 ν → 0 \nu\to0 ν → 0 q q q
Theorem 1.1. Let Ω ⊂ R 2 \Omega\subset\mathbb{R}^{2} Ω ⊂ R 2 T ∈ ( 0 , ∞ ) T \in(0,\infty) T ∈ ( 0 , ∞ ) ω 0 ∈ L p ( Ω ) \omega_{0} \in L^{p}(\Omega) ω 0 ∈ L p ( Ω ) p ∈ ( 1 , ∞ ) p \in(1,\infty) p ∈ ( 1 , ∞ ) u ν u^{\nu} u ν u ν ( 0 , ⋅ ) = K Ω ( ω 0 ) u^{\nu}(0,\cdot)=K_{\Omega}(\omega_{0}) u ν ( 0 , ⋅ ) = K Ω ( ω 0 ) α ν ∈ C 2 ( ∂ Ω ) \alpha^{\nu} \in C^{2}(\partial\Omega) α ν ∈ C 2 ( ∂ Ω ) ρ ∈ ( 1 , p ] \rho\in(1,p] ρ ∈ ( 1 , p ] q ≥ 0 q \geq0 q ≥ 0 ρ q < min { p − 1 2 p , 1 4 } \rho q < \min\left\{\frac{p-1}{2p},\frac{1}{4}\right\} ρq < min { 2 p p − 1 , 4 1 } ν k → 0 \nu_{k}\to0 ν k → 0
u ν k → u strongly in C ( [ 0 , T ] ; W 1 , ρ ( Ω ) ) as k → ∞ , u^{\nu_{k}} \to u \text{ strongly in } C([0,T];W^{1,\rho}(\Omega)) \text{ as } k\to\infty, u ν k → u strongly in C ([ 0 , T ] ; W 1 , ρ ( Ω )) as k → ∞ , and u ∈ C ( [ 0 , T ] ; W 1 , p ( Ω ) ) u \in C([0,T];W^{1,p}(\Omega)) u ∈ C ([ 0 , T ] ; W 1 , p ( Ω )) u ( 0 , ⋅ ) = K Ω ( ω 0 ) u(0,\cdot)=K_{\Omega}(\omega_{0}) u ( 0 , ⋅ ) = K Ω ( ω 0 )
ω ν k → curl u strongly in C ( [ 0 , T ] ; L ρ ( Ω ) ) as k → ∞ , \omega^{\nu_{k}} \to\operatorname{curl}u \text{ strongly in } C([0,T];L^{\rho}(\Omega)) \text{ as } k\to\infty, ω ν k → curl u strongly in C ([ 0 , T ] ; L ρ ( Ω )) as k → ∞ , and the limiting vorticity ω = curl u \omega=\operatorname{curl}u ω = curl u ∂ t ω + u ⋅ ∇ ω = 0 \partial_{t}\omega+u\cdot\nabla\omega=0 ∂ t ω + u ⋅ ∇ ω = 0 ω 0 \omega_{0} ω 0
∥ ω ( t ) ∥ L p ( Ω ) = ∥ ω 0 ∥ L p ( Ω ) for every t ∈ [ 0 , T ] . \|\omega(t)\|_{L^{p}(\Omega)}=\|\omega_{0}\|_{L^{p}(\Omega)} \quad\text{for every } t\in[0,T]. ∥ ω ( t ) ∥ L p ( Ω ) = ∥ ω 0 ∥ L p ( Ω ) for every t ∈ [ 0 , T ] . The underlying estimates also yield conservation of kinetic energy for the limiting Euler solution.
Analogous results for vanishing-viscosity limits without physical boundaries were established in [5 , 6 ] .
Corollary 1.2. Under the same assumptions as in Theorem 1.1 , we have
ν ∫ 0 T ∥ ∇ u ν ( t ) ∥ L 2 ( Ω ) 2 d t + ν ∫ 0 T ∫ ∂ Ω ( ∣ κ ∣ + ∣ α ν ∣ ) ∣ u ν ( t ) ⋅ τ ∣ 2 d S d t → 0 as ν → 0. \nu\int_{0}^{T}\|\nabla u^{\nu}(t)\|_{L^{2}(\Omega)}^{2}\,\mathrm{d}t+\nu\int_{0}^{T}\int_{\partial\Omega}(|\kappa|+|\alpha^{\nu}|)|u^{\nu}(t)\cdot\tau|^{2}\,\mathrm{d}S\,\mathrm{d}t\to0 \text{ as } \nu\to0. ν ∫ 0 T ∥∇ u ν ( t ) ∥ L 2 ( Ω ) 2 d t + ν ∫ 0 T ∫ ∂ Ω ( ∣ κ ∣ + ∣ α ν ∣ ) ∣ u ν ( t ) ⋅ τ ∣ 2 d S d t → 0 as ν → 0. In particular, every limiting Euler solution satisfies
∥ u ( t ) ∥ L 2 ( Ω ) = ∥ K Ω ( ω 0 ) ∥ L 2 ( Ω ) for every t ∈ [ 0 , T ] . \|u(t)\|_{L^{2}(\Omega)}=\|K_{\Omega}(\omega_{0})\|_{L^{2}(\Omega)} \quad\text{for every } t\in[0,T]. ∥ u ( t ) ∥ L 2 ( Ω ) = ∥ K Ω ( ω 0 ) ∥ L 2 ( Ω ) for every t ∈ [ 0 , T ] . For viscosity-dependent Navier friction, strong convergence of velocity towards sufficiently regular Euler solutions has been established in [17 , 18 , 29 ] by classical energy estimates. For analytic data in the half-plane, short-time convergence of velocity at the critical scale q = 1 q=1 q = 1 [27 ] . Also, Kato-type convergence criteria have been developed in [23 ] . Regarding weak Euler solutions, a conditional convergence result for velocity was shown in [10 ] , assuming viscosity-independent interior bounds on the viscous flows.
For initial vorticity in L p ( Ω ) L^{p}(\Omega) L p ( Ω ) p > 1 p>1 p > 1 q < min { p − 1 2 p , 1 4 } q<\min\left\{\frac{p-1}{2p},\frac{1}{4}\right\} q < min { 2 p p − 1 , 4 1 } C ( [ 0 , T ] ; L ρ ( Ω ) ) C([0,T];L^{\rho}(\Omega)) C ([ 0 , T ] ; L ρ ( Ω )) ρ > 1 \rho>1 ρ > 1 q = 1 2 q=\frac{1}{2} q = 2 1 [16 ] : strong vorticity convergence implies strong convergence of the tangential velocities at the boundary, which fails in our counterexample. The restriction on q q q Theorem 1.1 originates from the methods and is not expected to be optimal. We now explain how we control the vorticity generated at the boundary.
Proof strategy
A main theme in our argument is to divide the vorticity ω ν \omega^\nu ω ν a ν a^\nu a ν
{ ∂ t a ν + u ν ⋅ ∇ a ν = ν Δ a ν in ( 0 , T ) × Ω , a ν = 0 on ( 0 , T ) × ∂ Ω , a ν ( 0 ) = ω 0 in Ω , (5) \begin{cases}
\partial_t a^\nu+ u^\nu\cdot\nabla a^\nu= \nu\Delta a^\nu& \text{in } (0,T) \times\Omega, \\
a^\nu= 0 & \text{on } (0,T) \times\partial\Omega,\quad a^\nu(0) = \omega_0 \text{ in } \Omega, \tag*{(5)}
\end{cases} { ∂ t a ν + u ν ⋅ ∇ a ν = ν Δ a ν a ν = 0 in ( 0 , T ) × Ω , on ( 0 , T ) × ∂ Ω , a ν ( 0 ) = ω 0 in Ω , (5) and a boundary contribution b ν b^\nu b ν
{ ∂ t b ν + u ν ⋅ ∇ b ν = ν Δ b ν in ( 0 , T ) × Ω , b ν = ( 2 κ − α ν ) u ν ⋅ τ on ( 0 , T ) × ∂ Ω , b ν ( 0 ) = 0 in Ω . (6) \begin{cases}
\partial_t b^\nu+ u^\nu\cdot\nabla b^\nu= \nu\Delta b^\nu& \text{in } (0,T) \times\Omega, \\
b^\nu= (2\kappa- \alpha^\nu)u^\nu\cdot\tau& \text{on } (0,T) \times\partial\Omega,\quad b^\nu(0) = 0 \text{ in } \Omega. \tag*{(6)}
\end{cases} { ∂ t b ν + u ν ⋅ ∇ b ν = ν Δ b ν b ν = ( 2 κ − α ν ) u ν ⋅ τ in ( 0 , T ) × Ω , on ( 0 , T ) × ∂ Ω , b ν ( 0 ) = 0 in Ω. (6) With u ν u^\nu u ν ω ν = a ν + b ν \omega^\nu= a^\nu+ b^\nu ω ν = a ν + b ν a ν a^\nu a ν Lemma 4.2 ) and parabolic smoothing (Lemma 4.3 ). Therefore, the main challenge lies in controlling b ν b^\nu b ν b ν b^\nu b ν
First, we establish an estimate combining renormalization with control of the boundary for advection-diffusion equations with divergence-free drift tangent to the boundary (Lemma 4.1 ). We assume a suitable growth bound on the inward normal component of the drift in a boundary collar (4.2), which lets us replace the uncontrolled diffusive boundary flux (see also [12 Section 4, Remark 2] ) in the renormalization identity by a manageable boundary contribution. Second, we will show that u ν u^\nu u ν
∣ u ν ( s ) ⋅ ∇ d ∣ ≤ C ( ν s ) − 1 / r ∥ curl u 0 ∥ L r ( Ω ) d γ , s ∈ ( 0 , T ) , (7) |u^\nu(s) \cdot\nabla d| \le C(\nu s)^{-1/r}\|\operatorname{curl} u_0\|_{L^r(\Omega)}d^\gamma,\qquad s \in(0,T), \tag*{(7)} ∣ u ν ( s ) ⋅ ∇ d ∣ ≤ C ( ν s ) − 1/ r ∥ curl u 0 ∥ L r ( Ω ) d γ , s ∈ ( 0 , T ) , (7) where r ∈ ( 1 , min { 2 , p } ) r \in(1,\min\{2,p\}) r ∈ ( 1 , min { 2 , p }) γ ∈ ( 0 , 1 ) \gamma\in(0,1) γ ∈ ( 0 , 1 ) d ( x ) = dist ( x , ∂ Ω ) d(x) = \operatorname{dist}(x,\partial\Omega) d ( x ) = dist ( x , ∂ Ω ) (7) , we first establish parabolic smoothing estimates for the vorticity (Lemma 5.2 ). Then, we use the Biot–Savart law and Morrey’s inequality to deduce Hölder estimates for the velocity, which imply the desired estimate thanks to the impermeability condition (Corollary 5.5). The restriction r < 2 r < 2 r < 2 Lemma 5.2 from combining the interpolation estimate (Lemma 5.1 ) with a uniform L 1 ( Ω ) L^1(\Omega) L 1 ( Ω )
With these tools, we obtain a uniform vorticity bound (Proposition 6.1 and Lemma 6.2), and consequently, for ρ ∈ ( 1 , p ] \rho\in(1,p] ρ ∈ ( 1 , p ]
sup t ∈ [ 0 , T ] ∥ b ν ( t ) ∥ L ρ ( Ω ) ≤ C ∥ ω 0 ∥ L ρ ( Ω ) ( ν 1 / 2 − ρ q + ν ( γ − 1 / r ) / ( 1 + γ ) − ρ q ∥ ω 0 ∥ L r ( Ω ) 1 / ( 1 + γ ) ) . \sup_{t\in[0,T]}\|b^\nu(t)\|_{L^\rho(\Omega)} \leq C\|\omega_0\|_{L^\rho(\Omega)}\left(\nu^{1/2-\rho q}+\nu^{(\gamma-1/r)/(1+\gamma)-\rho q}\|\omega_0\|_{L^r(\Omega)}^{1/(1+\gamma)}\right). t ∈ [ 0 , T ] sup ∥ b ν ( t ) ∥ L ρ ( Ω ) ≤ C ∥ ω 0 ∥ L ρ ( Ω ) ( ν 1/2 − ρq + ν ( γ − 1/ r ) / ( 1 + γ ) − ρq ∥ ω 0 ∥ L r ( Ω ) 1/ ( 1 + γ ) ) . The first summand comes from diffusion and requires ρ q < 1 / 2 \rho q<1/2 ρq < 1/2 ρ > 1 \rho>1 ρ > 1 q < 1 / 2 q<1/2 q < 1/2 q = 1 / 2 q=1/2 q = 1/2
ρ q < sup γ ∈ ( 0 , 1 ) r ∈ ( 1 , min { 2 , p } ) γ − 1 / r 1 + γ = min { p − 1 2 p , 1 4 } . \rho q<\sup_{\substack{\gamma\in(0,1)\\r\in(1,\min\{2,p\})}}\frac{\gamma-1/r}{1+\gamma}=\min\left\{\frac{p-1}{2p},\frac{1}{4}\right\}. ρq < γ ∈ ( 0 , 1 ) r ∈ ( 1 , m i n { 2 , p }) sup 1 + γ γ − 1/ r = min { 2 p p − 1 , 4 1 } . Consequently, b ν b^\nu b ν L ρ ( Ω ) L^\rho(\Omega) L ρ ( Ω ) a ν a^\nu a ν
Declaration of AI Use
The author directed the mathematical work on this project. Large language models, including OpenAI’s GPT-5.5 Pro and GPT-5.6 Sol, served as research assistants, advising the author on specific questions that arose in developing the proof. The counterexample in Lemma B.1 was found by OpenAI’s GPT-6 Astra.
The author wrote the manuscript from scratch and takes full responsibility for the mathematical content and presentation of the paper.
Notation
In the following, x , y ∈ Ω ‾ x,y\in\overline{\Omega} x , y ∈ Ω t , s ∈ [ 0 , T ] t,s\in[0,T] t , s ∈ [ 0 , T ]
We denote with v ⋅ w v\cdot w v ⋅ w A : B A:B A : B
In two dimensions, we set
curl v : = ∂ x 1 v 2 − ∂ x 2 v 1 , ∇ ⊥ f : = ( − ∂ x 2 ∂ x 1 ) f . \operatorname{curl}v:=\partial_{x_1}v_2-\partial_{x_2}v_1,\qquad\nabla^\perp f:=\begin{pmatrix}-\partial_{x_2}\\ \partial_{x_1}\end{pmatrix}f. curl v := ∂ x 1 v 2 − ∂ x 2 v 1 , ∇ ⊥ f := ( − ∂ x 2 ∂ x 1 ) f . Regarding the boundary, we set τ = n ⊥ = ( − n 2 n 1 ) \tau=n^\perp=\begin{pmatrix}-n_2\\ n_1\end{pmatrix} τ = n ⊥ = ( − n 2 n 1 ) κ \kappa κ ∂ τ n = κ τ \partial_\tau n=\kappa\tau ∂ τ n = κ τ ∂ n f = ∇ f ⋅ n \partial_n f=\nabla f\cdot n ∂ n f = ∇ f ⋅ n d ( x ) = dist ( x , ∂ Ω ) d(x)=\operatorname{dist}(x,\partial\Omega) d ( x ) = dist ( x , ∂ Ω ) Ω ξ : = { x ∈ Ω : dist ( x , ∂ Ω ) < ξ } \Omega_\xi:=\{x\in\Omega:\operatorname{dist}(x,\partial\Omega)<\xi\} Ω ξ := { x ∈ Ω : dist ( x , ∂ Ω ) < ξ } ξ > 0 \xi>0 ξ > 0 Δ y G Ω ( x , ⋅ ) = δ x \Delta_y G_\Omega(x,\cdot)=\delta_x Δ y G Ω ( x , ⋅ ) = δ x G Ω ( x , ⋅ ) = 0 G_\Omega(x,\cdot)=0 G Ω ( x , ⋅ ) = 0 ∂ Ω \partial\Omega ∂ Ω [8 ] , we introduce the following Hilbert spaces
H = { v ∈ L 2 ( Ω ) : div v = 0 in Ω and v ⋅ n = 0 on ∂ Ω } , H=\{v\in L^2(\Omega):\operatorname{div}v=0\text{ in }\Omega\text{ and }v\cdot n=0\text{ on }\partial\Omega\}, H = { v ∈ L 2 ( Ω ) : div v = 0 in Ω and v ⋅ n = 0 on ∂ Ω } , V = { v ∈ H 1 ( Ω ) : div v = 0 in Ω and v ⋅ n = 0 on ∂ Ω } , V=\{v\in H^1(\Omega):\operatorname{div}v=0\text{ in }\Omega\text{ and }v\cdot n=0\text{ on }\partial\Omega\}, V = { v ∈ H 1 ( Ω ) : div v = 0 in Ω and v ⋅ n = 0 on ∂ Ω } , W = { v ∈ H 2 ( Ω ) ∩ V : 2 ( D v ) S n ⋅ τ + α ν v ⋅ τ = 0 on ∂ Ω } . W=\{v\in H^2(\Omega)\cap V:2(Dv)Sn\cdot\tau+\alpha^\nu v\cdot\tau=0\text{ on }\partial\Omega\}. W = { v ∈ H 2 ( Ω ) ∩ V : 2 ( D v ) S n ⋅ τ + α ν v ⋅ τ = 0 on ∂ Ω } . For W W W α ν \alpha^\nu α ν γ ∈ ( 0 , 1 ) \gamma\in(0,1) γ ∈ ( 0 , 1 )
[ v ] C 0 , γ ( Ω ‾ ) : = sup x , y ∈ Ω ‾ x ≠ y ∣ v ( x ) − v ( y ) ∣ ∣ x − y ∣ γ . [v]_{C^{0,\gamma}(\overline{\Omega})}:=\sup_{\substack{x,y\in\overline{\Omega}\\x\ne y}}\frac{|v(x)-v(y)|}{|x-y|^\gamma}. [ v ] C 0 , γ ( Ω ) := x , y ∈ Ω x = y sup ∣ x − y ∣ γ ∣ v ( x ) − v ( y ) ∣ . We will use the same notation for scalar- and vector-valued function spaces. Finally, values of constants in an inequality may change from line to line and may depend on the domain and also on functions of the domain.
Weak and strong solutions for Navier friction
For the convenience of the reader, we repeat well-posedness and regularity results for Navier boundary conditions. Like in the Dirichlet case, there exists a unique Leray–Hopf solution for u 0 ∈ H u_0\in H u 0 ∈ H
Theorem 3.1 (Theorem 6.1, [15 ] ). For ν > 0 \nu>0 ν > 0 u 0 ∈ H u_0\in H u 0 ∈ H u ν u^\nu u ν
u ν ∈ C ( [ 0 , T ] ; H ) ∩ L 2 ( 0 , T ; V ) u^\nu\in C([0,T];H)\cap L^2(0,T;V) u ν ∈ C ([ 0 , T ] ; H ) ∩ L 2 ( 0 , T ; V ) with u ν ( 0 , ⋅ ) = u 0 u^\nu(0,\cdot)=u_0 u ν ( 0 , ⋅ ) = u 0
d d t ∫ Ω u ν ⋅ v d x + ∫ Ω ( u ν ⋅ ∇ u ν ) ⋅ v d x + ν ∫ Ω ∇ u ν : ∇ v d x = ν ∫ ∂ Ω ( κ − α ν ) u ν ⋅ v d S , (8) \frac{d}{dt}\int_\Omega u^\nu\cdot v\,dx+\int_\Omega(u^\nu\cdot\nabla u^\nu)\cdot v\,dx+\nu\int_\Omega\nabla u^\nu:\nabla v\,dx=\nu\int_{\partial\Omega}(\kappa-\alpha^\nu)u^\nu\cdot v\,dS, \tag*{(8)} d t d ∫ Ω u ν ⋅ v d x + ∫ Ω ( u ν ⋅ ∇ u ν ) ⋅ v d x + ν ∫ Ω ∇ u ν : ∇ v d x = ν ∫ ∂ Ω ( κ − α ν ) u ν ⋅ v d S , (8) for every v ∈ V v\in V v ∈ V ∂ t u ν ∈ L 2 ( 0 , T ; V ′ ) \partial_tu^\nu\in L^2(0,T;V') ∂ t u ν ∈ L 2 ( 0 , T ; V ′ ) t ∈ [ 0 , T ] t\in[0,T] t ∈ [ 0 , T ]
∥ u ν ( t ) ∥ L 2 ( Ω ) 2 + 2 ν ∫ 0 t ∥ ∇ u ν ∥ L 2 ( Ω ) 2 d s = ∥ u 0 ∥ L 2 ( Ω ) 2 + 2 ν ∫ 0 t ∫ ∂ Ω ( κ − α ν ) ∣ u ν ⋅ τ ∣ 2 d S d s . (9) \|u^\nu(t)\|_{L^2(\Omega)}^2+2\nu\int_0^t\|\nabla u^\nu\|_{L^2(\Omega)}^2\,ds=\|u_0\|_{L^2(\Omega)}^2+2\nu\int_0^t\int_{\partial\Omega}(\kappa-\alpha^\nu)|u^\nu\cdot\tau|^2\,dS\,ds. \tag*{(9)} ∥ u ν ( t ) ∥ L 2 ( Ω ) 2 + 2 ν ∫ 0 t ∥∇ u ν ∥ L 2 ( Ω ) 2 d s = ∥ u 0 ∥ L 2 ( Ω ) 2 + 2 ν ∫ 0 t ∫ ∂ Ω ( κ − α ν ) ∣ u ν ⋅ τ ∣ 2 d S d s . (9) Remark 3.2. For α ν ≥ 0 \alpha^\nu\ge0 α ν ≥ 0
∥ u ν ( t ) ∥ L 2 ( Ω ) 2 + ν ∫ 0 t ∥ ∇ u ν ∥ L 2 ( Ω ) 2 d s ≤ e C ν t ∥ u 0 ∥ L 2 ( Ω ) 2 . \|u^\nu(t)\|_{L^2(\Omega)}^2+\nu\int_0^t\|\nabla u^\nu\|_{L^2(\Omega)}^2\,ds\le e^{C\nu t}\|u_0\|_{L^2(\Omega)}^2. ∥ u ν ( t ) ∥ L 2 ( Ω ) 2 + ν ∫ 0 t ∥∇ u ν ∥ L 2 ( Ω ) 2 d s ≤ e C ν t ∥ u 0 ∥ L 2 ( Ω ) 2 . For positive times, the weak solution of Theorem 3.1 satisfies
u ν ∈ C ( [ ε , T ] ; V ) ∩ L 2 ( ε , T ; W ) u^\nu\in C([\varepsilon,T];V)\cap L^2(\varepsilon,T;W) u ν ∈ C ([ ε , T ] ; V ) ∩ L 2 ( ε , T ; W ) for every ε ∈ ( 0 , T ) \varepsilon\in(0,T) ε ∈ ( 0 , T )
Since u ν ∈ L 2 ( 0 , T ; V ) u^\nu\in L^2(0,T;V) u ν ∈ L 2 ( 0 , T ; V ) s ∈ ( 0 , ε ) s\in(0,\varepsilon) s ∈ ( 0 , ε ) u ν ( s ) ∈ V u^\nu(s)\in V u ν ( s ) ∈ V [9 Theorem 1.1] starting at time s s s
A common idea, which refers back to [12 ] , is to consider approximations of the initial data that satisfy the Navier boundary condition in order to work with strong solutions.
Lemma 3.3 ([12 ] ). Let ω 0 ∈ L p ( Ω ) \omega_0\in L^p(\Omega) ω 0 ∈ L p ( Ω ) p > 1 p>1 p > 1 { ω 0 , n } \{\omega_{0,n}\} { ω 0 , n } { ω 0 , n } ⊂ H 1 ( Ω ) ∩ L ∞ ( Ω ) \{\omega_{0,n}\}\subset H^1(\Omega)\cap L^\infty(\Omega) { ω 0 , n } ⊂ H 1 ( Ω ) ∩ L ∞ ( Ω ) K Ω ( ω 0 , n ) ∈ W K_\Omega(\omega_{0,n})\in W K Ω ( ω 0 , n ) ∈ W ω 0 \omega_0 ω 0 L p ( Ω ) L^p(\Omega) L p ( Ω )
Theorem 3.4 (Theorem 2.3, [8 ] ; Theorem 7.1, [15 ] ). For ν > 0 \nu>0 ν > 0 ω 0 ∈ H 1 ( Ω ) ∩ L ∞ ( Ω ) \omega_0\in H^1(\Omega)\cap L^\infty(\Omega) ω 0 ∈ H 1 ( Ω ) ∩ L ∞ ( Ω ) u 0 = K Ω ( ω 0 ) ∈ W u_0=K_\Omega(\omega_0)\in W u 0 = K Ω ( ω 0 ) ∈ W u ν ∈ C ( [ 0 , T ] ; W ) u^\nu\in C([0,T];W) u ν ∈ C ([ 0 , T ] ; W ) u ν ( 0 , ⋅ ) = K Ω ( ω 0 ) u^\nu(0,\cdot)=K_\Omega(\omega_0) u ν ( 0 , ⋅ ) = K Ω ( ω 0 ) ∂ t u ν ∈ L 2 ( 0 , T ; V ) ∩ C ( [ 0 , T ] ; H ) \partial_tu^\nu\in L^2(0,T;V)\cap C([0,T];H) ∂ t u ν ∈ L 2 ( 0 , T ; V ) ∩ C ([ 0 , T ] ; H ) ω ν = curl u ν \omega^\nu=\operatorname{curl}u^\nu ω ν = curl u ν ω ν ∈ C ( [ 0 , T ] ; H 1 ( Ω ) ) ∩ L ∞ ( ( 0 , T ) × Ω ) \omega^\nu\in C([0,T];H^1(\Omega))\cap L^\infty((0,T)\times\Omega) ω ν ∈ C ([ 0 , T ] ; H 1 ( Ω )) ∩ L ∞ (( 0 , T ) × Ω )
A priori estimates for the advection-diffusion equation
If we split the vorticity ω ν \omega^\nu ω ν a ν a^\nu a ν b ν b^\nu b ν
{ ∂ t θ ν + v ν ⋅ ∇ θ ν = ν Δ θ ν in ( 0 , T ) × Ω , div v ν = 0 in ( 0 , T ) × Ω , v ν ⋅ n = 0 on ( 0 , T ) × ∂ Ω , θ ν = g ν on ( 0 , T ) × ∂ Ω . (10) \begin{cases}
\partial_t\theta^\nu+v^\nu\cdot\nabla\theta^\nu=\nu\Delta\theta^\nu& \text{in }(0,T)\times\Omega,\\
\operatorname{div}v^\nu=0 & \text{in }(0,T)\times\Omega,\\
v^\nu\cdot n=0 & \text{on }(0,T)\times\partial\Omega,\\
\theta^\nu=g^\nu& \text{on }(0,T)\times\partial\Omega. \tag*{(10)}
\end{cases} ⎩ ⎨ ⎧ ∂ t θ ν + v ν ⋅ ∇ θ ν = ν Δ θ ν div v ν = 0 v ν ⋅ n = 0 θ ν = g ν in ( 0 , T ) × Ω , in ( 0 , T ) × Ω , on ( 0 , T ) × ∂ Ω , on ( 0 , T ) × ∂ Ω. (10) In this section, we quantify the effect of the boundary data g ν g^\nu g ν v ν v^\nu v ν Lemma 4.1 ) and establish well-known a priori estimates in the case g ν = 0 g^\nu=0 g ν = 0 Lemma 4.2 and Lemma 4.3 ). For a simpler notation and as all statements are for fixed ν ∈ ( 0 , 1 ) \nu\in(0,1) ν ∈ ( 0 , 1 ) ν \nu ν
Lemma 4.1. Fix ν ∈ ( 0 , 1 ) \nu\in(0,1) ν ∈ ( 0 , 1 ) γ > 0 \gamma>0 γ > 0 β ∈ C ( R ; [ 0 , ∞ ) ) \beta\in C(\mathbb{R};[0,\infty)) β ∈ C ( R ; [ 0 , ∞ )) v ∈ L l o c 2 ( 0 , T ; H ) v\in L^2_{\mathrm{loc}}(0,T;H) v ∈ L loc 2 ( 0 , T ; H ) g ∈ L l o c 2 ( 0 , T ; H 1 / 2 ( ∂ Ω ) ) g\in L^2_{\mathrm{loc}}(0,T;H^{1/2}(\partial\Omega)) g ∈ L loc 2 ( 0 , T ; H 1/2 ( ∂ Ω )) β ( g ) ∈ L ∞ ( 0 , T ; L 1 ( ∂ Ω ) ) \beta(g)\in L^\infty(0,T;L^1(\partial\Omega)) β ( g ) ∈ L ∞ ( 0 , T ; L 1 ( ∂ Ω )) θ \theta θ
θ ∈ C ( [ 0 , T ] ; L 1 ( Ω ) ) ∩ L l o c 2 ( 0 , T ; H 1 ( Ω ) ) \theta\in C([0,T];L^1(\Omega))\cap L^2_{\mathrm{loc}}(0,T;H^1(\Omega)) θ ∈ C ([ 0 , T ] ; L 1 ( Ω )) ∩ L loc 2 ( 0 , T ; H 1 ( Ω )) with β ( θ ( 0 ) ) ∈ L 1 ( Ω ) \beta(\theta(0))\in L^1(\Omega) β ( θ ( 0 )) ∈ L 1 ( Ω )
Assume also that there exists a strictly positive, nonincreasing B ∈ W l o c 1 , 1 ( 0 , T ) B \in W_{\mathrm{loc}}^{1,1}(0,T) B ∈ W loc 1 , 1 ( 0 , T ) B 1 / ( 1 + γ ) ∈ L 1 ( 0 , T ) B^{1/(1+\gamma)} \in L^1(0,T) B 1/ ( 1 + γ ) ∈ L 1 ( 0 , T )
v ( s , x ) ⋅ ∇ d ( x ) ≤ B ( s ) d ( x ) γ for almost every ( s , x ) ∈ ( 0 , T ) × Ω ξ , (11) v(s,x)\cdot\nabla d(x) \le B(s)d(x)^\gamma\quad\text{for almost every } (s,x)\in(0,T)\times\Omega_\xi, \tag*{(11)} v ( s , x ) ⋅ ∇ d ( x ) ≤ B ( s ) d ( x ) γ for almost every ( s , x ) ∈ ( 0 , T ) × Ω ξ , (11) where the collar width ξ > 0 \xi>0 ξ > 0 v v v ν \nu ν d ∈ C 2 ( Ω ξ ) d\in C^2(\Omega_\xi) d ∈ C 2 ( Ω ξ ) ξ ∥ Δ d ∥ L ∞ ( { d < ξ } ) ≤ 1 4 \xi\|\Delta d\|_{L^\infty(\{d<\xi\})}\le\frac14 ξ ∥Δ d ∥ L ∞ ({ d < ξ }) ≤ 4 1 0 ≤ s < t ≤ T 0\le s<t\le T 0 ≤ s < t ≤ T
∫ Ω β ( θ ( t ) ) d x ≤ ∫ Ω β ( θ ( s ) ) d x + C ∫ s t ( ν t − r + ν γ / ( 1 + γ ) B ( r ) 1 / ( 1 + γ ) ) ∫ ∂ Ω β ( g ( r ) ) d S d r , \int_\Omega\beta(\theta(t))\,dx\le\int_\Omega\beta(\theta(s))\,dx+C\int_s^t\left(\sqrt{\frac{\nu}{t-r}}+\nu^{\gamma/(1+\gamma)}B(r)^{1/(1+\gamma)}\right)\int_{\partial\Omega}\beta(g(r))\,dS\,dr, ∫ Ω β ( θ ( t )) d x ≤ ∫ Ω β ( θ ( s )) d x + C ∫ s t ( t − r ν + ν γ / ( 1 + γ ) B ( r ) 1/ ( 1 + γ ) ) ∫ ∂ Ω β ( g ( r )) d S d r , where the constant C > 0 C>0 C > 0 v v v ν \nu ν
Proof. The idea is to test the advection-diffusion equation of (10) with β ′ ( θ ) φ \beta'(\theta)\varphi β ′ ( θ ) φ (11) to choose φ \varphi φ β ∈ C 2 ( R ; [ 0 , ∞ ) ) \beta\in C^2(\mathbb{R};[0,\infty)) β ∈ C 2 ( R ; [ 0 , ∞ )) β ′ , β ′ ′ ∈ L ∞ ( R ) \beta',\beta''\in L^\infty(\mathbb{R}) β ′ , β ′′ ∈ L ∞ ( R ) β ′ ( θ ) φ \beta'(\theta)\varphi β ′ ( θ ) φ 0 ≤ s < t ≤ T 0\le s<t\le T 0 ≤ s < t ≤ T s < a < b < t s<a<b<t s < a < b < t ρ ∈ C c ∞ ( ( − 1 , 1 ) ) \rho\in C^\infty_c((-1,1)) ρ ∈ C c ∞ (( − 1 , 1 )) ∫ R ρ = 1 \int_{\mathbb{R}}\rho=1 ∫ R ρ = 1 ε ∈ ( 0 , min { a , T − b } ) \varepsilon\in(0,\min\{a,T-b\}) ε ∈ ( 0 , min { a , T − b }) r ∈ [ a , b ] r\in[a,b] r ∈ [ a , b ]
f ε ( r ) : = ∫ 0 T ρ ε ( r − τ ) f ( τ ) d τ , ρ ε ( r ) : = 1 ε ρ ( r ε ) . f_\varepsilon(r):=\int_0^T\rho_\varepsilon(r-\tau)f(\tau)\,d\tau,\qquad\rho_\varepsilon(r):=\frac{1}{\varepsilon}\rho\left(\frac{r}{\varepsilon}\right). f ε ( r ) := ∫ 0 T ρ ε ( r − τ ) f ( τ ) d τ , ρ ε ( r ) := ε 1 ρ ( ε r ) . The mollified θ ε \theta_\varepsilon θ ε
∂ r θ ε + ( v ⋅ ∇ θ ) ε = ν Δ θ ε . (12) \partial_r\theta_\varepsilon+(v\cdot\nabla\theta)_\varepsilon=\nu\Delta\theta_\varepsilon. \tag*{(12)} ∂ r θ ε + ( v ⋅ ∇ θ ) ε = ν Δ θ ε . (12) For nonnegative φ ∈ W 1 , 1 ( a , b ; L ∞ ( Ω ) ) ∩ L 2 ( a , b ; H 0 1 ( Ω ) ) \varphi\in W^{1,1}(a,b;L^\infty(\Omega))\cap L^2(a,b;H_0^1(\Omega)) φ ∈ W 1 , 1 ( a , b ; L ∞ ( Ω )) ∩ L 2 ( a , b ; H 0 1 ( Ω )) Δ φ ∈ L ∞ ( ( a , b ) × Ω ) \Delta\varphi\in L^\infty((a,b)\times\Omega) Δ φ ∈ L ∞ (( a , b ) × Ω ) (12) with β ′ ( θ ε ) φ \beta'(\theta_\varepsilon)\varphi β ′ ( θ ε ) φ
∫ Ω β ( θ ε ( b ) ) φ ( b ) d x − ∫ Ω β ( θ ε ( a ) ) φ ( a ) d x − ∫ a b ∫ Ω β ( θ ε ) ∂ r φ d x d r + ∫ a b ∫ Ω ( v ⋅ ∇ θ ) ε β ′ ( θ ε ) φ d x d r = − ν ∫ a b ∫ Ω ∇ β ( θ ε ) ⋅ ∇ φ d x d r − ν ∫ a b ∫ Ω ∣ ∇ θ ε ∣ 2 β ′ ′ ( θ ε ) φ d x d r . \begin{aligned}
&\int_\Omega\beta(\theta_\varepsilon(b))\varphi(b)\,dx-\int_\Omega\beta(\theta_\varepsilon(a))\varphi(a)\,dx-\int_a^b\int_\Omega\beta(\theta_\varepsilon)\partial_r\varphi\,dx\,dr\\
&\quad+\int_a^b\int_\Omega(v\cdot\nabla\theta)_\varepsilon\beta'(\theta_\varepsilon)\varphi\,dx\,dr\\
&=-\nu\int_a^b\int_\Omega\nabla\beta(\theta_\varepsilon)\cdot\nabla\varphi\,dx\,dr-\nu\int_a^b\int_\Omega|\nabla\theta_\varepsilon|^2\beta''(\theta_\varepsilon)\varphi\,dx\,dr.
\end{aligned} ∫ Ω β ( θ ε ( b )) φ ( b ) d x − ∫ Ω β ( θ ε ( a )) φ ( a ) d x − ∫ a b ∫ Ω β ( θ ε ) ∂ r φ d x d r + ∫ a b ∫ Ω ( v ⋅ ∇ θ ) ε β ′ ( θ ε ) φ d x d r = − ν ∫ a b ∫ Ω ∇ β ( θ ε ) ⋅ ∇ φ d x d r − ν ∫ a b ∫ Ω ∣∇ θ ε ∣ 2 β ′′ ( θ ε ) φ d x d r . From the convexity of β \beta β β ′ ′ ≥ 0 \beta''\ge0 β ′′ ≥ 0 L 1 ( Ω ) L^1(\Omega) L 1 ( Ω ) θ \theta θ
θ ε → θ in L 2 ( a , b ; H 1 ( Ω ) ) ∩ C ( [ a , b ] ; L 1 ( Ω ) ) and ( v ⋅ ∇ θ ) ε → v ⋅ ∇ θ in L 1 ( ( a , b ) × Ω ) . \theta_\varepsilon\to\theta\ \text{in }L^2(a,b;H^1(\Omega))\cap C([a,b];L^1(\Omega))\ \text{and}\ (v\cdot\nabla\theta)_\varepsilon\to v\cdot\nabla\theta\ \text{in }L^1((a,b)\times\Omega). θ ε → θ in L 2 ( a , b ; H 1 ( Ω )) ∩ C ([ a , b ] ; L 1 ( Ω )) and ( v ⋅ ∇ θ ) ε → v ⋅ ∇ θ in L 1 (( a , b ) × Ω ) . For a subsequence, we also have θ ε → θ \theta_\varepsilon\to\theta θ ε → θ β \beta β
∫ Ω β ( θ ( b ) ) φ ( b ) d x − ∫ Ω β ( θ ( a ) ) φ ( a ) d x − ∫ a b ∫ Ω β ( θ ) ∂ r φ d x d r + ∫ a b ∫ Ω ( v ⋅ ∇ θ ) β ′ ( θ ) φ d x d r ≤ − ν ∫ a b ∫ Ω ∇ β ( θ ) ⋅ ∇ φ d x d r . \int_{\Omega}\beta(\theta(b))\varphi(b)\,dx-\int_{\Omega}\beta(\theta(a))\varphi(a)\,dx-\int_a^b\int_{\Omega}\beta(\theta)\partial_r\varphi\,dx\,dr+\int_a^b\int_{\Omega}(v\cdot\nabla\theta)\beta'(\theta)\varphi\,dx\,dr
\leq-\nu\int_a^b\int_{\Omega}\nabla\beta(\theta)\cdot\nabla\varphi\,dx\,dr. ∫ Ω β ( θ ( b )) φ ( b ) d x − ∫ Ω β ( θ ( a )) φ ( a ) d x − ∫ a b ∫ Ω β ( θ ) ∂ r φ d x d r + ∫ a b ∫ Ω ( v ⋅ ∇ θ ) β ′ ( θ ) φ d x d r ≤ − ν ∫ a b ∫ Ω ∇ β ( θ ) ⋅ ∇ φ d x d r . With integration by parts and the Dirichlet boundary condition of (10) , this yields
∫ Ω β ( θ ( b ) ) φ ( b ) d x ≤ ∫ Ω β ( θ ( a ) ) φ ( a ) d x + ∫ a b ∫ Ω β ( θ ) [ ∂ r φ + v ⋅ ∇ φ + ν Δ φ ] d x d r − ν ∫ a b ∫ ∂ Ω β ( g ) ∂ n φ d S d r . \int_{\Omega}\beta(\theta(b))\varphi(b)\,dx\leq\int_{\Omega}\beta(\theta(a))\varphi(a)\,dx+\int_a^b\int_{\Omega}\beta(\theta)[\partial_r\varphi+v\cdot\nabla\varphi+\nu\Delta\varphi]\,dx\,dr-\nu\int_a^b\int_{\partial\Omega}\beta(g)\partial_n\varphi\,dS\,dr. ∫ Ω β ( θ ( b )) φ ( b ) d x ≤ ∫ Ω β ( θ ( a )) φ ( a ) d x + ∫ a b ∫ Ω β ( θ ) [ ∂ r φ + v ⋅ ∇ φ + ν Δ φ ] d x d r − ν ∫ a b ∫ ∂ Ω β ( g ) ∂ n φ d S d r . Now, we want to construct φ \varphi φ φ ∈ W 1 , 1 ( a , b ; L ∞ ( Ω ) ) ∩ L 2 ( a , b ; H 0 1 ( Ω ) ) \varphi\in W^{1,1}(a,b;L^\infty(\Omega))\cap L^2(a,b;H_0^1(\Omega)) φ ∈ W 1 , 1 ( a , b ; L ∞ ( Ω )) ∩ L 2 ( a , b ; H 0 1 ( Ω )) Δ φ ∈ L ∞ ( ( a , b ) × Ω ) \Delta\varphi\in L^\infty((a,b)\times\Omega) Δ φ ∈ L ∞ (( a , b ) × Ω ) ∂ r φ + v ⋅ ∇ φ + ν Δ φ ≤ 0 \partial_r\varphi+v\cdot\nabla\varphi+\nu\Delta\varphi\leq0 ∂ r φ + v ⋅ ∇ φ + ν Δ φ ≤ 0 ( s , t ) × Ω (s,t)\times\Omega ( s , t ) × Ω Ω \Omega Ω r → t r\to t r → t − ν ∂ n φ -\nu\partial_n\varphi − ν ∂ n φ
φ ( r , x ) : = { 2 d ( x ) l ( r ) − d ( x ) 2 l ( r ) 2 , d ( x ) < l ( r ) , 1 , d ( x ) ≥ l ( r ) , with φ = 0 and − ν ∂ n φ = 2 ν l ( r ) on ∂ Ω . \varphi(r,x):=
\begin{cases}
2\frac{d(x)}{l(r)}-\frac{d(x)^2}{l(r)^2}, & d(x)<l(r),\\
1, & d(x)\geq l(r),
\end{cases}
\qquad\text{with }\varphi=0\text{ and }-\nu\partial_n\varphi=\frac{2\nu}{l(r)}\text{ on }\partial\Omega. φ ( r , x ) := { 2 l ( r ) d ( x ) − l ( r ) 2 d ( x ) 2 , 1 , d ( x ) < l ( r ) , d ( x ) ≥ l ( r ) , with φ = 0 and − ν ∂ n φ = l ( r ) 2 ν on ∂ Ω. where
l ( r ) : = min { ξ , ν ( t − r ) , ( ν 4 B ( r ) ) 1 / ( 1 + γ ) } . l(r):=\min\left\{\xi,\sqrt{\nu(t-r)},\left(\frac{\nu}{4B(r)}\right)^{1/(1+\gamma)}\right\}. l ( r ) := min { ξ , ν ( t − r ) , ( 4 B ( r ) ν ) 1/ ( 1 + γ ) } . Let us explain the choice of l ( r ) l(r) l ( r ) l ( r ) ≤ ξ l(r)\leq\xi l ( r ) ≤ ξ d d d ν Δ d \nu\Delta d ν Δ d l ( r ) ≤ ν ( t − r ) l(r)\leq\sqrt{\nu(t-r)} l ( r ) ≤ ν ( t − r ) lim r → t φ ( r , x ) = 1 \lim_{r\to t}\varphi(r,x)=1 lim r → t φ ( r , x ) = 1 x ∈ Ω x\in\Omega x ∈ Ω − l ′ ( r ) ≤ ν 2 l ( r ) -l'(r)\leq\frac{\nu}{2l(r)} − l ′ ( r ) ≤ 2 l ( r ) ν d ( x ) < l ( r ) d(x)<l(r) d ( x ) < l ( r ) (11) :
∂ r φ + v ⋅ ∇ φ + ν Δ φ = 2 l ( 1 − d l ) ( − d l l ′ + v ⋅ ∇ d + ν Δ d ) − 2 ν l 2 ≤ 2 l B d γ − ν 2 l 2 ≤ 0 , \partial_r\varphi+v\cdot\nabla\varphi+\nu\Delta\varphi
=\frac{2}{l}\left(1-\frac{d}{l}\right)\left(-\frac{d}{l}l'+v\cdot\nabla d+\nu\Delta d\right)-\frac{2\nu}{l^2}
\leq\frac{2}{l}Bd^\gamma-\frac{\nu}{2l^2}\leq0, ∂ r φ + v ⋅ ∇ φ + ν Δ φ = l 2 ( 1 − l d ) ( − l d l ′ + v ⋅ ∇ d + ν Δ d ) − l 2 2 ν ≤ l 2 B d γ − 2 l 2 ν ≤ 0 , due to the third term
2 l B d γ ≤ ν 2 l 2 ⟺ l d γ ≤ ν 4 B . \frac{2}{l}Bd^\gamma\leq\frac{\nu}{2l^2}\Longleftrightarrow ld^\gamma\leq\frac{\nu}{4B}. l 2 B d γ ≤ 2 l 2 ν ⟺ l d γ ≤ 4 B ν . For d ( x ) ≥ l ( r ) d(x) \ge l(r) d ( x ) ≥ l ( r )
∫ Ω β ( θ ( b ) ) φ ( b ) d x ≤ ∫ Ω β ( θ ( a ) ) φ ( a ) d x + ν ∫ a b 2 l ∫ ∂ Ω β ( g ) d S d r ≤ ∫ Ω β ( θ ( a ) ) d x + 2 ν ∫ a b ( 1 ξ + 1 ν ( t − r ) + ( 4 B ( r ) ν ) 1 / ( 1 + γ ) ) ∫ ∂ Ω β ( g ) d S d r . ≤ ∫ Ω β ( θ ( a ) ) d x + C ∫ s t ( ν t − r + ν γ / ( 1 + γ ) B ( r ) 1 / ( 1 + γ ) ) ∫ ∂ Ω β ( g ) d S d r . \begin{aligned}
\int_{\Omega} \beta(\theta(b))\varphi(b)\,dx \le\int_{\Omega} \beta(\theta(a))\varphi(a)\,dx + \nu\int_a^b \frac{2}{l}\int_{\partial\Omega}\beta(g)\,dS\,dr \\
&\le\int_{\Omega}\beta(\theta(a))\,dx+2\nu\int_a^b\left(\frac{1}{\xi}+\frac{1}{\sqrt{\nu(t-r)}}+\left(\frac{4B(r)}{\nu}\right)^{1/(1+\gamma)}\right)\int_{\partial\Omega}\beta(g)\,dS\,dr. \\
&\le\int_{\Omega}\beta(\theta(a))\,dx+C\int_s^t\left(\sqrt{\frac{\nu}{t-r}}+\nu^{\gamma/(1+\gamma)}B(r)^{1/(1+\gamma)}\right)\int_{\partial\Omega}\beta(g)\,dS\,dr.
\end{aligned} ∫ Ω β ( θ ( b )) φ ( b ) d x ≤ ∫ Ω β ( θ ( a )) φ ( a ) d x + ν ∫ a b l 2 ∫ ∂ Ω β ( g ) d S d r ≤ ∫ Ω β ( θ ( a )) d x + 2 ν ∫ a b ( ξ 1 + ν ( t − r ) 1 + ( ν 4 B ( r ) ) 1/ ( 1 + γ ) ) ∫ ∂ Ω β ( g ) d S d r . ≤ ∫ Ω β ( θ ( a )) d x + C ∫ s t ( t − r ν + ν γ / ( 1 + γ ) B ( r ) 1/ ( 1 + γ ) ) ∫ ∂ Ω β ( g ) d S d r . The Lipschitz continuity of β \beta β β ( θ ) ∈ C ( [ s , t ] ; L 1 ( Ω ) ) \beta(\theta)\in C([s,t];L^1(\Omega)) β ( θ ) ∈ C ([ s , t ] ; L 1 ( Ω ))
lim a → s ∫ Ω β ( θ ( a ) ) d x = ∫ Ω β ( θ ( s ) ) d x and lim b → t ∫ Ω β ( θ ( b ) ) φ ( b ) d x = ∫ Ω β ( θ ( t ) ) d x . \lim_{a\to s}\int_{\Omega}\beta(\theta(a))\,dx=\int_{\Omega}\beta(\theta(s))\,dx \quad\text{and}\quad\lim_{b\to t}\int_{\Omega}\beta(\theta(b))\varphi(b)\,dx=\int_{\Omega}\beta(\theta(t))\,dx. a → s lim ∫ Ω β ( θ ( a )) d x = ∫ Ω β ( θ ( s )) d x and b → t lim ∫ Ω β ( θ ( b )) φ ( b ) d x = ∫ Ω β ( θ ( t )) d x . In total, we have shown
∫ Ω β ( θ ( t ) ) d x ≤ ∫ Ω β ( θ ( s ) ) d x + C ∫ s t ( ν t − r + ν γ / ( 1 + γ ) B ( r ) 1 / ( 1 + γ ) ) ∫ ∂ Ω β ( g ) d S d r , (13) \int_{\Omega}\beta(\theta(t))\,dx\le\int_{\Omega}\beta(\theta(s))\,dx+C\int_s^t\left(\sqrt{\frac{\nu}{t-r}}+\nu^{\gamma/(1+\gamma)}B(r)^{1/(1+\gamma)}\right)\int_{\partial\Omega}\beta(g)\,dS\,dr, \tag*{(13)} ∫ Ω β ( θ ( t )) d x ≤ ∫ Ω β ( θ ( s )) d x + C ∫ s t ( t − r ν + ν γ / ( 1 + γ ) B ( r ) 1/ ( 1 + γ ) ) ∫ ∂ Ω β ( g ) d S d r , (13) so it remains to remove the extra smoothness assumptions on β \beta β β ∈ C ( R ; [ 0 , ∞ ) ) \beta\in C(\mathbb{R};[0,\infty)) β ∈ C ( R ; [ 0 , ∞ )) β \beta β β k ∈ C 2 ( R ; [ 0 , ∞ ) ) \beta_k\in C^2(\mathbb{R};[0,\infty)) β k ∈ C 2 ( R ; [ 0 , ∞ )) β k ′ , β k ′ ′ ∈ L ∞ ( R ) \beta_k',\beta_k''\in L^\infty(\mathbb{R}) β k ′ , β k ′′ ∈ L ∞ ( R ) 0 ≤ β k ≤ β + 1 k 0\le\beta_k\le\beta+\frac{1}{k} 0 ≤ β k ≤ β + k 1 β k → β \beta_k\to\beta β k → β β k \beta_k β k s = 0 s=0 s = 0 β ( θ ( 0 ) ) ∈ L 1 ( Ω ) \beta(\theta(0))\in L^1(\Omega) β ( θ ( 0 )) ∈ L 1 ( Ω ) β ( θ ( t ) ) ∈ L 1 ( Ω ) \beta(\theta(t))\in L^1(\Omega) β ( θ ( t )) ∈ L 1 ( Ω ) t ∈ [ 0 , T ] t\in[0,T] t ∈ [ 0 , T ] s ∈ [ 0 , t ) s\in[0,t) s ∈ [ 0 , t )
For the sake of completeness, we will repeat some properties of the advection-diffusion equation with homogeneous Dirichlet data, i.e. θ = 0 \theta=0 θ = 0 ( 0 , T ) × ∂ Ω (0,T)\times\partial\Omega ( 0 , T ) × ∂ Ω L p ( Ω ) L^p(\Omega) L p ( Ω )
Lemma 4.2. Fix ν ∈ ( 0 , 1 ) \nu\in(0,1) ν ∈ ( 0 , 1 ) p ∈ [ 1 , ∞ ] p\in[1,\infty] p ∈ [ 1 , ∞ ] v ∈ L 2 ( 0 , T ; H ) v\in L^2(0,T;H) v ∈ L 2 ( 0 , T ; H ) θ ∈ C ( [ 0 , T ] ; L 1 ( Ω ) ) ∩ L l o c 2 ( 0 , T ; H 0 1 ( Ω ) ) \theta\in C([0,T];L^1(\Omega))\cap L^2_{\mathrm{loc}}(0,T;H_0^1(\Omega)) θ ∈ C ([ 0 , T ] ; L 1 ( Ω )) ∩ L loc 2 ( 0 , T ; H 0 1 ( Ω )) θ ( 0 ) ∈ L p ( Ω ) \theta(0)\in L^p(\Omega) θ ( 0 ) ∈ L p ( Ω ) g = 0 g=0 g = 0 0 ≤ s < t ≤ T 0\le s<t\le T 0 ≤ s < t ≤ T
∥ θ ( t ) ∥ L p ( Ω ) ≤ ∥ θ ( s ) ∥ L p ( Ω ) . \|\theta(t)\|_{L^p(\Omega)}\le\|\theta(s)\|_{L^p(\Omega)}. ∥ θ ( t ) ∥ L p ( Ω ) ≤ ∥ θ ( s ) ∥ L p ( Ω ) . Proof. Let β ∈ C 2 ( R ; [ 0 , ∞ ) ) \beta\in C^2(\mathbb{R};[0,\infty)) β ∈ C 2 ( R ; [ 0 , ∞ )) β ′ , β ′ ′ ∈ L ∞ ( R ) \beta',\beta''\in L^\infty(\mathbb{R}) β ′ , β ′′ ∈ L ∞ ( R ) β ′ ( 0 ) = 0 \beta'(0)=0 β ′ ( 0 ) = 0 β ′ ( θ ) \beta'(\theta) β ′ ( θ ) β ′ ( θ ) \beta'(\theta) β ′ ( θ )
d d r ∫ Ω β ( θ ( r ) ) d x = − ν ∫ Ω β ′ ′ ( θ ( r ) ) ∣ ∇ θ ( r ) ∣ 2 d x ≤ 0 in D ′ ( 0 , T ) . (14) \frac{d}{dr}\int_{\Omega}\beta(\theta(r))\,dx=-\nu\int_{\Omega}\beta''(\theta(r))|\nabla\theta(r)|^2\,dx\le0\quad\text{in }\mathcal{D}'(0,T). \tag*{(14)} d r d ∫ Ω β ( θ ( r )) d x = − ν ∫ Ω β ′′ ( θ ( r )) ∣∇ θ ( r ) ∣ 2 d x ≤ 0 in D ′ ( 0 , T ) . (14) Since β \beta β β ( θ ) ∈ C ( [ 0 , T ] ; L 1 ( Ω ) ) \beta(\theta) \in C([0,T];L^1(\Omega)) β ( θ ) ∈ C ([ 0 , T ] ; L 1 ( Ω ))
∫ Ω β ( θ ( t ) ) d x ≤ ∫ Ω β ( θ ( s ) ) d x for every 0 ≤ s < t ≤ T . (15) \int_{\Omega} \beta(\theta(t))\,dx \le\int_{\Omega} \beta(\theta(s))\,dx \quad\text{for every } 0 \le s < t \le T. \tag*{(15)} ∫ Ω β ( θ ( t )) d x ≤ ∫ Ω β ( θ ( s )) d x for every 0 ≤ s < t ≤ T . (15) For p ∈ [ 1 , ∞ ) p \in[1,\infty) p ∈ [ 1 , ∞ ) β k ∈ C 2 ( R ; [ 0 , ∞ ) ) \beta_k \in C^2(\mathbb{R};[0,\infty)) β k ∈ C 2 ( R ; [ 0 , ∞ ))
β k ′ ( 0 ) = 0 , β k ′ , β k ′ ′ ∈ L ∞ ( R ) , 0 ≤ β k ( z ) ≤ ∣ z ∣ p + 1 k , β k → ∣ ⋅ ∣ p . \beta_k'(0)=0,\qquad\beta_k',\beta_k'' \in L^\infty(\mathbb{R}),\qquad0 \le\beta_k(z) \le|z|^p+\frac{1}{k},\qquad\beta_k \to|\cdot|^p. β k ′ ( 0 ) = 0 , β k ′ , β k ′′ ∈ L ∞ ( R ) , 0 ≤ β k ( z ) ≤ ∣ z ∣ p + k 1 , β k → ∣ ⋅ ∣ p . Like before, we get with Fatou’s lemma and dominated convergence that
∫ Ω ∣ θ ( t ) ∣ p d x ≤ lim inf k → ∞ ∫ Ω β k ( θ ( t ) ) d x ≤ lim k → ∞ ∫ Ω β k ( θ ( 0 ) ) d x = ∫ Ω ∣ θ ( 0 ) ∣ p d x , \int_{\Omega} |\theta(t)|^p\,dx \le\liminf_{k\to\infty}\int_{\Omega}\beta_k(\theta(t))\,dx \le\lim_{k\to\infty}\int_{\Omega}\beta_k(\theta(0))\,dx = \int_{\Omega}|\theta(0)|^p\,dx, ∫ Ω ∣ θ ( t ) ∣ p d x ≤ k → ∞ lim inf ∫ Ω β k ( θ ( t )) d x ≤ k → ∞ lim ∫ Ω β k ( θ ( 0 )) d x = ∫ Ω ∣ θ ( 0 ) ∣ p d x , hence θ ( t ) ∈ L p ( Ω ) \theta(t) \in L^p(\Omega) θ ( t ) ∈ L p ( Ω ) t ∈ [ 0 , T ] t \in[0,T] t ∈ [ 0 , T ] s ∈ [ 0 , t ) s \in[0,t) s ∈ [ 0 , t )
∥ θ ( t ) ∥ L p ( Ω ) ≤ ∥ θ ( s ) ∥ L p ( Ω ) . (16) \|\theta(t)\|_{L^p(\Omega)} \le\|\theta(s)\|_{L^p(\Omega)}. \tag*{(16)} ∥ θ ( t ) ∥ L p ( Ω ) ≤ ∥ θ ( s ) ∥ L p ( Ω ) . (16) For p = ∞ p=\infty p = ∞ θ ( 0 ) ∈ L q ( Ω ) \theta(0) \in L^q(\Omega) θ ( 0 ) ∈ L q ( Ω ) q q q ∣ Ω ∣ < ∞ |\Omega|<\infty ∣Ω∣ < ∞ q → ∞ q\to\infty q → ∞ L q L^q L q L ∞ L^\infty L ∞
For positive times, we even gain integrability thanks to parabolic smoothing estimates, whose underlying argument goes back to Nash [25 ] . On the whole space, similar parabolic smoothing estimates have also been shown in [2 , 31 ] .
Lemma 4.3. Fix ν ∈ ( 0 , 1 ) \nu\in(0,1) ν ∈ ( 0 , 1 ) p ∈ [ 1 , ∞ ] p \in[1,\infty] p ∈ [ 1 , ∞ ] v ∈ L 2 ( 0 , T ; H ) v \in L^2(0,T;H) v ∈ L 2 ( 0 , T ; H )
θ ∈ C ( [ 0 , T ] ; L 1 ( Ω ) ) ∩ L l o c 2 ( 0 , T ; H 0 1 ( Ω ) ) \theta\in C([0,T];L^1(\Omega)) \cap L^2_{\mathrm{loc}}(0,T;H_0^1(\Omega)) θ ∈ C ([ 0 , T ] ; L 1 ( Ω )) ∩ L loc 2 ( 0 , T ; H 0 1 ( Ω )) be a corresponding weak solution to (4.1) with g = 0 g=0 g = 0 a n d $ 1 ≤ m ≤ p and \$1 \le m \le p an d $1 ≤ m ≤ p θ ( s ) ∈ L m ( Ω ) \theta(s) \in L^m(\Omega) θ ( s ) ∈ L m ( Ω ) θ ( t ) ∈ L p ( Ω ) \theta(t) \in L^p(\Omega) θ ( t ) ∈ L p ( Ω )
∥ θ ( t ) ∥ L p ( Ω ) ≤ C [ ν ( t − s ) ] − ( 1 / m − 1 / p ) ∥ θ ( s ) ∥ L m ( Ω ) . \|\theta(t)\|_{L^p(\Omega)} \le C[\nu(t-s)]^{-(1/m-1/p)}\|\theta(s)\|_{L^m(\Omega)}. ∥ θ ( t ) ∥ L p ( Ω ) ≤ C [ ν ( t − s ) ] − ( 1/ m − 1/ p ) ∥ θ ( s ) ∥ L m ( Ω ) . Here 1 / ∞ = 0 1/\infty=0 1/∞ = 0 C > 0 C>0 C > 0 v v v ν \nu ν
Proof. For m = ∞ m=\infty m = ∞ m ∈ [ 1 , ∞ ) m \in[1,\infty) m ∈ [ 1 , ∞ ) θ ( s ) ∈ L ∞ ( Ω ) \theta(s) \in L^\infty(\Omega) θ ( s ) ∈ L ∞ ( Ω ) θ ∈ C ( [ 0 , T ] ; L 1 ( Ω ) ) \theta\in C([0,T];L^1(\Omega)) θ ∈ C ([ 0 , T ] ; L 1 ( Ω )) θ ∈ C ( [ s , T ] ; L q ( Ω ) ) \theta\in C([s,T];L^q(\Omega)) θ ∈ C ([ s , T ] ; L q ( Ω )) q ∈ [ 1 , ∞ ) q \in[1,\infty) q ∈ [ 1 , ∞ ) q ≥ 2 q \ge2 q ≥ 2 β \beta β ∣ ⋅ ∣ q |\cdot|^q ∣ ⋅ ∣ q
d d r ∥ θ ( r ) ∥ L q ( Ω ) q + 4 ν ( q − 1 ) q ∥ ∇ ∣ θ ( r ) ∣ q / 2 ∥ L 2 ( Ω ) 2 = 0 for almost every r ∈ ( s , T ) . (17) \frac{d}{dr}\|\theta(r)\|_{L^q(\Omega)}^q+\frac{4\nu(q-1)}{q}\left\|\nabla|\theta(r)|^{q/2}\right\|_{L^2(\Omega)}^2=0 \quad\text{for almost every } r \in(s,T). \tag*{(17)} d r d ∥ θ ( r ) ∥ L q ( Ω ) q + q 4 ν ( q − 1 ) ∇∣ θ ( r ) ∣ q /2 L 2 ( Ω ) 2 = 0 for almost every r ∈ ( s , T ) . (17) Nash’s inequality [25 p. 936] , applied to the zero extension of f ∈ H 0 1 ( Ω ) f \in H^1_0(\Omega) f ∈ H 0 1 ( Ω )
∥ f ∥ L 2 ( Ω ) 4 ≤ C ∥ f ∥ L 1 ( Ω ) 2 ∥ ∇ f ∥ L 2 ( Ω ) 2 . \|f\|_{L^2(\Omega)}^4 \leq C\|f\|_{L^1(\Omega)}^2\|\nabla f\|_{L^2(\Omega)}^2. ∥ f ∥ L 2 ( Ω ) 4 ≤ C ∥ f ∥ L 1 ( Ω ) 2 ∥∇ f ∥ L 2 ( Ω ) 2 . If θ ( t ) = 0 \theta(t)=0 θ ( t ) = 0 θ ( r ) ≠ 0 \theta(r)\ne0 θ ( r ) = 0 r ∈ [ s , t ] r\in[s,t] r ∈ [ s , t ] s ≤ a < b ≤ t s\leq a<b\leq t s ≤ a < b ≤ t r ∈ ( a , b ) r\in(a,b) r ∈ ( a , b ) f = ∣ θ ( r ) ∣ q / 2 ∈ H 0 1 ( Ω ) f=|\theta(r)|^{q/2}\in H^1_0(\Omega) f = ∣ θ ( r ) ∣ q /2 ∈ H 0 1 ( Ω ) L q / 2 L^{q/2} L q /2
∥ θ ( r ) ∥ L q ( Ω ) 2 q ≤ C ∥ θ ( a ) ∥ L q / 2 ( Ω ) q ∥ ∇ ∣ θ ( r ) ∣ q / 2 ∥ L 2 ( Ω ) 2 . \|\theta(r)\|_{L^q(\Omega)}^{2q}\leq C\|\theta(a)\|_{L^{q/2}(\Omega)}^q\left\|\nabla|\theta(r)|^{q/2}\right\|_{L^2(\Omega)}^2. ∥ θ ( r ) ∥ L q ( Ω ) 2 q ≤ C ∥ θ ( a ) ∥ L q /2 ( Ω ) q ∇∣ θ ( r ) ∣ q /2 L 2 ( Ω ) 2 . With (4.8), we obtain, for almost every r ∈ ( a , b ) r\in(a,b) r ∈ ( a , b )
d d r ( ∥ θ ( r ) ∥ L q ( Ω ) − q ) = − d d r ∥ θ ( r ) ∥ L q ( Ω ) q ∥ θ ( r ) ∥ L q ( Ω ) 2 q = 4 ν ( q − 1 ) q ∥ ∇ ∣ θ ( r ) ∣ q / 2 ∥ L 2 ( Ω ) 2 ∥ θ ( r ) ∥ L q ( Ω ) 2 q ≥ 4 ν ( q − 1 ) q 1 C ∥ θ ( a ) ∥ L q / 2 ( Ω ) q ≥ 2 ν C ∥ θ ( a ) ∥ L q / 2 ( Ω ) q . \begin{aligned}
\frac{d}{dr}\left(\|\theta(r)\|_{L^q(\Omega)}^{-q}\right)=-\frac{\frac{d}{dr}\|\theta(r)\|_{L^q(\Omega)}^q}{\|\theta(r)\|_{L^q(\Omega)}^{2q}}=\frac{4\nu(q-1)}{q}\frac{\left\|\nabla|\theta(r)|^{q/2}\right\|_{L^2(\Omega)}^2}{\|\theta(r)\|_{L^q(\Omega)}^{2q}} \\
&\geq\frac{4\nu(q-1)}{q}\frac{1}{C\|\theta(a)\|_{L^{q/2}(\Omega)}^q}\geq\frac{2\nu}{C\|\theta(a)\|_{L^{q/2}(\Omega)}^q}.
\end{aligned} d r d ( ∥ θ ( r ) ∥ L q ( Ω ) − q ) = − ∥ θ ( r ) ∥ L q ( Ω ) 2 q d r d ∥ θ ( r ) ∥ L q ( Ω ) q = q 4 ν ( q − 1 ) ∥ θ ( r ) ∥ L q ( Ω ) 2 q ∇∣ θ ( r ) ∣ q /2 L 2 ( Ω ) 2 ≥ q 4 ν ( q − 1 ) C ∥ θ ( a ) ∥ L q /2 ( Ω ) q 1 ≥ C ∥ θ ( a ) ∥ L q /2 ( Ω ) q 2 ν . Integrating from a a a b b b
∥ θ ( b ) ∥ L q ( Ω ) ≤ ( C ν ( b − a ) ) 1 / q ∥ θ ( a ) ∥ L q / 2 ( Ω ) . (18) \|\theta(b)\|_{L^q(\Omega)}\leq\left(\frac{C}{\nu(b-a)}\right)^{1/q}\|\theta(a)\|_{L^{q/2}(\Omega)}. \tag*{(18)} ∥ θ ( b ) ∥ L q ( Ω ) ≤ ( ν ( b − a ) C ) 1/ q ∥ θ ( a ) ∥ L q /2 ( Ω ) . (18) The constant C C C q q q q k = 2 k m q_k=2^k m q k = 2 k m t k = s + ( t − s ) ( 1 − 2 − k ) t_k=s+(t-s)(1-2^{-k}) t k = s + ( t − s ) ( 1 − 2 − k ) k ≥ 0 k\geq0 k ≥ 0 q = q j q=q_j q = q j [ t j − 1 , t j ] [t_{j-1},t_j] [ t j − 1 , t j ] j ∈ { 1 , … , k } j\in\{1,\ldots,k\} j ∈ { 1 , … , k }
∥ θ ( t ) ∥ L q k ( Ω ) ≤ ∥ θ ( t k ) ∥ L q k ( Ω ) ≤ ∥ θ ( s ) ∥ L m ( Ω ) ∏ j = 1 k ( C 2 j ν ( t − s ) ) 1 / ( 2 j m ) . \|\theta(t)\|_{L^{q_k}(\Omega)}\leq\|\theta(t_k)\|_{L^{q_k}(\Omega)}\leq\|\theta(s)\|_{L^m(\Omega)}\prod_{j=1}^{k}\left(\frac{C2^j}{\nu(t-s)}\right)^{1/(2^j m)}. ∥ θ ( t ) ∥ L q k ( Ω ) ≤ ∥ θ ( t k ) ∥ L q k ( Ω ) ≤ ∥ θ ( s ) ∥ L m ( Ω ) j = 1 ∏ k ( ν ( t − s ) C 2 j ) 1/ ( 2 j m ) . By letting k → ∞ k\to\infty k → ∞
∥ θ ( t ) ∥ L ∞ ( Ω ) ≤ ∥ θ ( s ) ∥ L m ( 4 C ν ( t − s ) ) 1 / m , \|\theta(t)\|_{L^\infty(\Omega)}\leq\|\theta(s)\|_{L^m}\left(\frac{4C}{\nu(t-s)}\right)^{1/m}, ∥ θ ( t ) ∥ L ∞ ( Ω ) ≤ ∥ θ ( s ) ∥ L m ( ν ( t − s ) 4 C ) 1/ m , where we used for the right-hand side the series ∑ j = 1 ∞ 2 − j = 1 \sum_{j=1}^{\infty}2^{-j}=1 ∑ j = 1 ∞ 2 − j = 1 ∑ j = 1 ∞ j 2 − j = 2 \sum_{j=1}^{\infty}j2^{-j}=2 ∑ j = 1 ∞ j 2 − j = 2 p p p
∥ θ ( t ) ∥ L p ( Ω ) ≤ ∥ θ ( t ) ∥ L m ( Ω ) m / p ∥ θ ( t ) ∥ L ∞ ( Ω ) 1 − m / p ≤ C [ ν ( t − s ) ] − ( 1 / m − 1 / p ) ∥ θ ( s ) ∥ L m ( Ω ) . \|\theta(t)\|_{L^p(\Omega)}\leq\|\theta(t)\|_{L^m(\Omega)}^{m/p}\|\theta(t)\|_{L^\infty(\Omega)}^{1-m/p}\leq C[\nu(t-s)]^{-(1/m-1/p)}\|\theta(s)\|_{L^m(\Omega)}. ∥ θ ( t ) ∥ L p ( Ω ) ≤ ∥ θ ( t ) ∥ L m ( Ω ) m / p ∥ θ ( t ) ∥ L ∞ ( Ω ) 1 − m / p ≤ C [ ν ( t − s ) ] − ( 1/ m − 1/ p ) ∥ θ ( s ) ∥ L m ( Ω ) . It remains to remove the assumption θ ( s ) ∈ L ∞ ( Ω ) \theta(s)\in L^\infty(\Omega) θ ( s ) ∈ L ∞ ( Ω ) θ ( s ) ∈ L m ( Ω ) \theta(s)\in L^m(\Omega) θ ( s ) ∈ L m ( Ω ) θ ( s ) \theta(s) θ ( s ) L m ( Ω ) L^m(\Omega) L m ( Ω ) θ n \theta_n θ n
sup r ∈ [ s , t ] ∥ θ n ( r ) − θ ( r ) ∥ L m ( Ω ) ≤ ∥ θ n ( s ) − θ ( s ) ∥ L m ( Ω ) → 0. \sup_{r\in[s,t]}\|\theta_n(r)-\theta(r)\|_{L^m(\Omega)}\leq\|\theta_n(s)-\theta(s)\|_{L^m(\Omega)}\to0. r ∈ [ s , t ] sup ∥ θ n ( r ) − θ ( r ) ∥ L m ( Ω ) ≤ ∥ θ n ( s ) − θ ( s ) ∥ L m ( Ω ) → 0. The assertion follows by lower semicontinuity of the L p ( Ω ) L^p(\Omega) L p ( Ω )
Parabolic smoothing for Navier friction
In this section, we show that parabolic smoothing also holds for the vorticity ω ν \omega^\nu ω ν ν \nu ν
Lemma 5.1. Let f ∈ L ∞ ( Ω ) f \in L^\infty(\Omega) f ∈ L ∞ ( Ω )
∥ K Ω ( f ) ⋅ τ ∥ L m ( ∂ Ω ) ≤ C ∥ f ∥ L 1 ( Ω ) 1 / 2 ∥ f ∥ L m ( Ω ) 1 / 2 for all m ∈ [ 1 , ∞ ) , \|K_\Omega(f)\cdot\tau\|_{L^m(\partial\Omega)} \leq C\|f\|_{L^1(\Omega)}^{1/2}\|f\|_{L^m(\Omega)}^{1/2} \quad\text{for all }m\in[1,\infty), ∥ K Ω ( f ) ⋅ τ ∥ L m ( ∂ Ω ) ≤ C ∥ f ∥ L 1 ( Ω ) 1/2 ∥ f ∥ L m ( Ω ) 1/2 for all m ∈ [ 1 , ∞ ) , and
∥ K Ω ( f ) ⋅ τ ∥ L ∞ ( ∂ Ω ) ≤ C ( m ) ∥ f ∥ L m ( Ω ) m / 2 ∥ f ∥ L ∞ ( Ω ) 1 − m / 2 for all m ∈ [ 1 , 2 ) . \|K_\Omega(f)\cdot\tau\|_{L^\infty(\partial\Omega)} \leq C(m)\|f\|_{L^m(\Omega)}^{m/2}\|f\|_{L^\infty(\Omega)}^{1-m/2} \quad\text{for all }m\in[1,2). ∥ K Ω ( f ) ⋅ τ ∥ L ∞ ( ∂ Ω ) ≤ C ( m ) ∥ f ∥ L m ( Ω ) m /2 ∥ f ∥ L ∞ ( Ω ) 1 − m /2 for all m ∈ [ 1 , 2 ) . Proof. Without loss of generality, we will assume f ≢ 0 f \not\equiv0 f ≡ 0 P Ω : = ∂ n G Ω P_\Omega:= \partial_nG_\Omega P Ω := ∂ n G Ω [20 Section 1.3] ). For y ∈ ∂ Ω y \in\partial\Omega y ∈ ∂ Ω
K Ω ( f ) ( y ) ⋅ τ ( y ) = ∫ Ω ∇ y G Ω ( x , y ) ⋅ τ ( y ) f ( x ) d x = ∫ Ω ∂ n G Ω ( x , y ) f ( x ) d x = ∫ Ω P Ω ( x , y ) f ( x ) d x . K_\Omega(f)(y)\cdot\tau(y)=\int_\Omega\nabla_yG_\Omega(x,y)\cdot\tau(y)f(x)\,dx=\int_\Omega\partial_nG_\Omega(x,y)f(x)\,dx=\int_\Omega P_\Omega(x,y)f(x)\,dx. K Ω ( f ) ( y ) ⋅ τ ( y ) = ∫ Ω ∇ y G Ω ( x , y ) ⋅ τ ( y ) f ( x ) d x = ∫ Ω ∂ n G Ω ( x , y ) f ( x ) d x = ∫ Ω P Ω ( x , y ) f ( x ) d x . From [19 Theorem 1] , we know that
0 ≤ P Ω ( x , y ) ≤ C d ( x ) ∣ x − y ∣ 2 ≤ C ∣ x − y ∣ . (19) 0\leq P_\Omega(x,y)\leq\frac{Cd(x)}{|x-y|^2}\leq\frac{C}{|x-y|}. \tag*{(19)} 0 ≤ P Ω ( x , y ) ≤ ∣ x − y ∣ 2 C d ( x ) ≤ ∣ x − y ∣ C . (19) which implies, for every R > 0 R>0 R > 0
sup y ∈ ∂ Ω ∫ Ω ∩ B R ( y ) P Ω ( x , y ) d x ≤ C ∫ B R ( 0 ) 1 ∣ z ∣ d z ≤ C R . (20) \sup_{y\in\partial\Omega}\int_{\Omega\cap B_R(y)}P_\Omega(x,y)\,dx\leq C\int_{B_R(0)}\frac{1}{|z|}\,dz\leq CR. \tag*{(20)} y ∈ ∂ Ω sup ∫ Ω ∩ B R ( y ) P Ω ( x , y ) d x ≤ C ∫ B R ( 0 ) ∣ z ∣ 1 d z ≤ C R . (20) Write K Ω ( f ) ⋅ τ = N R f + F R f K_\Omega(f)\cdot\tau=N_Rf+F_Rf K Ω ( f ) ⋅ τ = N R f + F R f
( N R f ) ( y ) : = ∫ Ω ∩ B R ( y ) P Ω ( x , y ) f ( x ) d x and ( F R f ) ( y ) : = ∫ Ω ∖ B R ( y ) P Ω ( x , y ) f ( x ) d x . (N_Rf)(y):=\int_{\Omega\cap B_R(y)}P_\Omega(x,y)f(x)\,dx
\qquad\text{and}\qquad
(F_Rf)(y):=\int_{\Omega\setminus B_R(y)}P_\Omega(x,y)f(x)\,dx. ( N R f ) ( y ) := ∫ Ω ∩ B R ( y ) P Ω ( x , y ) f ( x ) d x and ( F R f ) ( y ) := ∫ Ω ∖ B R ( y ) P Ω ( x , y ) f ( x ) d x . For m ∈ ( 1 , ∞ ) m\in(1,\infty) m ∈ ( 1 , ∞ )
∥ N R f ∥ L m ( ∂ Ω ) m ≤ ( C R ) m − 1 ∫ Ω ∣ f ( x ) ∣ m ∫ ∂ Ω P Ω ( x , y ) d S y ⏟ = 1 d x = ( C R ) m − 1 ∥ f ∥ L m ( Ω ) m . \|N_Rf\|_{L^m(\partial\Omega)}^m\leq(CR)^{m-1}\int_\Omega|f(x)|^m\underbrace{\int_{\partial\Omega}P_\Omega(x,y)\,dS_y}_{=1}\,dx=(CR)^{m-1}\|f\|_{L^m(\Omega)}^m. ∥ N R f ∥ L m ( ∂ Ω ) m ≤ ( C R ) m − 1 ∫ Ω ∣ f ( x ) ∣ m = 1 ∫ ∂ Ω P Ω ( x , y ) d S y d x = ( C R ) m − 1 ∥ f ∥ L m ( Ω ) m . For the far region, using (5.1) gives P Ω m ≤ ( C / R ) m − 1 P Ω P_\Omega^m\leq(C/R)^{m-1}P_\Omega P Ω m ≤ ( C / R ) m − 1 P Ω
∥ F R f ∥ L m ( ∂ Ω ) ≤ ∫ Ω ∣ f ( x ) ∣ ( ∫ ∂ Ω ∖ B R ( x ) P Ω ( x , y ) m d S y ) 1 / m d x ≤ C R 1 / m − 1 ∥ f ∥ L 1 ( Ω ) . \|F_Rf\|_{L^m(\partial\Omega)}\leq\int_\Omega|f(x)|\left(\int_{\partial\Omega\setminus B_R(x)}P_\Omega(x,y)^m\,dS_y\right)^{1/m}dx\leq CR^{1/m-1}\|f\|_{L^1(\Omega)}. ∥ F R f ∥ L m ( ∂ Ω ) ≤ ∫ Ω ∣ f ( x ) ∣ ( ∫ ∂ Ω ∖ B R ( x ) P Ω ( x , y ) m d S y ) 1/ m d x ≤ C R 1/ m − 1 ∥ f ∥ L 1 ( Ω ) . Choosing R = ( ∥ f ∥ L 1 ( Ω ) ∥ f ∥ L m ( Ω ) ) m / ( 2 m − 2 ) R = \left(\frac{\lVert f\rVert_{L^1(\Omega)}}{\lVert f\rVert_{L^m(\Omega)}}\right)^{m/(2m-2)} R = ( ∥ f ∥ L m ( Ω ) ∥ f ∥ L 1 ( Ω ) ) m / ( 2 m − 2 ) m > 1 m > 1 m > 1 m = 1 m = 1 m = 1
For the second estimate, fix m ∈ ( 1 , 2 ) m \in(1,2) m ∈ ( 1 , 2 ) y ∈ ∂ Ω y \in\partial\Omega y ∈ ∂ Ω (20) , we directly get
∣ ( N R f ) ( y ) ∣ ≤ ∥ f ∥ L ∞ ( Ω ) ∫ Ω ∩ B R ( y ) P Ω ( x , y ) d x ≤ C R ∥ f ∥ L ∞ ( Ω ) . \lvert(N_Rf)(y)\rvert\leq\lVert f\rVert_{L^\infty(\Omega)} \int_{\Omega\cap B_R(y)} P_\Omega(x,y)\,\mathrm{d}x \leq CR\lVert f\rVert_{L^\infty(\Omega)}. ∣( N R f ) ( y )∣ ≤ ∥ f ∥ L ∞ ( Ω ) ∫ Ω ∩ B R ( y ) P Ω ( x , y ) d x ≤ C R ∥ f ∥ L ∞ ( Ω ) . For the far region, (19) , Hölder’s inequality and polar coordinates give
∣ ( F R f ) ( y ) ∣ ≤ C ∫ Ω ∖ B R ( y ) ∣ f ( x ) ∣ ∣ x − y ∣ d x ≤ C ∥ f ∥ L m ( Ω ) ( 2 π ∫ R ∞ r 1 − m / ( m − 1 ) d r ) ( m − 1 ) / m ≤ C ( m ) R 1 − 2 / m ∥ f ∥ L m ( Ω ) . \begin{aligned}
\lvert(F_Rf)(y)\rvert\leq C\int_{\Omega\setminus B_R(y)}\frac{\lvert f(x)\rvert}{\lvert x-y\rvert}\,\mathrm{d}x \leq C\lVert f\rVert_{L^m(\Omega)}\left(2\pi\int_R^\infty r^{1-m/(m-1)}\,\mathrm{d}r\right)^{(m-1)/m} \\
&\leq C(m)R^{1-2/m}\lVert f\rVert_{L^m(\Omega)}.
\end{aligned} ∣( F R f ) ( y )∣ ≤ C ∫ Ω ∖ B R ( y ) ∣ x − y ∣ ∣ f ( x )∣ d x ≤ C ∥ f ∥ L m ( Ω ) ( 2 π ∫ R ∞ r 1 − m / ( m − 1 ) d r ) ( m − 1 ) / m ≤ C ( m ) R 1 − 2/ m ∥ f ∥ L m ( Ω ) . As a remark, we want point out that C ( m ) C(m) C ( m ) m → 2 m\to2 m → 2 m = 1 m=1 m = 1 (19) . We conclude by choosing
R = ( ∥ f ∥ L m ( Ω ) ∥ f ∥ L ∞ ( Ω ) ) m / 2 . R=\left(\frac{\lVert f\rVert_{L^m(\Omega)}}{\lVert f\rVert_{L^\infty(\Omega)}}\right)^{m/2}. R = ( ∥ f ∥ L ∞ ( Ω ) ∥ f ∥ L m ( Ω ) ) m /2 . The following lemma is the main driver of the article. It establishes parabolic smoothing rates for ω ν \omega^\nu ω ν α ν \alpha^\nu α ν ν \nu ν Lemma 4.1 on a startup time interval. We then consider sufficiently small ν \nu ν [ 0 , T ] [0,T] [ 0 , T ]
Lemma 5.2. Let ω 0 ∈ H 1 ( Ω ) ∩ L ∞ ( Ω ) \omega_0\in H^1(\Omega)\cap L^\infty(\Omega) ω 0 ∈ H 1 ( Ω ) ∩ L ∞ ( Ω ) u 0 = K Ω ( ω 0 ) ∈ W u_0=K_\Omega(\omega_0)\in W u 0 = K Ω ( ω 0 ) ∈ W ω ν = curl u ν \omega^\nu=\operatorname{curl}u^\nu ω ν = curl u ν u ν u^\nu u ν (1) for u ν ( 0 , ⋅ ) = u 0 u^\nu(0,\cdot)=u_0 u ν ( 0 , ⋅ ) = u 0 r ∈ ( 1 , 2 ) r\in(1,2) r ∈ ( 1 , 2 ) q ∈ [ 0 , r − 1 2 r ) q\in\left[0,\frac{r-1}{2r}\right) q ∈ [ 0 , 2 r r − 1 ) σ ∈ ( q , r − 1 2 r ) \sigma\in\left(q,\frac{r-1}{2r}\right) σ ∈ ( q , 2 r r − 1 ) ν 0 > 0 \nu_0>0 ν 0 > 0 ∥ curl u 0 ∥ L r ( Ω ) \lVert\operatorname{curl}u_0\rVert_{L^r(\Omega)} ∥ curl u 0 ∥ L r ( Ω ) ν ∈ ( 0 , ν 0 ) \nu\in(0,\nu_0) ν ∈ ( 0 , ν 0 ) t ∈ ( 0 , T ] t\in(0,T] t ∈ ( 0 , T ]
∥ ω ν ( t ) ∥ L p ( Ω ) ≤ C ν − 2 max { r q − σ , 0 } / p ( ν t ) − ( 1 / r − 1 / p ) ∥ curl u 0 ∥ L r ( Ω ) , r ≤ p ≤ ∞ . \lVert\omega^\nu(t)\rVert_{L^p(\Omega)}\leq C\nu^{-2\max\{rq-\sigma,0\}/p}(\nu t)^{-(1/r-1/p)}\lVert\operatorname{curl}u_0\rVert_{L^r(\Omega)},\qquad r\leq p\leq\infty. ∥ ω ν ( t ) ∥ L p ( Ω ) ≤ C ν − 2 m a x { r q − σ , 0 } / p ( ν t ) − ( 1/ r − 1/ p ) ∥ curl u 0 ∥ L r ( Ω ) , r ≤ p ≤ ∞. Here 1 / ∞ = 0 1/\infty=0 1/∞ = 0 C > 0 C>0 C > 0 ν \nu ν
Proof. In the following, we will show the estimates
∥ ω ν ( t ) ∥ L r ( Ω ) r ≤ C ν − 2 max { r q − σ , 0 } ∥ ω ν ( 0 ) ∥ L r ( Ω ) r , ( ν t ) 1 / r ∥ ω ν ( t ) ∥ L ∞ ( Ω ) ≤ C ∥ ω ν ( 0 ) ∥ L r ( Ω ) . (21) \lVert\omega^\nu(t)\rVert_{L^r(\Omega)}^r\leq C\nu^{-2\max\{rq-\sigma,0\}}\lVert\omega^\nu(0)\rVert_{L^r(\Omega)}^r,\qquad(\nu t)^{1/r}\lVert\omega^\nu(t)\rVert_{L^\infty(\Omega)}\leq C\lVert\omega^\nu(0)\rVert_{L^r(\Omega)}. \tag*{(21)} ∥ ω ν ( t ) ∥ L r ( Ω ) r ≤ C ν − 2 m a x { r q − σ , 0 } ∥ ω ν ( 0 ) ∥ L r ( Ω ) r , ( ν t ) 1/ r ∥ ω ν ( t ) ∥ L ∞ ( Ω ) ≤ C ∥ ω ν ( 0 ) ∥ L r ( Ω ) . (21) which imply the statement by interpolation
∥ ω ν ( t ) ∥ L p ( Ω ) ≤ ∥ ω ν ( t ) ∥ L r ( Ω ) r / p ∥ ω ν ( t ) ∥ L ∞ ( Ω ) 1 − r / p ≤ C ν − 2 max { r q − σ , 0 } / p ( ν t ) − ( 1 / r − 1 / p ) ∥ curl u 0 ∥ L r ( Ω ) . \lVert\omega^\nu(t)\rVert_{L^p(\Omega)}\leq\lVert\omega^\nu(t)\rVert_{L^r(\Omega)}^{r/p}\lVert\omega^\nu(t)\rVert_{L^\infty(\Omega)}^{1-r/p}\leq C\nu^{-2\max\{rq-\sigma,0\}/p}(\nu t)^{-(1/r-1/p)}\lVert\operatorname{curl}u_0\rVert_{L^r(\Omega)}. ∥ ω ν ( t ) ∥ L p ( Ω ) ≤ ∥ ω ν ( t ) ∥ L r ( Ω ) r / p ∥ ω ν ( t ) ∥ L ∞ ( Ω ) 1 − r / p ≤ C ν − 2 m a x { r q − σ , 0 } / p ( ν t ) − ( 1/ r − 1/ p ) ∥ curl u 0 ∥ L r ( Ω ) . For better readability, we will set
X ( t ) : = sup s ∈ ( 0 , t ] ( ν s ) 1 / r ∥ ω ν ( s ) ∥ L ∞ ( Ω ) , X ( 0 ) : = 0. X(t) := \sup_{s\in(0,t]} (\nu s)^{1/r} \|\omega^\nu(s)\|_{L^\infty(\Omega)}, \qquad X(0) := 0. X ( t ) := s ∈ ( 0 , t ] sup ( ν s ) 1/ r ∥ ω ν ( s ) ∥ L ∞ ( Ω ) , X ( 0 ) := 0. Observe that X ∈ C ( [ 0 , T ] ) X \in C([0,T]) X ∈ C ([ 0 , T ]) ν > 0 \nu> 0 ν > 0 ω ν \omega^\nu ω ν X ( t ) → 0 X(t) \to0 X ( t ) → 0 t → 0 t \to0 t → 0 L ∞ ( Ω ) L^\infty(\Omega) L ∞ ( Ω ) [21 Corollary 6.26] ) and the continuity of the boundary data in L ∞ ( ∂ Ω ) L^\infty(\partial\Omega) L ∞ ( ∂ Ω ) γ ∈ ( 1 / r , 1 ) \gamma\in(1/r,1) γ ∈ ( 1/ r , 1 ) σ < γ − 1 / r 1 + γ \sigma< \frac{\gamma-1/r}{1+\gamma} σ < 1 + γ γ − 1/ r
γ − 1 r − ( 1 + γ ) σ > 0. \gamma-\frac{1}{r}-(1+\gamma)\sigma> 0. γ − r 1 − ( 1 + γ ) σ > 0. Define the stopping time
τ ν : = sup { t ∈ ( 0 , T ] : ν γ − 1 r − ( 1 + γ ) σ X ( t ) ≤ 1 } , \tau_\nu:= \sup\left\{t \in(0,T]: \nu^{\gamma-\frac{1}{r}-(1+\gamma)\sigma}X(t) \le1\right\}, τ ν := sup { t ∈ ( 0 , T ] : ν γ − r 1 − ( 1 + γ ) σ X ( t ) ≤ 1 } , which is well defined due to X ∈ C ( [ 0 , T ] ) X \in C([0,T]) X ∈ C ([ 0 , T ]) X ( 0 ) = 0 X(0)=0 X ( 0 ) = 0 ( 0 , τ ν ] (0,\tau_\nu] ( 0 , τ ν ] ( 0 , min { ν , τ ν } ] (0,\min\{\nu,\tau_\nu\}] ( 0 , min { ν , τ ν }] L r ( Ω ) L^r(\Omega) L r ( Ω ) ( 0 , τ ν ] (0,\tau_\nu] ( 0 , τ ν ] τ ν < T \tau_\nu<T τ ν < T
We want to use Lemma 4.1 on [ 0 , τ ν ] [0,\tau_\nu] [ 0 , τ ν ] B B B ξ > 0 \xi>0 ξ > 0 d ∈ C 2 ( Ω ξ ) d\in C^2(\Omega_\xi) d ∈ C 2 ( Ω ξ ) y ∈ ∂ Ω y\in\partial\Omega y ∈ ∂ Ω x ∈ Ω ξ x\in\Omega_\xi x ∈ Ω ξ d ( x ) = ∣ x − y ∣ d(x)=|x-y| d ( x ) = ∣ x − y ∣ ∇ d ( x ) = − n ( y ) \nabla d(x)=-n(y) ∇ d ( x ) = − n ( y ) ( s , x ) ∈ ( 0 , τ ν ) × Ω ξ (s,x)\in(0,\tau_\nu)\times\Omega_\xi ( s , x ) ∈ ( 0 , τ ν ) × Ω ξ Lemma A.2 , that
∣ u ν ( s , x ) ⋅ ∇ d ( x ) ∣ = ∣ ( u ν ( s , x ) − u ν ( s , y ) ) ⋅ ∇ d ( x ) ∣ ≤ ∣ u ν ( s , x ) − u ν ( s , y ) ∣ ≤ [ u ν ( s ) ] C 0 , γ ∣ x − y ∣ γ ≤ C ∥ u ν ( s ) ∥ W 1 , 2 / ( 1 − γ ) ( Ω ) d ( x ) γ ≤ C ∥ ω ν ( s ) ∥ L 2 / ( 1 − γ ) ( Ω ) d ( x ) γ ≤ C ∥ ω ν ( s ) ∥ L ∞ ( Ω ) d ( x ) γ ≤ C ( ν s ) − 1 / r X ( τ ν ) d ( x ) γ . (22) \begin{aligned}
|u^\nu(s,x)\cdot\nabla d(x)|
&= |(u^\nu(s,x)-u^\nu(s,y))\cdot\nabla d(x)| \le|u^\nu(s,x)-u^\nu(s,y)| \\
&\le[u^\nu(s)]_{C^{0,\gamma}}|x-y|^\gamma\le C\|u^\nu(s)\|_{W^{1,2/(1-\gamma)}(\Omega)}d(x)^\gamma\\
&\le C\|\omega^\nu(s)\|_{L^{2/(1-\gamma)}(\Omega)}d(x)^\gamma\le C\|\omega^\nu(s)\|_{L^\infty(\Omega)}d(x)^\gamma\\
&\le C(\nu s)^{-1/r}X(\tau_\nu)d(x)^\gamma. \tag*{(22)}
\end{aligned} ∣ u ν ( s , x ) ⋅ ∇ d ( x ) ∣ = ∣ ( u ν ( s , x ) − u ν ( s , y )) ⋅ ∇ d ( x ) ∣ ≤ ∣ u ν ( s , x ) − u ν ( s , y ) ∣ ≤ [ u ν ( s ) ] C 0 , γ ∣ x − y ∣ γ ≤ C ∥ u ν ( s ) ∥ W 1 , 2/ ( 1 − γ ) ( Ω ) d ( x ) γ ≤ C ∥ ω ν ( s ) ∥ L 2/ ( 1 − γ ) ( Ω ) d ( x ) γ ≤ C ∥ ω ν ( s ) ∥ L ∞ ( Ω ) d ( x ) γ ≤ C ( ν s ) − 1/ r X ( τ ν ) d ( x ) γ . (22) By the stopping condition, we have
ν γ / ( 1 + γ ) B ( s ) 1 / ( 1 + γ ) ≤ C 1 / ( 1 + γ ) ( ν γ − 1 r − ( 1 + γ ) σ X ( τ ν ) ) 1 / ( 1 + γ ) ν σ s − 1 / ( r + r γ ) ≤ C ν σ s − 1 / ( r + r γ ) . \nu^{\gamma/(1+\gamma)}B(s)^{1/(1+\gamma)}
\le C^{1/(1+\gamma)}
\left(\nu^{\gamma-\frac{1}{r}-(1+\gamma)\sigma}X(\tau_\nu)\right)^{1/(1+\gamma)}
\nu^\sigma s^{-1/(r+r\gamma)}
\le C\nu^\sigma s^{-1/(r+r\gamma)}. ν γ / ( 1 + γ ) B ( s ) 1/ ( 1 + γ ) ≤ C 1/ ( 1 + γ ) ( ν γ − r 1 − ( 1 + γ ) σ X ( τ ν ) ) 1/ ( 1 + γ ) ν σ s − 1/ ( r + r γ ) ≤ C ν σ s − 1/ ( r + r γ ) . Therefore, we get with Lemma 4.1 for convex β ∈ C ( R ; [ 0 , ∞ ) ) \beta\in C(\mathbb{R};[0,\infty)) β ∈ C ( R ; [ 0 , ∞ ))
∫ Ω β ( ω ν ( t ) ) d x ≤ ∫ Ω β ( ω ν ( s ) ) d x + C ∫ s t ( ν t − z + C ν σ z − 1 / ( r + r γ ) ) ∫ ∂ Ω β ( ω ν ( z ) ) d S d z , 0 ≤ s < t ≤ τ ν . (23) \int_{\Omega}\beta(\omega^\nu(t))\,dx \leq\int_{\Omega}\beta(\omega^\nu(s))\,dx + C\int_s^t\left(\sqrt{\frac{\nu}{t-z}}+C\nu^\sigma z^{-1/(r+r\gamma)}\right)\int_{\partial\Omega}\beta(\omega^\nu(z))\,dS\,dz,\quad0\leq s<t\leq\tau_\nu. \tag*{(23)} ∫ Ω β ( ω ν ( t )) d x ≤ ∫ Ω β ( ω ν ( s )) d x + C ∫ s t ( t − z ν + C ν σ z − 1/ ( r + r γ ) ) ∫ ∂ Ω β ( ω ν ( z )) d S d z , 0 ≤ s < t ≤ τ ν . (23) For the boundary integral, we later want the interpolation estimates of Lemma 5.1. Therefore, we start with an L 1 ( Ω ) L^1(\Omega) L 1 ( Ω ) β = ∣ ⋅ ∣ \beta=|\cdot| β = ∣ ⋅ ∣ m = 1 m=1 m = 1 t ∈ ( 0 , τ ν ] t\in(0,\tau_\nu] t ∈ ( 0 , τ ν ]
∥ ω ν ( t ) ∥ L 1 ( Ω ) ≤ ∥ ω ν ( 0 ) ∥ L 1 ( Ω ) + C ∫ 0 t ( ν t − z + C ν σ z − 1 / ( r + r γ ) ) ∥ ω ν ( z ) ∥ L 1 ( ∂ Ω ) d z ≤ ∥ ω ν ( 0 ) ∥ L 1 ( Ω ) + C ( ν 1 / 2 − q + ν σ − q ) sup s ∈ [ 0 , τ ν ] ∥ ω ν ( s ) ∥ L 1 ( Ω ) , \begin{aligned}
\|\omega^\nu(t)\|_{L^1(\Omega)}\leq\|\omega^\nu(0)\|_{L^1(\Omega)}+C\int_0^t\left(\sqrt{\frac{\nu}{t-z}}+C\nu^\sigma z^{-1/(r+r\gamma)}\right)\|\omega^\nu(z)\|_{L^1(\partial\Omega)}\,dz \\
&\leq\|\omega^\nu(0)\|_{L^1(\Omega)}+C\left(\nu^{1/2-q}+\nu^{\sigma-q}\right)\sup_{s\in[0,\tau_\nu]}\|\omega^\nu(s)\|_{L^1(\Omega)},
\end{aligned} ∥ ω ν ( t ) ∥ L 1 ( Ω ) ≤ ∥ ω ν ( 0 ) ∥ L 1 ( Ω ) + C ∫ 0 t ( t − z ν + C ν σ z − 1/ ( r + r γ ) ) ∥ ω ν ( z ) ∥ L 1 ( ∂ Ω ) d z ≤ ∥ ω ν ( 0 ) ∥ L 1 ( Ω ) + C ( ν 1/2 − q + ν σ − q ) s ∈ [ 0 , τ ν ] sup ∥ ω ν ( s ) ∥ L 1 ( Ω ) , which yields
sup s ∈ [ 0 , τ ν ] ∥ ω ν ( s ) ∥ L 1 ( Ω ) ≤ C ∥ ω ν ( 0 ) ∥ L 1 ( Ω ) . (24) \sup_{s\in[0,\tau_\nu]}\|\omega^\nu(s)\|_{L^1(\Omega)}\leq C\|\omega^\nu(0)\|_{L^1(\Omega)}. \tag*{(24)} s ∈ [ 0 , τ ν ] sup ∥ ω ν ( s ) ∥ L 1 ( Ω ) ≤ C ∥ ω ν ( 0 ) ∥ L 1 ( Ω ) . (24) by choosing ν 0 > 0 \nu_0>0 ν 0 > 0
C ( ν 1 / 2 − q + ν σ − q ) ≤ 1 2 for all ν ∈ ( 0 , ν 0 ) . C\left(\nu^{1/2-q}+\nu^{\sigma-q}\right)\leq\frac{1}{2}\quad\text{for all }\nu\in(0,\nu_0). C ( ν 1/2 − q + ν σ − q ) ≤ 2 1 for all ν ∈ ( 0 , ν 0 ) . Next, we want to examine (5.5) for β = ∣ ⋅ ∣ r \beta=|\cdot|^r β = ∣ ⋅ ∣ r (24) to estimate the boundary integral
∥ ω ν ( s ) ∥ L r ( ∂ Ω ) ≤ ∥ 2 κ − α ν ∥ L ∞ ( ∂ Ω ) ∥ K Ω ( ω ν ( s ) ) ⋅ τ ∥ L r ( ∂ Ω ) ≤ C ν − r q ∥ ω ν ( s ) ∥ L 1 ( Ω ) ∥ ω ν ( s ) ∥ L r ( Ω ) ≤ C ν − r q ∥ ω ν ( 0 ) ∥ L 1 ( Ω ) r / 2 ∥ ω ν ( s ) ∥ L r ( Ω ) r / 2 . \begin{aligned}
\|\omega^\nu(s)\|_{L^r(\partial\Omega)}\leq\|2\kappa-\alpha^\nu\|_{L^\infty(\partial\Omega)}\|K_\Omega(\omega^\nu(s))\cdot\tau\|_{L^r(\partial\Omega)}
\leq C\nu^{-rq}\|\omega^\nu(s)\|_{L^1(\Omega)}\|\omega^\nu(s)\|_{L^r(\Omega)} \\
&\leq C\nu^{-rq}\|\omega^\nu(0)\|_{L^1(\Omega)}^{r/2}\|\omega^\nu(s)\|_{L^r(\Omega)}^{r/2}.
\end{aligned} ∥ ω ν ( s ) ∥ L r ( ∂ Ω ) ≤ ∥2 κ − α ν ∥ L ∞ ( ∂ Ω ) ∥ K Ω ( ω ν ( s )) ⋅ τ ∥ L r ( ∂ Ω ) ≤ C ν − r q ∥ ω ν ( s ) ∥ L 1 ( Ω ) ∥ ω ν ( s ) ∥ L r ( Ω ) ≤ C ν − r q ∥ ω ν ( 0 ) ∥ L 1 ( Ω ) r /2 ∥ ω ν ( s ) ∥ L r ( Ω ) r /2 . Importantly, the estimate is sublinear in ∥ ω ν ∥ L r ( Ω ) \|\omega^\nu\|_{L^r(\Omega)} ∥ ω ν ∥ L r ( Ω ) t ∈ ( 0 , τ ν ] t\in(0,\tau_\nu] t ∈ ( 0 , τ ν ] s ∈ ( 0 , t ] s\in(0,t] s ∈ ( 0 , t ]
∥ ω ν ( s ) ∥ L r ( Ω ) r ≤ ∥ ω ν ( 0 ) ∥ L r ( Ω ) r + C ν − r q ∥ ω ν ( 0 ) ∥ L 1 ( Ω ) r / 2 sup s ∈ [ 0 , t ] ∥ ω ν ( s ) ∥ L r ( Ω ) r / 2 ∫ 0 s ( ν s − z + C ν σ z − 1 / ( r + r γ ) ) d z ≤ ∥ ω ν ( 0 ) ∥ L r ( Ω ) r + C ∥ ω ν ( 0 ) ∥ L 1 ( Ω ) r / 2 sup s ∈ [ 0 , t ] ∥ ω ν ( s ) ∥ L r ( Ω ) r / 2 ( ν 1 / 2 − r q t 1 / 2 + ν σ − r q t 1 − 1 / ( r + r γ ) ) ≤ ∥ ω ν ( 0 ) ∥ L r ( Ω ) r + C ∥ ω ν ( 0 ) ∥ L 1 ( Ω ) r / 2 ( ν 1 / 2 − r q t 1 / 2 + ν σ − r q t 1 − 1 / ( r + r γ ) ) 2 + 1 2 sup s ∈ [ 0 , t ] ∥ ω ν ( s ) ∥ L r ( Ω ) r . \begin{aligned}
\|\omega^\nu(s)\|_{L^r(\Omega)}^r \\
&\leq\|\omega^\nu(0)\|_{L^r(\Omega)}^r+C\nu^{-rq}\|\omega^\nu(0)\|_{L^1(\Omega)}^{r/2}\sup_{s\in[0,t]}\|\omega^\nu(s)\|_{L^r(\Omega)}^{r/2}\int_0^s\left(\sqrt{\frac{\nu}{s-z}}+C\nu^\sigma z^{-1/(r+r\gamma)}\right)\,dz \\
&\leq\|\omega^\nu(0)\|_{L^r(\Omega)}^r+C\|\omega^\nu(0)\|_{L^1(\Omega)}^{r/2}\sup_{s\in[0,t]}\|\omega^\nu(s)\|_{L^r(\Omega)}^{r/2}\left(\nu^{1/2-rq}t^{1/2}+\nu^{\sigma-rq}t^{1-1/(r+r\gamma)}\right) \\
&\leq\|\omega^\nu(0)\|_{L^r(\Omega)}^r+C\|\omega^\nu(0)\|_{L^1(\Omega)}^{r/2}\left(\nu^{1/2-rq}t^{1/2}+\nu^{\sigma-rq}t^{1-1/(r+r\gamma)}\right)^2+\frac{1}{2}\sup_{s\in[0,t]}\|\omega^\nu(s)\|_{L^r(\Omega)}^r.
\end{aligned} ∥ ω ν ( s ) ∥ L r ( Ω ) r ≤ ∥ ω ν ( 0 ) ∥ L r ( Ω ) r + C ν − r q ∥ ω ν ( 0 ) ∥ L 1 ( Ω ) r /2 s ∈ [ 0 , t ] sup ∥ ω ν ( s ) ∥ L r ( Ω ) r /2 ∫ 0 s ( s − z ν + C ν σ z − 1/ ( r + r γ ) ) d z ≤ ∥ ω ν ( 0 ) ∥ L r ( Ω ) r + C ∥ ω ν ( 0 ) ∥ L 1 ( Ω ) r /2 s ∈ [ 0 , t ] sup ∥ ω ν ( s ) ∥ L r ( Ω ) r /2 ( ν 1/2 − r q t 1/2 + ν σ − r q t 1 − 1/ ( r + r γ ) ) ≤ ∥ ω ν ( 0 ) ∥ L r ( Ω ) r + C ∥ ω ν ( 0 ) ∥ L 1 ( Ω ) r /2 ( ν 1/2 − r q t 1/2 + ν σ − r q t 1 − 1/ ( r + r γ ) ) 2 + 2 1 s ∈ [ 0 , t ] sup ∥ ω ν ( s ) ∥ L r ( Ω ) r . Recall that r q < 1 2 rq < \frac{1}{2} r q < 2 1 1 r + r γ < 1 2 \frac{1}{r+r\gamma} < \frac{1}{2} r + r γ 1 < 2 1 1 + ν 2 ( σ − r q ) ≤ 2 ν − 2 max { r q − σ , 0 } 1+\nu^{2(\sigma-rq)} \le2\nu^{-2\max\{rq-\sigma,0\}} 1 + ν 2 ( σ − r q ) ≤ 2 ν − 2 m a x { r q − σ , 0 } (21) on [ 0 , τ ν ] [0,\tau_\nu] [ 0 , τ ν ]
sup s ∈ [ 0 , τ ν ] ∥ ω ν ( s ) ∥ L r ( Ω ) r ≤ C ∥ ω ν ( 0 ) ∥ L r ( Ω ) r ν − 2 max { r q − σ , 0 } \sup_{s\in[0,\tau_\nu]}\|\omega^\nu(s)\|_{L^r(\Omega)}^r \le C\|\omega^\nu(0)\|_{L^r(\Omega)}^r\nu^{-2\max\{rq-\sigma,0\}} s ∈ [ 0 , τ ν ] sup ∥ ω ν ( s ) ∥ L r ( Ω ) r ≤ C ∥ ω ν ( 0 ) ∥ L r ( Ω ) r ν − 2 m a x { r q − σ , 0 } As previously mentioned, we need an uniform-in-small-viscosity bound in L r ( Ω ) L^r(\Omega) L r ( Ω ) [ 0 , min { ν , τ ν } ] [0,\min\{\nu,\tau_\nu\}] [ 0 , min { ν , τ ν }] L ∞ ( Ω ) L^\infty(\Omega) L ∞ ( Ω ) t ≤ ν t\le\nu t ≤ ν
(5.7)
sup s ∈ [ 0 , min { ν , τ ν } ] ∥ ω ν ( s ) ∥ L r ( Ω ) r ≤ C ∥ ω ν ( 0 ) ∥ L r ( Ω ) r [ 1 + ( ν 1 − r q + ν σ + 1 − 1 r + r γ − r q ) 2 ] ≤ C ∥ ω ν ( 0 ) ∥ L r ( Ω ) r . (25) \sup_{s\in[0,\min\{\nu,\tau_\nu\}]}\|\omega^\nu(s)\|_{L^r(\Omega)}^r \le C\|\omega^\nu(0)\|_{L^r(\Omega)}^r\left[1+\left(\nu^{1-rq}+\nu^{\sigma+1-\frac{1}{r+r\gamma}-rq}\right)^2\right]\le C\|\omega^\nu(0)\|_{L^r(\Omega)}^r . \tag*{(25)} s ∈ [ 0 , m i n { ν , τ ν }] sup ∥ ω ν ( s ) ∥ L r ( Ω ) r ≤ C ∥ ω ν ( 0 ) ∥ L r ( Ω ) r [ 1 + ( ν 1 − r q + ν σ + 1 − r + r γ 1 − r q ) 2 ] ≤ C ∥ ω ν ( 0 ) ∥ L r ( Ω ) r . (25) To prove the L ∞ ( Ω ) L^\infty(\Omega) L ∞ ( Ω ) a ν a^\nu a ν
{ ∂ t a ν + u ν ⋅ ∇ a ν = ν Δ a ν in ( s , t ) × Ω , a ν = 0 on ( s , t ) × ∂ Ω , a ν ( s ) = ω ν ( s ) in Ω , \begin{cases}
\partial_ta^\nu+u^\nu\cdot\nabla a^\nu=\nu\Delta a^\nu& \text{in }(s,t)\times\Omega,\\
a^\nu=0 & \text{on }(s,t)\times\partial\Omega,\quad a^\nu(s)=\omega^\nu(s)\text{ in }\Omega,
\end{cases} { ∂ t a ν + u ν ⋅ ∇ a ν = ν Δ a ν a ν = 0 in ( s , t ) × Ω , on ( s , t ) × ∂ Ω , a ν ( s ) = ω ν ( s ) in Ω , and b ν b^\nu b ν
{ ∂ t b ν + u ν ⋅ ∇ b ν = ν Δ b ν in ( s , t ) × Ω , b ν = ( 2 κ − α ν ) u ν ⋅ τ on ( s , t ) × ∂ Ω , b ν ( s ) = 0 in Ω . \begin{cases}
\partial_tb^\nu+u^\nu\cdot\nabla b^\nu=\nu\Delta b^\nu& \text{in }(s,t)\times\Omega,\\
b^\nu=(2\kappa-\alpha^\nu)u^\nu\cdot\tau& \text{on }(s,t)\times\partial\Omega,\quad b^\nu(s)=0\text{ in }\Omega.
\end{cases} { ∂ t b ν + u ν ⋅ ∇ b ν = ν Δ b ν b ν = ( 2 κ − α ν ) u ν ⋅ τ in ( s , t ) × Ω , on ( s , t ) × ∂ Ω , b ν ( s ) = 0 in Ω. Then, by linearity and uniqueness for the advection-diffusion problem with drift u ν u^\nu u ν ω ν = a ν + b ν \omega^\nu=a^\nu+b^\nu ω ν = a ν + b ν [ s , t ] [s,t] [ s , t ] a ν a^\nu a ν Lemma 4.3 and b ν b^\nu b ν s ∈ ( 0 , T ] s\in(0,T] s ∈ ( 0 , T ] s / 2 s/2 s /2
∥ ω ν ( s ) ∥ L ∞ ( Ω ) ≤ ∥ a ν ( s ) ∥ L ∞ ( Ω ) + ∥ b ν ( s ) ∥ L ∞ ( Ω ) ≤ C ( ν s 2 ) − 1 / r ∥ a ν ( s / 2 ) ∥ L r ( Ω ) + sup z ∈ [ s / 2 , s ] ∥ ( 2 κ − α ν ( z ) ) u ν ⋅ τ ∥ L ∞ ( ∂ Ω ) ≤ C ( ν s ) − 1 / r ∥ ω ν ( s / 2 ) ∥ L r ( Ω ) + C ν − q sup z ∈ [ s / 2 , s ] ∥ K Ω ( ω ν ( z ) ) ⋅ τ ∥ L ∞ ( ∂ Ω ) ≤ C ( ν s ) − 1 / r ∥ ω ν ( s / 2 ) ∥ L r ( Ω ) + C ν − q sup z ∈ [ s / 2 , s ] ∥ ω ν ( z ) ∥ L r ( Ω ) r / 2 ∥ ω ν ( z ) ∥ L ∞ ( Ω ) 1 − r / 2 . \begin{aligned}
\|\omega^\nu(s)\|_{L^\infty(\Omega)}
&\le\|a^\nu(s)\|_{L^\infty(\Omega)}+\|b^\nu(s)\|_{L^\infty(\Omega)}\\
&\le C\left(\nu\frac{s}{2}\right)^{-1/r}\|a^\nu(s/2)\|_{L^r(\Omega)}
+\sup_{z\in[s/2,s]}\|(2\kappa-\alpha^\nu(z))u^\nu\cdot\tau\|_{L^\infty(\partial\Omega)}\\
&\le C(\nu s)^{-1/r}\|\omega^\nu(s/2)\|_{L^r(\Omega)}
+C\nu^{-q}\sup_{z\in[s/2,s]}\|K_\Omega(\omega^\nu(z))\cdot\tau\|_{L^\infty(\partial\Omega)}\\
&\le C(\nu s)^{-1/r}\|\omega^\nu(s/2)\|_{L^r(\Omega)}
+C\nu^{-q}\sup_{z\in[s/2,s]}\|\omega^\nu(z)\|_{L^r(\Omega)}^{r/2}\|\omega^\nu(z)\|_{L^\infty(\Omega)}^{1-r/2}.
\end{aligned} ∥ ω ν ( s ) ∥ L ∞ ( Ω ) ≤ ∥ a ν ( s ) ∥ L ∞ ( Ω ) + ∥ b ν ( s ) ∥ L ∞ ( Ω ) ≤ C ( ν 2 s ) − 1/ r ∥ a ν ( s /2 ) ∥ L r ( Ω ) + z ∈ [ s /2 , s ] sup ∥ ( 2 κ − α ν ( z )) u ν ⋅ τ ∥ L ∞ ( ∂ Ω ) ≤ C ( ν s ) − 1/ r ∥ ω ν ( s /2 ) ∥ L r ( Ω ) + C ν − q z ∈ [ s /2 , s ] sup ∥ K Ω ( ω ν ( z )) ⋅ τ ∥ L ∞ ( ∂ Ω ) ≤ C ( ν s ) − 1/ r ∥ ω ν ( s /2 ) ∥ L r ( Ω ) + C ν − q z ∈ [ s /2 , s ] sup ∥ ω ν ( z ) ∥ L r ( Ω ) r /2 ∥ ω ν ( z ) ∥ L ∞ ( Ω ) 1 − r /2 . With Lemma 5.1, we get for t ∈ ( 0 , min { ν , τ ν } ] t \in(0,\min\{\nu,\tau_\nu\}] t ∈ ( 0 , min { ν , τ ν }]
X ( t ) = sup s ∈ ( 0 , t ] ( ν s ) 1 / r ∥ ω ν ( s ) ∥ L ∞ ( Ω ) ≤ C sup s ∈ ( 0 , t ] ∥ ω ν ( s / 2 ) ∥ L r ( Ω ) + C ν − q sup s ∈ ( 0 , t ] ( ν s ) 1 / r ∥ ω ν ( s ) ∥ L r ( Ω ) ∥ ω ν ( s ) ∥ L ∞ ( Ω ) 1 − r / 2 ≤ C ∥ ω ν ( 0 ) ∥ L r ( Ω ) + C ν − q ∥ ω ν ( 0 ) ∥ L r ( Ω ) r / 2 ( ν t ) 1 / 2 ( sup s ∈ ( 0 , t ] ( ν s ) 1 / r ∥ ω ν ( s ) ∥ L ∞ ( Ω ) ) 1 − r / 2 ≤ C ∥ ω ν ( 0 ) ∥ L r ( Ω ) + C ν 1 − q ∥ ω ν ( 0 ) ∥ L r ( Ω ) r / 2 ( X ( t ) ) 1 − r / 2 ≤ C ∥ ω ν ( 0 ) ∥ L r ( Ω ) + C ν 2 ( 1 − q ) / r ∥ ω ν ( 0 ) ∥ L r ( Ω ) + 1 2 X ( t ) , \begin{aligned}
X(t) &= \sup_{s\in(0,t]}(\nu s)^{1/r}\|\omega^\nu(s)\|_{L^\infty(\Omega)} \\
&\le C\sup_{s\in(0,t]}\|\omega^\nu(s/2)\|_{L^r(\Omega)}+C\nu^{-q}\sup_{s\in(0,t]}(\nu s)^{1/r}\|\omega^\nu(s)\|_{L^r(\Omega)}\|\omega^\nu(s)\|_{L^\infty(\Omega)}^{1-r/2}\\
&\le C\|\omega^\nu(0)\|_{L^r(\Omega)}+C\nu^{-q}\|\omega^\nu(0)\|_{L^r(\Omega)}^{r/2}(\nu t)^{1/2}\left(\sup_{s\in(0,t]}(\nu s)^{1/r}\|\omega^\nu(s)\|_{L^\infty(\Omega)}\right)^{1-r/2}\\
&\le C\|\omega^\nu(0)\|_{L^r(\Omega)}+C\nu^{1-q}\|\omega^\nu(0)\|_{L^r(\Omega)}^{r/2}(X(t))^{1-r/2}\\
&\le C\|\omega^\nu(0)\|_{L^r(\Omega)}+C\nu^{2(1-q)/r}\|\omega^\nu(0)\|_{L^r(\Omega)}+\frac{1}{2}X(t),
\end{aligned} X ( t ) = s ∈ ( 0 , t ] sup ( ν s ) 1/ r ∥ ω ν ( s ) ∥ L ∞ ( Ω ) ≤ C s ∈ ( 0 , t ] sup ∥ ω ν ( s /2 ) ∥ L r ( Ω ) + C ν − q s ∈ ( 0 , t ] sup ( ν s ) 1/ r ∥ ω ν ( s ) ∥ L r ( Ω ) ∥ ω ν ( s ) ∥ L ∞ ( Ω ) 1 − r /2 ≤ C ∥ ω ν ( 0 ) ∥ L r ( Ω ) + C ν − q ∥ ω ν ( 0 ) ∥ L r ( Ω ) r /2 ( ν t ) 1/2 ( s ∈ ( 0 , t ] sup ( ν s ) 1/ r ∥ ω ν ( s ) ∥ L ∞ ( Ω ) ) 1 − r /2 ≤ C ∥ ω ν ( 0 ) ∥ L r ( Ω ) + C ν 1 − q ∥ ω ν ( 0 ) ∥ L r ( Ω ) r /2 ( X ( t ) ) 1 − r /2 ≤ C ∥ ω ν ( 0 ) ∥ L r ( Ω ) + C ν 2 ( 1 − q ) / r ∥ ω ν ( 0 ) ∥ L r ( Ω ) + 2 1 X ( t ) , which yields
X ( t ) ≤ C ∥ ω ν ( 0 ) ∥ L r ( Ω ) , t ∈ ( 0 , min { ν , τ ν } ] . X(t)\le C\|\omega^\nu(0)\|_{L^r(\Omega)},\qquad t\in(0,\min\{\nu,\tau_\nu\}]. X ( t ) ≤ C ∥ ω ν ( 0 ) ∥ L r ( Ω ) , t ∈ ( 0 , min { ν , τ ν }] . If τ ν ≥ ν \tau_\nu\ge\nu τ ν ≥ ν ν / 2 \nu/2 ν /2
sup s ∈ [ ν / 2 , τ ν ] ∥ b ν ( s ) ∥ L ∞ ( Ω ) ≤ C ν − q sup s ∈ [ ν / 2 , τ ν ] ∥ K Ω ( ω ν ( s ) ) ⋅ τ ∥ L ∞ ( ∂ Ω ) ≤ C ν − q ( sup s ∈ [ ν / 2 , τ ν ] ∥ K Ω ( a ν ( s ) ) ⋅ τ ∥ L ∞ ( ∂ Ω ) + sup s ∈ [ ν / 2 , τ ν ] ∥ K Ω ( b ν ( s ) ) ⋅ τ ∥ L ∞ ( ∂ Ω ) ) ≤ C ν − q ( sup s ∈ [ ν / 2 , τ ν ] ∥ a ν ( s ) ∥ L r ( Ω ) r / 2 ∥ a ν ( s ) ∥ L ∞ ( Ω ) 1 − r / 2 + sup s ∈ [ ν / 2 , τ ν ] ∥ b ν ( s ) ∥ L 1 ( Ω ) 1 / 2 ∥ b ν ( s ) ∥ L ∞ ( Ω ) 1 / 2 ) ≤ C ν − q ( ∥ ω ν ( 0 ) ∥ L r ( Ω ) r / 2 ∥ ω ν ( ν / 2 ) ∥ L ∞ ( Ω ) 1 − r / 2 + sup s ∈ [ ν / 2 , τ ν ] ( ∥ b ν ( s ) ∥ L 1 ( Ω ) + ∥ a ν ( s ) ∥ L 1 ( Ω ) ) 1 / 2 ∥ b ν ( s ) ∥ L ∞ ( Ω ) 1 / 2 ) ≤ C ν − q ( ∥ ω ν ( 0 ) ∥ L r ( Ω ) r / 2 ( ( ν 2 ) − 1 / r X ( ν / 2 ) ) 1 − r / 2 + ∥ ω ν ( 0 ) ∥ L 1 ( Ω ) 1 / 2 sup s ∈ [ ν / 2 , τ ν ] ∥ b ν ( s ) ∥ L ∞ ( Ω ) 1 / 2 ) ≤ C ν 1 − 2 / r − q ∥ ω ν ( 0 ) ∥ L r ( Ω ) + C ν − q ∥ ω ν ( 0 ) ∥ L 1 ( Ω ) 1 / 2 ( sup s ∈ [ ν / 2 , τ ν ] ∥ b ν ( s ) ∥ L ∞ ( Ω ) ) 1 / 2 . \begin{aligned}
\sup_{s\in[\nu/2,\tau_\nu]}\|b^\nu(s)\|_{L^\infty(\Omega)}
&\le C\nu^{-q}\sup_{s\in[\nu/2,\tau_\nu]}\|K_\Omega(\omega^\nu(s))\cdot\tau\|_{L^\infty(\partial\Omega)}\\
&\le C\nu^{-q}\left(\sup_{s\in[\nu/2,\tau_\nu]}\|K_\Omega(a^\nu(s))\cdot\tau\|_{L^\infty(\partial\Omega)}+\sup_{s\in[\nu/2,\tau_\nu]}\|K_\Omega(b^\nu(s))\cdot\tau\|_{L^\infty(\partial\Omega)}\right)\\
&\le C\nu^{-q}\left(\sup_{s\in[\nu/2,\tau_\nu]}\|a^\nu(s)\|_{L^r(\Omega)}^{r/2}\|a^\nu(s)\|_{L^\infty(\Omega)}^{1-r/2}+\sup_{s\in[\nu/2,\tau_\nu]}\|b^\nu(s)\|_{L^1(\Omega)}^{1/2}\|b^\nu(s)\|_{L^\infty(\Omega)}^{1/2}\right)\\
&\le C\nu^{-q}\left(\|\omega^\nu(0)\|_{L^r(\Omega)}^{r/2}\|\omega^\nu(\nu/2)\|_{L^\infty(\Omega)}^{1-r/2}+\sup_{s\in[\nu/2,\tau_\nu]}\left(\|b^\nu(s)\|_{L^1(\Omega)}+\|a^\nu(s)\|_{L^1(\Omega)}\right)^{1/2}\|b^\nu(s)\|_{L^\infty(\Omega)}^{1/2}\right)\\
&\le C\nu^{-q}\left(\|\omega^\nu(0)\|_{L^r(\Omega)}^{r/2}\left(\left(\frac{\nu}{2}\right)^{-1/r}X(\nu/2)\right)^{1-r/2}+\|\omega^\nu(0)\|_{L^1(\Omega)}^{1/2}\sup_{s\in[\nu/2,\tau_\nu]}\|b^\nu(s)\|_{L^\infty(\Omega)}^{1/2}\right)\\
&\le C\nu^{1-2/r-q}\|\omega^\nu(0)\|_{L^r(\Omega)}+C\nu^{-q}\|\omega^\nu(0)\|_{L^1(\Omega)}^{1/2}\left(\sup_{s\in[\nu/2,\tau_\nu]}\|b^\nu(s)\|_{L^\infty(\Omega)}\right)^{1/2}.
\end{aligned} s ∈ [ ν /2 , τ ν ] sup ∥ b ν ( s ) ∥ L ∞ ( Ω ) ≤ C ν − q s ∈ [ ν /2 , τ ν ] sup ∥ K Ω ( ω ν ( s )) ⋅ τ ∥ L ∞ ( ∂ Ω ) ≤ C ν − q ( s ∈ [ ν /2 , τ ν ] sup ∥ K Ω ( a ν ( s )) ⋅ τ ∥ L ∞ ( ∂ Ω ) + s ∈ [ ν /2 , τ ν ] sup ∥ K Ω ( b ν ( s )) ⋅ τ ∥ L ∞ ( ∂ Ω ) ) ≤ C ν − q ( s ∈ [ ν /2 , τ ν ] sup ∥ a ν ( s ) ∥ L r ( Ω ) r /2 ∥ a ν ( s ) ∥ L ∞ ( Ω ) 1 − r /2 + s ∈ [ ν /2 , τ ν ] sup ∥ b ν ( s ) ∥ L 1 ( Ω ) 1/2 ∥ b ν ( s ) ∥ L ∞ ( Ω ) 1/2 ) ≤ C ν − q ( ∥ ω ν ( 0 ) ∥ L r ( Ω ) r /2 ∥ ω ν ( ν /2 ) ∥ L ∞ ( Ω ) 1 − r /2 + s ∈ [ ν /2 , τ ν ] sup ( ∥ b ν ( s ) ∥ L 1 ( Ω ) + ∥ a ν ( s ) ∥ L 1 ( Ω ) ) 1/2 ∥ b ν ( s ) ∥ L ∞ ( Ω ) 1/2 ) ≤ C ν − q ( ∥ ω ν ( 0 ) ∥ L r ( Ω ) r /2 ( ( 2 ν ) − 1/ r X ( ν /2 ) ) 1 − r /2 + ∥ ω ν ( 0 ) ∥ L 1 ( Ω ) 1/2 s ∈ [ ν /2 , τ ν ] sup ∥ b ν ( s ) ∥ L ∞ ( Ω ) 1/2 ) ≤ C ν 1 − 2/ r − q ∥ ω ν ( 0 ) ∥ L r ( Ω ) + C ν − q ∥ ω ν ( 0 ) ∥ L 1 ( Ω ) 1/2 ( s ∈ [ ν /2 , τ ν ] sup ∥ b ν ( s ) ∥ L ∞ ( Ω ) ) 1/2 . Hence, we have with Young’s inequality
sup s ∈ [ ν / 2 , τ ν ] ∥ b ν ( s ) ∥ L ∞ ( Ω ) ≤ C ( ν 1 − 2 / r − q + ν − 2 q ) ∥ ω ν ( 0 ) ∥ L r ( Ω ) . \sup_{s\in[\nu/2,\tau_\nu]}\|b^\nu(s)\|_{L^\infty(\Omega)}
\le C\left(\nu^{1-2/r-q}+\nu^{-2q}\right)\|\omega^\nu(0)\|_{L^r(\Omega)}. s ∈ [ ν /2 , τ ν ] sup ∥ b ν ( s ) ∥ L ∞ ( Ω ) ≤ C ( ν 1 − 2/ r − q + ν − 2 q ) ∥ ω ν ( 0 ) ∥ L r ( Ω ) . For t ∈ [ ν , τ ν ] t \in[\nu,\tau_\nu] t ∈ [ ν , τ ν ] t − ν / 2 ≥ t / 2 t-\nu/2 \ge t/2 t − ν /2 ≥ t /2
( ν t ) 1 / r ∥ ω ν ( t ) ∥ L ∞ ( Ω ) ≤ ( ν t ) 1 / r ( C ( ν ( t − ν 2 ) ) − 1 / r ∥ ω ν ( ν / 2 ) ∥ L r ( Ω ) + ∥ b ν ( t ) ∥ L ∞ ( Ω ) ) ≤ C [ 1 + ν 1 − 1 / r − q + ν 1 / r − 2 q ] ∥ ω ν ( 0 ) ∥ L r ( Ω ) ≤ C ∥ ω ν ( 0 ) ∥ L r ( Ω ) . \begin{aligned}
(\nu t)^{1/r}\lVert\omega^\nu(t)\rVert_{L^\infty(\Omega)}
\leq(\nu t)^{1/r}\left(C\left(\nu\left(t-\frac{\nu}{2}\right)\right)^{-1/r}\lVert\omega^\nu(\nu/2)\rVert_{L^r(\Omega)}+\lVert b^\nu(t)\rVert_{L^\infty(\Omega)}\right) \\
&\leq C\left[1+\nu^{1-1/r-q}+\nu^{1/r-2q}\right]\lVert\omega^\nu(0)\rVert_{L^r(\Omega)} \\
&\leq C\lVert\omega^\nu(0)\rVert_{L^r(\Omega)}.
\end{aligned} ( ν t ) 1/ r ∥ ω ν ( t ) ∥ L ∞ ( Ω ) ≤ ( ν t ) 1/ r ( C ( ν ( t − 2 ν ) ) − 1/ r ∥ ω ν ( ν /2 ) ∥ L r ( Ω ) + ∥ b ν ( t ) ∥ L ∞ ( Ω ) ) ≤ C [ 1 + ν 1 − 1/ r − q + ν 1/ r − 2 q ] ∥ ω ν ( 0 ) ∥ L r ( Ω ) ≤ C ∥ ω ν ( 0 ) ∥ L r ( Ω ) . Recall that q < r − 1 2 r q<\frac{r-1}{2r} q < 2 r r − 1 r q < 1 2 rq<\frac{1}{2} r q < 2 1
X ( t ) ≤ C ∥ ω ν ( 0 ) ∥ L r ( Ω ) , t ∈ [ 0 , τ ν ] . X(t)\leq C\lVert\omega^\nu(0)\rVert_{L^r(\Omega)},\qquad t\in[0,\tau_\nu]. X ( t ) ≤ C ∥ ω ν ( 0 ) ∥ L r ( Ω ) , t ∈ [ 0 , τ ν ] . and thus the estimates (21) on ( 0 , τ ν ] (0,\tau_\nu] ( 0 , τ ν ] τ ν < T \tau_\nu<T τ ν < T τ ν < T \tau_\nu<T τ ν < T
ν γ − 1 r − ( 1 + γ ) σ X ( τ ν ) = 1. (26) \nu^{\gamma-\frac{1}{r}-(1+\gamma)\sigma}X(\tau_\nu)=1. \tag*{(26)} ν γ − r 1 − ( 1 + γ ) σ X ( τ ν ) = 1. (26) With the estimates (21) on ( 0 , τ ν ] (0,\tau_\nu] ( 0 , τ ν ] ν 0 \nu_0 ν 0
C ∥ curl u 0 ∥ L r ( Ω ) ν 0 γ − 1 r − ( 1 + γ ) σ ≤ 1 2 , C\lVert\operatorname{curl}u_0\rVert_{L^r(\Omega)}\nu_0^{\gamma-\frac{1}{r}-(1+\gamma)\sigma}\leq\frac{1}{2}, C ∥ curl u 0 ∥ L r ( Ω ) ν 0 γ − r 1 − ( 1 + γ ) σ ≤ 2 1 , thus contradicting (26) . Hence, we have τ ν = T \tau_\nu=T τ ν = T
When q = 0 q=0 q = 0
Corollary 5.3. Let ω 0 ∈ H 1 ( Ω ) ∩ L ∞ ( Ω ) \omega_0\in H^1(\Omega)\cap L^\infty(\Omega) ω 0 ∈ H 1 ( Ω ) ∩ L ∞ ( Ω ) u 0 = K Ω ( ω 0 ) ∈ W u_0=K_\Omega(\omega_0)\in W u 0 = K Ω ( ω 0 ) ∈ W ω ν = curl u ν \omega^\nu=\operatorname{curl}u^\nu ω ν = curl u ν u ν u^\nu u ν u ν ( 0 , ⋅ ) = u 0 u^\nu(0,\cdot)=u_0 u ν ( 0 , ⋅ ) = u 0 q = 0 q=0 q = 0 r ∈ ( 1 , ∞ ) r\in(1,\infty) r ∈ ( 1 , ∞ ) ν 0 > 0 \nu_0>0 ν 0 > 0 ∥ curl u 0 ∥ L r ( Ω ) \lVert\operatorname{curl}u_0\rVert_{L^r(\Omega)} ∥ curl u 0 ∥ L r ( Ω ) ν ∈ ( 0 , ν 0 ) \nu\in(0,\nu_0) ν ∈ ( 0 , ν 0 ) t ∈ ( 0 , T ] t\in(0,T] t ∈ ( 0 , T ]
∥ ω ν ( t ) ∥ L p ( Ω ) ≤ C ( ν t ) − ( 1 / r − 1 / p ) ∥ curl u 0 ∥ L r ( Ω ) , r ≤ p ≤ ∞ . \lVert\omega^\nu(t)\rVert_{L^p(\Omega)}
\leq C(\nu t)^{-(1/r-1/p)}\lVert\operatorname{curl}u_0\rVert_{L^r(\Omega)},\qquad r\leq p\leq\infty. ∥ ω ν ( t ) ∥ L p ( Ω ) ≤ C ( ν t ) − ( 1/ r − 1/ p ) ∥ curl u 0 ∥ L r ( Ω ) , r ≤ p ≤ ∞. Here 1 / ∞ = 0 1/\infty=0 1/∞ = 0 C > 0 C>0 C > 0 ν \nu ν
Proof. For r ∈ ( 1 , 2 ) r\in(1,2) r ∈ ( 1 , 2 ) r ∈ [ 2 , ∞ ) r\in[2,\infty) r ∈ [ 2 , ∞ )
m : = 2 r r + 1 ∈ ( 1 , 2 ) , so that 1 m − 1 2 = 1 2 r . m:=\frac{2r}{r+1}\in(1,2),\qquad\text{so that}\qquad\frac{1}{m}-\frac{1}{2}=\frac{1}{2r}. m := r + 1 2 r ∈ ( 1 , 2 ) , so that m 1 − 2 1 = 2 r 1 . Since q = 0 q=0 q = 0
∥ ω ν ∥ L r ( ∂ Ω ) ≤ C ∥ ω ν ∥ L 1 ( Ω ) 1 / 2 ∥ ω ν ∥ L r ( Ω ) 1 / 2 ≤ C ∥ ω ν ∥ L r ( Ω ) . \lVert\omega^\nu\rVert_{L^r(\partial\Omega)}
\leq C\lVert\omega^\nu\rVert_{L^1(\Omega)}^{1/2}\lVert\omega^\nu\rVert_{L^r(\Omega)}^{1/2}
\leq C\lVert\omega^\nu\rVert_{L^r(\Omega)}. ∥ ω ν ∥ L r ( ∂ Ω ) ≤ C ∥ ω ν ∥ L 1 ( Ω ) 1/2 ∥ ω ν ∥ L r ( Ω ) 1/2 ≤ C ∥ ω ν ∥ L r ( Ω ) . The argument of Lemma 5.2 holds for m m m σ ∈ ( 0 , r − 1 4 r ) \sigma\in\left(0,\frac{r-1}{4r}\right) σ ∈ ( 0 , 4 r r − 1 ) β = ∣ ⋅ ∣ r \beta=|\cdot|^r β = ∣ ⋅ ∣ r
sup s ∈ [ 0 , T ] ∥ ω ν ( s ) ∥ L r ( Ω ) r ≤ ∥ ω ν ( 0 ) ∥ L r ( Ω ) r + C ( ν 1 / 2 + ν σ ) sup s ∈ [ 0 , T ] ∥ ω ν ( s ) ∥ L r ( Ω ) r (27) \sup_{s\in[0,T]}\lVert\omega^\nu(s)\rVert_{L^r(\Omega)}^r
\leq\lVert\omega^\nu(0)\rVert_{L^r(\Omega)}^r
+C(\nu^{1/2}+\nu^\sigma)\sup_{s\in[0,T]}\lVert\omega^\nu(s)\rVert_{L^r(\Omega)}^r \tag*{(27)} s ∈ [ 0 , T ] sup ∥ ω ν ( s ) ∥ L r ( Ω ) r ≤ ∥ ω ν ( 0 ) ∥ L r ( Ω ) r + C ( ν 1/2 + ν σ ) s ∈ [ 0 , T ] sup ∥ ω ν ( s ) ∥ L r ( Ω ) r (27) By choosing ν 0 \nu_0 ν 0 C ( ν 1 / 2 + ν σ ) ≤ 1 2 C(\nu^{1/2}+\nu^\sigma)\le\frac{1}{2} C ( ν 1/2 + ν σ ) ≤ 2 1
sup s ∈ [ 0 , T ] ∥ ω ν ( s ) ∥ L r ( Ω ) ≤ 2 ∥ ω ν ( 0 ) ∥ L r ( Ω ) . \sup_{s\in[0,T]} \|\omega^\nu(s)\|_{L^r(\Omega)} \le2\|\omega^\nu(0)\|_{L^r(\Omega)} . s ∈ [ 0 , T ] sup ∥ ω ν ( s ) ∥ L r ( Ω ) ≤ 2∥ ω ν ( 0 ) ∥ L r ( Ω ) . Regarding the L ∞ ( Ω ) L^\infty(\Omega) L ∞ ( Ω ) s ∈ ( 0 , T ] s\in(0,T] s ∈ ( 0 , T ]
∥ K Ω ( ω ν ( s ) ) ⋅ τ ∥ L ∞ ( ∂ Ω ) ≤ C ∥ ω ν ( s ) ∥ L m ( Ω ) m / 2 ∥ ω ν ( s ) ∥ L ∞ ( Ω ) 1 − m / 2 ≤ C ( ν s ) − ( 1 / m − 1 / 2 ) ∥ ω ν ( 0 ) ∥ L m ( Ω ) ≤ C ( ν s ) − 1 / ( 2 r ) ∥ ω ν ( 0 ) ∥ L r ( Ω ) . \begin{aligned}
\|K_\Omega(\omega^\nu(s))\cdot\tau\|_{L^\infty(\partial\Omega)}
\le C\|\omega^\nu(s)\|_{L^m(\Omega)}^{m/2}\|\omega^\nu(s)\|_{L^\infty(\Omega)}^{1-m/2} \\
&\le C(\nu s)^{-(1/m-1/2)}\|\omega^\nu(0)\|_{L^m(\Omega)} \\
&\le C(\nu s)^{-1/(2r)}\|\omega^\nu(0)\|_{L^r(\Omega)} .
\end{aligned} ∥ K Ω ( ω ν ( s )) ⋅ τ ∥ L ∞ ( ∂ Ω ) ≤ C ∥ ω ν ( s ) ∥ L m ( Ω ) m /2 ∥ ω ν ( s ) ∥ L ∞ ( Ω ) 1 − m /2 ≤ C ( ν s ) − ( 1/ m − 1/2 ) ∥ ω ν ( 0 ) ∥ L m ( Ω ) ≤ C ( ν s ) − 1/ ( 2 r ) ∥ ω ν ( 0 ) ∥ L r ( Ω ) . Next, we start the operator splitting at t / 2 t/2 t /2
∥ ω ν ( t ) ∥ L ∞ ( Ω ) ≤ C ( ν t ) − 1 / r ∥ ω ν ( t / 2 ) ∥ L r ( Ω ) + C sup s ∈ [ t / 2 , t ] ∥ K Ω ( ω ν ( s ) ) ⋅ τ ∥ L ∞ ( ∂ Ω ) ≤ C ( ( ν t ) − 1 / r + ( ν t ) − 1 / ( 2 r ) ) ∥ ω ν ( 0 ) ∥ L r ( Ω ) ≤ C ( ν t ) − 1 / r ∥ ω ν ( 0 ) ∥ L r ( Ω ) , \begin{aligned}
\|\omega^\nu(t)\|_{L^\infty(\Omega)}
\le C(\nu t)^{-1/r}\|\omega^\nu(t/2)\|_{L^r(\Omega)}
+C\sup_{s\in[t/2,t]}\|K_\Omega(\omega^\nu(s))\cdot\tau\|_{L^\infty(\partial\Omega)} \\
&\le C\left((\nu t)^{-1/r}+(\nu t)^{-1/(2r)}\right)\|\omega^\nu(0)\|_{L^r(\Omega)} \\
&\le C(\nu t)^{-1/r}\|\omega^\nu(0)\|_{L^r(\Omega)} ,
\end{aligned} ∥ ω ν ( t ) ∥ L ∞ ( Ω ) ≤ C ( ν t ) − 1/ r ∥ ω ν ( t /2 ) ∥ L r ( Ω ) + C s ∈ [ t /2 , t ] sup ∥ K Ω ( ω ν ( s )) ⋅ τ ∥ L ∞ ( ∂ Ω ) ≤ C ( ( ν t ) − 1/ r + ( ν t ) − 1/ ( 2 r ) ) ∥ ω ν ( 0 ) ∥ L r ( Ω ) ≤ C ( ν t ) − 1/ r ∥ ω ν ( 0 ) ∥ L r ( Ω ) , where we have chosen ν 0 \nu_0 ν 0 ν T ≤ 1 \nu T\le1 ν T ≤ 1
∥ ω ν ( t ) ∥ L p ( Ω ) ≤ ∥ ω ν ( t ) ∥ L r ( Ω ) r / p ∥ ω ν ( t ) ∥ L ∞ ( Ω ) 1 − r / p ≤ C ( ν t ) − ( 1 / r − 1 / p ) ∥ curl u 0 ∥ L r ( Ω ) . \|\omega^\nu(t)\|_{L^p(\Omega)}
\le\|\omega^\nu(t)\|_{L^r(\Omega)}^{r/p}\|\omega^\nu(t)\|_{L^\infty(\Omega)}^{1-r/p}
\le C(\nu t)^{-(1/r-1/p)}\|\operatorname{curl}u_0\|_{L^r(\Omega)} . ∥ ω ν ( t ) ∥ L p ( Ω ) ≤ ∥ ω ν ( t ) ∥ L r ( Ω ) r / p ∥ ω ν ( t ) ∥ L ∞ ( Ω ) 1 − r / p ≤ C ( ν t ) − ( 1/ r − 1/ p ) ∥ curl u 0 ∥ L r ( Ω ) . Remark 5.4. If we take r = p r=p r = p
sup t ∈ [ 0 , T ] ∥ ω ν ( t ) ∥ L p ( Ω ) ≤ C ∥ curl u 0 ∥ L p ( Ω ) for all p > 1. \sup_{t\in[0,T]}\|\omega^\nu(t)\|_{L^p(\Omega)}
\le C\|\operatorname{curl}u_0\|_{L^p(\Omega)}
\quad\text{for all }p>1. t ∈ [ 0 , T ] sup ∥ ω ν ( t ) ∥ L p ( Ω ) ≤ C ∥ curl u 0 ∥ L p ( Ω ) for all p > 1. This extends the well-known bound for p > 2 p>2 p > 2 [12 Lemma 3] .
An important consequence of Lemma 5.2 is that it allows us to find a suitable B B B u ν u^\nu u ν
Corollary 5.5. Let ω 0 ∈ H 1 ( Ω ) ∩ L ∞ ( Ω ) \omega_0\in H^1(\Omega)\cap L^\infty(\Omega) ω 0 ∈ H 1 ( Ω ) ∩ L ∞ ( Ω ) u 0 = K Ω ( ω 0 ) ∈ W u_0=K_\Omega(\omega_0)\in W u 0 = K Ω ( ω 0 ) ∈ W ω ν = curl u ν \omega^\nu=\operatorname{curl}u^\nu ω ν = curl u ν u ν u^\nu u ν u ν ( 0 , ⋅ ) = u 0 u^\nu(0,\cdot)=u_0 u ν ( 0 , ⋅ ) = u 0 r ∈ ( 1 , 2 ) r\in(1,2) r ∈ ( 1 , 2 ) q ∈ [ 0 , r − 1 2 r ) q\in\left[0,\frac{r-1}{2r}\right) q ∈ [ 0 , 2 r r − 1 ) γ ∈ ( 0 , 1 ) \gamma\in(0,1) γ ∈ ( 0 , 1 ) ν 0 > 0 \nu_0>0 ν 0 > 0 ∥ curl u 0 ∥ L r ( Ω ) \|\operatorname{curl}u_0\|_{L^r(\Omega)} ∥ curl u 0 ∥ L r ( Ω ) ν ∈ ( 0 , ν 0 ) \nu\in(0,\nu_0) ν ∈ ( 0 , ν 0 )
∣ u ν ( s , x ) ⋅ ∇ d ( x ) ∣ ≤ C ( ν s ) − 1 / r ∥ curl u 0 ∥ L r ( Ω ) d ( x ) γ |u^\nu(s,x)\cdot\nabla d(x)|\le C(\nu s)^{-1/r}\|\operatorname{curl}u_0\|_{L^r(\Omega)}d(x)^\gamma ∣ u ν ( s , x ) ⋅ ∇ d ( x ) ∣ ≤ C ( ν s ) − 1/ r ∥ curl u 0 ∥ L r ( Ω ) d ( x ) γ for almost every ( s , x ) ∈ ( 0 , T ) × Ω ξ (s,x)\in(0,T)\times\Omega_\xi ( s , x ) ∈ ( 0 , T ) × Ω ξ ξ > 0 \xi>0 ξ > 0 d ∈ C 2 ( Ω ξ ) d\in C^2(\Omega_\xi) d ∈ C 2 ( Ω ξ ) C > 0 C>0 C > 0 ν \nu ν
Proof. Choose a small enough ξ > 0 \xi> 0 ξ > 0
∣ u ν ( s , x ) ⋅ ∇ d ( x ) ∣ ≤ C ∥ ω ν ( s ) ∥ L ∞ ( Ω ) d ( x ) γ ≤ C ( ν s ) − 1 / r ∥ curl u 0 ∥ L r ( Ω ) d ( x ) γ . |u^\nu(s,x)\cdot\nabla d(x)|\le C\|\omega^\nu(s)\|_{L^\infty(\Omega)}d(x)^\gamma\le C(\nu s)^{-1/r}\|\operatorname{curl}u_0\|_{L^r(\Omega)}d(x)^\gamma. ∣ u ν ( s , x ) ⋅ ∇ d ( x ) ∣ ≤ C ∥ ω ν ( s ) ∥ L ∞ ( Ω ) d ( x ) γ ≤ C ( ν s ) − 1/ r ∥ curl u 0 ∥ L r ( Ω ) d ( x ) γ . Proposition 6.1. Let ω 0 ∈ H 1 ( Ω ) ∩ L ∞ ( Ω ) \omega_0\in H^1(\Omega)\cap L^\infty(\Omega) ω 0 ∈ H 1 ( Ω ) ∩ L ∞ ( Ω ) u 0 = K Ω ( ω 0 ) ∈ W u_0=K_\Omega(\omega_0)\in W u 0 = K Ω ( ω 0 ) ∈ W ω ν = curl u ν \omega^\nu=\operatorname{curl}u^\nu ω ν = curl u ν u ν u^\nu u ν u ν ( 0 , ⋅ ) = u 0 u^\nu(0,\cdot)=u_0 u ν ( 0 , ⋅ ) = u 0 r ∈ ( 1 , 2 ) r\in(1,2) r ∈ ( 1 , 2 ) ρ ∈ ( 1 , ∞ ) \rho\in(1,\infty) ρ ∈ ( 1 , ∞ ) q ≥ 0 q\ge0 q ≥ 0 ρ q < r − 1 2 r \rho q<\frac{r-1}{2r} ρq < 2 r r − 1 ν 0 > 0 \nu_0>0 ν 0 > 0 ∥ curl u 0 ∥ L r ( Ω ) \|\operatorname{curl}u_0\|_{L^r(\Omega)} ∥ curl u 0 ∥ L r ( Ω ) ν ∈ ( 0 , ν 0 ) \nu\in(0,\nu_0) ν ∈ ( 0 , ν 0 )
sup t ∈ [ 0 , T ] ∥ ω ν ( t ) ∥ L ρ ( Ω ) ≤ C ∥ ω ν ( 0 ) ∥ L ρ ( Ω ) , \sup_{t\in[0,T]}\|\omega^\nu(t)\|_{L^\rho(\Omega)}\le C\|\omega^\nu(0)\|_{L^\rho(\Omega)}, t ∈ [ 0 , T ] sup ∥ ω ν ( t ) ∥ L ρ ( Ω ) ≤ C ∥ ω ν ( 0 ) ∥ L ρ ( Ω ) , where the constant C > 0 C>0 C > 0 ν \nu ν
Proof. The idea is to use Lemma 4.1, which is justified due to Corollary 5.5, and pick a small enough ν 0 \nu_0 ν 0 ν 0 \nu_0 ν 0 ∥ curl u 0 ∥ L r ( Ω ) \|\operatorname{curl}u_0\|_{L^r(\Omega)} ∥ curl u 0 ∥ L r ( Ω ) ρ q < r − 1 2 r \rho q<\frac{r-1}{2r} ρq < 2 r r − 1 γ ∈ ( 0 , 1 ) \gamma\in(0,1) γ ∈ ( 0 , 1 )
ρ q + 1 / r 1 − ρ q < γ < 1. \frac{\rho q+1/r}{1-\rho q}<\gamma<1. 1 − ρq ρq + 1/ r < γ < 1. This ensures
γ − 1 / r 1 + γ − ρ q > 0. (28) \frac{\gamma-1/r}{1+\gamma}-\rho q>0. \tag*{(28)} 1 + γ γ − 1/ r − ρq > 0. (28) With Lemma 4.1 for β = ∣ ⋅ ∣ ρ \beta=|\cdot|^\rho β = ∣ ⋅ ∣ ρ t ∈ ( 0 , T ] t\in(0,T] t ∈ ( 0 , T ]
∥ ω ν ( t ) ∥ L ρ ( Ω ) ρ ≤ ∥ ω ν ( 0 ) ∥ L ρ ( Ω ) ρ + C ∫ 0 t [ ν t − s + ν γ / ( 1 + γ ) ( C ( ν s ) − 1 / r ∥ curl u 0 ∥ L r ( Ω ) ) 1 / ( 1 + γ ) ] ∥ ( 2 κ − α ν ) u ν ( s ) ⋅ τ ∥ L ρ ( ∂ Ω ) ρ d s . \|\omega^\nu(t)\|_{L^\rho(\Omega)}^\rho\le\|\omega^\nu(0)\|_{L^\rho(\Omega)}^\rho+C\int_0^t\left[\sqrt{\frac{\nu}{t-s}}+\nu^{\gamma/(1+\gamma)}\left(C(\nu s)^{-1/r}\|\operatorname{curl}u_0\|_{L^r(\Omega)}\right)^{1/(1+\gamma)}\right]\|(2\kappa-\alpha^\nu)u^\nu(s)\cdot\tau\|_{L^\rho(\partial\Omega)}^\rho\,ds. ∥ ω ν ( t ) ∥ L ρ ( Ω ) ρ ≤ ∥ ω ν ( 0 ) ∥ L ρ ( Ω ) ρ + C ∫ 0 t [ t − s ν + ν γ / ( 1 + γ ) ( C ( ν s ) − 1/ r ∥ curl u 0 ∥ L r ( Ω ) ) 1/ ( 1 + γ ) ] ∥ ( 2 κ − α ν ) u ν ( s ) ⋅ τ ∥ L ρ ( ∂ Ω ) ρ d s . Regarding the boundary term, we use Lemma 5.1 to get
∥ ( 2 κ − α ν ) u ν ( s ) ⋅ τ ∥ L ρ ( ∂ Ω ) ρ = ∥ ( 2 κ − α ν ) K Ω ( ω ν ) ( s ) ⋅ τ ∥ L ρ ( ∂ Ω ) ρ ≤ ∥ 2 κ − α ν ∥ L ∞ ( ∂ Ω ) ρ ∥ K Ω ( ω ν ) ( s ) ⋅ τ ∥ L ρ ( ∂ Ω ) ρ ≤ C ν − ρ q ∥ ω ν ( s ) ∥ L ρ ( Ω ) ρ , \begin{aligned}
\|(2\kappa-\alpha^\nu)u^\nu(s)\cdot\tau\|_{L^\rho(\partial\Omega)}^\rho=\|(2\kappa-\alpha^\nu)K_\Omega(\omega^\nu)(s)\cdot\tau\|_{L^\rho(\partial\Omega)}^\rho\\
&\le\|2\kappa-\alpha^\nu\|_{L^\infty(\partial\Omega)}^\rho\|K_\Omega(\omega^\nu)(s)\cdot\tau\|_{L^\rho(\partial\Omega)}^\rho\\
&\le C\nu^{-\rho q}\|\omega^\nu(s)\|_{L^\rho(\Omega)}^\rho,
\end{aligned} ∥ ( 2 κ − α ν ) u ν ( s ) ⋅ τ ∥ L ρ ( ∂ Ω ) ρ = ∥ ( 2 κ − α ν ) K Ω ( ω ν ) ( s ) ⋅ τ ∥ L ρ ( ∂ Ω ) ρ ≤ ∥2 κ − α ν ∥ L ∞ ( ∂ Ω ) ρ ∥ K Ω ( ω ν ) ( s ) ⋅ τ ∥ L ρ ( ∂ Ω ) ρ ≤ C ν − ρq ∥ ω ν ( s ) ∥ L ρ ( Ω ) ρ , which yields for the time integral
∫ 0 t [ ν t − s + ν γ / ( 1 + γ ) ( C ( ν s ) − 1 / r ∥ curl u 0 ∥ L r ( Ω ) ) 1 / ( 1 + γ ) ] ∥ ( 2 κ − α ν ) u ν ( s ) ⋅ τ ∥ L ρ ( ∂ Ω ) ρ d s ≤ C ν − ρ q sup t ∈ [ 0 , T ] ∥ ω ν ( t ) ∥ L ρ ( Ω ) ρ ∫ 0 t ( ν t − s + ν γ / ( 1 + γ ) ( ν s ) − 1 / ( r + r γ ) ∥ curl u 0 ∥ L r ( Ω ) 1 / ( 1 + γ ) ) d s ≤ C sup t ∈ [ 0 , T ] ∥ ω ν ( t ) ∥ L ρ ( Ω ) ρ ( ν 1 / 2 − ρ q t 1 / 2 + ν ( γ − 1 / r ) / ( 1 + γ ) − ρ q ∥ curl u 0 ∥ L r ( Ω ) 1 / ( 1 + γ ) t 1 − 1 / ( r + r γ ) ) ≤ C sup t ∈ [ 0 , T ] ∥ ω ν ( t ) ∥ L ρ ( Ω ) ρ [ ν 1 / 2 − ρ q + ν ( γ − 1 / r ) / ( 1 + γ ) − ρ q ∥ curl u 0 ∥ L r ( Ω ) 1 / ( 1 + γ ) ] . \begin{aligned}
\int_{0}^{t}\left[\sqrt{\frac{\nu}{t-s}}+\nu^{\gamma/(1+\gamma)}\left(C(\nu s)^{-1/r}\|\operatorname{curl}u_{0}\|_{L^{r}(\Omega)}\right)^{1/(1+\gamma)}\right]\|(2\kappa-\alpha^{\nu})u^{\nu}(s)\cdot\tau\|_{L^{\rho}(\partial\Omega)}^{\rho}\,ds \\
&\leq C\nu^{-\rho q}\sup_{t\in[0,T]}\|\omega^{\nu}(t)\|_{L^{\rho}(\Omega)}^{\rho}\int_{0}^{t}\left(\sqrt{\frac{\nu}{t-s}}+\nu^{\gamma/(1+\gamma)}(\nu s)^{-1/(r+r\gamma)}\|\operatorname{curl}u_{0}\|_{L^{r}(\Omega)}^{1/(1+\gamma)}\right)\,ds \\
&\leq C\sup_{t\in[0,T]}\|\omega^{\nu}(t)\|_{L^{\rho}(\Omega)}^{\rho}\left(\nu^{1/2-\rho q}t^{1/2}+\nu^{(\gamma-1/r)/(1+\gamma)-\rho q}\|\operatorname{curl}u_{0}\|_{L^{r}(\Omega)}^{1/(1+\gamma)}t^{1-1/(r+r\gamma)}\right) \\
&\leq C\sup_{t\in[0,T]}\|\omega^{\nu}(t)\|_{L^{\rho}(\Omega)}^{\rho}\left[\nu^{1/2-\rho q}+\nu^{(\gamma-1/r)/(1+\gamma)-\rho q}\|\operatorname{curl}u_{0}\|_{L^{r}(\Omega)}^{1/(1+\gamma)}\right].
\end{aligned} ∫ 0 t [ t − s ν + ν γ / ( 1 + γ ) ( C ( ν s ) − 1/ r ∥ curl u 0 ∥ L r ( Ω ) ) 1/ ( 1 + γ ) ] ∥ ( 2 κ − α ν ) u ν ( s ) ⋅ τ ∥ L ρ ( ∂ Ω ) ρ d s ≤ C ν − ρq t ∈ [ 0 , T ] sup ∥ ω ν ( t ) ∥ L ρ ( Ω ) ρ ∫ 0 t ( t − s ν + ν γ / ( 1 + γ ) ( ν s ) − 1/ ( r + r γ ) ∥ curl u 0 ∥ L r ( Ω ) 1/ ( 1 + γ ) ) d s ≤ C t ∈ [ 0 , T ] sup ∥ ω ν ( t ) ∥ L ρ ( Ω ) ρ ( ν 1/2 − ρq t 1/2 + ν ( γ − 1/ r ) / ( 1 + γ ) − ρq ∥ curl u 0 ∥ L r ( Ω ) 1/ ( 1 + γ ) t 1 − 1/ ( r + r γ ) ) ≤ C t ∈ [ 0 , T ] sup ∥ ω ν ( t ) ∥ L ρ ( Ω ) ρ [ ν 1/2 − ρq + ν ( γ − 1/ r ) / ( 1 + γ ) − ρq ∥ curl u 0 ∥ L r ( Ω ) 1/ ( 1 + γ ) ] . The first exponent of ν \nu ν ρ q < 1 4 \rho q<\frac{1}{4} ρq < 4 1 ν 0 > 0 \nu_{0}>0 ν 0 > 0 ν ∈ ( 0 , ν 0 ) \nu\in(0,\nu_{0}) ν ∈ ( 0 , ν 0 )
C [ ν 1 / 2 − ρ q + ν ( γ − 1 / r ) / ( 1 + γ ) − ρ q ∥ curl u 0 ∥ L r ( Ω ) 1 / ( 1 + γ ) ] ≤ 1 2 . (29) C\left[\nu^{1/2-\rho q}+\nu^{(\gamma-1/r)/(1+\gamma)-\rho q}\|\operatorname{curl}u_{0}\|_{L^{r}(\Omega)}^{1/(1+\gamma)}\right]\leq\frac{1}{2}. \tag*{(29)} C [ ν 1/2 − ρq + ν ( γ − 1/ r ) / ( 1 + γ ) − ρq ∥ curl u 0 ∥ L r ( Ω ) 1/ ( 1 + γ ) ] ≤ 2 1 . (29) As the right-hand side is independent of t ∈ ( 0 , T ] t\in(0,T] t ∈ ( 0 , T ]
sup t ∈ [ 0 , T ] ∥ ω ν ( t ) ∥ L ρ ( Ω ) ρ ≤ ∥ ω ν ( 0 ) ∥ L ρ ( Ω ) ρ + 1 2 sup t ∈ [ 0 , T ] ∥ ω ν ( t ) ∥ L ρ ( Ω ) ρ , \sup_{t\in[0,T]}\|\omega^{\nu}(t)\|_{L^{\rho}(\Omega)}^{\rho}\leq\|\omega^{\nu}(0)\|_{L^{\rho}(\Omega)}^{\rho}+\frac{1}{2}\sup_{t\in[0,T]}\|\omega^{\nu}(t)\|_{L^{\rho}(\Omega)}^{\rho}, t ∈ [ 0 , T ] sup ∥ ω ν ( t ) ∥ L ρ ( Ω ) ρ ≤ ∥ ω ν ( 0 ) ∥ L ρ ( Ω ) ρ + 2 1 t ∈ [ 0 , T ] sup ∥ ω ν ( t ) ∥ L ρ ( Ω ) ρ , which concludes the proof. □
We use an Aubin–Lions argument, as suggested in [12 ] , to transfer the uniform vorticity estimate, the parabolic smoothing estimate, and the normal-drift estimate to weak solutions.
Lemma 6.2. Let ω 0 ∈ L p ( Ω ) \omega_{0}\in L^{p}(\Omega) ω 0 ∈ L p ( Ω ) p ∈ ( 1 , ∞ ) p\in(1,\infty) p ∈ ( 1 , ∞ ) ω ν = curl u ν \omega^{\nu}=\operatorname{curl}u^{\nu} ω ν = curl u ν u ν u^{\nu} u ν u ν ( 0 , ⋅ ) = K Ω ( ω 0 ) u^{\nu}(0,\cdot)=K_{\Omega}(\omega_{0}) u ν ( 0 , ⋅ ) = K Ω ( ω 0 ) ρ ∈ ( 1 , p ] \rho\in(1,p] ρ ∈ ( 1 , p ] q ≥ 0 q\geq0 q ≥ 0 ρ q < min { p − 1 2 p , 1 4 } \rho q<\min\left\{\frac{p-1}{2p},\frac{1}{4}\right\} ρq < min { 2 p p − 1 , 4 1 } ν 0 > 0 \nu_{0}>0 ν 0 > 0 ∥ ω 0 ∥ L p ( Ω ) \|\omega_{0}\|_{L^{p}(\Omega)} ∥ ω 0 ∥ L p ( Ω ) ν ∈ ( 0 , ν 0 ) \nu\in(0,\nu_{0}) ν ∈ ( 0 , ν 0 )
sup t ∈ [ 0 , T ] ∥ ω ν ( t ) ∥ L ρ ( Ω ) ≤ C ∥ ω 0 ∥ L ρ ( Ω ) . \sup_{t\in[0,T]}\|\omega^{\nu}(t)\|_{L^{\rho}(\Omega)}\leq C\|\omega_{0}\|_{L^{\rho}(\Omega)}. t ∈ [ 0 , T ] sup ∥ ω ν ( t ) ∥ L ρ ( Ω ) ≤ C ∥ ω 0 ∥ L ρ ( Ω ) . Furthermore, for every γ ∈ ( 0 , 1 ) \gamma\in(0,1) γ ∈ ( 0 , 1 ) r ∈ ( 1 , min { p , 2 } ) r\in(1,\min\{p,2\}) r ∈ ( 1 , min { p , 2 }) ρ q < r − 1 2 r \rho q<\frac{r-1}{2r} ρq < 2 r r − 1 ν > 0 \nu>0 ν > 0
∥ ω ν ( t ) ∥ L ∞ ( Ω ) ≤ C ( ν t ) − 1 / r ∥ ω 0 ∥ L r ( Ω ) for every t ∈ ( 0 , T ] , \|\omega^{\nu}(t)\|_{L^{\infty}(\Omega)}\leq C(\nu t)^{-1/r}\|\omega_{0}\|_{L^{r}(\Omega)}\quad\text{for every }t\in(0,T], ∥ ω ν ( t ) ∥ L ∞ ( Ω ) ≤ C ( ν t ) − 1/ r ∥ ω 0 ∥ L r ( Ω ) for every t ∈ ( 0 , T ] , and
∣ u ν ( s , x ) ⋅ ∇ d ( x ) ∣ ≤ C ( ν s ) − 1 / r ∥ ω 0 ∥ L r ( Ω ) d ( x ) γ for almost every ( s , x ) ∈ ( 0 , T ) × Ω ξ , |u^{\nu}(s,x)\cdot\nabla d(x)|\leq C(\nu s)^{-1/r}\|\omega_{0}\|_{L^{r}(\Omega)}d(x)^{\gamma}\quad\text{for almost every }(s,x)\in(0,T)\times\Omega_{\xi}, ∣ u ν ( s , x ) ⋅ ∇ d ( x ) ∣ ≤ C ( ν s ) − 1/ r ∥ ω 0 ∥ L r ( Ω ) d ( x ) γ for almost every ( s , x ) ∈ ( 0 , T ) × Ω ξ , where the collar width ξ > 0 \xi>0 ξ > 0 d ∈ C 2 ( Ω ξ ) d\in C^{2}(\Omega_{\xi}) d ∈ C 2 ( Ω ξ ) C > 0 C>0 C > 0 ν \nu ν
Proof. Let ω 0 , n ν ∈ H 1 ( Ω ) ∩ L ∞ ( Ω ) \omega_{0,n}^{\nu} \in H^{1}(\Omega) \cap L^{\infty}(\Omega) ω 0 , n ν ∈ H 1 ( Ω ) ∩ L ∞ ( Ω )
ω 0 , n ν → ω 0 in L p ( Ω ) , \omega_{0,n}^{\nu} \to\omega_{0} \text{ in } L^{p}(\Omega), ω 0 , n ν → ω 0 in L p ( Ω ) , and denote by u n ν u_{n}^{\nu} u n ν ω n ν \omega_{n}^{\nu} ω n ν ν \nu ν α ν \alpha^{\nu} α ν Proposition 6.1 , we choose r ∈ ( 1 , min { 2 , p } ) r \in(1,\min\{2,p\}) r ∈ ( 1 , min { 2 , p }) ρ q < r − 1 2 r \rho q < \frac{r-1}{2r} ρq < 2 r r − 1 ν 0 \nu_{0} ν 0 ∥ ω 0 ∥ L p ( Ω ) \|\omega_{0}\|_{L^{p}(\Omega)} ∥ ω 0 ∥ L p ( Ω )
sup t ∈ [ 0 , T ] ∥ ω n ν ( t ) ∥ L ρ ( Ω ) ≤ C ∥ ω 0 , n ν ∥ L ρ ( Ω ) , ν ∈ ( 0 , ν 0 ) , (30) \sup_{t\in[0,T]}\|\omega_{n}^{\nu}(t)\|_{L^{\rho}(\Omega)} \leq C\|\omega_{0,n}^{\nu}\|_{L^{\rho}(\Omega)}, \qquad\nu\in(0,\nu_{0}), \tag*{(30)} t ∈ [ 0 , T ] sup ∥ ω n ν ( t ) ∥ L ρ ( Ω ) ≤ C ∥ ω 0 , n ν ∥ L ρ ( Ω ) , ν ∈ ( 0 , ν 0 ) , (30) where the constant C > 0 C>0 C > 0 n n n ν \nu ν ν ∈ ( 0 , ν 0 ) \nu\in(0,\nu_{0}) ν ∈ ( 0 , ν 0 ) (30) , Lemma A.2 , and Poincaré’s inequality that
sup n ∥ u n ν ∥ L ∞ ( 0 , T ; W 1 , ρ ( Ω ) ) ≤ C ∥ ω 0 ∥ L p ( Ω ) . \sup_{n}\|u_{n}^{\nu}\|_{L^{\infty}(0,T;W^{1,\rho}(\Omega))} \leq C\|\omega_{0}\|_{L^{p}(\Omega)}. n sup ∥ u n ν ∥ L ∞ ( 0 , T ; W 1 , ρ ( Ω )) ≤ C ∥ ω 0 ∥ L p ( Ω ) . By the argument in the proof of [12 Proposition 1] , with the suggested adaptations, it follows from the energy inequality that
sup n ∥ ∂ t u n ν ∥ L 2 ( 0 , T ; H − 1 ( Ω ) ) ≤ C ( ν , ∥ K Ω ( ω 0 ) ∥ L 2 ( Ω ) ) ≤ C ( ν , ∥ ω 0 ∥ L p ( Ω ) ) . \sup_{n}\|\partial_{t}u_{n}^{\nu}\|_{L^{2}(0,T;H^{-1}(\Omega))} \leq C\left(\nu,\|K_{\Omega}(\omega_{0})\|_{L^{2}(\Omega)}\right) \leq C\left(\nu,\|\omega_{0}\|_{L^{p}(\Omega)}\right). n sup ∥ ∂ t u n ν ∥ L 2 ( 0 , T ; H − 1 ( Ω )) ≤ C ( ν , ∥ K Ω ( ω 0 ) ∥ L 2 ( Ω ) ) ≤ C ( ν , ∥ ω 0 ∥ L p ( Ω ) ) . We use the Aubin–Lions lemma A.3 with X 0 = W 1 , ρ ( Ω ) X_{0}=W^{1,\rho}(\Omega) X 0 = W 1 , ρ ( Ω ) X = L 2 ( Ω ) X=L^{2}(\Omega) X = L 2 ( Ω ) X 1 = H − 1 X_{1}=H^{-1} X 1 = H − 1 n k → ∞ n_{k}\to\infty n k → ∞
u n k ν → v strongly in C ( [ 0 , T ] ; L 2 ( Ω ) ) as k → ∞ , u_{n_{k}}^{\nu} \to v \text{ strongly in } C([0,T];L^{2}(\Omega)) \text{ as } k\to\infty, u n k ν → v strongly in C ([ 0 , T ] ; L 2 ( Ω )) as k → ∞ , for some v ∈ C ( [ 0 , T ] ; L 2 ( Ω ) ) v\in C([0,T];L^{2}(\Omega)) v ∈ C ([ 0 , T ] ; L 2 ( Ω )) L 2 ( 0 , T ; H 1 ( Ω ) ) L^{2}(0,T;H^{1}(\Omega)) L 2 ( 0 , T ; H 1 ( Ω )) v ∈ L 2 ( 0 , T ; H 1 ( Ω ) ) v\in L^{2}(0,T;H^{1}(\Omega)) v ∈ L 2 ( 0 , T ; H 1 ( Ω )) (8) to conclude that v = u ν v=u^{\nu} v = u ν
ω n k ν ( t ) = curl u n k ν ( t ) → curl u ν ( t ) in distributions for t ∈ [ 0 , T ] , \omega_{n_{k}}^{\nu}(t)=\operatorname{curl}u_{n_{k}}^{\nu}(t)\to\operatorname{curl}u^{\nu}(t) \text{ in distributions for } t\in[0,T], ω n k ν ( t ) = curl u n k ν ( t ) → curl u ν ( t ) in distributions for t ∈ [ 0 , T ] , and hence also weak convergence in L ρ ( Ω ) L^{\rho}(\Omega) L ρ ( Ω ) (30) . With lower semicontinuity, we obtain that
∥ ω ν ( t ) ∥ L ρ ( Ω ) ≤ lim inf k → ∞ ∥ ω n k ν ( t ) ∥ L ρ ( Ω ) ≤ C lim k → ∞ ∥ ω 0 , n k ν ∥ L ρ ( Ω ) = C ∥ ω 0 ∥ L ρ ( Ω ) . \|\omega^{\nu}(t)\|_{L^{\rho}(\Omega)}\leq\liminf_{k\to\infty}\|\omega_{n_{k}}^{\nu}(t)\|_{L^{\rho}(\Omega)}\leq C\lim_{k\to\infty}\|\omega_{0,n_{k}}^{\nu}\|_{L^{\rho}(\Omega)}=C\|\omega_{0}\|_{L^{\rho}(\Omega)}. ∥ ω ν ( t ) ∥ L ρ ( Ω ) ≤ k → ∞ lim inf ∥ ω n k ν ( t ) ∥ L ρ ( Ω ) ≤ C k → ∞ lim ∥ ω 0 , n k ν ∥ L ρ ( Ω ) = C ∥ ω 0 ∥ L ρ ( Ω ) . After decreasing ν 0 \nu_{0} ν 0 Lemma 5.2 yields an L ∞ ( Ω ) L^{\infty}(\Omega) L ∞ ( Ω ) k k k t ∈ ( 0 , T ] t\in(0,T] t ∈ ( 0 , T ]
∥ ω ν ( t ) ∥ L ∞ ( Ω ) ≤ lim inf k → ∞ ∥ ω n k ν ( t ) ∥ L ∞ ( Ω ) ≤ C ( ν t ) − 1 / r lim k → ∞ ∥ ω 0 , n k ν ∥ L r ( Ω ) = C ( ν t ) − 1 / r ∥ ω 0 ∥ L r ( Ω ) . \|\omega^{\nu}(t)\|_{L^{\infty}(\Omega)}\leq\liminf_{k\to\infty}\|\omega_{n_{k}}^{\nu}(t)\|_{L^{\infty}(\Omega)}\leq C(\nu t)^{-1/r}\lim_{k\to\infty}\|\omega_{0,n_{k}}^{\nu}\|_{L^{r}(\Omega)}=C(\nu t)^{-1/r}\|\omega_{0}\|_{L^{r}(\Omega)}. ∥ ω ν ( t ) ∥ L ∞ ( Ω ) ≤ k → ∞ lim inf ∥ ω n k ν ( t ) ∥ L ∞ ( Ω ) ≤ C ( ν t ) − 1/ r k → ∞ lim ∥ ω 0 , n k ν ∥ L r ( Ω ) = C ( ν t ) − 1/ r ∥ ω 0 ∥ L r ( Ω ) . For a further subsequence, which we label with the same index, we also have u n k ν → u ν u^{\nu}_{n_k} \to u^{\nu} u n k ν → u ν ( 0 , T ) × Ω (0,T) \times\Omega ( 0 , T ) × Ω ( s , x ) ∈ ( 0 , T ) × Ω ξ (s,x) \in(0,T) \times\Omega_{\xi} ( s , x ) ∈ ( 0 , T ) × Ω ξ
∣ u ν ( s , x ) ⋅ ∇ d ( x ) ∣ = lim k → ∞ ∣ u n k ν ( s , x ) ⋅ ∇ d ( x ) ∣ ≤ lim k → ∞ C ( ν s ) − 1 / r ∥ ω n k 0 ∥ L r ( Ω ) d ( x ) γ = C ( ν s ) − 1 / r ∥ ω 0 ∥ L r ( Ω ) d ( x ) γ . |u^{\nu}(s,x) \cdot\nabla d(x)| = \lim_{k \to\infty} |u^{\nu}_{n_k}(s,x) \cdot\nabla d(x)| \le\lim_{k \to\infty} C(\nu s)^{-1/r} \|\omega^{0}_{n_k}\|_{L^r(\Omega)}d(x)^{\gamma}
= C(\nu s)^{-1/r}\|\omega_0\|_{L^r(\Omega)}d(x)^{\gamma}. ∣ u ν ( s , x ) ⋅ ∇ d ( x ) ∣ = k → ∞ lim ∣ u n k ν ( s , x ) ⋅ ∇ d ( x ) ∣ ≤ k → ∞ lim C ( ν s ) − 1/ r ∥ ω n k 0 ∥ L r ( Ω ) d ( x ) γ = C ( ν s ) − 1/ r ∥ ω 0 ∥ L r ( Ω ) d ( x ) γ . Strong convergence of vorticity
We are now in a position to show Theorem 1.1, which we will do in three steps:
u ν → u u^{\nu} \to u u ν → u C ( [ 0 , T ] ; L 2 ( Ω ) ) C([0,T];L^2(\Omega)) C ([ 0 , T ] ; L 2 ( Ω ))
b ν → 0 b^{\nu} \to0 b ν → 0 C ( [ 0 , T ] ; L ρ ( Ω ) ) C([0,T];L^{\rho}(\Omega)) C ([ 0 , T ] ; L ρ ( Ω ))
a ν → curl u a^{\nu} \to\operatorname{curl} u a ν → curl u C ( [ 0 , T ] ; L ρ ( Ω ) ) C([0,T];L^{\rho}(\Omega)) C ([ 0 , T ] ; L ρ ( Ω ))
Proof of Theorem 1.1. Throughout the proof, choose r ∈ ( 1 , min { p , 2 } ) r \in(1,\min\{p,2\}) r ∈ ( 1 , min { p , 2 }) ρ q < r − 1 2 r \rho q < \frac{r-1}{2r} ρq < 2 r r − 1
Step 1: u ν → u u^{\nu} \to u u ν → u C ( [ 0 , T ] ; L 2 ( Ω ) ) C([0,T];L^2(\Omega)) C ([ 0 , T ] ; L 2 ( Ω )) By Lemmas 6.2 and A.2, we have
sup ν ∈ ( 0 , ν 0 ) ∥ u ν ∥ L ∞ ( 0 , T ; W 1 , ρ ( Ω ) ) ≤ C ( ∥ ω 0 ∥ L ρ ) , \sup_{\nu\in(0,\nu_0)} \|u^{\nu}\|_{L^{\infty}(0,T;W^{1,\rho}(\Omega))} \le C\left(\|\omega_0\|_{L^{\rho}}\right), ν ∈ ( 0 , ν 0 ) sup ∥ u ν ∥ L ∞ ( 0 , T ; W 1 , ρ ( Ω )) ≤ C ( ∥ ω 0 ∥ L ρ ) , where the constant C > 0 C>0 C > 0 ν \nu ν ν 0 \nu_0 ν 0 ∥ ω 0 ∥ L p ( Ω ) \|\omega_0\|_{L^p(\Omega)} ∥ ω 0 ∥ L p ( Ω ) φ ∈ V ∩ W 1 , ∞ ( Ω ) \varphi\in V \cap W^{1,\infty}(\Omega) φ ∈ V ∩ W 1 , ∞ ( Ω )
∣ ∫ Ω ∂ t u ν ⋅ φ d x ∣ ≤ ∥ u ν ∥ L 2 ( Ω ) 2 ∥ ∇ φ ∥ L ∞ ( Ω ) + ν ∥ ∇ u ν ∥ L 1 ( Ω ) ∥ ∇ φ ∥ L ∞ ( Ω ) + ν ∥ κ − α ν ∥ L ∞ ( ∂ Ω ) ∥ u ν ∥ L 1 ( ∂ Ω ) ∥ φ ∥ L ∞ ( ∂ Ω ) ≤ C ( 1 + ν + ν 1 − q ) ∥ φ ∥ W 1 , ∞ ( Ω ) . \left|\int_{\Omega}\partial_tu^{\nu}\cdot\varphi\,dx\right|
\le\|u^{\nu}\|_{L^2(\Omega)}^2\|\nabla\varphi\|_{L^{\infty}(\Omega)}
+\nu\|\nabla u^{\nu}\|_{L^1(\Omega)}\|\nabla\varphi\|_{L^{\infty}(\Omega)}
+\nu\|\kappa-\alpha^{\nu}\|_{L^{\infty}(\partial\Omega)}
\|u^{\nu}\|_{L^1(\partial\Omega)}\|\varphi\|_{L^{\infty}(\partial\Omega)}
\le C(1+\nu+\nu^{1-q})\|\varphi\|_{W^{1,\infty}(\Omega)}. ∫ Ω ∂ t u ν ⋅ φ d x ≤ ∥ u ν ∥ L 2 ( Ω ) 2 ∥∇ φ ∥ L ∞ ( Ω ) + ν ∥∇ u ν ∥ L 1 ( Ω ) ∥∇ φ ∥ L ∞ ( Ω ) + ν ∥ κ − α ν ∥ L ∞ ( ∂ Ω ) ∥ u ν ∥ L 1 ( ∂ Ω ) ∥ φ ∥ L ∞ ( ∂ Ω ) ≤ C ( 1 + ν + ν 1 − q ) ∥ φ ∥ W 1 , ∞ ( Ω ) . Therefore, Aubin–Lions lemma A.3 gives, after passing to a subsequence, a limit u ∈ C ( [ 0 , T ] ; H ) ∩ L ∞ ( 0 , T ; W 1 , ρ ( Ω ) ) u \in C([0,T];H) \cap L^{\infty}(0,T;W^{1,\rho}(\Omega)) u ∈ C ([ 0 , T ] ; H ) ∩ L ∞ ( 0 , T ; W 1 , ρ ( Ω ))
u ν → u strongly in C ( [ 0 , T ] ; L 2 ( Ω ) ) as ν → 0. u^{\nu} \to u \text{ strongly in } C([0,T];L^2(\Omega)) \text{ as } \nu\to0. u ν → u strongly in C ([ 0 , T ] ; L 2 ( Ω )) as ν → 0. Also, the convergence and uniform bounds are sufficient to pass to the limit in (3.1); thus u u u
Step 2: b ν → 0 b^\nu\to0 b ν → 0 C ( [ 0 , T ] ; L ρ ( Ω ) ) C([0,T];L^\rho(\Omega)) C ([ 0 , T ] ; L ρ ( Ω )) Let a ν a^\nu a ν b ν = ω ν − a ν b^\nu=\omega^\nu-a^\nu b ν = ω ν − a ν
∥ b ν ∥ L ∞ ( 0 , T ; L ρ ( ∂ Ω ) ) ≤ C ν − ρ q ∥ u ν ∥ L ∞ ( 0 , T ; W 1 , ρ ( Ω ) ) ≤ C ν − ρ q ∥ ω 0 ∥ L ρ ( Ω ) . (31) \|b^\nu\|_{L^\infty(0,T;L^\rho(\partial\Omega))}\le C\nu^{-\rho q}\|u^\nu\|_{L^\infty(0,T;W^{1,\rho}(\Omega))}\le C\nu^{-\rho q}\|\omega_0\|_{L^\rho(\Omega)}. \tag*{(31)} ∥ b ν ∥ L ∞ ( 0 , T ; L ρ ( ∂ Ω )) ≤ C ν − ρq ∥ u ν ∥ L ∞ ( 0 , T ; W 1 , ρ ( Ω )) ≤ C ν − ρq ∥ ω 0 ∥ L ρ ( Ω ) . (31) As in Proposition 6.1, we choose a γ ∈ ( ρ q + 1 / r 1 − ρ q , 1 ) \gamma\in\left(\frac{\rho q+1/r}{1-\rho q},1\right) γ ∈ ( 1 − ρq ρq + 1/ r , 1 )
γ − 1 / r 1 + γ − ρ q > 0. \frac{\gamma-1/r}{1+\gamma}-\rho q>0. 1 + γ γ − 1/ r − ρq > 0. Thanks to the normal-drift estimate in Lemma 6.2, we can also use Lemma 4.1 with β = ∣ ⋅ ∣ ρ \beta=|\cdot|^\rho β = ∣ ⋅ ∣ ρ b ν b^\nu b ν
sup t ∈ [ 0 , T ] ∥ b ν ( t ) ∥ L ρ ( Ω ) ρ ≤ C ∥ ω 0 ∥ L ρ ( Ω ) ρ ( ν 1 / 2 − ρ q T 1 / 2 + ν ( γ − 1 / r ) / ( 1 + γ ) − ρ q ∥ ω 0 ∥ L r ( Ω ) 1 / ( 1 + γ ) T 1 − 1 / ( r + r γ ) ) . \sup_{t\in[0,T]}\|b^\nu(t)\|_{L^\rho(\Omega)}^\rho\le C\|\omega_0\|_{L^\rho(\Omega)}^\rho\left(\nu^{1/2-\rho q}T^{1/2}+\nu^{(\gamma-1/r)/(1+\gamma)-\rho q}\|\omega_0\|_{L^r(\Omega)}^{1/(1+\gamma)}T^{1-1/(r+r\gamma)}\right). t ∈ [ 0 , T ] sup ∥ b ν ( t ) ∥ L ρ ( Ω ) ρ ≤ C ∥ ω 0 ∥ L ρ ( Ω ) ρ ( ν 1/2 − ρq T 1/2 + ν ( γ − 1/ r ) / ( 1 + γ ) − ρq ∥ ω 0 ∥ L r ( Ω ) 1/ ( 1 + γ ) T 1 − 1/ ( r + r γ ) ) . As both powers of ν \nu ν
sup t ∈ [ 0 , T ] ∥ b ν ( t ) ∥ L ρ ( Ω ) ρ → 0 as ν → 0. \sup_{t\in[0,T]}\|b^\nu(t)\|_{L^\rho(\Omega)}^\rho\to0\text{ as }\nu\to0. t ∈ [ 0 , T ] sup ∥ b ν ( t ) ∥ L ρ ( Ω ) ρ → 0 as ν → 0. Step 3: a ν → curl u a^\nu\to\operatorname{curl}u a ν → curl u C ( [ 0 , T ] ; L ρ ( Ω ) ) C([0,T];L^\rho(\Omega)) C ([ 0 , T ] ; L ρ ( Ω )) By Step 1, we can use Lemma A.4 to obtain
a ν → ω strongly in C ( [ 0 , T ] ; L ρ ( Ω ) ) , a^\nu\to\omega\text{ strongly in }C([0,T];L^\rho(\Omega)), a ν → ω strongly in C ([ 0 , T ] ; L ρ ( Ω )) , where ω ∈ C ( [ 0 , T ] ; L ρ ( Ω ) ) \omega\in C([0,T];L^\rho(\Omega)) ω ∈ C ([ 0 , T ] ; L ρ ( Ω )) u u u ω 0 \omega_0 ω 0 ω = curl u \omega=\operatorname{curl}u ω = curl u
∥ u ν − K Ω ( ω ) ∥ C ( [ 0 , T ] ; W 1 , ρ ( Ω ) ) ≤ C ∥ a ν − ω ∥ C ( [ 0 , T ] ; L ρ ( Ω ) ) + C ∥ b ν ∥ C ( [ 0 , T ] ; L ρ ( Ω ) ) → 0 as ν → 0. \|u^\nu-K_\Omega(\omega)\|_{C([0,T];W^{1,\rho}(\Omega))}\le C\|a^\nu-\omega\|_{C([0,T];L^\rho(\Omega))}+C\|b^\nu\|_{C([0,T];L^\rho(\Omega))}\to0\text{ as }\nu\to0. ∥ u ν − K Ω ( ω ) ∥ C ([ 0 , T ] ; W 1 , ρ ( Ω )) ≤ C ∥ a ν − ω ∥ C ([ 0 , T ] ; L ρ ( Ω )) + C ∥ b ν ∥ C ([ 0 , T ] ; L ρ ( Ω )) → 0 as ν → 0. By uniqueness of limits, this yields K Ω ( ω ) = u K_\Omega(\omega)=u K Ω ( ω ) = u ω = curl u \omega=\operatorname{curl}u ω = curl u
∥ ω ( t ) ∥ L p ( Ω ) = ∥ ω 0 ∥ L p ( Ω ) for every t ∈ [ 0 , T ] . (32) \|\omega(t)\|_{L^p(\Omega)}=\|\omega_0\|_{L^p(\Omega)}\quad\text{for every }t\in[0,T]. \tag*{(32)} ∥ ω ( t ) ∥ L p ( Ω ) = ∥ ω 0 ∥ L p ( Ω ) for every t ∈ [ 0 , T ] . (32) Then, continuity in L p ( Ω ) L^p(\Omega) L p ( Ω ) L p ( Ω ) L^p(\Omega) L p ( Ω ) L p ( Ω ) L^p(\Omega) L p ( Ω ) u ∈ C ( [ 0 , T ] ; W 1 , p ( Ω ) ) u\in C([0,T];W^{1,p}(\Omega)) u ∈ C ([ 0 , T ] ; W 1 , p ( Ω ))
The energy conservation property of the limit is a straightforward consequence of Lemma 6.2.
Proof of Corollary 1.2. Choose r ∈ ( 1 , min { p , 2 } ) r \in(1,\min\{p,2\}) r ∈ ( 1 , min { p , 2 }) ρ q < r − 1 2 r \rho_q < \frac{r-1}{2r} ρ q < 2 r r − 1
ν ∫ 0 T ∥ ∇ v ν ( t ) ∥ L 2 ( Ω ) 2 d t ≤ C ν ∫ 0 T ∥ ω ν ( t ) ∥ L 2 ( Ω ) 2 d t ≤ C ν ∫ 0 T ∥ ω ν ( t ) ∥ L 1 ( Ω ) ∥ ω ν ( t ) ∥ L ∞ ( Ω ) d t ≤ C ν 1 − 1 / r ∫ 0 T t − 1 / r d t → 0 as ν → 0. \begin{aligned}
\nu\int_0^T \|\nabla v^\nu(t)\|_{L^2(\Omega)}^2\,dt \le C\nu\int_0^T \|\omega^\nu(t)\|_{L^2(\Omega)}^2\,dt \le C\nu\int_0^T \|\omega^\nu(t)\|_{L^1(\Omega)}\|\omega^\nu(t)\|_{L^\infty(\Omega)}\,dt \\
&\le C\nu^{1-1/r}\int_0^T t^{-1/r}\,dt \to0\text{ as }\nu\to0.
\end{aligned} ν ∫ 0 T ∥∇ v ν ( t ) ∥ L 2 ( Ω ) 2 d t ≤ C ν ∫ 0 T ∥ ω ν ( t ) ∥ L 2 ( Ω ) 2 d t ≤ C ν ∫ 0 T ∥ ω ν ( t ) ∥ L 1 ( Ω ) ∥ ω ν ( t ) ∥ L ∞ ( Ω ) d t ≤ C ν 1 − 1/ r ∫ 0 T t − 1/ r d t → 0 as ν → 0. Similarly, we obtain with Lemma 6.2 and Lemma 5.1 that
ν ∫ 0 T ∫ ∂ Ω ( ∣ κ ∣ + ∣ α ′ ∣ ) ∣ u ν ( t ) ⋅ τ ∣ 2 d S d t ≤ C ν 1 − q ∫ 0 T ∥ u ν ( t ) ⋅ τ ∥ L 2 ( ∂ Ω ) 2 d t ≤ C ν 1 − q ∫ 0 T ∥ ω ν ( t ) ∥ L 1 ( Ω ) ∥ ω ν ( t ) ∥ L 2 ( Ω ) d t ≤ C ν 1 − q − 1 / ( 2 r ) ∫ 0 T t − 1 / ( 2 r ) d t → 0 as ν → 0. \begin{aligned}
\nu\int_0^T\int_{\partial\Omega} (|\kappa|+|\alpha^\prime|)|u^\nu(t)\cdot\tau|^2\,dS\,dt \le C\nu^{1-q}\int_0^T \|u^\nu(t)\cdot\tau\|_{L^2(\partial\Omega)}^2\,dt \\
&\le C\nu^{1-q}\int_0^T \|\omega^\nu(t)\|_{L^1(\Omega)}\|\omega^\nu(t)\|_{L^2(\Omega)}\,dt \\
&\le C\nu^{1-q-1/(2r)}\int_0^T t^{-1/(2r)}\,dt \to0\text{ as }\nu\to0.
\end{aligned} ν ∫ 0 T ∫ ∂ Ω ( ∣ κ ∣ + ∣ α ′ ∣ ) ∣ u ν ( t ) ⋅ τ ∣ 2 d S d t ≤ C ν 1 − q ∫ 0 T ∥ u ν ( t ) ⋅ τ ∥ L 2 ( ∂ Ω ) 2 d t ≤ C ν 1 − q ∫ 0 T ∥ ω ν ( t ) ∥ L 1 ( Ω ) ∥ ω ν ( t ) ∥ L 2 ( Ω ) d t ≤ C ν 1 − q − 1/ ( 2 r ) ∫ 0 T t − 1/ ( 2 r ) d t → 0 as ν → 0. Thus, passing to the limit in (9) yields
∥ u ( t ) ∥ L 2 ( Ω ) = ∥ K Ω ( ω 0 ) ∥ L 2 ( Ω ) . \|u(t)\|_{L^2(\Omega)}=\|K_\Omega(\omega_0)\|_{L^2(\Omega)}. ∥ u ( t ) ∥ L 2 ( Ω ) = ∥ K Ω ( ω 0 ) ∥ L 2 ( Ω ) . Appendix A. Auxiliary results
Proposition A.1 (Morrey’s inequality [11 ] ). Let Ω ⊂ R 2 \Omega\subset\mathbb{R}^2 Ω ⊂ R 2 C 1 C^1 C 1 γ ∈ ( 0 , 1 ) \gamma\in(0,1) γ ∈ ( 0 , 1 ) C > 0 C>0 C > 0 v ∈ W 1 , 2 / ( 1 − γ ) ( Ω ) v\in W^{1,2/(1-\gamma)}(\Omega) v ∈ W 1 , 2/ ( 1 − γ ) ( Ω ) C 0 , γ ( Ω ‾ ) C^{0,\gamma}(\overline{\Omega}) C 0 , γ ( Ω )
[ v ] C 0 , γ ( Ω ‾ ) ≤ C ∥ v ∥ W 1 , 2 / ( 1 − γ ) ( Ω ) . [v]_{C^{0,\gamma}(\overline{\Omega})}\le C\|v\|_{W^{1,2/(1-\gamma)}(\Omega)}. [ v ] C 0 , γ ( Ω ) ≤ C ∥ v ∥ W 1 , 2/ ( 1 − γ ) ( Ω ) . Lemma A.2 (Biot–Savart estimate). Let Ω ⊂ R 2 \Omega\subset\mathbb{R}^2 Ω ⊂ R 2 C 2 C^2 C 2 p ∈ ( 1 , ∞ ) p\in(1,\infty) p ∈ ( 1 , ∞ )
K Ω : L p ( Ω ) → W 1 , p ( Ω ) ∩ H 1 K_\Omega:L^p(\Omega)\to W^{1,p}(\Omega)\cap H^1 K Ω : L p ( Ω ) → W 1 , p ( Ω ) ∩ H 1 is a bounded linear operator satisfying, for every f ∈ L p ( Ω ) f\in L^p(\Omega) f ∈ L p ( Ω )
curl K Ω ( f ) = f , ∥ K Ω ( f ) ∥ W 1 , p ( Ω ) ≤ C ∥ f ∥ L p ( Ω ) . \operatorname{curl}K_\Omega(f)=f,\qquad\|K_\Omega(f)\|_{W^{1,p}(\Omega)}\le C\|f\|_{L^p(\Omega)}. curl K Ω ( f ) = f , ∥ K Ω ( f ) ∥ W 1 , p ( Ω ) ≤ C ∥ f ∥ L p ( Ω ) . Proof. Set ψ ( y ) : = ∫ Ω G Ω ( x , y ) f ( x ) d x \psi(y):=\int_\Omega G_\Omega(x,y)f(x)\,dx ψ ( y ) := ∫ Ω G Ω ( x , y ) f ( x ) d x [13 ] , Theorem 9.15 and Lemma 9.17, we have
ψ ∈ W 2 , p ( Ω ) ∩ W 0 1 , p ( Ω ) , Δ ψ = f , \psi\in W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega),\qquad\Delta\psi=f, ψ ∈ W 2 , p ( Ω ) ∩ W 0 1 , p ( Ω ) , Δ ψ = f , and
∥ K Ω ( f ) ∥ W 1 , p ( Ω ) = ∥ ∇ ⊥ ψ ∥ W 1 , p ( Ω ) ≤ C ∥ ψ ∥ W 2 , p ( Ω ) ≤ C ∥ f ∥ L p ( Ω ) . \lVert K_{\Omega}(f)\rVert_{W^{1,p}(\Omega)}=\lVert\nabla^{\perp}\psi\rVert_{W^{1,p}(\Omega)}\leq C\lVert\psi\rVert_{W^{2,p}(\Omega)}\leq C\lVert f\rVert_{L^{p}(\Omega)}. ∥ K Ω ( f ) ∥ W 1 , p ( Ω ) = ∥ ∇ ⊥ ψ ∥ W 1 , p ( Ω ) ≤ C ∥ ψ ∥ W 2 , p ( Ω ) ≤ C ∥ f ∥ L p ( Ω ) . Since p > 1 p>1 p > 1 ∇ ⊥ ψ = K Ω ( f ) ∈ H \nabla^{\perp}\psi=K_{\Omega}(f)\in H ∇ ⊥ ψ = K Ω ( f ) ∈ H
Proposition A.3 (Aubin–Lions lemma [4 ] ). Let X 0 ⊂ X ⊂ X 1 X_{0}\subset X\subset X_{1} X 0 ⊂ X ⊂ X 1 Banach spaces such that the embedding X 0 ⊂ X X_{0}\subset X X 0 ⊂ X X ⊂ X 1 X\subset X_{1} X ⊂ X 1
{ u ∈ L p ( 0 , T ; X 0 ) , ∂ t u ∈ L q ( 0 , T ; X 1 ) } \left\{u\in L^{p}(0,T;X_{0}),\ \partial_{t}u\in L^{q}(0,T;X_{1})\right\} { u ∈ L p ( 0 , T ; X 0 ) , ∂ t u ∈ L q ( 0 , T ; X 1 ) } embeds compactly into
L p ( 0 , T ; X ) L^{p}(0,T;X) L p ( 0 , T ; X ) p < ∞ p<\infty p < ∞ q ∈ [ 1 , ∞ ] q\in[1,\infty] q ∈ [ 1 , ∞ ]
C ( [ 0 , T ] ; X ) C([0,T];X) C ([ 0 , T ] ; X ) p = ∞ p=\infty p = ∞ q ∈ ( 1 , ∞ ] q\in(1,\infty] q ∈ ( 1 , ∞ ]
The following lemma adapts the linear stability argument in [6 ] to bounded domains.
Lemma A.4. Let Ω ⊂ R 2 \Omega\subset\mathbb{R}^{2} Ω ⊂ R 2 bounded domain with C 2 C^{2} C 2 boundary , ρ ∈ ( 1 , ∞ ) \rho\in(1,\infty) ρ ∈ ( 1 , ∞ )
v ν ∈ C ( [ 0 , T ] ; H ) ∩ L 2 ( 0 , T ; V ) , v ∈ C ( [ 0 , T ] ; H ) ∩ L ∞ ( 0 , T ; W 1 , ρ ( Ω ) ) , v^{\nu}\in C([0,T];H)\cap L^{2}(0,T;V),\qquad v\in C([0,T];H)\cap L^{\infty}(0,T;W^{1,\rho}(\Omega)), v ν ∈ C ([ 0 , T ] ; H ) ∩ L 2 ( 0 , T ; V ) , v ∈ C ([ 0 , T ] ; H ) ∩ L ∞ ( 0 , T ; W 1 , ρ ( Ω )) , such that
v ν → v strongly in C ( [ 0 , T ] ; L 2 ( Ω ) ) as ν → 0. v^{\nu}\to v\ \text{strongly in }C([0,T];L^{2}(\Omega))\text{ as }\nu\to0. v ν → v strongly in C ([ 0 , T ] ; L 2 ( Ω )) as ν → 0. For θ 0 ∈ L ρ ( Ω ) \theta_{0}\in L^{\rho}(\Omega) θ 0 ∈ L ρ ( Ω ) θ ν ∈ C ( [ 0 , T ] ; L ρ ( Ω ) ) \theta^{\nu}\in C([0,T];L^{\rho}(\Omega)) θ ν ∈ C ([ 0 , T ] ; L ρ ( Ω )) (10) with drift v ν v^{\nu} v ν g ν = 0 g^{\nu}=0 g ν = 0 θ ν ( 0 ) = θ 0 \theta^{\nu}(0)=\theta_{0} θ ν ( 0 ) = θ 0
θ ν → θ strongly in C ( [ 0 , T ] ; L ρ ( Ω ) ) , \theta^{\nu}\to\theta\ \text{strongly in }C([0,T];L^{\rho}(\Omega)), θ ν → θ strongly in C ([ 0 , T ] ; L ρ ( Ω )) , where θ ∈ C ( [ 0 , T ] ; L ρ ( Ω ) ) \theta\in C([0,T];L^{\rho}(\Omega)) θ ∈ C ([ 0 , T ] ; L ρ ( Ω ))
{ ∂ t θ + v ⋅ ∇ θ = 0 in ( 0 , T ) × Ω , θ ( 0 ) = θ 0 in Ω . (33) \left\{
\begin{aligned}
\partial_{t}\theta+v\cdot\nabla\theta&=0 &&\text{in }(0,T)\times\Omega,\\
\theta(0)&=\theta_{0} &&\text{in }\Omega.
\end{aligned}
\right. \tag*{(33)} { ∂ t θ + v ⋅ ∇ θ θ ( 0 ) = 0 = θ 0 in ( 0 , T ) × Ω , in Ω. (33) Proof. First, we adapt the linear stability argument of [6 Lemma 3.3] for θ 0 ∈ L ∞ ( Ω ) \theta_{0}\in L^{\infty}(\Omega) θ 0 ∈ L ∞ ( Ω ) [6 proof of Theorem 3.2, Step 3] to generalize to θ 0 ∈ L ρ ( Ω ) \theta_{0}\in L^{\rho}(\Omega) θ 0 ∈ L ρ ( Ω )
Suppose that θ 0 ∈ L ∞ ( Ω ) \theta_{0}\in L^{\infty}(\Omega) θ 0 ∈ L ∞ ( Ω )
∥ θ ν ∥ L ∞ ( ( 0 , T ) × Ω ) ≤ ∥ θ 0 ∥ L ∞ ( Ω ) , sup s ∈ [ 0 , T ] ∥ θ ν ( s ) ∥ L 2 ( Ω ) ≤ ∥ θ 0 ∥ L 2 ( Ω ) . \lVert\theta^{\nu}\rVert_{L^{\infty}((0,T)\times\Omega)}\leq\lVert\theta_{0}\rVert_{L^{\infty}(\Omega)},\qquad\sup_{s\in[0,T]}\lVert\theta^{\nu}(s)\rVert_{L^{2}(\Omega)}\leq\lVert\theta_{0}\rVert_{L^{2}(\Omega)}. ∥ θ ν ∥ L ∞ (( 0 , T ) × Ω ) ≤ ∥ θ 0 ∥ L ∞ ( Ω ) , s ∈ [ 0 , T ] sup ∥ θ ν ( s ) ∥ L 2 ( Ω ) ≤ ∥ θ 0 ∥ L 2 ( Ω ) . By Banach–Alaoglu’s theorem, there exists a subsequence, which we label again as ν \nu ν
θ ν ⇀ ∗ θ weakly-* in L ∞ ( ( 0 , T ) × Ω ) , \theta^\nu\mathrel{\overset{*}{\rightharpoonup}} \theta\text{ weakly-* in } L^\infty((0,T)\times\Omega), θ ν ⇀ ∗ θ weakly-* in L ∞ (( 0 , T ) × Ω ) , for some θ ∈ L ∞ ( ( 0 , T ) × Ω ) \theta\in L^\infty((0,T)\times\Omega) θ ∈ L ∞ (( 0 , T ) × Ω ) θ \theta θ v ⋅ n = 0 v\cdot n=0 v ⋅ n = 0 [1 Theorem 3.1] to deduce that
θ ∈ C ( [ 0 , T ] ; L 2 ( Ω ) ) , ∥ θ ( t ) ∥ L 2 ( Ω ) = ∥ θ 0 ∥ L 2 ( Ω ) for all t ∈ [ 0 , T ] . \theta\in C([0,T];L^2(\Omega)),\qquad\|\theta(t)\|_{L^2(\Omega)}=\|\theta_0\|_{L^2(\Omega)}\quad\text{for all }t\in[0,T]. θ ∈ C ([ 0 , T ] ; L 2 ( Ω )) , ∥ θ ( t ) ∥ L 2 ( Ω ) = ∥ θ 0 ∥ L 2 ( Ω ) for all t ∈ [ 0 , T ] . Since every weak-* subsequential limit is this same solution, the weak-* convergence holds for the entire family. For φ ∈ C c ∞ ( Ω ) \varphi\in C^\infty_c(\Omega) φ ∈ C c ∞ ( Ω ) t → ∫ Ω θ ν ( t ) φ d x t\to\int_\Omega\theta^\nu(t)\varphi\,dx t → ∫ Ω θ ν ( t ) φ d x L 2 ( Ω ) L^2(\Omega) L 2 ( Ω )
sup s ∈ [ 0 , T ] ∣ ∫ Ω ( θ ν ( s ) − θ ( s ) ) φ d x ∣ → 0 for all φ ∈ L 2 ( Ω ) . (34) \sup_{s\in[0,T]}\left|\int_\Omega(\theta^\nu(s)-\theta(s))\varphi\,dx\right|\to0\quad\text{for all }\varphi\in L^2(\Omega). \tag*{(34)} s ∈ [ 0 , T ] sup ∫ Ω ( θ ν ( s ) − θ ( s )) φ d x → 0 for all φ ∈ L 2 ( Ω ) . (34) Assume the convergence in C ( [ 0 , T ] ; L 2 ( Ω ) ) C([0,T];L^2(\Omega)) C ([ 0 , T ] ; L 2 ( Ω )) ε > 0 \varepsilon>0 ε > 0 ν → 0 \nu\to0 ν → 0 t ν ∈ [ 0 , T ] t_\nu\in[0,T] t ν ∈ [ 0 , T ]
∥ θ ν ( t ν ) − θ ( t ν ) ∥ L 2 ( Ω ) ≥ ε . (35) \|\theta^\nu(t_\nu)-\theta(t_\nu)\|_{L^2(\Omega)}\geq\varepsilon. \tag*{(35)} ∥ θ ν ( t ν ) − θ ( t ν ) ∥ L 2 ( Ω ) ≥ ε . (35) From the compactness of [ 0 , T ] [0,T] [ 0 , T ] t ν → t ∈ [ 0 , T ] t_\nu\to t\in[0,T] t ν → t ∈ [ 0 , T ] ν \nu ν
θ ν ( t ν ) ⇀ θ ( t ) weakly in L 2 ( Ω ) , \theta^\nu(t_\nu)\rightharpoonup\theta(t)\text{ weakly in }L^2(\Omega), θ ν ( t ν ) ⇀ θ ( t ) weakly in L 2 ( Ω ) , where we used (A.2) and the continuity of θ \theta θ
∥ θ ( t ) ∥ L 2 ( Ω ) ≤ lim inf ν → 0 ∥ θ ν ( t ν ) ∥ L 2 ( Ω ) ≤ lim sup ν → 0 ∥ θ ν ( t ν ) ∥ L 2 ( Ω ) ≤ ∥ θ 0 ∥ L 2 ( Ω ) = ∥ θ ( t ) ∥ L 2 ( Ω ) , \|\theta(t)\|_{L^2(\Omega)}\leq\liminf_{\nu\to0}\|\theta^\nu(t_\nu)\|_{L^2(\Omega)}\leq\limsup_{\nu\to0}\|\theta^\nu(t_\nu)\|_{L^2(\Omega)}\leq\|\theta_0\|_{L^2(\Omega)}=\|\theta(t)\|_{L^2(\Omega)}, ∥ θ ( t ) ∥ L 2 ( Ω ) ≤ ν → 0 lim inf ∥ θ ν ( t ν ) ∥ L 2 ( Ω ) ≤ ν → 0 lim sup ∥ θ ν ( t ν ) ∥ L 2 ( Ω ) ≤ ∥ θ 0 ∥ L 2 ( Ω ) = ∥ θ ( t ) ∥ L 2 ( Ω ) , and therefore with Radon–Riesz
θ ν ( t ν ) → θ ( t ) strongly in L 2 ( Ω ) . \theta^\nu(t_\nu)\to\theta(t)\text{ strongly in }L^2(\Omega). θ ν ( t ν ) → θ ( t ) strongly in L 2 ( Ω ) . This yields a contradiction to (A.3):
0 < ε ≤ ∥ θ ν ( t ν ) − θ ( t ν ) ∥ L 2 ( Ω ) ≤ ∥ θ ν ( t ν ) − θ ( t ) ∥ L 2 ( Ω ) + ∥ θ ( t ) − θ ( t ν ) ∥ L 2 ( Ω ) → 0. 0<\varepsilon\leq\|\theta^\nu(t_\nu)-\theta(t_\nu)\|_{L^2(\Omega)}
\leq\|\theta^\nu(t_\nu)-\theta(t)\|_{L^2(\Omega)}
+\|\theta(t)-\theta(t_\nu)\|_{L^2(\Omega)}\to0. 0 < ε ≤ ∥ θ ν ( t ν ) − θ ( t ν ) ∥ L 2 ( Ω ) ≤ ∥ θ ν ( t ν ) − θ ( t ) ∥ L 2 ( Ω ) + ∥ θ ( t ) − θ ( t ν ) ∥ L 2 ( Ω ) → 0. So, we have
θ ν → θ strongly in C ( [ 0 , T ] ; L 2 ( Ω ) ) . \theta^\nu\to\theta\text{ strongly in }C([0,T];L^2(\Omega)). θ ν → θ strongly in C ([ 0 , T ] ; L 2 ( Ω )) . By the contraction property in L ∞ ( Ω ) L^\infty(\Omega) L ∞ ( Ω )
θ ν → θ strongly in C ( [ 0 , T ] ; L ρ ( Ω ) ) . \theta^\nu\to\theta\text{ strongly in }C([0,T];L^\rho(\Omega)). θ ν → θ strongly in C ([ 0 , T ] ; L ρ ( Ω )) . For θ 0 ∈ L ρ ( Ω ) \theta_0 \in L^\rho(\Omega) θ 0 ∈ L ρ ( Ω ) θ 0 , k ∈ L ∞ ( Ω ) \theta_{0,k} \in L^\infty(\Omega) θ 0 , k ∈ L ∞ ( Ω ) θ 0 , k → θ 0 \theta_{0,k} \to\theta_0 θ 0 , k → θ 0 L ρ L^\rho L ρ θ ν , θ k ν \theta^\nu,\theta_k^\nu θ ν , θ k ν
∥ θ ν − θ ∥ C ( [ 0 , T ] ; L ρ ( Ω ) ) ≤ ∥ θ ν − θ k ν ∥ C ( [ 0 , T ] ; L ρ ( Ω ) ) + ∥ θ k ν − θ k ∥ C ( [ 0 , T ] ; L ρ ( Ω ) ) + ∥ θ k − θ ∥ C ( [ 0 , T ] ; L ρ ( Ω ) ) \|\theta^\nu-\theta\|_{C([0,T];L^\rho(\Omega))} \leq\|\theta^\nu-\theta_k^\nu\|_{C([0,T];L^\rho(\Omega))}+\|\theta_k^\nu-\theta_k\|_{C([0,T];L^\rho(\Omega))}+\|\theta_k-\theta\|_{C([0,T];L^\rho(\Omega))} ∥ θ ν − θ ∥ C ([ 0 , T ] ; L ρ ( Ω )) ≤ ∥ θ ν − θ k ν ∥ C ([ 0 , T ] ; L ρ ( Ω )) + ∥ θ k ν − θ k ∥ C ([ 0 , T ] ; L ρ ( Ω )) + ∥ θ k − θ ∥ C ([ 0 , T ] ; L ρ ( Ω )) ≤ 2 ∥ θ 0 − θ 0 , k ∥ L ρ ( Ω ) + ∥ θ k ν − θ k ∥ C ( [ 0 , T ] ; L ρ ( Ω ) ) → 0 , \leq2\|\theta_0-\theta_{0,k}\|_{L^\rho(\Omega)}+\|\theta_k^\nu-\theta_k\|_{C([0,T];L^\rho(\Omega))}\to0, ≤ 2∥ θ 0 − θ 0 , k ∥ L ρ ( Ω ) + ∥ θ k ν − θ k ∥ C ([ 0 , T ] ; L ρ ( Ω )) → 0 , where we first take ν → 0 \nu\to0 ν → 0 k → ∞ k\to\infty k → ∞
Counterexample to strong vorticity convergence at q = 1 / 2 q=1/2 q = 1/2
The following counterexample shows that Theorem 1.1 cannot hold for any p ∈ ( 1 , ∞ ) p\in(1,\infty) p ∈ ( 1 , ∞ ) q = 1 2 q=\frac{1}{2} q = 2 1
Lemma B.1. Let Ω = B 1 ( 0 ) ⊂ R 2 \Omega=B_1(0)\subset\mathbb{R}^2 Ω = B 1 ( 0 ) ⊂ R 2 T > 0 T>0 T > 0 u ν u^\nu u ν u 0 ( x ) = ( − x 2 x 1 ) u_0(x)=\begin{pmatrix}-x_2\\x_1\end{pmatrix} u 0 ( x ) = ( − x 2 x 1 ) α ν = ν − 1 / 2 \alpha^\nu=\nu^{-1/2} α ν = ν − 1/2
u ν → u 0 strongly in C ( [ 0 , T ] ; L 2 ( Ω ) ) , u^\nu\to u_0\ \text{strongly in }C([0,T];L^2(\Omega)), u ν → u 0 strongly in C ([ 0 , T ] ; L 2 ( Ω )) , where u ( t ) = u 0 u(t)=u_0 u ( t ) = u 0 ω = 2 \omega=2 ω = 2 ρ ∈ ( 1 , ∞ ] \rho\in(1,\infty] ρ ∈ ( 1 , ∞ ] C > 0 C>0 C > 0 ν 0 > 0 \nu_0>0 ν 0 > 0
inf t ∈ [ T / 2 , T ] ∥ ω ν ( t ) − 2 ∥ L ρ ( Ω ) ≥ C ν − ( ρ − 1 ) / ( 2 ρ ) \inf_{t\in[T/2,T]}\|\omega^\nu(t)-2\|_{L^\rho(\Omega)}\geq C\nu^{-(\rho-1)/(2\rho)} t ∈ [ T /2 , T ] inf ∥ ω ν ( t ) − 2 ∥ L ρ ( Ω ) ≥ C ν − ( ρ − 1 ) / ( 2 ρ ) for ν ∈ ( 0 , ν 0 ) \nu\in(0,\nu_0) ν ∈ ( 0 , ν 0 )
Proof. The solution is given by u ν ( t , x ) = h ν ( t , ∣ x ∣ ) x ⊥ u^\nu(t,x)=h^\nu(t,|x|)x^\perp u ν ( t , x ) = h ν ( t , ∣ x ∣ ) x ⊥ h ν h^\nu h ν
{ ∂ t h ν = ν ( ∂ r r h ν + 3 r ∂ r h ν ) , ( t , r ) ∈ ( 0 , T ) × ( 0 , 1 ) , ∂ r h ν ( t , 1 ) + α ν h ν ( t , 1 ) = 0 , t ∈ ( 0 , T ) , ∂ r h ν ( t , 0 ) = 0 , h ν ( 0 , r ) = 1. \begin{cases}
\partial_t h^\nu=\nu\left(\partial_{rr}h^\nu+\dfrac{3}{r}\partial_r h^\nu\right), & (t,r)\in(0,T)\times(0,1),\\
\partial_rh^\nu(t,1)+\alpha^\nu h^\nu(t,1)=0, & t\in(0,T),\\
\partial_rh^\nu(t,0)=0,\qquad h^\nu(0,r)=1.
\end{cases} ⎩ ⎨ ⎧ ∂ t h ν = ν ( ∂ r r h ν + r 3 ∂ r h ν ) , ∂ r h ν ( t , 1 ) + α ν h ν ( t , 1 ) = 0 , ∂ r h ν ( t , 0 ) = 0 , h ν ( 0 , r ) = 1. ( t , r ) ∈ ( 0 , T ) × ( 0 , 1 ) , t ∈ ( 0 , T ) , With the maximum principle, we have 0 ≤ h ν ≤ 1 0\leq h^\nu\leq1 0 ≤ h ν ≤ 1 ( h ν − 1 ) r 3 (h^\nu-1)r^3 ( h ν − 1 ) r 3
1 2 d d t ∫ 0 1 ∣ h ν − 1 ∣ 2 r 3 d r + ν ∫ 0 1 ∣ ∂ r h ν ∣ 2 r 3 d r = ν h ν ( t , 1 ) ( 1 − h ν ( t , 1 ) ) ≤ ν 4 , \frac{1}{2}\frac{d}{dt}\int_0^1|h^\nu-1|^2r^3\,dr+\nu\int_0^1|\partial_rh^\nu|^2r^3\,dr=\sqrt{\nu}\,h^\nu(t,1)(1-h^\nu(t,1))\leq\frac{\sqrt{\nu}}{4}, 2 1 d t d ∫ 0 1 ∣ h ν − 1 ∣ 2 r 3 d r + ν ∫ 0 1 ∣ ∂ r h ν ∣ 2 r 3 d r = ν h ν ( t , 1 ) ( 1 − h ν ( t , 1 )) ≤ 4 ν , which implies
sup t ∈ [ 0 , T ] ∥ u ν ( t ) − u 0 ∥ L 2 ( B 1 ( 0 ) ) 2 = sup t ∈ [ 0 , T ] 2 π ∫ 0 1 ∣ h ν ( t , r ) − 1 ∣ 2 r 3 d r ≤ π T ν . \sup_{t\in[0,T]}\|u^\nu(t)-u_0\|_{L^2(B_1(0))}^2=\sup_{t\in[0,T]}2\pi\int_0^1|h^\nu(t,r)-1|^2r^3\,dr\leq\pi T\sqrt{\nu}. t ∈ [ 0 , T ] sup ∥ u ν ( t ) − u 0 ∥ L 2 ( B 1 ( 0 )) 2 = t ∈ [ 0 , T ] sup 2 π ∫ 0 1 ∣ h ν ( t , r ) − 1 ∣ 2 r 3 d r ≤ π T ν . We next show that the tangential boundary traces remain separated. For r ∈ ( 1 / 2 , 1 ) r\in(1/2,1) r ∈ ( 1/2 , 1 )
h ‾ ν ( t , r ) = 1 + A T − A T e t − ( 1 − r ) / ν , A T = ( 2 e T − 1 ) − 1 . \overline{h}^\nu(t,r)=1+A_T-A_Te^{t-(1-r)/\sqrt{\nu}},\qquad A_T=(2e^T-1)^{-1}. h ν ( t , r ) = 1 + A T − A T e t − ( 1 − r ) / ν , A T = ( 2 e T − 1 ) − 1 . Choose a ν 0 > 0 \nu_0>0 ν 0 > 0 ν T ≤ 1 2 \sqrt{\nu T}\le\frac{1}{2} ν T ≤ 2 1 ν ∈ ( 0 , ν 0 ) \nu\in(0,\nu_0) ν ∈ ( 0 , ν 0 )
h ‾ ν ( 0 , r ) ≥ 1 = h ν ( 0 , r ) , r ∈ [ 1 / 2 , 1 ] , \overline{h}^{\nu}(0,r)\ge1=h^{\nu}(0,r),\qquad r\in[1/2,1], h ν ( 0 , r ) ≥ 1 = h ν ( 0 , r ) , r ∈ [ 1/2 , 1 ] , h ‾ ν ( t , 1 / 2 ) ≥ 1 ≥ h ν ( t , 1 / 2 ) , t ∈ [ 0 , T ] . \overline{h}^{\nu}(t,1/2)\ge1\ge h^{\nu}(t,1/2),\qquad t\in[0,T]. h ν ( t , 1/2 ) ≥ 1 ≥ h ν ( t , 1/2 ) , t ∈ [ 0 , T ] . Moreover, h ‾ ν \overline{h}^{\nu} h ν ( 0 , T ) × ( 1 / 2 , 1 ) (0,T)\times(1/2,1) ( 0 , T ) × ( 1/2 , 1 )
( ∂ t − ν ∂ r r − 3 ν r ∂ r ) h ‾ ν = 3 A T ν r e t − ( 1 − r ) / ν ≥ 0 , \left(\partial_t-\nu\partial_{rr}-\frac{3\nu}{r}\partial_r\right)\overline{h}^{\nu}=\frac{3A_T\sqrt{\nu}}{r}e^{t-(1-r)/\sqrt{\nu}}\ge0, ( ∂ t − ν ∂ r r − r 3 ν ∂ r ) h ν = r 3 A T ν e t − ( 1 − r ) / ν ≥ 0 , and its boundary values at r = 1 r=1 r = 1
∂ r h ‾ ν ( t , 1 ) + α ν h ‾ ν ( t , 1 ) = α ν ( 1 + A T − 2 A T e t ) ≥ 0. \partial_r\overline{h}^{\nu}(t,1)+\alpha^\nu\overline{h}^{\nu}(t,1)=\alpha^\nu(1+A_T-2A_Te^t)\ge0. ∂ r h ν ( t , 1 ) + α ν h ν ( t , 1 ) = α ν ( 1 + A T − 2 A T e t ) ≥ 0. With the comparison principle, we therefore get h ν ≤ h ‾ ν h^\nu\le\overline{h}^{\nu} h ν ≤ h ν [ 0 , T ] × [ 1 / 2 , 1 ] [0,T]\times[1/2,1] [ 0 , T ] × [ 1/2 , 1 ] t ∈ [ T / 2 , T ] t\in[T/2,T] t ∈ [ T /2 , T ]
h ν ( t , 1 ) ≤ h ‾ ν ( t , 1 ) = 1 − A T ( e t − 1 ) , h^\nu(t,1)\le\overline{h}^{\nu}(t,1)=1-A_T(e^t-1), h ν ( t , 1 ) ≤ h ν ( t , 1 ) = 1 − A T ( e t − 1 ) , which implies
u ( t , x ) ⋅ τ ( x ) − u ν ( t , x ) ⋅ τ ( x ) = 1 − h ν ( t , 1 ) ≥ A T ( e t − 1 ) ≥ A T ( e T / 2 − 1 ) > 0 on ∂ Ω . u(t,x)\cdot\tau(x)-u^\nu(t,x)\cdot\tau(x)=1-h^\nu(t,1)\ge A_T(e^t-1)\ge A_T(e^{T/2}-1)>0\quad\text{on }\partial\Omega. u ( t , x ) ⋅ τ ( x ) − u ν ( t , x ) ⋅ τ ( x ) = 1 − h ν ( t , 1 ) ≥ A T ( e t − 1 ) ≥ A T ( e T /2 − 1 ) > 0 on ∂ Ω. Fix t ∈ [ T / 2 , T ] t\in[T/2,T] t ∈ [ T /2 , T ] w ( r ) : = r 2 ( 1 − h ν ( t , r ) ) w(r):=r^2(1-h^\nu(t,r)) w ( r ) := r 2 ( 1 − h ν ( t , r )) w ≥ 0 w\ge0 w ≥ 0 w ( 0 ) = 0 w(0)=0 w ( 0 ) = 0 w ( 1 ) ≥ A T ( e T / 2 − 1 ) > 0 w(1)\ge A_T(e^{T/2}-1)>0 w ( 1 ) ≥ A T ( e T /2 − 1 ) > 0 r ⋆ < 1 r_\star<1 r ⋆ < 1 w ( r ⋆ ) = 1 2 A T ( e T / 2 − 1 ) w(r_\star)=\frac{1}{2}A_T(e^{T/2}-1) w ( r ⋆ ) = 2 1 A T ( e T /2 − 1 ) w ( r ) ≥ 1 2 A T ( e T / 2 − 1 ) w(r)\ge\frac{1}{2}A_T(e^{T/2}-1) w ( r ) ≥ 2 1 A T ( e T /2 − 1 ) r ∈ [ r ⋆ , 1 ] r\in[r_\star,1] r ∈ [ r ⋆ , 1 ] A = { x ∈ B 1 ( 0 ) : r ⋆ < ∣ x ∣ < 1 } A=\{x\in B_1(0):r_\star<|x|<1\} A = { x ∈ B 1 ( 0 ) : r ⋆ < ∣ x ∣ < 1 }
( A T ( e T / 2 − 1 ) ) 2 4 ∣ A ∣ ≤ 2 π ∫ 0 1 w ( r ) 2 r d r = 2 π ∫ 0 1 ( 1 − h ν ( t , r ) ) 2 r 5 d r ≤ ∫ 0 1 ( 1 − h ν ( t , r ) ) 2 r 3 d r ≤ π T ν , \frac{\left(A_T(e^{T/2}-1)\right)^2}{4}|A|\le2\pi\int_0^1w(r)^2r\,\mathrm{d}r=2\pi\int_0^1(1-h^\nu(t,r))^2r^5\,\mathrm{d}r\le\int_0^1(1-h^\nu(t,r))^2r^3\,\mathrm{d}r\le\pi T\sqrt{\nu}, 4 ( A T ( e T /2 − 1 ) ) 2 ∣ A ∣ ≤ 2 π ∫ 0 1 w ( r ) 2 r d r = 2 π ∫ 0 1 ( 1 − h ν ( t , r ) ) 2 r 5 d r ≤ ∫ 0 1 ( 1 − h ν ( t , r ) ) 2 r 3 d r ≤ π T ν , which implies ∣ A ∣ ≤ C ν |A|\le C\sqrt{\nu} ∣ A ∣ ≤ C ν ω ν = curl u ν = 2 h ν + r ∂ r h ν \omega^\nu=\operatorname{curl}u^\nu=2h^\nu+r\partial_rh^\nu ω ν = curl u ν = 2 h ν + r ∂ r h ν w ′ ( r ) = r ( 2 − ω ν ( t , r ) ) w'(r)=r(2-\omega^\nu(t,r)) w ′ ( r ) = r ( 2 − ω ν ( t , r ))
∫ A ( 2 − ω ν ( t , x ) ) d x = 2 π ( w ( 1 ) − w ( r ⋆ ) ) ≥ π A T ( e T / 2 − 1 ) . \int_A(2-\omega^\nu(t,x))\,\mathrm{d}x=2\pi(w(1)-w(r_\star))\ge\pi A_T(e^{T/2}-1). ∫ A ( 2 − ω ν ( t , x )) d x = 2 π ( w ( 1 ) − w ( r ⋆ )) ≥ π A T ( e T /2 − 1 ) . We conclude with Hölder’s inequality, for every ρ ∈ ( 1 , ∞ ] \rho\in(1,\infty] ρ ∈ ( 1 , ∞ ]
∥ ω ν ( t ) − 2 ∥ L ρ ( Ω ) ≥ 1 ∣ A ∣ 1 − 1 / ρ ∫ A ( 2 − ω ν ( t , x ) ) d x ≥ π A T ( e T / 2 − 1 ) ∣ A ∣ 1 − 1 / ρ ≥ C ν − ( ρ − 1 ) / ( 2 ρ ) . \|\omega^\nu(t)-2\|_{L^\rho(\Omega)}\ge\frac{1}{|A|^{1-1/\rho}}\int_A(2-\omega^\nu(t,x))\,\mathrm{d}x\ge\frac{\pi A_T(e^{T/2}-1)}{|A|^{1-1/\rho}}\ge C\nu^{-(\rho-1)/(2\rho)}. ∥ ω ν ( t ) − 2 ∥ L ρ ( Ω ) ≥ ∣ A ∣ 1 − 1/ ρ 1 ∫ A ( 2 − ω ν ( t , x )) d x ≥ ∣ A ∣ 1 − 1/ ρ π A T ( e T /2 − 1 ) ≥ C ν − ( ρ − 1 ) / ( 2 ρ ) . References
Department of Mathematics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstr. 11, 91058 Erlangen, Germany
Email address: josef.demmel@fau.de
