Introduction
The Landau equation is a fundamental model in kinetic theory which can be used to describe the dynamics of collisional plasmas. It characterizes the evolution of a large system of interacting particles through the evolution of the phase-space density f ( t , x , v ) f(t,x,v) f ( t , x , v ) t ≥ 0 t \ge0 t ≥ 0 x , v ∈ R 3 x,v \in\mathbb{R}^{3} x , v ∈ R 3
∂ t f + v ⋅ ∇ x f = Q ( f ) , (1) \partial_t f + v \cdot\nabla_x f = Q(f), \tag*{(1)} ∂ t f + v ⋅ ∇ x f = Q ( f ) , (1) where the collision operator acts only on the velocity variable and is defined by
Q ( f ) ( v ) = div v ∫ R 3 Φ ( v − v ∗ ) ( f ( t , x , v ∗ ) ∇ v f ( t , x , v ) − f ( t , x , v ) ∇ v ∗ f ( t , x , v ∗ ) ) d v ∗ , (2) Q(f)(v) = \operatorname{div}_v \int_{\mathbb{R}^{3}} \Phi(v-v_*) \left(f(t,x,v_*)\nabla_v f(t,x,v) - f(t,x,v)\nabla_{v_*}f(t,x,v_*)\right)\,dv_*, \tag*{(2)} Q ( f ) ( v ) = div v ∫ R 3 Φ ( v − v ∗ ) ( f ( t , x , v ∗ ) ∇ v f ( t , x , v ) − f ( t , x , v ) ∇ v ∗ f ( t , x , v ∗ ) ) d v ∗ , (2) with
Φ ( z ) = ∣ z ∣ 2 + γ P ( z ) , P ( z ) = Id − z ⊗ z ∣ z ∣ 2 , (3) \Phi(z) = |z|^{2+\gamma}P(z), \qquad P(z) = \operatorname{Id} - \frac{z \otimes z}{|z|^2}, \tag*{(3)} Φ ( z ) = ∣ z ∣ 2 + γ P ( z ) , P ( z ) = Id − ∣ z ∣ 2 z ⊗ z , (3) where P ( z ) P(z) P ( z ) z z z γ \gamma γ γ ≥ − 3 \gamma\ge-3 γ ≥ − 3 γ = − 3 \gamma=-3 γ = − 3
The question of whether solutions to (1) develop finite-time singularities remains a challenging open problem. Global-in-time smooth solutions are known to exist only for specific classes of initial data. In the spatially homogeneous setting, this was established by Guillen and Silvestre [20 ] , with subsequent developments for non smooth initial data by Golding, Gualdani, and Loher [18 ] , Ji [26 ] ,
MPG is partially supported by NSF grant DMS-2511625 and by the CNS SPARK grant. and Desvillettes et al. [10 ] . In the perturbative regime near equilibrium, global well-posedness was proved by Guo [22 ] and later refined by Guo and Strain [37 ] . For initial data close to vacuum and interaction potentials satisfying γ > − 2 \gamma> -2 γ > − 2 [29 ] for soft potentials and by Chaturvedi [9 ] for hard potentials. On the other hand, the second author et al. [3 ] ruled out a class of Type I self-similar blow-up profiles. More recently, the second author et al. [2 ] proved the existence of finite-time singularities for interaction potentials γ ∈ ( 3 , 2 ] \gamma\in(\sqrt{3},2] γ ∈ ( 3 , 2 ]
The contribution of this manuscript is twofold: we show the existence of a class of smooth initial data for which the equation (1) admits a global-in-time smooth solution, and another class for which smooth solutions develop hydrodynamic singularities in finite time. Our results are collected in the following two theorems:
Theorem 1.1 (Global well-posedness). Let T > 0 T > 0 T > 0 γ ≥ − 3 \gamma\ge-3 γ ≥ − 3 L 0 ∈ R 3 × 3 L_0 \in\mathbb{R}^{3 \times3} L 0 ∈ R 3 × 3
det ( I + t L 0 ) > 0 (4) \det(I+tL_0)>0 \tag*{(4)} det ( I + t L 0 ) > 0 (4) for all 0 ≤ t ≤ T 0 \le t \le T 0 ≤ t ≤ T g i n ( ⋅ ) ≥ 0 g_{\mathrm{in}}(\cdot) \ge0 g in ( ⋅ ) ≥ 0 R 3 \mathbb{R}^3 R 3
The inhomogeneous Landau equation (1) with initial data
f i n ( x , v ) = g i n ( v − L 0 x ) , f_{\mathrm{in}}(x,v)=g_{\mathrm{in}}(v-L_0x), f in ( x , v ) = g in ( v − L 0 x ) , has a classical solution f ∈ C ∞ ( [ 0 , T ] × R 6 ) ∩ L t , x ∞ ( L v p ) f \in C^\infty([0,T]\times\mathbb{R}^6)\cap L^\infty_{t,x}(L^p_v) f ∈ C ∞ ([ 0 , T ] × R 6 ) ∩ L t , x ∞ ( L v p ) 1 ≤ p ≤ + ∞ 1 \le p \le+\infty 1 ≤ p ≤ + ∞
Our second result concerns formation of hydrodynamic singularities in finite time in the case L 0 = − I L_0=-I L 0 = − I
Theorem 1.2 (Hydrodynamic implosion). Let γ ≥ − 3 \gamma\ge-3 γ ≥ − 3 g i n ≥ 0 g_{\mathrm{in}} \ge0 g in ≥ 0 R 3 \mathbb{R}^3 R 3 ∫ R 3 w g i n ( w ) d w = 0 \int_{\mathbb{R}^3} w g_{\mathrm{in}}(w)\,\mathrm{d}w=0 ∫ R 3 w g in ( w ) d w = 0
f i n ( x , v ) : = g i n ( x + v ) . f_{\mathrm{in}}(x,v):=g_{\mathrm{in}}(x+v). f in ( x , v ) := g in ( x + v ) . Then the inhomogeneous Landau equation (1) with initial data f i n f_{\mathrm{in}} f in f f f
f f f t = 1 t=1 t = 1 f ∈ C ∞ ( [ 0 , 1 ) × R 6 ) ∩ L ∞ ( [ 0 , 1 ) × R 6 ) f \in C^\infty([0,1)\times\mathbb{R}^6)\cap L^\infty([0,1)\times\mathbb{R}^6) f ∈ C ∞ ([ 0 , 1 ) × R 6 ) ∩ L ∞ ([ 0 , 1 ) × R 6 )
The hydrodynamic quantities of f f f t = 1 t=1 t = 1 ρ \rho ρ u u u θ \theta θ f f f
ρ ( t , x ) : = ∫ R 3 f ( t , x , v ) d v = ( 1 − t ) − 3 ∫ R 3 g i n ( w ) d w , (5) \rho(t,x):=\int_{\mathbb{R}^3}f(t,x,v)\,\mathrm{d}v=(1-t)^{-3}\int_{\mathbb{R}^3}g_{\mathrm{in}}(w)\,\mathrm{d}w, \tag*{(5)} ρ ( t , x ) := ∫ R 3 f ( t , x , v ) d v = ( 1 − t ) − 3 ∫ R 3 g in ( w ) d w , (5) u ( t , x ) : = 1 ρ ( t , x ) ∫ R 3 v f ( t , x , v ) d v = − ( 1 − t ) − 1 x , (6) u(t,x):=\frac{1}{\rho(t,x)}\int_{\mathbb{R}^3}vf(t,x,v)\,\mathrm{d}v=-(1-t)^{-1}x, \tag*{(6)} u ( t , x ) := ρ ( t , x ) 1 ∫ R 3 v f ( t , x , v ) d v = − ( 1 − t ) − 1 x , (6) θ ( t , x ) : = 1 3 ∫ R 3 ∣ v − u ( t , x ) ∣ 2 f ( t , x , v ) d v ρ ( t , x ) = 1 3 ( 1 − t ) − 2 ∫ R 3 ∣ w ∣ 2 g i n ( w ) d w ∫ R 3 g i n ( w ) d w . (7) \theta(t,x):=\frac{1}{3}\frac{\int_{\mathbb{R}^3}|v-u(t,x)|^2f(t,x,v)\,\mathrm{d}v}{\rho(t,x)}
=\frac{1}{3}(1-t)^{-2}\frac{\int_{\mathbb{R}^3}|w|^2g_{\mathrm{in}}(w)\,\mathrm{d}w}{\int_{\mathbb{R}^3}g_{\mathrm{in}}(w)\,\mathrm{d}w}. \tag*{(7)} θ ( t , x ) := 3 1 ρ ( t , x ) ∫ R 3 ∣ v − u ( t , x ) ∣ 2 f ( t , x , v ) d v = 3 1 ( 1 − t ) − 2 ∫ R 3 g in ( w ) d w ∫ R 3 ∣ w ∣ 2 g in ( w ) d w . (7) The hydrodynamic fields (5) –(6) are a solution to the monoatomic compressible Euler equations (11) .
If γ ≥ − 2 \gamma\ge-2 γ ≥ − 2 t → 1 − t \to1^- t → 1 −
f ( t , x , v ) ∼ ρ ( t ) ( 2 π θ ( t ) ) 3 / 2 exp { − ∣ v − u ( t , x ) ∣ 2 2 θ ( t ) } . f(t,x,v)\sim\frac{\rho(t)}{(2\pi\theta(t))^{3/2}}\exp\left\{-\frac{|v-u(t,x)|^2}{2\theta(t)}\right\}. f ( t , x , v ) ∼ ( 2 π θ ( t ) ) 3/2 ρ ( t ) exp { − 2 θ ( t ) ∣ v − u ( t , x ) ∣ 2 } . If γ < − 2 \gamma< -2 γ < − 2 t → 1 − t \to1^- t → 1 − g i n g_{\mathrm{in}} g in
f ( t , x , v ) ∼ F ( s 0 , x + ( 1 − t ) v ) , s 0 = − 1 γ + 2 , f(t,x,v)\sim F(s_0,x+(1-t)v),\qquad s_0=-\frac{1}{\gamma+2}, f ( t , x , v ) ∼ F ( s 0 , x + ( 1 − t ) v ) , s 0 = − γ + 2 1 , where F ( s 0 , w ) F(s_0,w) F ( s 0 , w ) g i n ( w ) g_{\mathrm{in}}(w) g in ( w ) s = s 0 s=s_0 s = s 0
All statements made in Theorem 1.1 and Theorem 1.2 have a clear analog for the Boltzmann equation. In fact, already in 1963 Nikolskii observed in [32 ] that certain solutions to the Boltzmann equation exhibits a compressive behavior similar to the one of Theorem 1.2 . We have chosen to focus on the Landau equation for the sake of simplicity.
Some more comments are in order:
Initial data. The initial data f i n f_{\mathrm{in}} f in Theorem 1.1 and Theorem 1.2 are smooth and bounded, but not integrable in R 3 × R 3 \mathbb{R}^{3}\times\mathbb{R}^{3} R 3 × R 3 L x ∞ L v 1 ( R 3 ) ∩ L ∞ ( R 6 ) L_{x}^{\infty}L_{v}^{1}(\mathbb{R}^{3})\cap L^{\infty}(\mathbb{R}^{6}) L x ∞ L v 1 ( R 3 ) ∩ L ∞ ( R 6 ) ρ i n ( x ) \rho_{\mathrm{in}}(x) ρ in ( x ) θ i n ( x ) \theta_{\mathrm{in}}(x) θ in ( x ) x x x x x x f i n f_{\mathrm{in}} f in x x x
ρ i n u i n = ∫ v f i n ( x , v ) d v = c 1 + ρ i n L 0 x , \rho_{\mathrm{in}}u_{\mathrm{in}}=\int v f_{\mathrm{in}}(x,v)\,dv=c_{1}+\rho_{\mathrm{in}}L_{0}x, ρ in u in = ∫ v f in ( x , v ) d v = c 1 + ρ in L 0 x , M 2 ( f i n ) = ∫ ∣ v ∣ 2 f i n ( x , v ) d v = c 2 + ∣ L 0 x ∣ 2 ρ i n + 2 L 0 x ⋅ c 1 , M_{2}(f_{\mathrm{in}})=\int|v|^{2}f_{\mathrm{in}}(x,v)\,dv=c_{2}+|L_{0}x|^{2}\rho_{\mathrm{in}}+2L_{0}x\cdot c_{1}, M 2 ( f in ) = ∫ ∣ v ∣ 2 f in ( x , v ) d v = c 2 + ∣ L 0 x ∣ 2 ρ in + 2 L 0 x ⋅ c 1 , where
c 1 : = ∫ w g i n ( w ) d w , c 2 : = ∫ ∣ w ∣ 2 g i n d w , c_{1}:=\int w g_{\mathrm{in}}(w)\,dw,\qquad c_{2}:=\int|w|^{2}g_{\mathrm{in}}\,dw, c 1 := ∫ w g in ( w ) d w , c 2 := ∫ ∣ w ∣ 2 g in d w , and we remind that L 0 ∈ R 3 × 3 L_{0}\in\mathbb{R}^{3\times3} L 0 ∈ R 3 × 3 ∫ R 3 w g i n ( w ) d w = 0 \int_{\mathbb{R}^{3}}w g_{\mathrm{in}}(w)\,dw=0 ∫ R 3 w g in ( w ) d w = 0 g i n g_{\mathrm{in}} g in L 0 L_{0} L 0
The main results. The key observation in the proof of Theorem 1.1 is that solutions to (1) with initial data of the form g i n ( v − L 0 x ) g_{\mathrm{in}}(v-L_{0}x) g in ( v − L 0 x )
f ( t , x , v ) = g ( t , v − L ( t ) x ) , L ( t ) = L 0 ( I + t L 0 ) − 1 (8) f(t,x,v)=g(t,v-L(t)x),\qquad L(t)=L_{0}(I+tL_{0})^{-1} \tag*{(8)} f ( t , x , v ) = g ( t , v − L ( t ) x ) , L ( t ) = L 0 ( I + t L 0 ) − 1 (8) and that g g g (16) below). Hence, analyzing (1) with initial data of the form g i n ( v − L 0 x ) g_{\mathrm{in}}(v-L_{0}x) g in ( v − L 0 x ) (16) with general initial data. Our solution f f f Theorem 1.1 is stated for smooth initial data, we expect the same conclusion to hold for g i n ∈ L 1 g_{\mathrm{in}}\in L^{1} g in ∈ L 1 [26 ] or [10 ] for the homogeneous Landau equation. Examples of matrices L 0 L_{0} L 0 (4) globally in time are L 0 = I L_{0}=I L 0 = I L 0 = diag ( λ 1 , λ 2 , λ 3 ) L_{0}=\operatorname{diag}(\lambda_{1},\lambda_{2},\lambda_{3}) L 0 = diag ( λ 1 , λ 2 , λ 3 ) λ i ≥ 0 \lambda_{i}\ge0 λ i ≥ 0
L 0 = ( 0 K 0 0 0 0 0 0 0 ) , K ∈ R + . (9) L_{0}=\begin{pmatrix}0 & K & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{pmatrix},\qquad K\in\mathbb{R}_{+}. \tag*{(9)} L 0 = 0 0 0 K 0 0 0 0 0 , K ∈ R + . (9) More examples are given in Appendix § 5 .
The proof of Theorem 1.2 has a particularly simple structure. We will show that f f f
f ( t , x , v ) = h ( s ( t ) , ( 1 − t ) v + x ) , s ( t ) : = ∫ 0 t ( 1 − τ ) − 3 − γ d τ , (10) f(t,x,v)=h\left(s(t),(1-t)v+x\right),\qquad s(t):=\int_{0}^{t}(1-\tau)^{-3-\gamma}\,d\tau, \tag*{(10)} f ( t , x , v ) = h ( s ( t ) , ( 1 − t ) v + x ) , s ( t ) := ∫ 0 t ( 1 − τ ) − 3 − γ d τ , (10) where h h h g i n g_{\mathrm{in}} g in Theorem 1.1 if one takes L 0 = − I L_{0}=-I L 0 = − I L 0 L_{0} L 0 det ( I + t L 0 ) > 0 \det(I+tL_{0})>0 det ( I + t L 0 ) > 0 t = 1 t=1 t = 1 L 0 = I L_{0}=I L 0 = I Theorem 1.1 , and solutions to (1) with initial data f i n ( x , v ) = g i n ( v − x ) f_{\mathrm{in}}(x,v)=g_{\mathrm{in}}(v-x) f in ( x , v ) = g in ( v − x ) f i n ( x , v ) = g i n ( v + x ) f_{\mathrm{in}}(x,v)=g_{\mathrm{in}}(v+x) f in ( x , v ) = g in ( v + x )
u ( t , x ) : = ∫ v f ( t , x , v ) d v ∫ f ( t , x , v ) d v . u(t,x):=\frac{\int v f(t,x,v)\,dv}{\int f(t,x,v)\,dv}. u ( t , x ) := ∫ f ( t , x , v ) d v ∫ v f ( t , x , v ) d v . For L 0 = I L_0=I L 0 = I t = 0 t=0 t = 0 u ( 0 , x ) = x u(0,x)=x u ( 0 , x ) = x L 0 = − I L_0=-I L 0 = − I u ( 0 , x ) = − x u(0,x)=-x u ( 0 , x ) = − x Figure 1 a and Figure 1 b, respectively.
Figure 1. Initial bulk velocities u 0 ( x ) = ± x u_0(x)=\pm x u 0 ( x ) = ± x f i n ( x , v ) = g i n ( v − L 0 x ) f_{\mathrm{in}}(x,v)=g_{\mathrm{in}}(v-L_0x) f in ( x , v ) = g in ( v − L 0 x ) L 0 = ± I L_0=\pm I L 0 = ± I
For γ = − 3 \gamma=-3 γ = − 3 f f f Theorem 1.2 actually stays regular in time until t = 1 t=1 t = 1 f ∈ C ∞ ( [ 0 , 1 ] × R 6 ) f\in C^\infty([0,1]\times\mathbb{R}^6) f ∈ C ∞ ([ 0 , 1 ] × R 6 ) γ ≠ − 3 \gamma\ne-3 γ = − 3 f f f (35) for more details. The hydrodynamic fields (5) –(6) are a solution to the monoatomic compressible Euler equations
{ ∂ t ρ + div ( ρ u ) = 0 , ∂ t ( ρ u ) + div ( ρ ( u ⊗ u ) + p I ) = 0 , ∂ t ( ρ ( 1 2 ∣ u ∣ 2 + 3 2 θ ) ) + div ( ( ρ ( 1 2 ∣ u ∣ 2 + 3 2 θ ) + p ) u ) = 0 , p = ρ θ , (11) \left\{
\begin{aligned}
\partial_t\rho+\operatorname{div}(\rho u)&=0,\\
\partial_t(\rho u)+\operatorname{div}\bigl(\rho(u\otimes u)+pI\bigr)&=0,\\
\partial_t\left(\rho\left(\frac{1}{2}|u|^2+\frac{3}{2}\theta\right)\right)+\operatorname{div}\left(\left(\rho\left(\frac{1}{2}|u|^2+\frac{3}{2}\theta\right)+p\right)u\right)&=0,
\end{aligned}
\right.
\qquad p=\rho\theta, \tag*{(11)} ⎩ ⎨ ⎧ ∂ t ρ + div ( ρ u ) ∂ t ( ρ u ) + div ( ρ ( u ⊗ u ) + p I ) ∂ t ( ρ ( 2 1 ∣ u ∣ 2 + 2 3 θ ) ) + div ( ( ρ ( 2 1 ∣ u ∣ 2 + 2 3 θ ) + p ) u ) = 0 , = 0 , = 0 , p = ρθ , (11) where θ \theta θ ( ρ , u , θ ) ∣ t = 0 = ( ρ i n , − x , θ i n ) (\rho,u,\theta)|_{t=0}=(\rho_{\mathrm{in}},-x,\theta_{\mathrm{in}}) ( ρ , u , θ ) ∣ t = 0 = ( ρ in , − x , θ in ) ρ i n , θ i n > 0 \rho_{\mathrm{in}},\theta_{\mathrm{in}}>0 ρ in , θ in > 0
General deformation matrices L 0 L_0 L 0 det ( I + t ∗ L 0 ) = 0 \det(I+t_*L_0)=0 det ( I + t ∗ L 0 ) = 0
ρ ( t ) = ρ ( 0 ) det ( I + t L 0 ) ≳ ( t ∗ − t ) − n λ , \rho(t)=\frac{\rho(0)}{\det(I+tL_0)}\gtrsim(t_*-t)^{-n_\lambda}, ρ ( t ) = det ( I + t L 0 ) ρ ( 0 ) ≳ ( t ∗ − t ) − n λ , where n λ n_\lambda n λ λ = − 1 t ∗ \lambda=-\frac{1}{t_*} λ = − t ∗ 1 L 0 L_0 L 0
Comparison with conditional regularity results. Theorem 1.2 , to the best of our knowledge, provides the first example of a solution to the Landau equation that exhibits finite-time singularities in the range γ ∈ [ − 3 , 3 ] \gamma\in[-3,\sqrt{3}] γ ∈ [ − 3 , 3 ] γ > 3 \gamma>\sqrt{3} γ > 3 [2 ] by exploiting the connection of the Landau equation with the compressible Euler system.
As in [2 ] , also in our case the singularity occurs due to tail fattening, meaning the function f f f [19 ] . There is no contradiction, however, between our work and [19 ] : the a priori estimate in [19 ] applies only to initial data with velocity decay of the form
f i n ( x , v ) ≤ C ( 1 + ∣ v ∣ ) − m , f_{\mathrm{in}}(x,v)\le C(1+|v|)^{-m}, f in ( x , v ) ≤ C ( 1 + ∣ v ∣ ) − m , for some m > 0 m>0 m > 0 [36 ] do not apply here because the initial data is not localized as required by their theorem.
Homoenergetic solutions. Solutions arising from Theorems 1.1 and 1.2 are of homoenergetic type. Homoenergetic solutions to kinetic equations were first introduced by Galkin and Truesdell [17 , 39 , 38 ] in the context of the Boltzmann equation. Their defining feature is that even if the function f f f v v v x x x ρ ( t , x ) \rho(t,x) ρ ( t , x ) θ ( t , x ) \theta(t,x) θ ( t , x ) t t t [8 ] studied homoenergetic affine flows for Boltzmann with hard-sphere and angular-cutoff interactions and showed that, depending on the initial data, solutions exist globally in time or exhibit an implosion with the density becoming unbounded in finite time.
A systematic mathematical theory for the long time regime of the homoenergetic Boltzmann equation was subsequently developed in [23 , 24 , 25 , 4 , 34 , 33 ] . These works highlight how the presence of the deformation matrix affects the long time asymptotics.
Several subsequent works have broadened this framework: [27 , 28 ] addressed non-cutoff Maxwell molecules and hard potentials; [12 , 13 ] treated uniform shear flow, deriving sharp regularity and stability results; [30 , 31 ] studied the linear Rayleigh–Boltzmann analogue; and [7 ] recently extended the theory to dissipative granular collisions under shear heating. More details about the long time asymptotics together with the matrix classification are provided in Appendix § 5 .
Structure of the paper. In § 2 , we present a motivation of the homoenergetic ansatz and derive the equation (16) satisfied by the function g g g (8) . In § 3 we show well posedness of solutions to (16) . In § 4 , we prove Theorem 1.2 . Finally, in Appendix § 5 , we recall a classification of deformation matrices which remain regular for all times and provide a heuristic argument that predicts the long time behavior of solutions.
Notation. Throughout, I I I 3 × 3 3\times3 3 × 3 L ∈ R 3 × 3 L\in\mathbb{R}^{3\times3} L ∈ R 3 × 3 ∣ L ∣ |L| ∣ L ∣ ⟨ w ⟩ : = ( 1 + ∣ w ∣ 2 ) 1 / 2 \langle w\rangle:=(1+|w|^2)^{1/2} ⟨ w ⟩ := ( 1 + ∣ w ∣ 2 ) 1/2 s s s
H ( g ) = ∫ g log g d w M s ( g ) = ∫ ∣ w ∣ s g d w , ρ ( t ) = M 0 ( g ( t ) ) . \mathcal{H}(g)=\int g\log g\,\mathrm{d}w \qquad M_s(g)=\int|w|^s g\,\mathrm{d}w,\qquad\rho(t)=M_0(g(t)). H ( g ) = ∫ g log g d w M s ( g ) = ∫ ∣ w ∣ s g d w , ρ ( t ) = M 0 ( g ( t )) . Fisher information is denoted
I ( g ) = ∫ ∣ ∇ g ∣ 2 g d w = 4 ∫ ∣ ∇ g ∣ 2 d w = ∫ ∣ ∇ log g ∣ 2 g d w . (12) I(g)=\int\frac{|\nabla g|^2}{g}\,\mathrm{d}w=4\int|\nabla\sqrt{g}|^2\,\mathrm{d}w=\int|\nabla\log g|^2g\,\mathrm{d}w. \tag*{(12)} I ( g ) = ∫ g ∣∇ g ∣ 2 d w = 4 ∫ ∣∇ g ∣ 2 d w = ∫ ∣∇ log g ∣ 2 g d w . (12) The Gateaux derivative of a functional E ( g ) \mathcal{E}(g) E ( g ) g g g h h h ⟨ E ′ ( g ) , h ⟩ \langle\mathcal{E}'(g),h\rangle ⟨ E ′ ( g ) , h ⟩
AI statement. AI tools were used only to improve the exposition and to proofread the final version of the manuscript.
The homoenergetic flow
While the early works of Truesdell and Galkin studied homoenergetic flows at the level of the moment equations, the equation governing g g g [8 ] and later by James, the third author, and Velazquez in [24 ] . In the same spirit, we begin with the classical inhomogeneous Landau (or Boltzmann) equation
∂ t f + v ⋅ ∇ x f = Q ( f ) , \partial_t f+v\cdot\nabla_x f=Q(f), ∂ t f + v ⋅ ∇ x f = Q ( f ) , and plug in the ansatz
f ( t , x , v ) = g ( t , v − ξ ( t , x ) ) , f(t,x,v)=g\left(t,v-\xi(t,x)\right), f ( t , x , v ) = g ( t , v − ξ ( t , x ) ) , which yields the following equation for g g g
∂ t g − ( D x ξ ⋅ w + D x ξ ⋅ ξ + ∂ t ξ ) ⋅ ∇ w g = Q ( g ) . \partial_tg-\left(D_x\xi\cdot w+D_x\xi\cdot\xi+\partial_t\xi\right)\cdot\nabla_wg=Q(g). ∂ t g − ( D x ξ ⋅ w + D x ξ ⋅ ξ + ∂ t ξ ) ⋅ ∇ w g = Q ( g ) . Here
w : = v − ξ ( t , x ) . w:=v-\xi(t,x). w := v − ξ ( t , x ) . We are looking for deformations ξ ( t , x ) \xi(t,x) ξ ( t , x ) ∂ t ξ + D x ξ ⋅ w + D x ξ ⋅ ξ \partial_t\xi+D_x\xi\cdot w+D_x\xi\cdot\xi ∂ t ξ + D x ξ ⋅ w + D x ξ ⋅ ξ t t t x x x v v v w w w
∂ t ξ + D x ξ ⋅ w + D x ξ ⋅ ξ = Z ( t , w ) . (13) \partial_t \xi+ D_x\xi\cdot w + D_x\xi\cdot\xi= Z(t,w). \tag*{(13)} ∂ t ξ + D x ξ ⋅ w + D x ξ ⋅ ξ = Z ( t , w ) . (13) We show that the only Z ( t , w ) Z(t,w) Z ( t , w ) (13) is a vector affine in w w w Z ( t , w ) − D x ξ ⋅ w Z(t,w)-D_x\xi\cdot w Z ( t , w ) − D x ξ ⋅ w w w w
D w Z ( t , w ) = D x ξ ( t , x ) . D_wZ(t,w)=D_x\xi(t,x). D w Z ( t , w ) = D x ξ ( t , x ) . Therefore
Z ( t , w ) = D x ξ ( t , x ) w + a ( t ) , D x ξ ( t , x ) = L ( t ) , Z(t,w)=D_x\xi(t,x)w+a(t), \qquad D_x\xi(t,x)=L(t), Z ( t , w ) = D x ξ ( t , x ) w + a ( t ) , D x ξ ( t , x ) = L ( t ) , for some functions a ( t ) a(t) a ( t ) L ( t ) ∈ R 3 × 3 L(t)\in\mathbb{R}^{3\times3} L ( t ) ∈ R 3 × 3
ξ ( t , x ) = L ( t ) x + b ( t ) , (14) \xi(t,x)=L(t)x+b(t), \tag*{(14)} ξ ( t , x ) = L ( t ) x + b ( t ) , (14) and
Z ( t , w ) = L ( t ) w + a ( t ) . (15) Z(t,w)=L(t)w+a(t). \tag*{(15)} Z ( t , w ) = L ( t ) w + a ( t ) . (15) Plugging (14) and (15) into (13) yields
( L ′ ( t ) + L ( t ) 2 ) x + b ′ ( t ) + L ( t ) b ( t ) = a ( t ) , \left(L'(t)+L(t)^2\right)x+b'(t)+L(t)b(t)=a(t), ( L ′ ( t ) + L ( t ) 2 ) x + b ′ ( t ) + L ( t ) b ( t ) = a ( t ) , which implies
L ′ ( t ) = − L ( t ) 2 , b ′ ( t ) = a ( t ) − L ( t ) b ( t ) . L'(t)=-L(t)^2, \qquad b'(t)=a(t)-L(t)b(t). L ′ ( t ) = − L ( t ) 2 , b ′ ( t ) = a ( t ) − L ( t ) b ( t ) . The equation for L L L
L ( t ) = L 0 ( I + t L 0 ) − 1 , L(t)=L_0(I+tL_0)^{-1}, L ( t ) = L 0 ( I + t L 0 ) − 1 , which leaves us with the final simplification
f ( t , x , v ) = g ( t , v − L ( t ) x − b ( t ) ) , ∂ t g − ( L ( t ) w + a ( t ) ) ⋅ ∇ w g = Q ( g ) . f(t,x,v)=g(t,v-L(t)x-b(t)), \qquad\partial_tg-(L(t)w+a(t))\cdot\nabla_wg=Q(g). f ( t , x , v ) = g ( t , v − L ( t ) x − b ( t )) , ∂ t g − ( L ( t ) w + a ( t )) ⋅ ∇ w g = Q ( g ) . Finally, the change of variables g ~ ( t , w ) = g ( t , w − b ( t ) ) \widetilde{g}(t,w)=g(t,w-b(t)) g ( t , w ) = g ( t , w − b ( t )) a ( t ) = b ( t ) = 0 a(t)=b(t)=0 a ( t ) = b ( t ) = 0
∂ t g = Q ( g ) + L ( t ) w ⋅ ∇ w g . (16) \partial_tg=Q(g)+L(t)w\cdot\nabla_wg. \tag*{(16)} ∂ t g = Q ( g ) + L ( t ) w ⋅ ∇ w g . (16) Well posedness for the homoenergetic flow
A priori estimates and Fisher information
This section contains a-priori estimates for solutions to (16) with initial data g i n ( w ) g_{\mathrm{in}}(w) g in ( w ) L ( t ) = L 0 ( I + t L 0 ) − 1 L(t)=L_0(I+tL_0)^{-1} L ( t ) = L 0 ( I + t L 0 ) − 1 L 0 L_0 L 0 [ 0 , T ] [0,T] [ 0 , T ]
det ( I + t L 0 ) > 0 for all t ∈ [ 0 , T ] \det(I+tL_0)>0 \qquad\text{for all }t\in[0,T] det ( I + t L 0 ) > 0 for all t ∈ [ 0 , T ] and for any γ ≥ − 3 \gamma\geq-3 γ ≥ − 3
Lemma 3.1 (Moments and Entropy). Let g g g (16) with M 2 ( g i n ) < ∞ M_2(g_{\mathrm{in}})<\infty M 2 ( g in ) < ∞ H ( g i n ) < ∞ \mathcal{H}(g_{\mathrm{in}})<\infty H ( g in ) < ∞ g g g
(1) The mass density ρ ( t ) \rho(t) ρ ( t )
ρ ( t ) = ρ ( 0 ) exp { − ∫ 0 t tr L ( s ) d s } = ρ ( 0 ) det ( I + t L 0 ) , ρ ( 0 ) : = ∫ R 3 g i n d w . (17) \rho(t)=\rho(0)\exp\left\{-\int_0^t\operatorname{tr}L(s)\,\mathrm{d}s\right\}=\frac{\rho(0)}{\det(I+tL_0)}, \qquad\rho(0):=\int_{\mathbb{R}^3}g_{\mathrm{in}}\,\mathrm{d}w. \tag*{(17)} ρ ( t ) = ρ ( 0 ) exp { − ∫ 0 t tr L ( s ) d s } = det ( I + t L 0 ) ρ ( 0 ) , ρ ( 0 ) := ∫ R 3 g in d w . (17) (2) The first moment M ⃗ 1 ( g ) : = ∫ w g ( w ) d w \vec{M}_1(g):=\int w g(w)\,\mathrm{d}w M 1 ( g ) := ∫ w g ( w ) d w
d d t M ⃗ 1 ( g ) = − tr L ( t ) M ⃗ 1 ( g ) − L ( t ) M ⃗ 1 ( g ) , (18) \frac{\mathrm{d}}{\mathrm{d}t}\vec{M}_1(g)=-\operatorname{tr}L(t)\vec{M}_1(g)-L(t)\vec{M}_1(g), \tag*{(18)} d t d M 1 ( g ) = − tr L ( t ) M 1 ( g ) − L ( t ) M 1 ( g ) , (18) In particular, if M ⃗ 1 ( g i n ) = 0 \vec{M}_1(g_{\mathrm{in}})=0 M 1 ( g in ) = 0 M ⃗ 1 ( g ( t ) ) = 0 \vec{M}_1(g(t))=0 M 1 ( g ( t )) = 0 t t t
(3) The second moment satisfies the estimate
M 2 ( g ) ≤ M 2 ( g i n ) exp { ∫ 0 t ( 2 ∣ L ( s ) ∣ − tr L ( s ) ) d s } . (19) M_2(g)\leq M_2(g_{\mathrm{in}})\exp\left\{\int_0^t\left(2\lvert L(s)\rvert-\operatorname{tr}L(s)\right)\,\mathrm{d}s\right\}. \tag*{(19)} M 2 ( g ) ≤ M 2 ( g in ) exp { ∫ 0 t ( 2 ∣ L ( s )∣ − tr L ( s ) ) d s } . (19) (4) The entropy satisfies the identity
H ( g ( t ) ) + ρ ( t ) ∫ 0 t ρ ( s ) − 1 D h ( g ( s ) ) d s = ρ ( t ) ρ ( 0 ) H ( g i n ) , (20) \mathcal{H}(g(t))+\rho(t)\int_{0}^{t}\rho(s)^{-1}D_h(g(s))\,\mathrm{d}s=\frac{\rho(t)}{\rho(0)}\mathcal{H}(g_{\mathrm{in}}), \tag*{(20)} H ( g ( t )) + ρ ( t ) ∫ 0 t ρ ( s ) − 1 D h ( g ( s )) d s = ρ ( 0 ) ρ ( t ) H ( g in ) , (20) where D h ( g ) D_h(g) D h ( g )
D h ( g ) : = − ∫ Q ( g ) log g d w ≥ 0. D_h(g):=-\int Q(g)\log g\,\mathrm{d}w\ge0. D h ( g ) := − ∫ Q ( g ) log g d w ≥ 0. Proof. We now prove statements (1)-(4) sequentially.
(1) For the mass, we simply observe that
ρ ′ ( t ) = ∫ L ( t ) w ⋅ ∇ g g d w = − tr L ( t ) ρ ( t ) . \rho'(t)=\int L(t)w\cdot\nabla_g g\,\mathrm{d}w=-\operatorname{tr}L(t)\rho(t). ρ ′ ( t ) = ∫ L ( t ) w ⋅ ∇ g g d w = − tr L ( t ) ρ ( t ) . Hence
ρ ( t ) = ρ ( 0 ) exp { − ∫ 0 t tr L ( s ) d s } . \rho(t)=\rho(0)\exp\left\{-\int_{0}^{t}\operatorname{tr}L(s)\,\mathrm{d}s\right\}. ρ ( t ) = ρ ( 0 ) exp { − ∫ 0 t tr L ( s ) d s } . Since
tr L ( t ) = tr ( L 0 ( I + t L 0 ) − 1 ) = d d t log det ( I + t L 0 ) , \operatorname{tr}L(t)=\operatorname{tr}\left(L_0(I+tL_0)^{-1}\right)=\frac{\mathrm{d}}{\mathrm{d}t}\log\det(I+tL_0), tr L ( t ) = tr ( L 0 ( I + t L 0 ) − 1 ) = d t d log det ( I + t L 0 ) , we have the second formula for ρ ( t ) \rho(t) ρ ( t )
ρ ( 0 ) exp { − ∫ 0 t tr L ( s ) d s } = ρ ( 0 ) det ( I + t L 0 ) . \rho(0)\exp\left\{-\int_{0}^{t}\operatorname{tr}L(s)\,\mathrm{d}s\right\}=\frac{\rho(0)}{\det(I+tL_0)}. ρ ( 0 ) exp { − ∫ 0 t tr L ( s ) d s } = det ( I + t L 0 ) ρ ( 0 ) . (2) The evolution of the first moments in (18) is obtained similarly, by differentiating ∫ w i g ( w ) d w \int w_i g(w)\,\mathrm{d}w ∫ w i g ( w ) d w
(3) For the energy, we compute
d d t M 2 ( g ) = − ∫ div w ( L ( t ) w ∣ w ∣ 2 ) g = − ∫ ( 2 L ( t ) w ⋅ w + tr L ( t ) ∣ w ∣ 2 ) g \frac{\mathrm{d}}{\mathrm{d}t}M_2(g)=-\int\operatorname{div}_w\left(L(t)w|w|^2\right)g=-\int\left(2L(t)w\cdot w+\operatorname{tr}L(t)|w|^2\right)g d t d M 2 ( g ) = − ∫ div w ( L ( t ) w ∣ w ∣ 2 ) g = − ∫ ( 2 L ( t ) w ⋅ w + tr L ( t ) ∣ w ∣ 2 ) g which gives (19) .
(4) Finally, the entropy identity (20) follows by solving the differential equality
d d t H ( g ) = ∫ ( log g + 1 ) ∂ t g = − D h ( g ) + ∫ ( log g + 1 ) L ( t ) w ⋅ ∇ g = − D h ( g ) + ∫ L ( t ) w ⋅ ∇ ( g log g ) = − D h ( g ) − tr L ( t ) H ( g ) . \begin{aligned}
\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{H}(g)=\int(\log g+1)\partial_tg=-D_h(g)+\int(\log g+1)L(t)w\cdot\nabla g \\
&=-D_h(g)+\int L(t)w\cdot\nabla(g\log g) \\
&=-D_h(g)-\operatorname{tr}L(t)\mathcal{H}(g).
\end{aligned} d t d H ( g ) = ∫ ( log g + 1 ) ∂ t g = − D h ( g ) + ∫ ( log g + 1 ) L ( t ) w ⋅ ∇ g = − D h ( g ) + ∫ L ( t ) w ⋅ ∇ ( g log g ) = − D h ( g ) − tr L ( t ) H ( g ) . In the next lemma, we establish estimates for the propagation and generation of moments:
Lemma 3.2. Let g g g (16) with M s ( g i n ) < ∞ M_s(g_{\mathrm{in}})<\infty M s ( g in ) < ∞ s ≥ 2 s\ge2 s ≥ 2 H ( g i n ) < ∞ \mathcal{H}(g_{\mathrm{in}})<\infty H ( g in ) < ∞
sup t ∈ [ 0 , T ] M s ( g ) ≤ C , (21) \sup_{t\in[0,T]}M_s(g)\le C, \tag*{(21)} t ∈ [ 0 , T ] sup M s ( g ) ≤ C , (21) where C C C s s s T T T M s ( g i n ) M_s(g_{\mathrm{in}}) M s ( g in ) H ( g i n ) \mathcal{H}(g_{\mathrm{in}}) H ( g in ) ∫ 0 T ∣ L ( t ) ∣ d t \int_0^T|L(t)|\,\mathrm{d}t ∫ 0 T ∣ L ( t ) ∣ d t
Moreover, if γ > 0 \gamma>0 γ > 0 M γ ( g i n ) < ∞ M_\gamma(g_{\mathrm{in}})<\infty M γ ( g in ) < ∞ s ≥ 2 s\ge2 s ≥ 2
M s ( g ) ≤ C t − s γ , (22) M_s(g)\le Ct^{-\frac{s}{\gamma}}, \tag*{(22)} M s ( g ) ≤ C t − γ s , (22) for small values of t t t M s ( g i n ) = ∞ M_s(g_{\mathrm{in}})=\infty M s ( g in ) = ∞
Proof. We decompose the equation into its transport and collision parts,
∂ t g = T g + Q ( g ) , \partial_tg=\mathcal{T}g+Q(g), ∂ t g = T g + Q ( g ) , and estimate the contribution of each term separately:
d d t M s ( g ) = ⟨ M s ′ ( g ) , T g ⟩ + ⟨ M s ′ ( g ) , Q ( g ) ⟩ . \frac{\mathrm{d}}{\mathrm{d}t}M_s(g)=\langle M_s'(g),\mathcal{T}g\rangle+\langle M_s'(g),Q(g)\rangle. d t d M s ( g ) = ⟨ M s ′ ( g ) , T g ⟩ + ⟨ M s ′ ( g ) , Q ( g )⟩ . The transport contribution is easily estimated:
⟨ M s ′ ( g ) , T g ⟩ = ∫ ∣ w ∣ s L ( t ) w ⋅ ∇ w g = − ∫ div w ( L ( t ) w ∣ w ∣ s ) g = − ∫ ( s L ( t ) w ⋅ w ∣ w ∣ s − 2 + tr L ( t ) ∣ w ∣ s ) g ≤ C ∣ L ( t ) ∣ M s ( g ) . \begin{aligned}
\langle M_s'(g),\mathcal{T}g\rangle=\int|w|^sL(t)w\cdot\nabla_wg=-\int\operatorname{div}_w\left(L(t)w|w|^s\right)g \\
&= - \int\left(s L(t)w \cdot w |w|^{s-2} + \operatorname{tr}L(t)|w|^s\right)g \le C|L(t)|M_s(g).
\end{aligned} ⟨ M s ′ ( g ) , T g ⟩ = ∫ ∣ w ∣ s L ( t ) w ⋅ ∇ w g = − ∫ div w ( L ( t ) w ∣ w ∣ s ) g = − ∫ ( s L ( t ) w ⋅ w ∣ w ∣ s − 2 + tr L ( t ) ∣ w ∣ s ) g ≤ C ∣ L ( t ) ∣ M s ( g ) . The effect of collisions has already been thoroughly studied. For soft potentials − 3 ≤ γ ≤ 0 -3 \le\gamma\le0 − 3 ≤ γ ≤ 0 [6 ] show that
⟨ M s ′ ( g ) , Q ( g ) ⟩ ≤ ( D h ( g ) + C ) M s ( g ) + C . \langle M_s'(g),Q(g)\rangle\le(D_h(g)+C)M_s(g)+C. ⟨ M s ′ ( g ) , Q ( g )⟩ ≤ ( D h ( g ) + C ) M s ( g ) + C . Combining the previous estimates, we obtain
d d t M s ( g ) ≤ ( D h ( g ) + C T ) M s ( g ) + C . \frac{d}{dt}M_s(g) \le(D_h(g)+C_T)M_s(g)+C. d t d M s ( g ) ≤ ( D h ( g ) + C T ) M s ( g ) + C . Applying Grönwall’s inequality yields
M s ( g ( t ) ) ≤ ( M s ( g i n ) + C t ) exp { ∫ 0 t ( D h ( g ( s ) ) + C ) d s } , M_s(g(t)) \le(M_s(g_{\mathrm{in}})+Ct)\exp\left\{\int_0^t(D_h(g(s))+C)\,ds\right\}, M s ( g ( t )) ≤ ( M s ( g in ) + C t ) exp { ∫ 0 t ( D h ( g ( s )) + C ) d s } , which proves the claim for − 3 ≤ γ ≤ 0 -3 \le\gamma\le0 − 3 ≤ γ ≤ 0 (20) .
For hard potentials, it only remains to estimate the collision term. This was established for 0 < γ ≤ 1 0 < \gamma\le1 0 < γ ≤ 1 [11 ] . We briefly sketch the argument for γ > 1 \gamma> 1 γ > 1
⟨ M s ′ ( g ) , Q ( g ) ⟩ ≤ s ∬ ∣ w − w ∗ ∣ γ ( − ∣ w ∣ s − ∣ w ∗ ∣ s + s 2 ∣ w ∣ s − 2 ∣ w ∗ ∣ 2 + s 2 ∣ w ∗ ∣ s − 2 ∣ w ∣ 2 ) g ( w ) g ( w ∗ ) d w d w ∗ . \langle M_s'(g),Q(g)\rangle\le s\iint|w-w_*|^\gamma\left(-|w|^s-|w_*|^s+\frac{s}{2}|w|^{s-2}|w_*|^2+\frac{s}{2}|w_*|^{s-2}|w|^2\right)g(w)g(w_*)\,dw\,dw_*. ⟨ M s ′ ( g ) , Q ( g )⟩ ≤ s ∬ ∣ w − w ∗ ∣ γ ( − ∣ w ∣ s − ∣ w ∗ ∣ s + 2 s ∣ w ∣ s − 2 ∣ w ∗ ∣ 2 + 2 s ∣ w ∗ ∣ s − 2 ∣ w ∣ 2 ) g ( w ) g ( w ∗ ) d w d w ∗ . We now bound the right hand side with a Povzner-type inequality. This inequality was proved in Lemma 2.5 of [21 ] for 0 < γ ≤ 1 0 < \gamma\le1 0 < γ ≤ 1 γ > 0 \gamma> 0 γ > 0 s > 2 s > 2 s > 2
( − ∣ w ∣ s − ∣ w ∗ ∣ s + s 2 ∣ w ∣ s − 2 ∣ w ∗ ∣ 2 + s 2 ∣ w ∗ ∣ s − 2 ∣ w ∣ 2 ) ∣ w − w ∗ ∣ γ ≤ − 2 − ( γ + 1 ) ∣ w ∣ s + γ − 2 − ( γ + 1 ) ∣ w ∗ ∣ s + γ + ∣ w ∣ s ∣ w ∗ ∣ γ + ∣ w ∣ γ ∣ w ∗ ∣ s + C γ , s ( ∣ w ∣ s − 2 + γ ∣ w ∗ ∣ 2 + ∣ w ∣ 2 ∣ w ∗ ∣ s − 2 + γ ) . \begin{aligned}
&\left(-|w|^s-|w_*|^s+\frac{s}{2}|w|^{s-2}|w_*|^2+\frac{s}{2}|w_*|^{s-2}|w|^2\right)|w-w_*|^\gamma\\
&\le-2^{-(\gamma+1)}|w|^{s+\gamma}-2^{-(\gamma+1)}|w_*|^{s+\gamma}+|w|^s|w_*|^\gamma+|w|^\gamma|w_*|^s \\
&\qquad+C_{\gamma,s}\left(|w|^{s-2+\gamma}|w_*|^2+|w|^2|w_*|^{s-2+\gamma}\right).
\end{aligned} ( − ∣ w ∣ s − ∣ w ∗ ∣ s + 2 s ∣ w ∣ s − 2 ∣ w ∗ ∣ 2 + 2 s ∣ w ∗ ∣ s − 2 ∣ w ∣ 2 ) ∣ w − w ∗ ∣ γ ≤ − 2 − ( γ + 1 ) ∣ w ∣ s + γ − 2 − ( γ + 1 ) ∣ w ∗ ∣ s + γ + ∣ w ∣ s ∣ w ∗ ∣ γ + ∣ w ∣ γ ∣ w ∗ ∣ s + C γ , s ( ∣ w ∣ s − 2 + γ ∣ w ∗ ∣ 2 + ∣ w ∣ 2 ∣ w ∗ ∣ s − 2 + γ ) . For s ≥ 4 s \ge4 s ≥ 4 C γ , s = C γ s 1 + γ 2 C_{\gamma,s}=C_\gamma s^{1+\frac{\gamma}{2}} C γ , s = C γ s 1 + 2 γ
⟨ M s ′ ( g ) , Q ( g ) ⟩ ≤ − ρ ( t ) s 2 − γ M s + γ ( g ) + 2 s M γ ( g ) M s ( g ) + 2 s C γ , s M 2 ( g ) M s − 2 + γ ( g ) . (23) \langle M_s'(g),Q(g)\rangle\le-\rho(t)s2^{-\gamma}M_{s+\gamma}(g)+2sM_\gamma(g)M_s(g)+2sC_{\gamma,s}M_2(g)M_{s-2+\gamma}(g). \tag*{(23)} ⟨ M s ′ ( g ) , Q ( g )⟩ ≤ − ρ ( t ) s 2 − γ M s + γ ( g ) + 2 s M γ ( g ) M s ( g ) + 2 s C γ , s M 2 ( g ) M s − 2 + γ ( g ) . (23) Inequality (23) yields boundedness of M s ( g ) M_s(g) M s ( g ) 0 < γ ≤ 2 0 < \gamma\le2 0 < γ ≤ 2 M γ ( g ) M_\gamma(g) M γ ( g ) M 2 ( g ) M_2(g) M 2 ( g ) M s − 2 + γ ( g ) M_{s-2+\gamma}(g) M s − 2 + γ ( g ) M s ( g ) M_s(g) M s ( g ) γ > 2 \gamma> 2 γ > 2 γ \gamma γ
⟨ M γ ′ ( g ) , Q ( g ) ⟩ ≤ − ρ ( t ) γ 2 − γ M 2 γ ( g ) + 2 γ M γ ( g ) 2 + C γ M 2 ( g ) M 2 ( γ − 1 ) ( g ) . \langle M_\gamma'(g),Q(g)\rangle\le-\rho(t)\gamma2^{-\gamma}M_{2\gamma}(g)+2\gamma M_\gamma(g)^2+C_\gamma M_2(g)M_{2(\gamma-1)}(g). ⟨ M γ ′ ( g ) , Q ( g )⟩ ≤ − ρ ( t ) γ 2 − γ M 2 γ ( g ) + 2 γ M γ ( g ) 2 + C γ M 2 ( g ) M 2 ( γ − 1 ) ( g ) . We estimate M 2 ( γ − 1 ) ( g ) M_{2(\gamma-1)}(g) M 2 ( γ − 1 ) ( g ) L 1 L^1 L 1 M 2 γ ( g ) M_{2\gamma}(g) M 2 γ ( g ) M γ ( g ) M_\gamma(g) M γ ( g ) M 2 ( g ) M_2(g) M 2 ( g ) M 2 γ ( g ) M_{2\gamma}(g) M 2 γ ( g )
C γ M 2 ( g ) M 2 ( γ − 1 ) ( g ) ≤ C T , γ M 2 γ ( g ) 1 − 1 γ ≤ ε M 2 γ ( g ) + C T , γ , ε ; C_\gamma M_2(g)M_{2(\gamma-1)}(g)\le C_{T,\gamma}M_{2\gamma}(g)^{1-\frac{1}{\gamma}}\le\varepsilon M_{2\gamma}(g)+C_{T,\gamma,\varepsilon}; C γ M 2 ( g ) M 2 ( γ − 1 ) ( g ) ≤ C T , γ M 2 γ ( g ) 1 − γ 1 ≤ ε M 2 γ ( g ) + C T , γ , ε ; 2 γ M γ ( g ) 2 ≤ C T , γ M 2 γ ( g ) γ − 1 γ ≤ ε M 2 γ ( g ) + C T , γ , ε . 2\gamma M_\gamma(g)^2\le C_{T,\gamma}M_{2\gamma}(g)^{\frac{\gamma-1}{\gamma}}\le\varepsilon M_{2\gamma}(g)+C_{T,\gamma,\varepsilon}. 2 γ M γ ( g ) 2 ≤ C T , γ M 2 γ ( g ) γ γ − 1 ≤ ε M 2 γ ( g ) + C T , γ , ε . Choosing ε > 0 \varepsilon>0 ε > 0
d d t M γ ( g ) ≤ C T , γ , \frac{d}{dt}M_\gamma(g)\le C_{T,\gamma}, d t d M γ ( g ) ≤ C T , γ , which gives a uniform bound on M γ ( g ) M_\gamma(g) M γ ( g ) (23) , we obtain
⟨ M s ′ ( g ) , Q ( g ) ⟩ ≤ − ρ ( t ) s 2 − γ M s + γ ( g ) + C T , γ M s ( g ) + 2 s C γ , s M 2 ( g ) M s − 2 + γ ( g ) ≤ − ρ ( t ) s 2 − γ M s + γ ( g ) + C T , γ M s ( g ) + 2 s C γ , s M 2 ( g ) ρ ( t ) − 2 s + γ M s + γ ( g ) 1 − 2 s + γ ≤ − ρ ( t ) s 2 − ( γ + 1 ) M s + γ ( g ) + C T , γ M s ( g ) + C T , γ . \begin{aligned}
\langle M_s'(g),Q(g)\rangle&\le-\rho(t)s2^{-\gamma}M_{s+\gamma}(g)+C_{T,\gamma}M_s(g)+2sC_{\gamma,s}M_2(g)M_{s-2+\gamma}(g)\\
&\le-\rho(t)s2^{-\gamma}M_{s+\gamma}(g)+C_{T,\gamma}M_s(g)+2sC_{\gamma,s}M_2(g)\rho(t)^{-\frac{2}{s+\gamma}}M_{s+\gamma}(g)^{1-\frac{2}{s+\gamma}}\\
&\le-\rho(t)s2^{-(\gamma+1)}M_{s+\gamma}(g)+C_{T,\gamma}M_s(g)+C_{T,\gamma}.
\end{aligned} ⟨ M s ′ ( g ) , Q ( g )⟩ ≤ − ρ ( t ) s 2 − γ M s + γ ( g ) + C T , γ M s ( g ) + 2 s C γ , s M 2 ( g ) M s − 2 + γ ( g ) ≤ − ρ ( t ) s 2 − γ M s + γ ( g ) + C T , γ M s ( g ) + 2 s C γ , s M 2 ( g ) ρ ( t ) − s + γ 2 M s + γ ( g ) 1 − s + γ 2 ≤ − ρ ( t ) s 2 − ( γ + 1 ) M s + γ ( g ) + C T , γ M s ( g ) + C T , γ . This proves the boundedness of moments. Finally, the generation estimate (22) follows from the inequality
M s + γ ( g ) ≥ ρ ( t ) − γ s M s ( g ) s + γ s , M_{s+\gamma}(g)\ge\rho(t)^{-\frac{\gamma}{s}}M_s(g)^{\frac{s+\gamma}{s}}, M s + γ ( g ) ≥ ρ ( t ) − s γ M s ( g ) s s + γ , which reduces the problem to solving a differential inequality of the form y ˙ ≤ − y 1 + γ s + C \dot y\le-y^{1+\frac{\gamma}{s}}+C y ˙ ≤ − y 1 + s γ + C
From the previous lemma we get information on the moments and energy for the original function f f f
Corollary 3.1. Assume for simplicity that ∫ w g i n ( w ) d w = 0 \int w_{\mathrm{gin}}(w)\,dw=0 ∫ w gin ( w ) d w = 0 Then, the bounds (22) and (21) on the moments of g g g translate to the following bounds for f f f solution to (1) :
∫ R 3 f ( t , x , v ) d v = ∫ R 3 f i n ( x , v ) d v det ( I + t L 0 ) , \int_{\mathbb{R}^{3}} f(t,x,v)\,dv=\frac{\int_{\mathbb{R}^{3}}f_{\mathrm{in}}(x,v)\,dv}{\det(I+tL_{0})}, ∫ R 3 f ( t , x , v ) d v = det ( I + t L 0 ) ∫ R 3 f in ( x , v ) d v , sup t ∈ [ 0 , T ] sup x ∫ ∣ v − u ( t , x ) ∣ s f ( t , x , v ) d v < ∞ , \sup_{t\in[0,T]}\sup_{x}\int|v-u(t,x)|^{s}f(t,x,v)\,dv<\infty, t ∈ [ 0 , T ] sup x sup ∫ ∣ v − u ( t , x ) ∣ s f ( t , x , v ) d v < ∞ , where u ( t , x ) u(t,x) u ( t , x ) denotes the bulk velocity
u ( t , x ) : = 1 ρ ∫ R 3 v f ( t , x , v ) d v = 1 ρ ∫ R 3 ( w + L ( t ) x ) g ( w ) d w = L ( t ) x . u(t,x):=\frac{1}{\rho}\int_{\mathbb{R}^{3}}v f(t,x,v)\,dv=\frac{1}{\rho}\int_{\mathbb{R}^{3}}(w+L(t)x)g(w)\,dw=L(t)x. u ( t , x ) := ρ 1 ∫ R 3 v f ( t , x , v ) d v = ρ 1 ∫ R 3 ( w + L ( t ) x ) g ( w ) d w = L ( t ) x . Moreover
M 2 ( f ( t , x , ⋅ ) ) = ∫ R 3 ∣ v ∣ 2 f ( t , x , v ) d v = ∫ ∣ v − u ( t , x ) ∣ 2 f ( t , x , v ) d v ⏟ < + ∞ + ∣ u ( t , x ) ∣ 2 ρ ( t , x ) M_{2}(f(t,x,\cdot))=\int_{\mathbb{R}^{3}}|v|^{2}f(t,x,v)\,dv=\underbrace{\int|v-u(t,x)|^{2}f(t,x,v)\,dv}_{<+\infty}+|u(t,x)|^{2}\rho(t,x) M 2 ( f ( t , x , ⋅ )) = ∫ R 3 ∣ v ∣ 2 f ( t , x , v ) d v = < + ∞ ∫ ∣ v − u ( t , x ) ∣ 2 f ( t , x , v ) d v + ∣ u ( t , x ) ∣ 2 ρ ( t , x ) = C ( t ) + ∣ L ( t ) x ∣ 2 ρ ( 0 ) det ( I + t L 0 ) . =C(t)+|L(t)x|^{2}\frac{\rho(0)}{\det(I+tL_{0})}. = C ( t ) + ∣ L ( t ) x ∣ 2 det ( I + t L 0 ) ρ ( 0 ) . Remark 3.1. The bounds of Corollary 3.1 show that implosion for the mass and energy of f f f occurs at the first time t ∗ t_{*} t ∗ such that det ( I + t ∗ L 0 ) = 0 \det(I+t_{*}L_{0})=0 det ( I + t ∗ L 0 ) = 0 If L 0 = − I L_{0}=-I L 0 = − I one is able to also provide the behavior of the function f f f This is the content of Theorem 1.2 .
The next lemma shows that the matrix
A [ g ] ( w ) = ∫ Φ ( w − z ) g ( z ) d z (24) A[g](w)=\int\Phi(w-z)g(z)\,dz \tag*{(24)} A [ g ] ( w ) = ∫ Φ ( w − z ) g ( z ) d z (24) is uniformly bounded from below.
Lemma 3.3. Let g g g be a solution of (16) such that g i n g_{\mathrm{in}} g in has M 2 ( g i n ) < ∞ M_{2}(g_{\mathrm{in}})<\infty M 2 ( g in ) < ∞ H ( g i n ) < ∞ \mathcal{H}(g_{\mathrm{in}})<\infty H ( g in ) < ∞ and let A [ g ] ( w ) A[g](w) A [ g ] ( w ) be defined as in (24) . Then
A [ g ] ( w ) ≥ λ ( t ) ⟨ w ⟩ γ I , A[g](w)\geq\lambda(t)\langle w\rangle^{\gamma}I, A [ g ] ( w ) ≥ λ ( t ) ⟨ w ⟩ γ I , where I I I denotes the identity, and the constant λ ( t ) > 0 \lambda(t)>0 λ ( t ) > 0 depends on ρ ( t ) \rho(t) ρ ( t ) M 2 ( g ( t ) ) M_{2}(g(t)) M 2 ( g ( t )) H ( g ( t ) ) \mathcal{H}(g(t)) H ( g ( t ))
Recall that, by Lemma 3.1, all these quantities remain bounded as long as ∫ 0 T ∣ L ( t ) ∣ d t < ∞ \int_{0}^{T}|L(t)|\,dt<\infty ∫ 0 T ∣ L ( t ) ∣ d t < ∞
Proof. The bound for soft potentials is a direct consequence of the analysis performed in [35 ] , where it is shown that if g g g
M 0 ≤ ∫ g ( w ) d w ≤ M 0 − 1 , ∫ ∣ w ∣ 2 g ( w ) d w ≤ E 0 , ∫ g log ( 1 + g ) ≤ H 0 , M_{0}\leq\int g(w)\,dw\leq M_{0}^{-1},\qquad\int|w|^{2}g(w)\,dw\leq E_{0},\qquad\int g\log(1+g)\leq H_{0}, M 0 ≤ ∫ g ( w ) d w ≤ M 0 − 1 , ∫ ∣ w ∣ 2 g ( w ) d w ≤ E 0 , ∫ g log ( 1 + g ) ≤ H 0 , then
A [ g ] ( w ) ≥ λ ⟨ w ⟩ γ I , A[g](w)\geq\lambda\langle w\rangle^{\gamma}I, A [ g ] ( w ) ≥ λ ⟨ w ⟩ γ I , where λ \lambda λ M 0 M_{0} M 0 E 0 E_{0} E 0 H 0 H_{0} H 0 γ ∈ [ 0 , 1 ] \gamma\in[0,1] γ ∈ [ 0 , 1 ] [11 ] , which can be generalized to any γ > 0 \gamma>0 γ > 0 □ \square □
Boundedness of Fisher information. Boundedness of the Fisher information is the key tool for establishing well-posedness of homoenergetic solutions for very soft potentials. This follows rather directly from the fact that the collision operator dissipates Fisher information. Moreover, the same argument also gives control of the Fisher dissipation functional introduced in [20 ] , namely
D i ( g ) : = ∑ k = 1 3 ∬ ∣ ∇ w , w ∗ ( ∣ w − w ∗ ∣ γ 2 b ~ k ⋅ ∇ w , w ∗ log G ) ∣ 2 G d w d w ∗ , G ( w , w ∗ ) = g ( w ) g ( w ∗ ) . (25) D_{i}(g):=\sum_{k=1}^{3}\iint\left|\nabla_{w,w_{*}}\left(|w-w_{*}|^{\frac{\gamma}{2}}\widetilde{b}_{k}\cdot\nabla_{w,w_{*}}\log G\right)\right|^{2}G\,dw\,dw_{*},\qquad G(w,w_{*})=g(w)g(w_{*}). \tag*{(25)} D i ( g ) := k = 1 ∑ 3 ∬ ∇ w , w ∗ ( ∣ w − w ∗ ∣ 2 γ b k ⋅ ∇ w , w ∗ log G ) 2 G d w d w ∗ , G ( w , w ∗ ) = g ( w ) g ( w ∗ ) . (25) where the vector fields b ~ k = b ~ k ( w − w ∗ ) ∈ R 6 \widetilde{b}_k=\widetilde{b}_k(w-w_*)\in\mathbb{R}^6 b k = b k ( w − w ∗ ) ∈ R 6
b 1 ( w − w ∗ ) : = ( 0 w ∗ 3 − w 3 w 2 − w ∗ 2 ) , b 2 ( w − w ∗ ) : = ( w 3 − w ∗ 3 0 w ∗ 1 − w 1 ) , b 3 ( w − w ∗ ) : = ( w 2 − w ∗ 2 w ∗ 1 − w 1 0 ) . b_1(w-w_*):=\begin{pmatrix}0\\w_*^3-w^3\\w^2-w_*^2\end{pmatrix},\qquad b_2(w-w_*):=\begin{pmatrix}w^3-w_*^3\\0\\w_*^1-w^1\end{pmatrix},\qquad b_3(w-w_*):=\begin{pmatrix}w^2-w_*^2\\w_*^1-w^1\\0\end{pmatrix}. b 1 ( w − w ∗ ) := 0 w ∗ 3 − w 3 w 2 − w ∗ 2 , b 2 ( w − w ∗ ) := w 3 − w ∗ 3 0 w ∗ 1 − w 1 , b 3 ( w − w ∗ ) := w 2 − w ∗ 2 w ∗ 1 − w 1 0 . and
b ~ k ( w − w ∗ ) : = ( b k ( w − w ∗ ) − b k ( w − w ∗ ) ) , k = 1 , 2 , 3. \widetilde{b}_k(w-w_*):=\begin{pmatrix}b_k(w-w_*)\\-b_k(w-w_*)\end{pmatrix},\qquad k=1,2,3. b k ( w − w ∗ ) := ( b k ( w − w ∗ ) − b k ( w − w ∗ ) ) , k = 1 , 2 , 3. We refer the reader to [26 ] for lower-bounds on D i ( g ) D_i(g) D i ( g )
Proposition 3.1. Let γ ∈ [ − 3 , 22 ] \gamma\in[-3,\sqrt{22}] γ ∈ [ − 3 , 22 ] det ( I + t L 0 ) > 0 \det(I+tL_0)>0 det ( I + t L 0 ) > 0 0 ≤ t ≤ T 0\leq t\leq T 0 ≤ t ≤ T g g g (16) with I ( g i n ) < ∞ I(g_{\mathrm{in}})<\infty I ( g in ) < ∞
e − ℓ ( t ) I ( g ( t ) ) + c γ ∫ 0 t e − ℓ ( s ) D i ( g ( s ) ) d s ≤ I ( g i n ) . (26) e^{-\ell(t)}I(g(t))+c_\gamma\int_0^t e^{-\ell(s)}D_i(g(s))\,\mathrm{d}s\leq I(g_{\mathrm{in}}). \tag*{(26)} e − ℓ ( t ) I ( g ( t )) + c γ ∫ 0 t e − ℓ ( s ) D i ( g ( s )) d s ≤ I ( g in ) . (26) Here c γ > 0 c_\gamma>0 c γ > 0 ∣ γ ∣ < 22 |\gamma|<\sqrt{22} ∣ γ ∣ < 22 c 22 = 0 c_{\sqrt{22}}=0 c 22 = 0
The growth rate ℓ ( t ) \ell(t) ℓ ( t )
ℓ ( t ) : = ∫ 0 t ( 2 ∣ L ( s ) ∣ − tr ( L ( s ) ) ) d s . \ell(t):=\int_0^t\left(2|L(s)|-\operatorname{tr}(L(s))\right)\,\mathrm{d}s. ℓ ( t ) := ∫ 0 t ( 2∣ L ( s ) ∣ − tr ( L ( s )) ) d s . Proof. We decompose the equation into its transport and collision parts,
∂ t g = T g + Q ( g ) , \partial_tg=\mathcal{T}g+Q(g), ∂ t g = T g + Q ( g ) , so that one can study the effect of each term on the Fisher information:
d d t I ( g ) = ⟨ I ′ ( g ) , T g ⟩ + ⟨ I ′ ( g ) , Q ( g ) ⟩ . \frac{\mathrm{d}}{\mathrm{d}t}I(g)=\langle I'(g),\mathcal{T}g\rangle+\langle I'(g),Q(g)\rangle. d t d I ( g ) = ⟨ I ′ ( g ) , T g ⟩ + ⟨ I ′ ( g ) , Q ( g )⟩ . In the breakthrough paper [20 ] , followed by [26 ] , it was shown that the second term is negative, with dissipation given by (25) . We thus have
d d t I ( g ) ≤ − c γ D i ( g ) + ⟨ I ′ ( g ) , T g ⟩ , \frac{\mathrm{d}}{\mathrm{d}t}I(g)\leq-c_\gamma D_i(g)+\langle I'(g),\mathcal{T}g\rangle, d t d I ( g ) ≤ − c γ D i ( g ) + ⟨ I ′ ( g ) , T g ⟩ , and it suffices to bound the effect of the extra transport term on Fisher information. In the following computations, we use an Einstein-like summation convention so that each index should be understood as having a summation in front:
T g = L i j w i ∂ w j g . \mathcal{T}g=L_{ij}w_i\partial_{w_j}g. T g = L ij w i ∂ w j g . Then we have
⟨ I ′ ( g ) , T g ⟩ = 2 ∫ ∂ k g ∂ k [ T g ] g − ∫ ( ∂ k g g ) 2 T g = : I 1 + I 2 . \langle I'(g),\mathcal{T}g\rangle=2\int\frac{\partial_kg\partial_k[\mathcal{T}g]}{g}-\int\left(\frac{\partial_kg}{g}\right)^2\mathcal{T}g=:I_1+I_2. ⟨ I ′ ( g ) , T g ⟩ = 2 ∫ g ∂ k g ∂ k [ T g ] − ∫ ( g ∂ k g ) 2 T g =: I 1 + I 2 . We expand both I 1 I_1 I 1 I 2 I_2 I 2 g \sqrt{g} g ∇ g = 2 g ∇ g \nabla g=2\sqrt{g}\nabla\sqrt{g} ∇ g = 2 g ∇ g
I 1 = 2 L i j ∫ ∂ k g ∂ k [ w i ∂ j g ] g = 8 L i j ∫ ∂ k g ∂ k [ w i g ∂ j g ] g = 8 L i j ∫ ∂ i g ∂ j g + 8 L i j ∫ ( ∂ k g ) 2 ∂ j g g w i + 8 L i j ∫ ∂ k g ∂ k j g w i . \begin{aligned}
I_1=2L_{ij}\int\frac{\partial_kg\partial_k[w_i\partial_jg]}{g} \\
&=8L_{ij}\int\frac{\partial_k\sqrt{g}\partial_k[w_i\sqrt{g}\partial_j\sqrt{g}]}{\sqrt{g}} \\
&=8L_{ij}\int\partial_i\sqrt{g}\partial_j\sqrt{g}+8L_{ij}\int\frac{(\partial_k\sqrt{g})^2\partial_j\sqrt{g}}{\sqrt{g}}w_i+8L_{ij}\int\partial_k\sqrt{g}\partial_{kj}\sqrt{g}w_i.
\end{aligned} I 1 = 2 L ij ∫ g ∂ k g ∂ k [ w i ∂ j g ] = 8 L ij ∫ g ∂ k g ∂ k [ w i g ∂ j g ] = 8 L ij ∫ ∂ i g ∂ j g + 8 L ij ∫ g ( ∂ k g ) 2 ∂ j g w i + 8 L ij ∫ ∂ k g ∂ k j g w i . Further note that
8 L i j ∫ ∂ k g ∂ k j g w i = 4 L i j ∫ ∂ j [ ( ∂ k g ) 2 ] w i = − 4 L i i ∫ ( ∂ k g ) 2 = − tr ( L ) I ( g ) . \begin{aligned}
8L_{ij}\int\partial_k\sqrt{g}\partial_{kj}\sqrt{g}w_i=4L_{ij}\int\partial_j\left[(\partial_k\sqrt{g})^2\right]w_i \\
&=-4L_{ii}\int(\partial_k\sqrt{g})^2=-\operatorname{tr}(L)I(g).
\end{aligned} 8 L ij ∫ ∂ k g ∂ k j g w i = 4 L ij ∫ ∂ j [ ( ∂ k g ) 2 ] w i = − 4 L ii ∫ ( ∂ k g ) 2 = − tr ( L ) I ( g ) . Therefore,
I 1 = 8 ∫ L ∇ g ⋅ ∇ g − tr ( L ) I ( g ) + 8 L i j ∫ ( ∂ k g ) 2 ∂ j g g w i . I_1 = 8\int L\nabla\sqrt{g}\cdot\nabla\sqrt{g}-\operatorname{tr}(L)I(g)+8L_{ij}\int\frac{(\partial_k\sqrt{g})^2\partial_j\sqrt{g}}{\sqrt{g}}w_i. I 1 = 8 ∫ L ∇ g ⋅ ∇ g − tr ( L ) I ( g ) + 8 L ij ∫ g ( ∂ k g ) 2 ∂ j g w i . As for the second term I 2 I_2 I 2
I 2 = − L i j ∫ ( ∂ k g g ) 2 w i ∂ w j g = − 8 L i j ∫ ( ∂ k g ) 2 ∂ j g g w i , I_2=-L_{ij}\int\left(\frac{\partial_k g}{g}\right)^2w_i\partial_{w_j}g=-8L_{ij}\int\frac{(\partial_k\sqrt{g})^2\partial_j\sqrt{g}}{\sqrt{g}}w_i, I 2 = − L ij ∫ ( g ∂ k g ) 2 w i ∂ w j g = − 8 L ij ∫ g ( ∂ k g ) 2 ∂ j g w i , which cancels with the last term of I 1 I_1 I 1
⟨ I ′ ( g ) , T g ⟩ = 8 ∫ L ∇ g ⋅ ∇ g − tr ( L ) I ( g ) ≤ ( 2 ∣ L ∣ − tr L ) I ( g ) . \langle I'(g),Tg\rangle=8\int L\nabla\sqrt{g}\cdot\nabla\sqrt{g}-\operatorname{tr}(L)I(g)\leq\left(2|L|-\operatorname{tr}L\right)I(g). ⟨ I ′ ( g ) , T g ⟩ = 8 ∫ L ∇ g ⋅ ∇ g − tr ( L ) I ( g ) ≤ ( 2∣ L ∣ − tr L ) I ( g ) . This shows the differential inequality
d d t I ( g ) ≤ − c γ D i ( g ) + ( 2 ∣ L ( t ) ∣ − tr ( L ( t ) ) ) I ( g ) , (27) \frac{d}{dt}I(g)\leq-c_\gamma D_i(g)+\left(2|L(t)|-\operatorname{tr}(L(t))\right)I(g), \tag*{(27)} d t d I ( g ) ≤ − c γ D i ( g ) + ( 2∣ L ( t ) ∣ − tr ( L ( t )) ) I ( g ) , (27) which implies the stated estimate. □
The qualitative behavior of the Fisher information of g g g L ( t ) L(t) L ( t ) (27) . If L 0 = diag ( λ 1 , λ 2 , λ 3 ) L_0=\operatorname{diag}(\lambda_1,\lambda_2,\lambda_3) L 0 = diag ( λ 1 , λ 2 , λ 3 ) λ i > 0 \lambda_i>0 λ i > 0 t → ∞ t\to\infty t → ∞
L ( t ) = 1 t I + O ( 1 t 2 ) , L(t)=\frac{1}{t}I+O\left(\frac{1}{t^2}\right), L ( t ) = t 1 I + O ( t 2 1 ) , which gives
d d t I ( g ) ≤ ( 2 ∣ L ( t ) ∣ − tr L ( t ) ) I ( g ) = ( 2 t − 3 t + O ( t − 2 ) ) I ( g ) . \frac{d}{dt}I(g)\leq\left(2|L(t)|-\operatorname{tr}L(t)\right)I(g)=\left(\frac{2}{t}-\frac{3}{t}+O(t^{-2})\right)I(g). d t d I ( g ) ≤ ( 2∣ L ( t ) ∣ − tr L ( t ) ) I ( g ) = ( t 2 − t 3 + O ( t − 2 ) ) I ( g ) . Thus, after a transient period, the Fisher information becomes monotone decreasing. By contrast, if L 0 L_0 L 0 (9) , there is no apparent reason to expect such monotonicity.
A similar phenomenon occurs for the second moment M 2 ( g ) M_2(g) M 2 ( g ) L ( t ) L(t) L ( t )
Remark 3.2. Estimate (26) yields the following bound for any f f f solution to (1) :
sup x ∈ R 3 ∫ ∣ ∇ v f ∣ 2 f d v = I ( g ( t ) ) , sup x ∈ R 3 ∫ ∣ ∇ x f ∣ 2 f d v ≤ ∣ L ( t ) ∣ 2 I ( g ( t ) ) . \sup_{x\in\mathbb{R}^3}\int\frac{|\nabla_v f|^2}{f}\,dv=I(g(t)),\qquad\sup_{x\in\mathbb{R}^3}\int\frac{|\nabla_x f|^2}{f}\,dv\leq|L(t)|^2I(g(t)). x ∈ R 3 sup ∫ f ∣ ∇ v f ∣ 2 d v = I ( g ( t )) , x ∈ R 3 sup ∫ f ∣ ∇ x f ∣ 2 d v ≤ ∣ L ( t ) ∣ 2 I ( g ( t )) . Global-in-time existence of smooth solution
The main goal of this section is to prove the following L ∞ L^\infty L ∞
Theorem 3.1. Assume det ( I + t L 0 ) > 0 \det(I+tL_0)>0 det ( I + t L 0 ) > 0 for all 0 ≤ t ≤ T 0\leq t\leq T 0 ≤ t ≤ T Let g g g be a solution of (16) with g i n ∈ L ∞ ∩ L 1 g_{\mathrm{in}}\in L^\infty\cap L^1 g in ∈ L ∞ ∩ L 1 M 2 ( g i n ) < ∞ M_2(g_{\mathrm{in}})<\infty M 2 ( g in ) < ∞ Further assume
(1) In the soft potential case − 3 ≤ γ < 0 -3\leq\gamma<0 − 3 ≤ γ < 0 that I ( g i n ) < ∞ I(g_{\mathrm{in}})<\infty I ( g in ) < ∞ and M s ( g i n ) < ∞ M_s(g_{\mathrm{in}})<\infty M s ( g in ) < ∞ for some
s > max { 2 ∣ γ ∣ 4 + γ , ∣ γ ∣ , 2 } . s>\max\left\{\frac{2|\gamma|}{4+\gamma},|\gamma|,2\right\}. s > max { 4 + γ 2∣ γ ∣ , ∣ γ ∣ , 2 } . Then
sup 0 ≤ t ≤ T ∥ g ( t ) ∥ L ∞ ≤ C , \sup_{0\leq t\leq T}\|g(t)\|_{L^\infty}\leq C, 0 ≤ t ≤ T sup ∥ g ( t ) ∥ L ∞ ≤ C , where C C C depends on γ \gamma γ T T T ∫ 0 T ∣ L ( t ) ∣ d t \int_0^T|L(t)|\,dt ∫ 0 T ∣ L ( t ) ∣ d t s s s ρ ( 0 ) \rho(0) ρ ( 0 ) M s ( g i n ) M_s(g_{\mathrm{in}}) M s ( g in ) I ( g i n ) I(g_{\mathrm{in}}) I ( g in ) and ∥ g i n ∥ L ∞ \|g_{\mathrm{in}}\|_{L^\infty} ∥ g in ∥ L ∞
(2) In the hard potential case γ ≥ 0 \gamma\geq0 γ ≥ 0 that M 3 γ 2 ( g i n ) < ∞ M_{\frac{3\gamma}{2}}(g_{\mathrm{in}})<\infty M 2 3 γ ( g in ) < ∞ Then
sup 0 ≤ t ≤ T ∥ g ( t ) ∥ L ∞ ≤ C , \sup_{0\leq t\leq T}\|g(t)\|_{L^\infty}\leq C, 0 ≤ t ≤ T sup ∥ g ( t ) ∥ L ∞ ≤ C , where C C C depends on γ \gamma γ T T T ∫ 0 T ∣ L ( t ) ∣ d t \int_0^T|L(t)|\,dt ∫ 0 T ∣ L ( t ) ∣ d t ρ ( 0 ) \rho(0) ρ ( 0 ) M 2 ( g i n ) M_2(g_{\mathrm{in}}) M 2 ( g in ) M 3 γ 2 ( g i n ) M_{\frac{3\gamma}{2}}(g_{\mathrm{in}}) M 2 3 γ ( g in ) and ∥ g i n ∥ L ∞ \|g_{\mathrm{in}}\|_{L^\infty} ∥ g in ∥ L ∞
Note that H ( g i n ) \mathcal{H}(g_{\mathrm{in}}) H ( g in ) γ ≥ − 3 \gamma\geq-3 γ ≥ − 3 ε \varepsilon ε
Lemma 3.4. Let − 3 ≤ γ < 0 -3 \le\gamma< 0 − 3 ≤ γ < 0
c γ [ g ] ( w ) : = { 2 ( 3 + γ ) ( g ∗ ∣ ⋅ ∣ γ ) ( w ) , − 3 < γ < 0 , 8 π g , γ = − 3. (28) c_{\gamma}[g](w) := \begin{cases} 2(3+\gamma)(g \ast\lvert\cdot\rvert^\gamma)(w), & -3 < \gamma< 0, \\ 8\pi g, & \gamma=-3. \end{cases} \tag*{(28)} c γ [ g ] ( w ) := { 2 ( 3 + γ ) ( g ∗ ∣ ⋅ ∣ γ ) ( w ) , 8 π g , − 3 < γ < 0 , γ = − 3. (28) Assume that g g g I ( g ) < ∞ I(g)<\infty I ( g ) < ∞ M m ( g ) < ∞ M_m(g)<\infty M m ( g ) < ∞ m > max { 2 ∣ γ ∣ 4 + γ , ∣ γ ∣ } m>\max\left\{\frac{2\lvert\gamma\rvert}{4+\gamma},\lvert\gamma\rvert\right\} m > max { 4 + γ 2 ∣ γ ∣ , ∣ γ ∣ } 0 < ε < 1 0<\varepsilon<1 0 < ε < 1
∫ g p c γ [ g ] ≤ ε ∫ ∣ ∇ g p 2 ∣ 2 ⟨ w ⟩ γ + C ε − M ∫ g p ⟨ w ⟩ γ , \int g^p c_\gamma[g] \le\varepsilon\int\lvert\nabla g^{\frac{p}{2}}\rvert^2\langle w\rangle^\gamma+C\varepsilon^{-M}\int g^p\langle w\rangle^\gamma, ∫ g p c γ [ g ] ≤ ε ∫ ∣ ∇ g 2 p ∣ 2 ⟨ w ⟩ γ + C ε − M ∫ g p ⟨ w ⟩ γ , where M M M γ \gamma γ C C C m m m g g g M m ( g ) M_m(g) M m ( g ) I ( g ) I(g) I ( g )
Proof. The starting point is the ε \varepsilon ε [1 ] , which states that
∫ ϕ 2 c γ [ g ] ≤ ε ∫ ∣ ∇ ( ⟨ w ⟩ γ 2 ϕ ) ∣ 2 + C 0 ( ∥ g ∥ L 1 + ε − s 1 − s ∥ ⟨ ⋅ ⟩ − γ g ∥ L q 1 1 − s ) ∫ ϕ 2 ⟨ w ⟩ γ , (29) \int\phi^2 c_\gamma[g] \le\varepsilon\int\left\lvert\nabla\left(\langle w\rangle^{\frac{\gamma}{2}}\phi\right)\right\rvert^2+C_0\left(\lVert g\rVert_{L^1}+\varepsilon^{-\frac{s}{1-s}}\lVert\langle\cdot\rangle^{-\gamma}g\rVert_{L^q}^{\frac{1}{1-s}}\right)\int\phi^2\langle w\rangle^\gamma, \tag*{(29)} ∫ ϕ 2 c γ [ g ] ≤ ε ∫ ∇ ( ⟨ w ⟩ 2 γ ϕ ) 2 + C 0 ( ∥ g ∥ L 1 + ε − 1 − s s ∥⟨ ⋅ ⟩ − γ g ∥ L q 1 − s 1 ) ∫ ϕ 2 ⟨ w ⟩ γ , (29) where max { 3 5 + γ , 1 } < q < 3 3 + γ \max\left\{\frac{3}{5+\gamma},1\right\}<q<\frac{3}{3+\gamma} max { 5 + γ 3 , 1 } < q < 3 + γ 3 s = 3 − q ( 3 + γ ) 2 q ∈ ( 0 , 1 ) s=\frac{3-q(3+\gamma)}{2q}\in(0,1) s = 2 q 3 − q ( 3 + γ ) ∈ ( 0 , 1 ) ⟨ w ⟩ γ 2 \langle w\rangle^{\frac{\gamma}{2}} ⟨ w ⟩ 2 γ
∫ ∣ ∇ ( ⟨ w ⟩ γ 2 ϕ ) ∣ 2 = ∫ ∣ ⟨ w ⟩ γ 2 ∇ ϕ + ϕ ∇ ⟨ w ⟩ γ 2 ∣ 2 = ∫ ∣ ∇ ϕ ∣ 2 ⟨ w ⟩ γ + 2 ∫ ( ∇ ϕ ⋅ ∇ ⟨ w ⟩ γ 2 ) ⟨ w ⟩ γ 2 ϕ + ∫ ∣ ∇ ⟨ w ⟩ γ 2 ∣ 2 ϕ 2 = ∫ ∣ ∇ ϕ ∣ 2 ⟨ w ⟩ γ + ∫ ( ∣ ∇ ⟨ w ⟩ γ 2 ∣ 2 − 1 2 Δ ⟨ w ⟩ γ ) ϕ 2 ≤ ∫ ∣ ∇ ϕ ∣ 2 ⟨ w ⟩ γ + 15 ∫ ϕ 2 ⟨ w ⟩ γ . \begin{aligned}
\int\left\lvert\nabla\left(\langle w\rangle^{\frac{\gamma}{2}}\phi\right)\right\rvert^2
&=\int\left\lvert\langle w\rangle^{\frac{\gamma}{2}}\nabla\phi+\phi\nabla\langle w\rangle^{\frac{\gamma}{2}}\right\rvert^2\\
&=\int\lvert\nabla\phi\rvert^2\langle w\rangle^\gamma+2\int\left(\nabla\phi\cdot\nabla\langle w\rangle^{\frac{\gamma}{2}}\right)\langle w\rangle^{\frac{\gamma}{2}}\phi+\int\left\lvert\nabla\langle w\rangle^{\frac{\gamma}{2}}\right\rvert^2\phi^2\\
&=\int\lvert\nabla\phi\rvert^2\langle w\rangle^\gamma+\int\left(\left\lvert\nabla\langle w\rangle^{\frac{\gamma}{2}}\right\rvert^2-\frac{1}{2}\Delta\langle w\rangle^\gamma\right)\phi^2\\
&\le\int\lvert\nabla\phi\rvert^2\langle w\rangle^\gamma+15\int\phi^2\langle w\rangle^\gamma.
\end{aligned} ∫ ∇ ( ⟨ w ⟩ 2 γ ϕ ) 2 = ∫ ⟨ w ⟩ 2 γ ∇ ϕ + ϕ ∇ ⟨ w ⟩ 2 γ 2 = ∫ ∣ ∇ ϕ ∣ 2 ⟨ w ⟩ γ + 2 ∫ ( ∇ ϕ ⋅ ∇ ⟨ w ⟩ 2 γ ) ⟨ w ⟩ 2 γ ϕ + ∫ ∇ ⟨ w ⟩ 2 γ 2 ϕ 2 = ∫ ∣ ∇ ϕ ∣ 2 ⟨ w ⟩ γ + ∫ ( ∇ ⟨ w ⟩ 2 γ 2 − 2 1 Δ ⟨ w ⟩ γ ) ϕ 2 ≤ ∫ ∣ ∇ ϕ ∣ 2 ⟨ w ⟩ γ + 15 ∫ ϕ 2 ⟨ w ⟩ γ . Plugging this into (29) and setting ϕ = g p 2 \phi=g^{\frac{p}{2}} ϕ = g 2 p
∫ g p c γ [ g ] ≤ ε ∫ ∣ ∇ ϕ ∣ 2 ⟨ w ⟩ γ + C 0 ( ∥ g ∥ L 1 + ε − s 1 − s ∥ ⟨ ⋅ ⟩ − γ g ∥ L q 1 1 − s + 15 ) ∫ g p ⟨ w ⟩ γ . \int g^p c_\gamma[g]\le\varepsilon\int\lvert\nabla\phi\rvert^2\langle w\rangle^\gamma+C_0\left(\lVert g\rVert_{L^1}+\varepsilon^{-\frac{s}{1-s}}\lVert\langle\cdot\rangle^{-\gamma}g\rVert_{L^q}^{\frac{1}{1-s}}+15\right)\int g^p\langle w\rangle^\gamma. ∫ g p c γ [ g ] ≤ ε ∫ ∣ ∇ ϕ ∣ 2 ⟨ w ⟩ γ + C 0 ( ∥ g ∥ L 1 + ε − 1 − s s ∥⟨ ⋅ ⟩ − γ g ∥ L q 1 − s 1 + 15 ) ∫ g p ⟨ w ⟩ γ . We already have control of the mass of g g g ∥ ⟨ ⋅ ⟩ − γ g ∥ L q \lVert\langle\cdot\rangle^{-\gamma}g\rVert_{L^q} ∥⟨ ⋅ ⟩ − γ g ∥ L q
max { 3 5 + γ , 1 } < q < min { 3 , 3 3 + γ } (30) \max\left\{\frac{3}{5+\gamma},1\right\}<q<\min\left\{3,\frac{3}{3+\gamma}\right\} \tag*{(30)} max { 5 + γ 3 , 1 } < q < min { 3 , 3 + γ 3 } (30) and interpolating ∥ ⟨ ⋅ ⟩ − γ g ∥ L q \lVert\langle\cdot\rangle^{-\gamma}g\rVert_{L^q} ∥⟨ ⋅ ⟩ − γ g ∥ L q L 3 L^3 L 3 L m 1 L^1_m L m 1 L 3 L^3 L 3 g g g
∥ g ∥ L 3 = ∥ g ∥ L 6 2 ≤ C S o b 2 ∥ ∇ g ∥ L 2 2 = C S o b 2 4 I ( g ) . \lVert g\rVert_{L^3}=\lVert\sqrt{g}\rVert_{L^6}^2\le C_{\mathrm{Sob}}^2\lVert\nabla\sqrt{g}\rVert_{L^2}^2=\frac{C_{\mathrm{Sob}}^2}{4}I(g). ∥ g ∥ L 3 = ∥ g ∥ L 6 2 ≤ C Sob 2 ∥ ∇ g ∥ L 2 2 = 4 C Sob 2 I ( g ) . A simple computation shows that for this choice to be possible, one only needs to have bounded moments of order m > max { 2 ∣ γ ∣ 4 + γ , ∣ γ ∣ } m>\max\left\{\frac{2\lvert\gamma\rvert}{4+\gamma},\lvert\gamma\rvert\right\} m > max { 4 + γ 2 ∣ γ ∣ , ∣ γ ∣ }
Next, we perform a simple L p L^p L p
Lemma 3.5. Assume the setting of Theorem 3.1 and let p ≥ 2 p\ge2 p ≥ 2 0 ≤ t ≤ T 0\le t\le T 0 ≤ t ≤ T
If γ ≥ 0 \gamma\ge0 γ ≥ 0
d d t ∫ g p + λ ∫ ⟨ w ⟩ γ ∣ ∇ g p 2 ∣ 2 ≤ C p ∫ ⟨ w ⟩ γ g p . \frac{d}{dt}\int g^p+\lambda\int\langle w\rangle^\gamma\left\lvert\nabla g^{\frac{p}{2}}\right\rvert^2\le C_p\int\langle w\rangle^\gamma g^p. d t d ∫ g p + λ ∫ ⟨ w ⟩ γ ∇ g 2 p 2 ≤ C p ∫ ⟨ w ⟩ γ g p . If γ < 0 \gamma<0 γ < 0
d d t ∫ g p + λ ∫ ⟨ w ⟩ γ ∣ ∇ g p 2 ∣ 2 ≤ C ∫ g p + C p M ∫ ⟨ w ⟩ γ g p . \frac{d}{dt}\int g^p+\lambda\int\langle w\rangle^\gamma\left\lvert\nabla g^{\frac{p}{2}}\right\rvert^2\le C\int g^p+Cp^M\int\langle w\rangle^\gamma g^p. d t d ∫ g p + λ ∫ ⟨ w ⟩ γ ∇ g 2 p 2 ≤ C ∫ g p + C p M ∫ ⟨ w ⟩ γ g p . Here C C C M M M γ \gamma γ C C C p p p
Proof. We start with a direct computation
d d t ∫ g p = p ∫ g p − 1 ( L ( t ) w ⋅ ∇ g ) + p ∫ g p − 1 Q ( g ) . \frac{d}{dt}\int g^p=p\int g^{p-1}(L(t)w\cdot\nabla g)+p\int g^{p-1}Q(g). d t d ∫ g p = p ∫ g p − 1 ( L ( t ) w ⋅ ∇ g ) + p ∫ g p − 1 Q ( g ) . The contribution of the drift term is
p ∫ g p − 1 ( L ( t ) w ⋅ ∇ g ) = ∫ ( L ( t ) w ) ⋅ ∇ g p = − tr L ( t ) ∫ g p . p\int g^{p-1}(L(t)w\cdot\nabla g)=\int(L(t)w)\cdot\nabla g^p=-\operatorname{tr}L(t)\int g^p. p ∫ g p − 1 ( L ( t ) w ⋅ ∇ g ) = ∫ ( L ( t ) w ) ⋅ ∇ g p = − tr L ( t ) ∫ g p . To estimate the contribution of the collision operator, we write it as
Q ( g ) = div ( A [ g ] ∇ g − b [ g ] g ) , b [ g ] ( w ) : = − 2 ∫ ∣ w − w ∗ ∣ γ ( w − w ∗ ) g ( w ∗ ) d w ∗ . Q(g)=\operatorname{div}\left(A[g]\nabla g-b[g]g\right),\qquad b[g](w):=-2\int|w-w_*|^\gamma(w-w_*)g(w_*)\,dw_*. Q ( g ) = div ( A [ g ] ∇ g − b [ g ] g ) , b [ g ] ( w ) := − 2 ∫ ∣ w − w ∗ ∣ γ ( w − w ∗ ) g ( w ∗ ) d w ∗ . Then
p ∫ g p − 1 Q ( g ) = − p ( p − 1 ) ∫ ( A [ g ] ∇ g ⋅ ∇ g ) g p − 2 + p ( p − 1 ) ∫ b [ g ] ⋅ ∇ g g p − 1 = − 4 ( p − 1 ) p ∫ ( A [ g ] ∇ g p 2 ⋅ ∇ g p 2 ) − ( p − 1 ) ∫ div b [ g ] g p . \begin{aligned}
p\int g^{p-1}Q(g)=-p(p-1)\int\left(A[g]\nabla g\cdot\nabla g\right)g^{p-2}+p(p-1)\int b[g]\cdot\nabla g\,g^{p-1} \\
&=-\frac{4(p-1)}{p}\int\left(A[g]\nabla g^{\frac p2}\cdot\nabla g^{\frac p2}\right)-(p-1)\int\operatorname{div}b[g]\,g^p.
\end{aligned} p ∫ g p − 1 Q ( g ) = − p ( p − 1 ) ∫ ( A [ g ] ∇ g ⋅ ∇ g ) g p − 2 + p ( p − 1 ) ∫ b [ g ] ⋅ ∇ g g p − 1 = − p 4 ( p − 1 ) ∫ ( A [ g ] ∇ g 2 p ⋅ ∇ g 2 p ) − ( p − 1 ) ∫ div b [ g ] g p . Now we use that
− div b [ g ] = c γ [ g ] = { 2 ( γ + 3 ) ∣ w ∣ γ ∗ g , γ > − 3 , 8 π g , γ = − 3. -\operatorname{div}b[g]=c_\gamma[g]=
\begin{cases}
2(\gamma+3)|w|^\gamma*g,&\gamma>-3,\\
8\pi g,&\gamma=-3.
\end{cases} − div b [ g ] = c γ [ g ] = { 2 ( γ + 3 ) ∣ w ∣ γ ∗ g , 8 π g , γ > − 3 , γ = − 3. where c γ [ g ] c_\gamma[g] c γ [ g ] Lemma 3.4 for γ < 0 \gamma<0 γ < 0 γ ≥ − 3 \gamma\ge-3 γ ≥ − 3 Lemma 3.3 and putting everything together, we obtain
d d t ∫ g p + 2 λ ∫ ⟨ w ⟩ γ ∣ ∇ g p 2 ∣ 2 ≤ C ∫ g p + p ∫ c γ [ g ] g p . \frac{d}{dt}\int g^p+2\lambda\int\langle w\rangle^\gamma\left|\nabla g^{\frac p2}\right|^2\le C\int g^p+p\int c_\gamma[g]g^p. d t d ∫ g p + 2 λ ∫ ⟨ w ⟩ γ ∇ g 2 p 2 ≤ C ∫ g p + p ∫ c γ [ g ] g p . Case of hard potentials γ ≥ 0 \gamma\ge0 γ ≥ 0 We are done, since
c γ [ g ] = ∫ ∣ w − w ∗ ∣ γ g ( w ∗ ) ≤ C ∫ ( ∣ w ∣ γ + ∣ w ∗ ∣ γ ) g ( w ∗ ) ≤ C ⟨ w ⟩ γ . c_\gamma[g]=\int|w-w_*|^\gamma g(w_*)\le C\int\left(|w|^\gamma+|w_*|^\gamma\right)g(w_*)\le C\langle w\rangle^\gamma. c γ [ g ] = ∫ ∣ w − w ∗ ∣ γ g ( w ∗ ) ≤ C ∫ ( ∣ w ∣ γ + ∣ w ∗ ∣ γ ) g ( w ∗ ) ≤ C ⟨ w ⟩ γ . Case of soft potentials. Using the ε \varepsilon ε Lemma 3.4 gives
d d t ∫ g p + 2 λ ∫ ⟨ w ⟩ γ ∣ ∇ g p 2 ∣ 2 ≤ C ∫ g p + p ε ∫ ∣ ∇ g p 2 ∣ 2 ∣ w ∣ γ + C p ε − M ∫ g p ⟨ w ⟩ γ . \frac{d}{dt}\int g^p+2\lambda\int\langle w\rangle^\gamma\left|\nabla g^{\frac p2}\right|^2\le C\int g^p+p\varepsilon\int\left|\nabla g^{\frac p2}\right|^2|w|^\gamma+Cp\varepsilon^{-M}\int g^p\langle w\rangle^\gamma. d t d ∫ g p + 2 λ ∫ ⟨ w ⟩ γ ∇ g 2 p 2 ≤ C ∫ g p + pε ∫ ∇ g 2 p 2 ∣ w ∣ γ + C p ε − M ∫ g p ⟨ w ⟩ γ . Then, choosing ε = λ / p \varepsilon=\lambda/p ε = λ / p
As an immediate consequence, we have control of L p L^p L p p p p
Corollary 3.2. Assume the setting of Theorem 3.1 and let p ≥ 2 p\ge2 p ≥ 2
i ) i) i ) γ ≥ 0 \gamma\ge0 γ ≥ 0
sup t ∈ [ 0 , T ] ∥ g ∥ L p + ( ∫ 0 T ∫ ⟨ w ⟩ γ ∣ ∇ g p 2 ∣ 2 ) 1 p ≤ C ( ∥ g i n ∥ L p + p 3 2 M 3 2 γ ( g i n ) + 1 ) . \sup_{t\in[0,T]}\|g\|_{L^p}+\left(\int_0^T\int\langle w\rangle^\gamma\left|\nabla g^{\frac p2}\right|^2\right)^{\frac1p}\le C\left(\|g_{\mathrm{in}}\|_{L^p}+p^{\frac32}M_{\frac32\gamma}(g_{\mathrm{in}})+1\right). t ∈ [ 0 , T ] sup ∥ g ∥ L p + ( ∫ 0 T ∫ ⟨ w ⟩ γ ∇ g 2 p 2 ) p 1 ≤ C ( ∥ g in ∥ L p + p 2 3 M 2 3 γ ( g in ) + 1 ) . i i ) ii) ii ) γ < 0 \gamma<0 γ < 0
sup t ∈ [ 0 , T ] ∥ g ∥ L p + ( ∫ 0 T ∫ ⟨ w ⟩ γ ∣ ∇ g p 2 ∣ 2 ) 1 p ≤ C p M ∥ g i n ∥ L p . \sup_{t\in[0,T]}\|g\|_{L^p}+\left(\int_0^T\int\langle w\rangle^\gamma\left|\nabla g^{\frac p2}\right|^2\right)^{\frac1p}\le Cp^M\|g_{\mathrm{in}}\|_{L^p}. t ∈ [ 0 , T ] sup ∥ g ∥ L p + ( ∫ 0 T ∫ ⟨ w ⟩ γ ∇ g 2 p 2 ) p 1 ≤ C p M ∥ g in ∥ L p . The constant C C C M M M γ \gamma γ
Proof. The estimate is immediate for soft potentials. For hard potentials, we use Hölder inequality with exponents 3 p − 1 p − 1 \frac{3p-1}{p-1} p − 1 3 p − 1 3 p − 1 2 p \frac{3p-1}{2p} 2 p 3 p − 1 g g g
p C ∫ ⟨ w ⟩ γ g p ≤ C p ( ∫ ∣ w ∣ γ 3 p − 1 2 p g ) 2 p 3 p − 1 ( ∫ g 3 p ) p − 1 3 p − 1 . pC\int\langle w\rangle^\gamma g^p\le Cp\left(\int|w|^{\gamma\frac{3p-1}{2p}}g\right)^{\frac{2p}{3p-1}}\left(\int g^{3p}\right)^{\frac{p-1}{3p-1}}. pC ∫ ⟨ w ⟩ γ g p ≤ C p ( ∫ ∣ w ∣ γ 2 p 3 p − 1 g ) 3 p − 1 2 p ( ∫ g 3 p ) 3 p − 1 p − 1 . Next, we note that p − 1 3 p − 1 < 1 3 \frac{p-1}{3p-1} < \frac{1}{3} 3 p − 1 p − 1 < 3 1 3 p − 1 3 p − 3 , 3 p − 1 2 \frac{3p-1}{3p-3}, \frac{3p-1}{2} 3 p − 3 3 p − 1 , 2 3 p − 1
p C ∫ ∣ w ∣ γ g p ≤ ε ( ∫ g 3 p ) 1 3 + C 3 p − 1 2 ε − ( p − 1 ) p 3 2 p M γ 3 p − 1 2 p ( g ) p ≤ ε C S o b ∫ ⟨ w ⟩ γ ∣ ∇ g p 2 ∣ 2 + C 3 p − 1 2 ε − ( p − 1 ) p 3 2 p M γ 3 p − 1 2 p ( g ) p . \begin{aligned}
pC \int|w|^\gamma g^p \leq\varepsilon\left(\int g^{3p}\right)^{\frac{1}{3}} + C^{\frac{3p-1}{2}} \varepsilon^{-(p-1)} p^{\frac{3}{2}p} M_{\gamma\frac{3p-1}{2p}}(g)^p \\
&\leq\varepsilon C_{\mathrm{Sob}} \int\langle w\rangle^\gamma\left|\nabla g^{\frac{p}{2}}\right|^2 + C^{\frac{3p-1}{2}} \varepsilon^{-(p-1)} p^{\frac{3}{2}p} M_{\gamma\frac{3p-1}{2p}}(g)^p.
\end{aligned} pC ∫ ∣ w ∣ γ g p ≤ ε ( ∫ g 3 p ) 3 1 + C 2 3 p − 1 ε − ( p − 1 ) p 2 3 p M γ 2 p 3 p − 1 ( g ) p ≤ ε C Sob ∫ ⟨ w ⟩ γ ∇ g 2 p 2 + C 2 3 p − 1 ε − ( p − 1 ) p 2 3 p M γ 2 p 3 p − 1 ( g ) p . Now the proof is finished choosing ε = λ Λ 2 C S o b \varepsilon= \frac{\lambda_\Lambda}{2C_{\mathrm{Sob}}} ε = 2 C Sob λ Λ
Now we are ready to prove the main result of the section.
Proof of Theorem 3.1. Our main tool is the following inequality, which follows by applying Gronwall’s inequality on Lemma 3.5:
sup t ∈ [ 0 , T ] ∫ g p + ∫ 0 T ∫ ⟨ w ⟩ γ ∣ ∇ g p 2 ∣ 2 ≤ C ( ∫ g i n p + p M ∫ 0 T ∫ ⟨ w ⟩ γ g p ) . (31) \sup_{t\in[0,T]} \int g^p + \int_0^T \int\langle w\rangle^\gamma\left|\nabla g^{\frac{p}{2}}\right|^2 \leq C \left(\int g_{\mathrm{in}}^p + p^M \int_0^T \int\langle w\rangle^\gamma g^p\right). \tag*{(31)} t ∈ [ 0 , T ] sup ∫ g p + ∫ 0 T ∫ ⟨ w ⟩ γ ∇ g 2 p 2 ≤ C ( ∫ g in p + p M ∫ 0 T ∫ ⟨ w ⟩ γ g p ) . (31) The constant C C C p p p γ ≥ − 3 \gamma\geq-3 γ ≥ − 3
( ∫ ⟨ w ⟩ 3 γ g 3 p ) 1 3 ≤ C ∫ ∣ ∇ ( ⟨ w ⟩ γ 2 g p 2 ) ∣ 2 ≤ C ∫ ⟨ w ⟩ γ ∣ ∇ g p 2 ∣ 2 + C ∫ ⟨ w ⟩ γ g p . (32) \left(\int\langle w\rangle^{3\gamma} g^{3p}\right)^{\frac{1}{3}} \leq C \int\left|\nabla\left(\langle w\rangle^{\frac{\gamma}{2}}g^{\frac{p}{2}}\right)\right|^2 \leq C \int\langle w\rangle^\gamma\left|\nabla g^{\frac{p}{2}}\right|^2 + C \int\langle w\rangle^\gamma g^p. \tag*{(32)} ( ∫ ⟨ w ⟩ 3 γ g 3 p ) 3 1 ≤ C ∫ ∇ ( ⟨ w ⟩ 2 γ g 2 p ) 2 ≤ C ∫ ⟨ w ⟩ γ ∇ g 2 p 2 + C ∫ ⟨ w ⟩ γ g p . (32) Together, these inequalities provide control of sup t ∈ [ 0 , T ] ∫ g p \sup_{t\in[0,T]} \int g^p sup t ∈ [ 0 , T ] ∫ g p ∫ 0 T ∫ ⟨ w ⟩ 3 γ g 3 p \int_0^T \int\langle w\rangle^{3\gamma}g^{3p} ∫ 0 T ∫ ⟨ w ⟩ 3 γ g 3 p L t , w 5 3 p L_{t,w}^{\frac{5}{3}p} L t , w 3 5 p ( 3 , 3 2 ) (3,\frac{3}{2}) ( 3 , 2 3 )
∫ 0 T ∫ ⟨ w ⟩ γ g 5 3 p = ∫ 0 T ∫ ⟨ w ⟩ γ g p ⋅ g 2 3 p ≤ ∫ 0 T ( ∫ ⟨ w ⟩ 3 γ g 3 p ) 1 3 ( ∫ g p ) 2 3 d t ≤ sup t ∈ [ 0 , T ] ( ∫ g p ) 2 3 ∫ 0 T ( ∫ ⟨ w ⟩ 3 γ g 3 p ) 1 3 . \begin{aligned}
\int_0^T \int\langle w\rangle^\gamma g^{\frac{5}{3}p} = \int_0^T \int\langle w\rangle^\gamma g^p \cdot g^{\frac{2}{3}p} \\
&\leq\int_0^T \left(\int\langle w\rangle^{3\gamma}g^{3p}\right)^{\frac{1}{3}}\left(\int g^p\right)^{\frac{2}{3}}dt \\
&\leq\sup_{t\in[0,T]}\left(\int g^p\right)^{\frac{2}{3}} \int_0^T \left(\int\langle w\rangle^{3\gamma}g^{3p}\right)^{\frac{1}{3}}.
\end{aligned} ∫ 0 T ∫ ⟨ w ⟩ γ g 3 5 p = ∫ 0 T ∫ ⟨ w ⟩ γ g p ⋅ g 3 2 p ≤ ∫ 0 T ( ∫ ⟨ w ⟩ 3 γ g 3 p ) 3 1 ( ∫ g p ) 3 2 d t ≤ t ∈ [ 0 , T ] sup ( ∫ g p ) 3 2 ∫ 0 T ( ∫ ⟨ w ⟩ 3 γ g 3 p ) 3 1 . Now, using eqs. (31) and (32) yields
∫ 0 T ∫ ⟨ w ⟩ γ g 5 3 p ≤ C sup t ∈ [ 0 , T ] ( ∫ g p ) 2 3 ( ∫ 0 T ∫ ⟨ w ⟩ γ ∣ ∇ g p 2 ∣ 2 + ∫ 0 T ∫ ⟨ w ⟩ γ g p ) ≤ C ( ∫ g i n p + p M ∫ 0 T ∫ ⟨ w ⟩ γ g p ) 5 3 . \begin{aligned}
\int_0^T \int\langle w\rangle^\gamma g^{\frac{5}{3}p} \leq C \sup_{t\in[0,T]} \left(\int g^p\right)^{\frac{2}{3}}\left(\int_0^T \int\langle w\rangle^\gamma\left|\nabla g^{\frac{p}{2}}\right|^2 + \int_0^T \int\langle w\rangle^\gamma g^p\right) \\
&\leq C \left(\int g_{\mathrm{in}}^p + p^M \int_0^T \int\langle w\rangle^\gamma g^p\right)^{\frac{5}{3}}.
\end{aligned} ∫ 0 T ∫ ⟨ w ⟩ γ g 3 5 p ≤ C t ∈ [ 0 , T ] sup ( ∫ g p ) 3 2 ( ∫ 0 T ∫ ⟨ w ⟩ γ ∇ g 2 p 2 + ∫ 0 T ∫ ⟨ w ⟩ γ g p ) ≤ C ( ∫ g in p + p M ∫ 0 T ∫ ⟨ w ⟩ γ g p ) 3 5 . Raising to 3 5 p \frac{3}{5p} 5 p 3 ( ∫ g i n p ) 1 p ≤ max { 1 , ρ ( 0 ) } 1 p ∥ g i n ∥ L ∞ 1 − 1 p \left(\int g_{\mathrm{in}}^p\right)^{\frac{1}{p}} \leq\max\{1,\rho(0)\}^{\frac{1}{p}}\|g_{\mathrm{in}}\|_{L^\infty}^{1-\frac{1}{p}} ( ∫ g in p ) p 1 ≤ max { 1 , ρ ( 0 ) } p 1 ∥ g in ∥ L ∞ 1 − p 1 max { 1 , ρ ( 0 ) } 1 p \max\{1,\rho(0)\}^{\frac{1}{p}} max { 1 , ρ ( 0 ) } p 1 C 1 p C^{\frac{1}{p}} C p 1
( ∫ 0 T ∫ ⟨ w ⟩ γ g 5 3 p ) 3 5 p ≤ C 1 p p M p max { 1 , ∥ g i n ∥ L ∞ , ( ∫ 0 T ∫ ⟨ w ⟩ γ g p ) 1 p } . \left(\int_0^T \int\langle w\rangle^\gamma g^{\frac{5}{3}p}\right)^{\frac{3}{5p}} \leq C^{\frac{1}{p}}p^{\frac{M}{p}}\max\left\{1,\|g_{\mathrm{in}}\|_{L^\infty},\left(\int_0^T \int\langle w\rangle^\gamma g^p\right)^{\frac{1}{p}}\right\}. ( ∫ 0 T ∫ ⟨ w ⟩ γ g 3 5 p ) 5 p 3 ≤ C p 1 p p M max ⎩ ⎨ ⎧ 1 , ∥ g in ∥ L ∞ , ( ∫ 0 T ∫ ⟨ w ⟩ γ g p ) p 1 ⎭ ⎬ ⎫ . Here, if we define
p k = 2 ( 5 3 ) k , E k = ( ∫ 0 T ∫ ⟨ w ⟩ γ g p k ) 1 p k , p_k = 2\left(\frac{5}{3}\right)^k,\qquad E_k = \left(\int_0^T \int\langle w\rangle^\gamma g^{p_k}\right)^{\frac{1}{p_k}}, p k = 2 ( 3 5 ) k , E k = ( ∫ 0 T ∫ ⟨ w ⟩ γ g p k ) p k 1 , we obtain the recursive bound
E k + 1 ≤ C 1 p k p k M p k max { 1 , ∥ g i n ∥ L ∞ , E k } . E_{k+1} \leq C^{\frac{1}{p_k}}p_k^{\frac{M}{p_k}}\max\{1,\|g_{\mathrm{in}}\|_{L^\infty},E_k\}. E k + 1 ≤ C p k 1 p k p k M max { 1 , ∥ g in ∥ L ∞ , E k } . This implies
∥ g ∥ L ∞ ( [ 0 , T ] × R 3 ) = E ∞ ≤ ∏ k = 0 + ∞ C R k 1 p k M p k max { 1 , ∥ g i n ∥ L ∞ , E 0 } , \|g\|_{L^\infty([0,T]\times\mathbb{R}^3)}=E_\infty\leq\prod_{k=0}^{+\infty} C R_k^{\frac{1}{p_k}}\frac{M}{p_k}\max\{1,\|g_{\mathrm{in}}\|_{L^\infty},E_0\}, ∥ g ∥ L ∞ ([ 0 , T ] × R 3 ) = E ∞ ≤ k = 0 ∏ + ∞ C R k p k 1 p k M max { 1 , ∥ g in ∥ L ∞ , E 0 } , which reduces the problem to simply estimating
E 0 2 = ∫ 0 T ∫ ⟨ w ⟩ γ g 2 . E_0^2=\int_0^T\int\langle w\rangle^\gamma g^2. E 0 2 = ∫ 0 T ∫ ⟨ w ⟩ γ g 2 . For soft potentials, this is a consequence of Corollary 3.2, since ⟨ w ⟩ γ ≤ 1 \langle w\rangle^\gamma\leq1 ⟨ w ⟩ γ ≤ 1
∫ ⟨ w ⟩ γ g 2 ≤ ( ∫ g 4 ) 1 3 ( ∫ ⟨ w ⟩ 3 γ 2 g ) 2 3 , \int\langle w\rangle^\gamma g^2\leq\left(\int g^4\right)^{\frac{1}{3}}\left(\int\langle w\rangle^{\frac{3\gamma}{2}}g\right)^{\frac{2}{3}}, ∫ ⟨ w ⟩ γ g 2 ≤ ( ∫ g 4 ) 3 1 ( ∫ ⟨ w ⟩ 2 3 γ g ) 3 2 , and the right hand side is bounded uniformly in t ∈ [ 0 , T ] t\in[0,T] t ∈ [ 0 , T ] p = 4 p=4 p = 4
A comment on uniqueness. Uniqueness of solutions to (16) can be obtained by adapting the techniques developed for the spatially homogeneous equation. In the case of soft potentials − 3 ≤ γ ≤ 0 -3\leq\gamma\leq0 − 3 ≤ γ ≤ 0 [15 , 14 , 5 ] apply with minor modifications and yield uniqueness of weak solutions to (16) within the class L t 1 L w p γ ( R 3 ) L^1_tL^{p_\gamma}_w(\mathbb{R}^3) L t 1 L w p γ ( R 3 ) p γ = 3 3 + γ ∈ [ 1 , ∞ ] p_\gamma=\frac{3}{3+\gamma}\in[1,\infty] p γ = 3 + γ 3 ∈ [ 1 , ∞ ] [16 ] , where uniqueness of weak solutions is established under moment assumptions. By contrast, uniqueness for the inhomogeneous equation (1) is considerably more delicate, and we are not aware of any result that applies in our setting beyond the class of bounded smooth solutions.
Hydrodynamic blow-up for singular deformation matrices
We now consider the case in which
t ∗ : = inf { t > 0 : det ( I + t L 0 ) = 0 } < ∞ . t_*:=\inf\{t>0:\det(I+tL_0)=0\}<\infty. t ∗ := inf { t > 0 : det ( I + t L 0 ) = 0 } < ∞. This occurs precisely when L 0 L_0 L 0 λ < 0 \lambda<0 λ < 0 t ∗ = − λ − 1 t_*=-\lambda^{-1} t ∗ = − λ − 1 g g g f f f
ρ ( t ) = ρ ( 0 ) det ( I + t L 0 ) ≳ ∣ t − t ∗ ∣ − n λ , \rho(t)=\frac{\rho(0)}{\det(I+tL_0)}\gtrsim|t-t_*|^{-n_\lambda}, ρ ( t ) = det ( I + t L 0 ) ρ ( 0 ) ≳ ∣ t − t ∗ ∣ − n λ , where n λ n_\lambda n λ λ \lambda λ γ ≤ 3 \gamma\leq\sqrt{3} γ ≤ 3
A natural next step is to study the behavior of the solution as t → t ∗ t\to t_* t → t ∗
Is the singularity confined to the blow-up of the hydrodynamic quantities of f f f
As a first step toward this question, we focus on L 0 = − I L_0=-I L 0 = − I f i n ( x , v ) = g i n ( x + v ) f_{\mathrm{in}}(x,v)=g_{\mathrm{in}}(x+v) f in ( x , v ) = g in ( x + v ) § 2 (with L 0 = − I L_0=-I L 0 = − I ξ ( t , x ) = L ( t ) x = − x 1 − t \xi(t,x)=L(t)x=-\frac{x}{1-t} ξ ( t , x ) = L ( t ) x = − 1 − t x (1) takes the form
f ( t , x , v ) = g ( t , v + x 1 − t ) . f(t,x,v)=g\left(t,v+\frac{x}{1-t}\right). f ( t , x , v ) = g ( t , v + 1 − t x ) . Motivated by this observation we perform a self-similar ansatz
f ( t , x , v ) = F ( s ( t ) , x ( 1 − t ) c x , v ( 1 − t ) c v ) , X : = x ( 1 − t ) c x , V : = v ( 1 − t ) c v , f(t,x,v)=F\left(s(t),\frac{x}{(1-t)^{c_x}},\frac{v}{(1-t)^{c_v}}\right),\qquad X:=\frac{x}{(1-t)^{c_x}},\quad V:=\frac{v}{(1-t)^{c_v}}, f ( t , x , v ) = F ( s ( t ) , ( 1 − t ) c x x , ( 1 − t ) c v v ) , X := ( 1 − t ) c x x , V := ( 1 − t ) c v v , where s ( t ) s(t) s ( t )
∂ t f = ( 1 − t ) − 1 [ ( 1 − t ) s ′ ( t ) ∂ s F + c x X ⋅ ∇ X F + c v V ⋅ ∇ V F ] , \partial_t f=(1-t)^{-1}\left[(1-t)s'(t)\partial_sF+c_xX\cdot\nabla_XF+c_vV\cdot\nabla_VF\right], ∂ t f = ( 1 − t ) − 1 [ ( 1 − t ) s ′ ( t ) ∂ s F + c x X ⋅ ∇ X F + c v V ⋅ ∇ V F ] , v ⋅ ∇ x f = ( 1 − t ) c v − c x V ⋅ ∇ X F , v\cdot\nabla_xf=(1-t)^{c_v-c_x}V\cdot\nabla_XF, v ⋅ ∇ x f = ( 1 − t ) c v − c x V ⋅ ∇ X F , Q ( f ) = ( 1 − t ) c v ( 3 + γ ) Q ( F ) . Q(f)=(1-t)^{c_v(3+\gamma)}Q(F). Q ( f ) = ( 1 − t ) c v ( 3 + γ ) Q ( F ) . Imposing
c v − c x = − 1 c_v-c_x=-1 c v − c x = − 1 matches the scaling of the transport terms, leading to
( 1 − t ) s ′ ( t ) ∂ s F + c x X ⋅ ∇ X F + V ⋅ ( ∇ X + c v ∇ V ) F = ( 1 − t ) c v ( 3 + γ ) + 1 Q ( F ) . (1-t)s'(t)\partial_sF+c_xX\cdot\nabla_XF+V\cdot(\nabla_X+c_v\nabla_V)F=(1-t)^{c_v(3+\gamma)+1}Q(F). ( 1 − t ) s ′ ( t ) ∂ s F + c x X ⋅ ∇ X F + V ⋅ ( ∇ X + c v ∇ V ) F = ( 1 − t ) c v ( 3 + γ ) + 1 Q ( F ) . Now there are two natural choices for the parameters c x , c v c_x,c_v c x , c v c v = 0 , c x = 1 c_v=0,c_x=1 c v = 0 , c x = 1 § 2 and leads to the homogeneous equation with an extra hyperbolic term. One can also choose
c v = − 1 , c x = 0 ; c_v=-1,\qquad c_x=0; c v = − 1 , c x = 0 ; this choice allows us to connect (1) with the homogeneous equation. Indeed,
f ( t , x , v ) = F ( s ( t ) , x , ( 1 − t ) v ) = F ( s ( t ) , X , V ) , f(t,x,v)=F(s(t),x,(1-t)v)=F(s(t),X,V), f ( t , x , v ) = F ( s ( t ) , x , ( 1 − t ) v ) = F ( s ( t ) , X , V ) , and F F F
( 1 − t ) s ′ ( t ) ∂ s F + V ⋅ ( ∇ X − ∇ V ) F = ( 1 − t ) − 2 − γ Q ( F ) . (33) (1-t)s'(t)\partial_sF+V\cdot(\nabla_X-\nabla_V)F=(1-t)^{-2-\gamma}Q(F). \tag*{(33)} ( 1 − t ) s ′ ( t ) ∂ s F + V ⋅ ( ∇ X − ∇ V ) F = ( 1 − t ) − 2 − γ Q ( F ) . (33) We look for solutions of the form
F ( s , X , V ) = F ( s , X + V ) , F(s,X,V)=F(s,X+V), F ( s , X , V ) = F ( s , X + V ) , and (33) becomes
s ′ ( t ) ∂ s F = ( 1 − t ) − 3 − γ Q ( F ) . s'(t)\partial_sF=(1-t)^{-3-\gamma}Q(F). s ′ ( t ) ∂ s F = ( 1 − t ) − 3 − γ Q ( F ) . Choosing the reparameterization
s ( t ) = ∫ 0 t ( 1 − τ ) − 3 − γ d τ , s(t)=\int_0^t(1-\tau)^{-3-\gamma}\,d\tau, s ( t ) = ∫ 0 t ( 1 − τ ) − 3 − γ d τ , we obtain that
f ( t , x , v ) = F ( s ( t ) , W ) , W = x + ( 1 − t ) v , (34) f(t,x,v)=F(s(t),W),\qquad W=x+(1-t)v, \tag*{(34)} f ( t , x , v ) = F ( s ( t ) , W ) , W = x + ( 1 − t ) v , (34) where now F F F
{ ∂ s F = Q ( F ) , F ( 0 , W ) = g i n ( W ) . \begin{cases}
\partial_sF=Q(F),\\
F(0,W)=g_{\mathrm{in}}(W).
\end{cases} { ∂ s F = Q ( F ) , F ( 0 , W ) = g in ( W ) . With this setup, studying the behavior of f f f t → 1 − t\to1^- t → 1 −
Proof of Theorem 1.2. The fact that f ∈ C ∞ ( [ 0 , 1 ) × R 6 ) ∩ L ∞ ( [ 0 , 1 ) × R 6 ) f\in C^\infty([0,1)\times\mathbb{R}^6)\cap L^\infty([0,1)\times\mathbb{R}^6) f ∈ C ∞ ([ 0 , 1 ) × R 6 ) ∩ L ∞ ([ 0 , 1 ) × R 6 ) (34) and the fact that F F F (5) to (7) now follow by a direct computation:
ρ ( t , x ) = ∫ F ( s ( t ) , x + ( 1 − t ) v ) d v = ( 1 − t ) − 3 ∫ F ( s ( t ) , W ) d W = ( 1 − t ) − 3 ∫ g i n ( W ) d W . \begin{aligned}
\rho(t,x)=\int F(s(t),x+(1-t)v)\,dv \\
&=(1-t)^{-3}\int F(s(t),W)\,dW=(1-t)^{-3}\int g_{\mathrm{in}}(W)\,dW.
\end{aligned} ρ ( t , x ) = ∫ F ( s ( t ) , x + ( 1 − t ) v ) d v = ( 1 − t ) − 3 ∫ F ( s ( t ) , W ) d W = ( 1 − t ) − 3 ∫ g in ( W ) d W . Similarly, for the bulk velocity we have
u ( t , x ) = 1 ρ ( t , x ) ∫ v F ( s ( t ) , x + ( 1 − t ) v ) d v = ( 1 − t ) − 4 ρ ( t , x ) ∫ ( W − x ) F ( s ( t ) , W ) d W = − x ( 1 − t ) 4 ρ ( 0 ) ρ ( t ) = − x 1 − t , \begin{aligned}
u(t,x)=\frac{1}{\rho(t,x)}\int vF(s(t),x+(1-t)v)\,dv \\
&=\frac{(1-t)^{-4}}{\rho(t,x)}\int(W-x)F(s(t),W)\,dW \\
&=-\frac{x}{(1-t)^4}\frac{\rho(0)}{\rho(t)}=-\frac{x}{1-t},
\end{aligned} u ( t , x ) = ρ ( t , x ) 1 ∫ v F ( s ( t ) , x + ( 1 − t ) v ) d v = ρ ( t , x ) ( 1 − t ) − 4 ∫ ( W − x ) F ( s ( t ) , W ) d W = − ( 1 − t ) 4 x ρ ( t ) ρ ( 0 ) = − 1 − t x , where we used that ∫ W F ( W ) d W = 0 \int WF(W)\,dW=0 ∫ W F ( W ) d W = 0
θ ( t , x ) = 1 3 ρ ( t ) ∫ ∣ v − u ( t , x ) ∣ 2 F ( s ( t ) , x + ( 1 − t ) v ) d v = 1 3 ρ ( t ) ( 1 − t ) 2 ∫ ∣ ( 1 − t ) v + x ∣ 2 F ( s ( t ) , x + ( 1 − t ) v ) d v \begin{aligned}
\theta(t,x)=\frac{1}{3\rho(t)}\int|v-u(t,x)|^2F(s(t),x+(1-t)v)\,dv \\
&=\frac{1}{3\rho(t)(1-t)^2}\int|(1-t)v+x|^2F(s(t),x+(1-t)v)\,dv
\end{aligned} θ ( t , x ) = 3 ρ ( t ) 1 ∫ ∣ v − u ( t , x ) ∣ 2 F ( s ( t ) , x + ( 1 − t ) v ) d v = 3 ρ ( t ) ( 1 − t ) 2 1 ∫ ∣ ( 1 − t ) v + x ∣ 2 F ( s ( t ) , x + ( 1 − t ) v ) d v = 1 3 ρ ( t ) ( 1 − t ) 5 ∫ ∣ W ∣ 2 F ( s ( t ) , W ) d W = 1 3 ( 1 − t ) − 2 ∫ ∣ W ∣ 2 g i n ( W ) d w ∫ g i n ( W ) d w , \begin{aligned}
&= \frac{1}{3\rho(t)(1-t)^5}\int|W|^2 F(s(t),W)\,dW\\
&= \frac{1}{3}(1-t)^{-2}\frac{\int|W|^2g_{\mathrm{in}}(W)\,dw}{\int g_{\mathrm{in}}(W)\,dw},
\end{aligned} = 3 ρ ( t ) ( 1 − t ) 5 1 ∫ ∣ W ∣ 2 F ( s ( t ) , W ) d W = 3 1 ( 1 − t ) − 2 ∫ g in ( W ) d w ∫ ∣ W ∣ 2 g in ( W ) d w , which finishes the proof.
We recall that, in Theorem 1.2 , the solution takes the form (34) . Written more explicitly:
f ( t , x , v ) = F ( s ( t ) , x + ( 1 − t ) v ) , s ( t ) = ∫ 0 t ( 1 − τ ) − γ − 3 d τ . (35) f(t,x,v)=F\bigl(s(t),x+(1-t)v\bigr),\qquad s(t)=\int_0^t(1-\tau)^{-\gamma-3}\,d\tau. \tag*{(35)} f ( t , x , v ) = F ( s ( t ) , x + ( 1 − t ) v ) , s ( t ) = ∫ 0 t ( 1 − τ ) − γ − 3 d τ . (35) Note that, if we fix x x x t → 1 − t\to1^{-} t → 1 − f ( t , x , ⋅ ) f(t,x,\cdot) f ( t , x , ⋅ ) F ( s ( 1 − ) , x ) F(s(1-),x) F ( s ( 1 − ) , x ) ( ρ , u , θ ) (\rho,u,\theta) ( ρ , u , θ ) F ( s ( t ) , ⋅ ) F(s(t),\cdot) F ( s ( t ) , ⋅ ) γ \gamma γ
If γ < − 2 \gamma<-2 γ < − 2 s ( 1 − ) < ∞ s(1-)<\infty s ( 1 − ) < ∞ F ( s ( 1 − ) , ⋅ ) F(s(1-),\cdot) F ( s ( 1 − ) , ⋅ ) g i n g_{\mathrm{in}} g in γ = − 3 \gamma=-3 γ = − 3 s ( t ) = t s(t)=t s ( t ) = t s ( 1 − ) = 1 s(1-)=1 s ( 1 − ) = 1
If γ ≥ − 2 \gamma\ge-2 γ ≥ − 2 s ( 1 − ) = ∞ s(1-)=\infty s ( 1 − ) = ∞ F ( s ( 1 − ) , ⋅ ) F(s(1-),\cdot) F ( s ( 1 − ) , ⋅ ) t ∼ 1 t\sim1 t ∼ 1 (35) behaves approximately as
f ( t , x , v ) ∼ ρ ( 0 ) ( 2 π θ ( 0 ) ) 3 / 2 exp { − ∣ x + ( 1 − t ) v ∣ 2 2 θ ( 0 ) } = ρ ( t ) ( 2 π θ ( t ) ) 3 / 2 exp { − ∣ u ( t , x ) − v ∣ 2 2 θ ( t ) } . \begin{aligned}
f(t,x,v)\sim\frac{\rho(0)}{(2\pi\theta(0))^{3/2}}\exp\left\{-\frac{|x+(1-t)v|^2}{2\theta(0)}\right\} \\
&=\frac{\rho(t)}{(2\pi\theta(t))^{3/2}}\exp\left\{-\frac{|u(t,x)-v|^2}{2\theta(t)}\right\}.
\end{aligned} f ( t , x , v ) ∼ ( 2 π θ ( 0 ) ) 3/2 ρ ( 0 ) exp { − 2 θ ( 0 ) ∣ x + ( 1 − t ) v ∣ 2 } = ( 2 π θ ( t ) ) 3/2 ρ ( t ) exp { − 2 θ ( t ) ∣ u ( t , x ) − v ∣ 2 } . The distinction of the asymptotic profile between very soft and γ ≥ − 2 \gamma\ge-2 γ ≥ − 2 Q Q Q γ > − 2 \gamma>-2 γ > − 2 γ < − 2 \gamma<-2 γ < − 2 γ \gamma γ [32 ] for the Boltzmann equation.
Classification of long time asymptotics for deformation matrices
James, Nota, and Velázquez classified in [24 ] all possible asymptotic behaviors as t → + ∞ t\to+\infty t → + ∞
L ( t ) : = L 0 ( I + t L 0 ) − 1 L(t):=L_0(I+tL_0)^{-1} L ( t ) := L 0 ( I + t L 0 ) − 1 for any matrix L 0 L_0 L 0
det ( I + t L 0 ) > 0 , for all t ≥ 0. \det(I+tL_0)>0,\qquad\text{for all }t\ge0. det ( I + t L 0 ) > 0 , for all t ≥ 0. Here we provide the complete list:
L ( t ) = 1 t I + O ( 1 t 2 ) , t → ∞ . L(t)=\frac{1}{t}I+O\left(\frac{1}{t^2}\right),\qquad t\to\infty. L ( t ) = t 1 I + O ( t 2 1 ) , t → ∞. L ( t ) = 1 t ( 1 0 K 0 1 0 0 0 0 ) + O ( 1 t 2 ) , t → ∞ , L(t)=\frac{1}{t}
\begin{pmatrix}
1 & 0 & K\\
0 & 1 & 0\\
0 & 0 & 0
\end{pmatrix}
+O\left(\frac{1}{t^2}\right),\qquad t\to\infty, L ( t ) = t 1 1 0 0 0 1 0 K 0 0 + O ( t 2 1 ) , t → ∞ , when K ≠ 0 K\ne0 K = 0 K = 0 K=0 K = 0
L ( t ) = 1 t ( 0 0 0 0 0 K 0 0 1 ) + O ( 1 t 2 ) , t → ∞ . L(t)=\frac{1}{t}
\begin{pmatrix}
0 & 0 & 0\\
0 & 0 & K\\
0 & 0 & 1
\end{pmatrix}
+O\left(\frac{1}{t^2}\right),\qquad t\to\infty. L ( t ) = t 1 0 0 0 0 0 0 0 K 1 + O ( t 2 1 ) , t → ∞. L ( t ) = ( 0 K 0 0 0 0 0 0 0 ) = L 0 , K ≠ 0. L(t)=\begin{pmatrix} 0 & K & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}=L_0,\qquad K\ne0. L ( t ) = 0 0 0 K 0 0 0 0 0 = L 0 , K = 0. L ( t ) = ( 0 K 2 0 0 0 0 0 0 0 ) + 1 t ( 0 K 1 K 3 K 1 0 0 0 0 K 3 1 ) + O ( 1 t 2 ) , K 2 ≠ 0 , t → ∞ . L(t)=\begin{pmatrix} 0 & K_2 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}+\frac{1}{t}\begin{pmatrix} 0 & K_1K_3 & K_1 \\ 0 & 0 & 0 \\ 0 & K_3 & 1 \end{pmatrix}+O\left(\frac{1}{t^2}\right),\qquad K_2\ne0,\qquad t\to\infty. L ( t ) = 0 0 0 K 2 0 0 0 0 0 + t 1 0 0 0 K 1 K 3 0 K 3 K 1 0 1 + O ( t 2 1 ) , K 2 = 0 , t → ∞. L ( t ) = ( 0 K 3 K 2 − t K 1 K 3 0 0 K 1 0 0 0 ) , K 1 K 3 ≠ 0 , t → ∞ . L(t)=\begin{pmatrix} 0 & K_3 & K_2-tK_1K_3 \\ 0 & 0 & K_1 \\ 0 & 0 & 0 \end{pmatrix},\qquad K_1K_3\ne0,\qquad t\to\infty. L ( t ) = 0 0 0 K 3 0 0 K 2 − t K 1 K 3 K 1 0 , K 1 K 3 = 0 , t → ∞. A comment on the expected asymptotics.
The long-time behavior of solutions to (16) can be understood comparing the order of magnitude of the hyperbolic term L ( t ) w ⋅ ∇ w g L(t)w\cdot\nabla_wg L ( t ) w ⋅ ∇ w g Q ( g ) Q(g) Q ( g ) ρ ( t ) \rho(t) ρ ( t ) [ w ] ( t ) [w](t) [ w ] ( t ) [ L ] ( t ) [L](t) [ L ] ( t ) L ( t ) L(t) L ( t ) Q Q Q
ρ ( t ) [ w ] γ [ g ] , (36) \rho(t)[w]^\gamma[g], \tag*{(36)} ρ ( t ) [ w ] γ [ g ] , (36) where [ g ] [g] [ g ] g g g γ \gamma γ Φ \Phi Φ γ + 2 \gamma+2 γ + 2 γ \gamma γ L ( t ) L(t) L ( t ) t → ∞ t\to\infty t → ∞ t → ∞ t\to\infty t → ∞ γ \gamma γ L ( t ) L(t) L ( t )
The hyperbolic term dominates over the collision term.
The collision term dominates over the hyperbolic term.
Both terms remain of the same order of magnitude.
An outline of the expected behavior for homoenergetic solutions to the Boltzmann equation was presented in [24 ] . Here, we focus on the Landau equation and provide intuition regarding the long-time behavior of solutions to (16) . To this end, we rewrite the collision operator in non-divergence form, in components,
Q [ g ] = ∑ i , j a i j [ g ] ∂ i j g + c γ [ g ] g , (37) Q[g]=\sum_{i,j}a_{ij}[g]\partial_{ij}g+c_\gamma[g]g, \tag*{(37)} Q [ g ] = i , j ∑ a ij [ g ] ∂ ij g + c γ [ g ] g , (37) where a i j [ g ] a_{ij}[g] a ij [ g ] A [ g ] = ( a i j [ g ] ) i , j = 1 3 A[g]=(a_{ij}[g])_{i,j=1}^{3} A [ g ] = ( a ij [ g ] ) i , j = 1 3 (24) ), given by
a i j [ g ] ( w ) = ( Φ i j ∗ g ) ( w ) , Φ i j ( z ) = ∣ z ∣ γ + 2 ( δ i j − z i z j ∣ z ∣ 2 ) , (38) a_{ij}[g](w)=(\Phi_{ij}*g)(w),\qquad\Phi_{ij}(z)=|z|^{\gamma+2}\left(\delta_{ij}-\frac{z_i z_j}{|z|^2}\right), \tag*{(38)} a ij [ g ] ( w ) = ( Φ ij ∗ g ) ( w ) , Φ ij ( z ) = ∣ z ∣ γ + 2 ( δ ij − ∣ z ∣ 2 z i z j ) , (38) with Φ \Phi Φ (2) and c γ [ g ] c_\gamma[g] c γ [ g ] (28) , namely
c γ [ g ] ( w ) = { 2 ( 3 + γ ) ( g ∗ ∣ ⋅ ∣ γ ) ( w ) , γ > − 3 , 8 π g ( w ) , γ = − 3. (39) c_\gamma[g](w)=\begin{cases}2(3+\gamma)(g*|\cdot|^\gamma)(w), & \gamma>-3,\\ 8\pi g(w), & \gamma=-3.\end{cases} \tag*{(39)} c γ [ g ] ( w ) = { 2 ( 3 + γ ) ( g ∗ ∣ ⋅ ∣ γ ) ( w ) , 8 π g ( w ) , γ > − 3 , γ = − 3. (39) We now estimate the order of magnitude of the two terms on the right-hand side of (A.2):
a i j [ g ] ∼ ρ ( t ) [ w ] γ + 2 , ∂ i j g ∼ [ g ] [ w ] 2 ⇒ a i j [ g ] ∂ i j g ∼ ρ ( t ) [ w ] γ [ g ] ; (40) a_{ij}[g]\sim\rho(t)[w]^{\gamma+2},\qquad\partial_{ij}g\sim\frac{[g]}{[w]^2}\quad\Rightarrow\quad a_{ij}[g]\partial_{ij}g\sim\rho(t)[w]^\gamma[g]; \tag*{(40)} a ij [ g ] ∼ ρ ( t ) [ w ] γ + 2 , ∂ ij g ∼ [ w ] 2 [ g ] ⇒ a ij [ g ] ∂ ij g ∼ ρ ( t ) [ w ] γ [ g ] ; (40) c γ [ g ] ∼ ρ ( t ) [ w ] γ ⇒ c γ [ g ] g ∼ ρ ( t ) [ w ] γ [ g ] . (41) c_\gamma[g]\sim\rho(t)[w]^\gamma\quad\Rightarrow\quad c_\gamma[g]g\sim\rho(t)[w]^\gamma[g]. \tag*{(41)} c γ [ g ] ∼ ρ ( t ) [ w ] γ ⇒ c γ [ g ] g ∼ ρ ( t ) [ w ] γ [ g ] . (41) Since both contributions (cf. (A.5)–(A.6)) give the same scaling, we obtain that the collision term scales as
Q ( g ) ( w ) ∼ ρ ( t ) [ w ] γ [ g ] . (42) Q(g)(w)\sim\rho(t)[w]^\gamma[g]. \tag*{(42)} Q ( g ) ( w ) ∼ ρ ( t ) [ w ] γ [ g ] . (42) On the other hand, assuming that L ( t ) L(t) L ( t ) [ L ] ( t ) [L](t) [ L ] ( t )
L ( t ) w ⋅ ∇ w g ∼ [ L ] ( t ) [ g ] (43) L(t)\mathbf{w}\cdot\nabla_{\mathbf{w}}g\sim[L](t)[g] \tag*{(43)} L ( t ) w ⋅ ∇ w g ∼ [ L ] ( t ) [ g ] (43) Consequently, at a formal level, determining the dominant physical mechanism reduces to comparing (A.7) and (A.8), namely
[ L ] ( t ) and ρ ( t ) [ w ] γ . [L](t)\quad\text{and}\quad\rho(t)[w]^{\gamma}. [ L ] ( t ) and ρ ( t ) [ w ] γ . This formal argument allows one to identify the expected regime (hyperbolic-dominated, balanced or collision-dominated), which guides the study of the corresponding long-time behavior. We refer to [24 ] for the rigorous computations in the case of the Boltzmann equation. Table 1 collects the three most important cases, dilation, planar shear and simple shear.
Table 1. Expected asymptotic regimes for the Landau equation for dilation, planar, and simple shear.
The University of Texas at Austin, USA Email address: funai.caja@utexas.edu
The University of Texas at Austin, USA Email address: gualdani@math.utexas.edu
Gran Sasso Science Institute, L’Aquila, Italy Email address: alessia.nota@gssi.it
