in Bounded Domains Ariane Trescases CMLA, ENS Cachan

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1 CMLA, ENS Cachan Joint work with Yan GUO, Chanwoo KIM and Daniela TONON International Conference on Nonlinear Analysis: Boundary Phenomena for Evolutionnary PDE Academia Sinica December 21, 214

2 Outline The system Results Convex domains Non-

3 The system Results The system Results Convex domains Non-

4 The Boltzmann equation F (t, x, v): density distribution of gas particles at time t, at point x of space Ω R 3 and with velocity v R 3. It satisfies tf + v }{{ xf = Q(F, F ), }}{{} free transport collisions The system Results Q(F 1, F 2) = Q gain (F 1, F 2) Q loss (F 1, F 2) = v u κ {F 1(u )F 2(v ) F 1(u)F 2(v)}q (θ) dω du, R 3 S 2 where v = v + [(u v) ω] ω, where u = u [(u v) ω] ω. Assume κ 1 (hard potential), q C v u ω (angular cutoff). v u

5 Boundary conditions Ω is a bounded domain of R 3. We define in the phase space the outgoing/incoming/grazing boundary γ + = {(x, v) Ω R 3 : v n(x) > }, (x,v) γ + (x,v) γ Ω The system Results γ = {(x, v) Ω R 3 : v n(x) < }, γ = {(x, v) Ω R 3 : v n(x) = }. n(x) x (x,v) γ - Diffuse boundary condition: writing µ(v) = exp{ v 2 /2}, (x, v) γ, F (t, x, v) = c µ µ(v) F (t, x, u) u n(x) du. u n(x)>

6 The system: summary Define F (t, x, v) = µ(v) f (t, x, v). We rewrite the Cauchy problem: tf + v xf + ν[ µf ]f = Γ(f, f ), f t= = f, (1) f γ = c µ µ(v) µ(u)f (u)u n(x) du, u n(x)> The system Results where ν[ µf ] = Q loss ( µf, 1), Γ(f, f ) = Q gain( µf, µf ) µ(v).

7 Known results for weak solutions in general bounded domains: existence theory (H ): F = µf and e θ v 2 f < ( < θ < 1/4). Theorem (Existence/uniqueness, [Guo 1], [GKTT 13]) (H ). There exists a unique solution f of (1) on [, T ) with T = T (f ). Furthermore for some < θ < θ, T < T The system Results sup e θ v 2 f (t) T e θ v 2 f. t T If e θ v 2 {f µ} 1, then T =.

8 Known results for weak solutions in general bounded domains: regularity In : continuity away from the grazing boundary γ [Guo 1] In non-: discontinuity created on the grazing boundary and propagated along the (grazing) characteristics [Kim 11] The system Results Continuity Ω Continuity Ω Discontinuity

9 New results: regularity in [GKTT 13] Assume Ω is a smooth strictly convex domain. Assume (H ) and f satisfies the diffuse boundary condition. Theorem (, 1 < p < 2) Assume p (1, 2) and f L p (Ω R 3 ). Then T T < T, sup f p p(t) + f p v.n ds dv dt t T Ω R 3 The system Results with P a polynomial. T f p p + P( e θ v 2 f ), Note: we use the notation tf = v xf ν[ µf ]f + Γ(f, f ).

10 New results: regularity in [GKTT 13] Assume Ω is a smooth strictly convex domain. Assume (H ) and f satisfies the diffuse boundary condition. Theorem (Weighted, p [2, ]) Assume d 2 p f L p (Ω R 3 ). Then T < T, if p [2, [ T sup dp 2 f p p(t) + dp 2 f p v.n ds dv dt t T Ω R 3 The system Results T d 2 p f p p + P( e θ v 2 f ), and if p =, sup d 2 p f (t) T d 2 p f + P( e θ v 2 f ), where d 2 p = d 2 p (t, x, v) (defined later) vanishes on the grazing boundary γ.

11 New results: regularity in [GKTT 13] Assume Ω is a smooth strictly convex domain. Assume (H ) and f satisfies the diffuse boundary condition. Theorem (Weighted, p [2, ]) Assume d 2 p f L p (Ω R 3 ). Then T < T, if p [2, [ T sup dp 2 f p p(t) + dp 2 f p v.n ds dv dt t T Ω R 3 The system Results T d 2 p f p p + P( e θ v 2 f ), and if p =, sup d 2 p f (t) T d 2 p f + P( e θ v 2 f ), where d 2 p = d 2 p (t, x, v) (defined later) vanishes on the grazing boundary γ. Theorem (C 1 propagation) If d 2 f C ( Ω R 3 ) and tf satisfies the diffuse boundary condition, then f is C 1 away from the grazing boundary.

12 New results: regularity in non- [GKTT 14] Assume Ω is a smooth bounded domain. Assume (H ). Theorem (BV propagation) If f BV (Ω R 3 ), then T < T, The system Results sup f (t) BV T f BV + P( e θ v 2 f ), t T and x,v f γ is a Radon measure σ(t) on Ω R 3 such that T σ( Ω R 3 ) dt T f BV + P( e θ v 2 f ).

13 The system Results Convex domains Non-

14 Road map for a priori estimates 1. for f (uses convexity) T sup f (t) p p + f p γ +,p f p p + I Ω (T ) + J Ω (T ). t T 2. Estimates inside the domain { } J Ω T sup s t f p p + P( e θ v 2 f ). 3. Estimates on the boundary (uses p > 1) T I Ω O(ε) f p dγds + tp( e θ v 2 f ) γ + [ T ] +C ε,t f p p + J Ω + f (s) p p. We conclude choosing ε 1, T 1.

15 Computation of the traces: a model problem Consider the system th + v xh + νh = H, h t= = f, h γ = g. Define tf and xg = ( τ1 g, τ2 g, ng) by tf := v xf νf + H t=, ng := 1 { tg } (v τ i ) τi g νg + H. v n Proposition If H C ([, T ], L p (Ω R 3 )), f L p (Ω R 3 ) and g C ([, T ], L p (γ )), if H, f, g strongly decay in v and g() = f γ, then h t= = f and h γ = g.

16 Computation of the traces: characteristics Proof: solve the system explicitly (method of characteristics). t> t-t b Ω x b (x,v) If t > t b, h(t, x, v) = e t b ν g(t t b, x b, v) + tb e s ν H(t s, x sv, v) ds

17 Computation of the traces: characteristics Remark: singularity in non-. Ω (x,v) (x',v) x' b x b If t > t b, h(t, x, v) = e t b ν g(t t b, x b, v) + tb e s ν H(t s, x sv, v) ds

18 For [ t, x, v ] t f + v x f + ν[ µf ] f = Γ(f, f ) + v xf + f ν[ µf ], f t= = f, { µ(v) } f γ = µ(u)f (u)u n(x) du. u n(x)>

19 For [ t, x, v ] t f + v x f + ν[ µf ] f = Γ(f, f ) + v xf + f ν[ µf ], f t= = f, { µ(v) } f γ = µ(u)f (u)u n(x) du. u n(x)> : f (t) p p + + t t f p γ +,p + t ν 1/p f p p = f p p + t f p γ,p Ω R 3 { Γ(f, f ) + v xf + f ν[ µf ]} f p 2 f. where dγ = v n(x) dx dv and γ+,p = L p (γ +, dγ).

20 Road map for a priori estimates 1. for f (uses convexity) T sup f (t) p p + f p γ +,p f p p + I Ω (T ) + J Ω (T ). t T 2. Estimates inside the domain { } J Ω T sup s t f p p + P( e θ v 2 f ). 3. Estimates on the boundary (uses p > 1) T I Ω O(ε) f p dγds + tp( e θ v 2 f ) γ + [ T ] +C ε,t f p p + J Ω + f (s) p p. We conclude choosing ε 1, T 1.

21 We use Grad s estimates [Grad 62] to show for any h = h(t, x, v) Therefore J Ω = Γ(h, h) p e θ v 2 h [ h p + h p], h ν[ µh] p e θ v 2 h [ h p + h p]. T T { Γ(f, f ) + v xf + f ν[ µf ]} f p 2 f Ω R 3 Γ(f, f ) + v xf + f ν[ µf ] p f p 1 p P( sup e θ v 2 f ) t T { T } T + f p p.

22 Road map for a priori estimates 1. for f (uses convexity) T sup f (t) p p + f p γ +,p f p p + I Ω (T ) + J Ω (T ). t T 2. Estimates inside the domain J Ω P( sup e θ v 2 f ) t T { T } T + f p p. 3. Estimates on the boundary (uses p > 1) T I Ω O(ε) f p dγds + tp( e θ v 2 f ) γ + [ T ] +C ε,t f p p + J Ω + f (s) p p. We conclude choosing ε 1, T 1.

23 Diffuse boundary f γ = c µ µ(v) tf γ = c µ µ(v) τ f γ = c µ µ(v) + c µ µ(v) u n(x)> u n(x)> u n(x)> u n(x)> µ(u) f (u)u n(x) du. µ(u) tf (u)u n(x) du, µ(u) τ f (u)u n(x) du, µ(u) τ T t (x)t (x)u f (u)u n(x) du, nf γ = 1 { tf } (v τ i ) τi f νf + Γ(f, f ), v n v f γ = c µv µ(v) µ(u) f (u)u n(x) du. u n(x)>

24 Diffuse boundary f γ = c µ µ(v) tf γ = c µ µ(v) τ f γ = c µ µ(v) + c µ µ(v) u n(x)> u n(x)> u n(x)> u n(x)> µ(u) f (u)u n(x) du. µ(u) tf (u)u n(x) du, µ(u) τ f (u)u n(x) du, µ(u) τ T t (x)t (x)u f (u)u n(x) du, nf γ = 1 { tf } (v τ i ) τi f νf + Γ(f, f ), v n v f γ = c µv µ(v) µ(u) f (u)u n(x) du. u n(x)> f γ ( 1 + v n 1) µ(v) 1/4 u n(x)> µ(u) f (u) u n du + O(f )

25 Diffuse boundary f γ p O(f ) + ( 1 + v n(x) p) µ(v) p/4 p µ(u) f (u) u n(x) du. We integrate T u n(x)> f p v n(x) dv dx dt TO(f ) γ ( v n 1 p ) µ(v) p/8 dv R 3 T p µ(u) f (u) u n(x) du dx dt. Ω u n(x)> To control the singularity: p < 2.

26 Diffuse boundary f γ p O(f ) + ( 1 + v n(x) p) µ(v) p/4 p µ(u) f (u) u n(x) du. We integrate T u n(x)> f p v n(x) dv dx dt TO(f ) γ ( v n 1 p ) µ(v) p/8 dv R 3 T p µ(u) f (u) u n(x) du dx dt. Ω u n(x)> To control the singularity: p < 2.

27 Outgoing flow Idea: the outgoing flow is determined by "inside" quantities during the time [t t b, t]. t> t-s, <s<t b t-t b x b=x-vt b (x-vs,v) (x,v) Ω This is not satisfying when t b 1, i.e. on the almost grazing boundary γ ɛ + = {(x, v) Ω R 3 : < v n(x) < ɛ or v 1/ɛ}.

28 Outgoing flow Idea: the outgoing flow is determined by "inside" quantities during the time [t t b, t]. t> t-s, <s<t b t-t b x b=x-vt b (x-vs,v) (x,v') Ω (x,v) x' b This is not satisfying when t b 1, i.e. on the almost grazing boundary γ ɛ + = {(x, v) Ω R 3 : < v n(x) < ɛ or v 1/ɛ}.

29 Outgoing flow: grazing and non-grazing boundary Away from the almost grazing boundary, p µ(u) 1/2 f (u) u n(x) du dx dt Ω u:(x,u) γ +\γ+ ε [ T ] C ε,t f p p + J Ω + f (s) p p.

30 Outgoing flow: grazing and non-grazing boundary Away from the almost grazing boundary, p µ(u) 1/2 f (u) u n(x) du dx dt Ω u:(x,u) γ +\γ+ ε [ T ] C ε,t f p p + J Ω + f (s) p p. For the almost grazing part, by Hölder s inequality p µ(u) 1/2 f (u) u n(x) du dx dt Ω u:(x,u) γ+ ε O(ε) f p dγ. γ +

31 Outgoing flow: grazing and non-grazing boundary Away from the almost grazing boundary, p µ(u) 1/2 f (u) u n(x) du dx dt Ω u:(x,u) γ +\γ+ ε [ T ] C ε,t f p p + J Ω + f (s) p p. For the almost grazing part, by Hölder s inequality p µ(u) 1/2 f (u) u n(x) du dx dt Ω u:(x,u) γ+ ε O(ε) f p dγ. γ + Remark: this computation breaks at p = 1!

32 Road map for a priori estimates 1. for f (uses convexity) T sup f (t) p p + f p γ +,p f p p + I Ω (T ) + J Ω (T ). t T 2. Estimates inside the domain J Ω P( sup e θ v 2 f ) t T { T } T + f p p. 3. Estimates on the boundary (uses p > 1) T I Ω O(ε) f p dγds + TP( sup e θ v 2 f ) γ + t T [ T ] +C ε,t f p p + J Ω + f (s) p p.

33 Road map for a priori estimates 1. for f (uses convexity) T sup f (t) p p + f p γ +,p f p p + I Ω (T ) + J Ω (T ). t T 2. Estimates inside the domain J Ω P( sup e θ v 2 f ) t T { T } T + f p p. 3. Estimates on the boundary (uses p > 1) T I Ω O(ε) f p dγds + TP( sup e θ v 2 f ) γ + t T [ T ] +C ε,t f p p + J Ω + f (s) p p. To conclude: recall sup t T e θ v 2 f e θ v 2 f, choose ε 1 and use a Gronwall s argument.

34 The system Results Convex domains Non- Convex domains Non-

35 Weighted in The (strictly convex) domain Ω is defined by Ω = {x R 3 : ξ(x) < }. Define the kinetic distance α(x, v) := v xξ 2 2{v 2 xξv}ξ(x). Convex domains Non- Main features: "distance": vanishes exactly on the grazing boundary "kinetic": invariant along the characteristics (up to some quantity in v ).

36 Weighted in The (strictly convex) domain Ω is defined by Ω = {x R 3 : ξ(x) < }. Define the kinetic distance α(x, v) := v xξ 2 2{v 2 xξv}ξ(x). Convex domains Non- Main features: "distance": vanishes exactly on the grazing boundary "kinetic": invariant along the characteristics (up to some quantity in v ). We therefore consider, for ω 1, d 2 p (t, x, v) := e ω<v>t α(x, v) βp.

37 W 1, propagation in No energy identity: follow the stochastic backward characteristics and control the number of "bounces"... Ω Convex domains Non- (x,v ) 1 1 (x,v) (x,v ) 3 3 (x,v ) 4 4 (x,v ) 2 2

38 BV propagation in non- We get rid of the grazing trajectories S B := {(x, v) Ω R 3 : n(x b (x, v)) v = } by applying a cut-off on a tubular neigborhood of S B (much work!). Convex domains Non- Ω For the W 1,1 propagation: to estimate the grazing boundary term we use the boundary condition twice (double iteration scheme).

39 BV propagation in non- We get rid of the grazing trajectories S B := {(x, v) Ω R 3 : n(x b (x, v)) v = } by applying a cut-off on a tubular neigborhood of S B (much work!). Convex domains Non- Ω For the W 1,1 propagation: to estimate the grazing boundary term we use the boundary condition twice (double iteration scheme).

40 BV propagation in non- We get rid of the grazing trajectories S B := {(x, v) Ω R 3 : n(x b (x, v)) v = } by applying a cut-off on a tubular neigborhood of S B (much work!). x b (x,v) Ω Convex domains Non- For the W 1,1 propagation: to estimate the grazing boundary term we use the boundary condition twice (double iteration scheme).

41 The system Results Convex domains Non-

42 Recap: propagation of regularity In In non- C away from γ [Guo 1] Discontinuity created on γ and propagated along the W 1,p for 1 < p < 2 grazing trajectories [Kim 11] Ct-ex. (transport eq.) : xf / L 2 BV propagation Weighted W 1,p for p [2, ] C 1 away from γ Ct-ex. : 2 f / L 1

43 Further results Other boundary conditions [GKTT13] In, propagation of C 1 regularity away from γ (with the help of the kinetic distance) for the specular and bounce-back boundary conditions. Non-isothermal boundary [EGKM13] Results of existence, uniqueness and stability (exponential convergence towards the solution of the stationnary problem) for a (non too much) varying boundary temperature. Continuity propagation in convex domains.

44 Esposito, R.; Guo, Y.; Kim, C. ; Marra, R: Non-Isothermal Boundary in the Boltzmann Theory and Fourier Law, Comm. Math. Phys. 323 (213) Grad, H.: Asymptotic theory of the Boltzmann equation, II. Rarefied gas dynamics, Proceedings of the 3rd international Symposium 26 59, Paris, (1962). Guo, Y.: Decay and Continuity of in Bounded Domains. Arch. Rational Mech. Anal. 197 (21) Guo, Y.; Kim, C.; Tonon, D.; T. A.: Boltzmann Equation in Convex Domains. Submitted. Guo, Y.; Kim, C.; Tonon, D.; T. A.: BV-regularity of the in Non-Convex Domains. Submitted. Kim, C.: Formation and propagation of discontinuity for Boltzmann equation in non-. Comm. Math. Phys. 38 (211)

Fourier Law and Non-Isothermal Boundary in the Boltzmann Theory

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