Holder regularity for hypoelliptic kinetic equations

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1 Holder regularity for hypoelliptic kinetic equations Alexis F. Vasseur Joint work with François Golse, Cyril Imbert, and Clément Mouhot The University of Texas at Austin Kinetic Equations: Modeling, Analysis and Numerics Austin, September

2 Table of contents Main result The theorem Some comments Regularity results a la De Giorgi Original result De Giorgi for Integral operators Non-homogeneous kinetic equations Hypoellipticity in kinetic theory Averaging lemmas Ideas of the proof Ellipticity and averaging lemmas A Liouville type lemma

3 De Giorgi applied to Fokker-Planck We consider the following generalized Fokker-Planck equation t f + v x f = v (A(x, v, t) v f ) + s, where s = s(x, v, t) is a bounded measurable source term, and the d d symmetric matrix A satisfies the ellipticity condition 1 Λ I A(x, v, t) ΛI, (t, x, v) R+ R 2d. Theorem (Hölder continuity) Let f be a solution in a domain Q 0 and consider an other domain Q 1 such that Q 1 Q 0. Then f is α-hölder continuous with respect to (x, v, t) in Q 1 and f C α (Q 1 ) C( f L 2 (Q 0 ) + s L (Q 0 )) for some α universal (i.e. α = α(d, Λ)) and C = C(d, Λ, Q 0, Q 1 ).

4 Some comments The result has been later used by: Jinoh Kim, Yan Guo and Hyung Ju Hwang to construct solutions to the Landau equation close to Maxwellian. Imbert and Silvestre to obtain conditional regularity to solutions to Boltzmann equation without cutt-off. The main difficulty is that the operator is regularizing in v only. However the transport term is mixing. This is a typical case of hypoellipticity.

5 Hypoellipticity The linear kinetic Fokker-Planck equation t f + v x f = v ( v f + vf ) shows ellipticity in the v variable only. However, it was showed by Kolmogorov that the equation exhibits regularization effects in both the x and the v variables. It inspired Hörmander and his theory of hypoellipticity. The theory considers C regularity. The idea is that the regularization in v is spread in x by the mixing effect of the transport in x. The proof uses the careful study of commutators.

6 Original result of De Giorgi: the 19th Hilbert problem The 19th Hilbert problem consists in showing the smoothness of local minimizers of convex energy functionals of the form E(w) = F( w) dx, Ω where F is a smooth convex function from R N to R, and Ω is bounded open set of R N. This is a generalization of the Dirichlet integral w 2 dx. Ω Local minimizers of the Dirichlet integral verify the associate Euler-Lagrange equation which is nothing but the Laplace equation: w = 0.

7 The linear problem Denote u the derivative with respect to x i of the solution w to the Euler-Lagrange equation. It verifies div (F ( w) u) = 0. Convexity gives the ellipticity condition: 1 Λ I F ( w) ΛI. Forgetting about the dependence of A(x) = F ( w) on w, it can be rewritten as a classical linear elliptic equation in the divergence form: with the elliptic condition on A: div (A(x) u) = 0, (1) 1 I A(x) ΛI, x Ω. (2) Λ

8 De Giorgi Theorem De Giorgi showed the following theorem. Theorem Let u H 1 (Ω) be a weak solution to (1) with A verifying (2). Then u C α ( Ω) for any Ω Ω, with u C α ( Ω) C u L 2 (Ω). The constant α depends only on Λ and N. The constant C depends on Λ, N, Ω, and Ω. This gives that w C 1,α. Bootstrapping classical Schauder regularity theorem gives w C.

9 Extension to Integral operators The method has been extended to non local operators in various context. Quasi-Geostrophic equation (Caffarelli-V. Annals of Math. 2010), Nonlinear integral-differential operators (Caffarelli-Chan-V., J. Amer. Math. Soc., 2011), Fractional operators in time and space (Allen-Caffarelli-V., ARMA 2016)... Typical example: t θ(t, x) ψ (θ(t, y) θ(t, x))k s (x y)dy = 0, R N with ψ convex and K s (z) = C s z s+n.

10 General form of kinetic equations Kinetic equations are usually of the form t f + v x f = Q(f ), where Q(f ) is a collision kernel, usually an integral operator in v only, possibly with regulation effects (Boltzmann without cut-off). Even if we have some kind of ellipticity in v, we have nothing if this kind in x. Our result can be seen as a hypoellipticity regularization in C α : De GIorgi for hypoelliptic operators.

11 Hypoellipitcity in Kinetic theory: the averaging lemmas (1) Velocity averaging designates a special type of smoothing effect for solutions of the free transport equation ( t + v x )f = S observed for the first time by Golse, Perthame and Sentis, and independently by Agoshkov. It was later improved and generalized in several context (see for instance DiPerna Lions). This smoothing effect bears on averages of f in the velocity variable v, i.e. on expressions of the form R d f (t, x, v)φ(v)dv, say for C c test functions φ. Of course, no smoothing on f itself can be observed, since the transport operator is hyperbolic and propagates the singularities of the source term S.

12 Averaging lemmas (2) However, if S is of the form S = v (A(v) v f ) + s where s is a given source term in L 2, the smoothing effect of velocity averaging can be combined with the H 1 regularity in the v variable implied by the energy inequality, in order to obtain some amount of smoothing on the solution f itself. The equation is now t f + v x f = v (A(v) v f ) + s. Bouchut has obtained Sobolev regularity estimates in the variables t, x, v for such solutions bounded in L 2. This can been seen as a Sobolev version of the hypoelliptic regularity results.

13 Holder regularization effect of Fokker-Planck This result is an application of the De Giorgi techniques to the kinetic theory. Averaging lemmas are a key ingredient of the proof. Actually, we can see this result as a C α version of the Bouchut s averaging lemmas. The result can be seen as a C α version of the hypoellipticity theory.

14 Main steps of the proof L 2 (bounded energy) to L (uniformly bounded). Classically, it involves an intertwined application of the energy inequality and the Sobolev inequality. It is done recursively on a sequence of level set of energy, and a shrinking sequence of sets Q n. In the kinetic setting, the lack of ellipticity in t, x is compensated by the averaging lemmas. We augment the energy inequality with the averaging lemmas of Bouchut. L (uniformly bounded) to C α (modulus of continuity). Classically it is based on an isoperimetric result. The argument has been simplified via a compactness argument (see V. and Chan-V.). Boiled down to a Liouville type result: patch-like solutions are constant.

15 L bounds Figure: L bounds

16 L bounds We want to show that θ < M (M depending on t 0 and U 0 ) Figure: L bounds

17 L bounds We construct a sequence of C k converging to M. Figure: L bounds

18 L bounds We consider U k "Energy/dissipation of energy" at the level set k. Figure: L bounds

19 L bounds U k Ck Mt N 0 U β k 1, β > 1. (Sobolev, Chebychev inequality) Figure: L bounds

20 L bounds Figure: L bounds

21 The second lemma Lemma For all δ 1 (0, 1), δ 2 (0, 1), ω > 0 (universal) such that (1 2ω) d 1 δ 2, (3) there exist α > 0 and θ > 0 (universal) such that for any solution f in Q(0, 2) with f 1 and s 1 and we have {f 1 θ} Q ω δ 1 Q ω {f 0} ˆQ δ 2 ˆQ {0 < f < 1 θ} (Q(0, 1) ˆQ) α.

22 Thank you Bon anniversaire Irene!!

23 Happy Birthday