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Hypoelliptic regularity in kinetic equations. (English) Zbl 1045.35093

Summary: We establish new regularity estimates, in terms of Sobolev spaces, of the solution \(f\) to a kinetic equation. The right-hand side can contain partial derivatives in time, space and velocity, as in classical averaging, and \(f\) is assumed to have a certain amount of regularity in velocity. The result is that \(f\) is also regular in time and space, and this is related to a commutator identity introduced by Hörmander for hypoelliptic operators. In contrast with averaging, the number of derivatives does not depend on the \(L^p\) space considered.
Three type of proofs are provided: one relies on the Fourier transform, another one uses Hörmander’s commutators, and the last uses a characteristics commutator. Regularity of averages in velocity are deduced. We apply our method to the linear Fokker-Planck operator and recover the known optimal regularity, by direct estimates using Hörmander’s commutator.

MSC:

35Q82 PDEs in connection with statistical mechanics
82C40 Kinetic theory of gases in time-dependent statistical mechanics
35B65 Smoothness and regularity of solutions to PDEs
35H10 Hypoelliptic equations
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[1] Bézard, M., Régularité \(L^p\) précisée des moyennes dans les équations de transport, Bull. Soc. Math. France, 122, 29-76 (1994) · Zbl 0798.35025
[2] Bouchut, F., Smoothing effect for the non-linear Vlasov-Poisson-Fokker-Planck system, J. Differential Equations, 122, 225-238 (1995) · Zbl 0840.35053
[3] Bouchut, F.; Desvillettes, L., Averaging lemmas without time Fourier transform and application to discretized kinetic equations, Proc. Roy. Soc. Edinburgh, 129A, 19-36 (1999) · Zbl 0933.35159
[4] Bouchut, F.; Golse, F.; Pulvirenti, M., Kinetic Equations and Asymptotic Theory. Kinetic Equations and Asymptotic Theory, Series in Appl. Math. (2000), Gauthiers-Villars
[5] Bournaveas, N.; Perthame, B., Averages over spheres for kinetic transport equations; hyperbolic Sobolev spaces and Strichartz inequalities, J. Math. Pures Appl. (9), 80, 517-534 (2001) · Zbl 1036.82023
[6] Chaleyat-Maurel, M., La condition d’hypoellipticité d’Hörmander, Astérisque, 84-85, 189-202 (1981) · Zbl 0475.35033
[7] Cheverry, C., Regularizing effects for multidimensional scalar conservation laws, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17, 413-472 (2000) · Zbl 0966.35074
[8] Desvillettes, L.; Villani, C., On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation, Comm. Pure Appl. Math., 54, 1-42 (2001) · Zbl 1029.82032
[9] DiPerna, R. J.; Lions, P.-L; Meyer, Y., \(L^p\) regularity of velocity averages, Ann. Inst. H. Poincaré Anal. Non Linéaire, 8, 271-287 (1991) · Zbl 0763.35014
[10] Gérard, P., Microlocal defect measures, Comm. Partial Differential Equations, 16, 1761-1794 (1991) · Zbl 0770.35001
[11] Golse, F.; Lions, P.-L; Perthame, B.; Sentis, R., Regularity of the moments of the solution of a transport equation, J. Funct. Anal., 76, 110-125 (1988) · Zbl 0652.47031
[12] Hörmander, L., Hypoelliptic second order differential equations, Acta Math., 119, 147-171 (1967) · Zbl 0156.10701
[13] P.-E. Jabin, B. Perthame, Regularity in kinetic formulations via averaging lemmas, Preprint; P.-E. Jabin, B. Perthame, Regularity in kinetic formulations via averaging lemmas, Preprint · Zbl 1065.35185
[14] Lions, P.-L, Régularité optimale des moyennes en vitesses, C. R. Acad. Sci. Sér. I, 320, 911-915 (1995) · Zbl 0827.35110
[15] Perthame, B.; Souganidis, P. E., A limiting case for velocity averaging, Ann. Sci. École Norm. Sup. (4e), 31, 591-598 (1998) · Zbl 0956.45010
[16] Rothschild, L. P.; Stein, E. M., Hypoelliptic differential operators and nilpotent groups, Acta Math., 137, 247-320 (1976) · Zbl 0346.35030
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