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.


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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