Regularity in kinetic formulations via averaging lemmas

¹ École Normale Supérieure, Département de Mathématiques et Applications, UMR 8553 du CNRS, 45 rue d'Ulm, 75230 Paris Cedex 05, France; jabin@dma.ens.fr.
² École Normale Supérieure, Département de Mathématiques et Applications, UMR 8553 du CNRS, 45 rue d'Ulm, 75230 Paris Cedex 05, France; perthame@dma.ens.fr.

Received: 3 December 2001
Revised: 2 February 2002

Abstract

We present a new class of averaging lemmas directly motivated by the question of regularity for different nonlinear equations or variational problems which admit a kinetic formulation. In particular they improve the known regularity for systems like γ = 3 in isentropic gas dynamics or in some variational problems arising in thin micromagnetic films. They also allow to obtain directly the best known regularizing effect in multidimensional scalar conservation laws. The new ingredient here is to use velocity regularity for the solution to the transport equation under consideration. The method of proof is based on a decomposition of the density in Fourier space, combined with the K-method of real interpolation.