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regularizing effects

ESAIM: Control, Optimisation and Calculus of Variations (1)

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Regularity in kinetic formulations via averaging lemmas
Pierre-Emmanuel Jabin, Benoît Perthame
Journal:

ESAIM: Control, Optimisation and Calculus of Variations / Volume 8 / 2002

Published online by Cambridge University Press:

15 August 2002, pp. 761-774

Print publication:

2002
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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.