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Regularity and propagation of moments in some nonlinear Vlasov systems

Published online by Cambridge University Press:  11 July 2007

I. Gasser
Affiliation:
Institut für Angewandte Mathematik, Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany
P.-E. Jabin
Affiliation:
Département de Mathématiques et Applications, Ecole Normale Supérieure, 45 rue d'Ulm, 75230 Paris Cedex 05, France
B. Perthame
Affiliation:
Département de Mathématiques et Applications, Ecole Normale Supérieure, 45 rue d'Ulm, 75230 Paris Cedex 05, France

Abstract

We introduce a new variant to prove the regularity of solutions to transport equations of the Vlasov type. Our approach is mainly based on the proof of propagation of velocity moments, as in a previous paper by Lions and Perthame. We combine it with moment lemmas which assert that, locally in space, velocity moments can be gained from the kinetic equation itself. We apply our theory to two cases. First, to the Vlasov–Poisson system, and we solve a long-standing conjecture, namely the propagation of any moment larger than two. Next, to the Vlasov–Stokes system, where we prove the same result for fairly singular kernels.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2000

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