Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Regularity and propagation of moments in some nonlinear Vlasov systems

I. Gassera1, P.-E. Jabina2 and B. Perthamea2

a1 Institut für Angewandte Mathematik, Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany

a2 Département de Mathématiques et Applications, Ecole Normale Supérieure, 45 rue d'Ulm, 75230 Paris Cedex 05, France


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.

(Received June 23 1999)

(Accepted September 27 1999)