Vol. 12, No. 1, 2019

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On propagation of higher space regularity for nonlinear Vlasov equations

Daniel Han-Kwan

Vol. 12 (2019), No. 1, 189–244

This work is concerned with the broad question of propagation of regularity for smooth solutions to nonlinear Vlasov equations. For a class of equations (that includes Vlasov–Poisson and relativistic Vlasov–Maxwell systems), we prove that higher regularity in space is propagated, locally in time, into higher regularity for the moments in velocity of the solution. This in turn can be translated into some anisotropic Sobolev higher regularity for the solution itself, which can be interpreted as a kind of weak propagation of space regularity. To this end, we adapt the methods introduced by D. Han-Kwan and F. Rousset (Ann. Sci. École Norm. Sup. 49:6 (2016) 1445–1495) in the context of the quasineutral limit of the Vlasov–Poisson system.

kinetic transport equations, kinetic averaging lemmas
Mathematical Subject Classification 2010
Primary: 35Q83
Received: 12 October 2017
Revised: 19 March 2018
Accepted: 19 April 2018
Published: 2 August 2018
Daniel Han-Kwan
École polytechnique