On propagation of higher space regularity for non-linear Vlasov equations

This work is concerned with the broad question of propagation of regularity for smooth solutions to non-linear Vlasov equations. For a class of equations (that includes Vlasov-Poisson and relativistic Vlasov-Maxwell), 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 in the context of the quasineutral limit of the Vlasov-Poisson system in [D. Han-Kwan and F. Rousset, Ann. Sci. \'Ecole Norm. Sup., 2016].