Regularity and propagation of moments in some nonlinear Vlasov systems

We introduce a new approach 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 [16]. 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.