Small Data Solutions of the Vlasov-Poisson System and the Vector Field Method

Abstract

The aim of this article is to demonstrate how the vector field method of Klainerman can be adapted to the study of transport equations. After an illustration of the method for the free transport operator, we apply the vector field method to the Vlasov-Poisson system in dimension 3 or greater. The main results are optimal decay estimates and the propagation of global bounds for commuted fields associated with the conservation laws of the free transport operators, under some smallness assumption. Similar decay estimates had been obtained previously by Hwang, Rendall and Velázquez using the method of characteristics, but the results presented here are the first to contain the global bounds for commuted fields and the optimal spatial decay estimates. In dimension 4 or greater, it suffices to use the standard vector fields commuting with the free transport operator while in dimension 3, the rate of decay is such that these vector fields would generate a logarithmic loss. Instead, we construct modified vector fields where the modification depends on the solution itself. The methods of this paper, being based on commutation vector fields and conservation laws, are applicable in principle to a wide range of systems, including the Einstein-Vlasov and the Vlasov-Nordström system.

Vector field methods for kinetic equations with applications to classical and relativistic systems

“…Let us mention that the analogue of the above decay estimates for the classical transport operator ∂ t + v i ∂ x i was proven in [7].…”

“…In that case, the difficulty was resolved [7] by modifying the commutation vector fields, replacing the lifted vector fields Z by some Y = Z + Φ i ∂ x i , where the coefficients Φ i are functions in the variable (t, x, v), depending on the solution and constructed to cancel the worst error terms in the commutator formulae. See also [15] for previous results concerning sharp asymptotics for solutions of the Vlasov-Poisson system based on the method of characteristics.…”

“…To estimate those, we consider these products as solutions to transport equation, to which we can again apply energy estimate for Vlasov fields. This strategy was originally developed in [7].…”

A commuting-vector-field approach to some dispersive estimates

“…More recently, Fajman, Joudioux, and Smulevici observed that by properly lifting the symmetry actions to the relativistic phase space, a similar argument can yield the dispersive decay for the relativistic counterpart to the Vlasov equation [4]. The argument has been modified by Smulevici to apply to the classical Vlasov equation and was used to show small data global existence for the Vlasov-Poisson equations [13].…”

“…Before treating (5) more generally, let us focus first on the case of the classical Vlasov equation. This case has been previously treated by Smulevici [13], we include the discussion here to set the stage for the general case, and to showcase how the analysis simplifies due to the Galilean (instead of Lorentzian) symmetry of the problem. The t-weights in the weighted energy estimates that drive both the temporal decay for the linear wave equation in the original Klainerman-Sobolev estimate [6] …”