Uniqueness and stability for the Vlasov-Poisson system with spatial density in Orlicz spaces

Abstract

In this paper, we establish uniqueness of the solution of the Vlasov-Poisson system with spatial density belonging to a certain class of Orlicz spaces. This extends the uniqueness result of Loeper [11] (which holds for density in L ∞ ) and of the paper [15]. Uniqueness is a direct consequence of our main result, which provides a quantitative stability estimate for the Wasserstein distance between two weak solutions with spatial density in such Orlicz spaces, in the spirit of Dobrushin's proof of stability for mean-field PDEs. Our proofs are built on the second-order structure of the underlying characteristic system associated to the equation.

A New Perspective on Wasserstein Distances for Kinetic Problems

“…similar to the one obtained for the W 1 distance, see [38,Remark 1.7]. In that paper, the authors rely crucially on the second-order structure of the Vlasov equation, namely Ẍ = ∇U (t, X ).…”

“…Loeper's result has been generalized to less singular kernels [37], and it is the cornerstone for several other stability arguments [7,14,36,42,44,48,58]. Also, Loeper's uniqueness criterion for Vlasov-Poisson has been extended to solutions whose associated density belongs to some suitable Orlicz spaces [38,51]. In what follows, we will focus our attention on some applications of Loeper's stability estimate related to quasi-neutral limit for the Vlasov-Poisson equation [28,30,34,35].…”

“…To mention some concrete applications, our ideas could also be applied in the setting of quantum systems by suitably modifying the quantum Wasserstein distances introduced in [21,24]. Also, our new Loeper-type estimate may be helpful to obtain stability estimates in W 2 when the density belongs to some suitable Orlicz spaces, in analogy to [38] where stability estimates have been proved for W 1 .…”

Semiclassical Limit to the Vlasov Equation with Inverse Power Law Potentials

“…Well-posedness in dimension three has been treated by Bardos and Degond in [4] for small data and by Pfaffelmoser [37] and Lions and Perthame [29] for more general initial densities. More recently, less stringent uniqueness criteria have been provided in [30,32,22].…”

Lagrangian solutions to the Vlasov-Poisson system with a point charge

“…Recently Miot [27] generalised the latter condition to a class of solutions whose L p norms of spatial density grow at most linearly w.r.t. p, then extended to spatial densities belonging to some Orlicz space in [22]. Unfortunately, it seems that none of these conditions apply to our setting and new ideas are needed.…”