Abstract
We introduce a new class of Wasserstein-type distances specifically designed to tackle questions concerning stability and convergence to equilibria for kinetic equations. Thanks to these new distances, we improve some classical estimates by Loeper (J Math Pures Appl (9) 86(1):68–79, 2006) and Dobrushin (Funktsional Anal i Prilozhen 13:48–58, 1979) on Vlasov-type equations, and we present an application to quasi-neutral limits.
Highlights
The first celebrated result relying on Monge–Kantorovich–Wasserstein distances in non-collisional kinetic theory is the proof by Dobrushin [16] on the wellposedness for Vlasov equations with C1,1 potentials, where existence, uniqueness, and stability are proved via a fixed point argument in the bounded-Lipschitz or the 1-Wasserstein distance
As we have seen in the last two sections, suitably modifying Wasserstein distances can be useful in a kinetic setting to take advantage of the asymmetry between x and v
:= inf a|x − y|2 + 2b(x − y) · (v − w) + c|v − w|2 π ∈ (μ,ν) (X ×Rd )2 dπ(x, v, y, w))1/p, In this paper, we have introduced two different generalizations
Summary
Monge–Kantorovich distances, known as Wasserstein distances, play a central role in statistical mechanics, especially in the theory of propagation of chaos and studying large particle systems’ mean behavior. From the late 1970s, there have been many applications of Wasserstein distances in kinetic theory, as is beautifully described in the bibliographical notes of [60, Chapter 6] These distances are frequently used to prove the uniqueness and stability of solutions to kinetic equations, study singular limits, and measure convergence to equilibrium. In [21] the authors prove quantitative stability estimates that are reminiscent of Dobrushin’s, and they show that, in the case of C1,1 potentials, the mean-field limit of the quantum mechanics of N identical particles is uniform in the classical limit. This work aims to push further the idea that, when applied to kinetic problems, Wasserstein distances should be modified to reflect the natural anisotropy between position and momentum variables Since these metrics are used to measure the distance between PDEs’ solutions, we will introduce time-dependent counterparts that can vary along with the characteristic flow. Before stating our main results, let us emphasize that the idea of finding appropriate generalised Wasserstein distances has been used successfully in other contexts in the optimal transport and evolution PDE community; see, for instance, [17,19,45,46,54,55] and references therein
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