Three New Results on Continuation Criteria for the 3D Relativistic Vlasov-Maxwell System

TL;DR

This paper introduces three new mathematical criteria for ensuring the global existence and uniqueness of solutions to the 3D relativistic Vlasov-Maxwell system, improving previous bounds and extending results to non-compact momentum support.

Contribution

It presents three novel continuation criteria for the relativistic Vlasov-Maxwell system, refining bounds in the compact support case and extending global existence results to non-compact momentum support with dynamic support planes.

Findings

Improved continuation criterion for compact support: $ orm{p_0^{rac{18}{5r}-1+eta}f}_{L^{}_t L^r_x L^1_p} \

Enhanced global existence condition for non-compact support: $ orm{p_0^{ heta}f}_{L^{1}_x L^{1}_p} \

Established well-posedness with time-varying momentum support planes.

Abstract

In this paper, we consider sufficient conditions, called continuation criteria, for global existence and uniqueness of classical solutions to the three-dimensional relativistic Vlasov-Maxwell system. In the compact momentum support setting, we prove that $\|p_{0}^{\frac{18}{5r} - 1+\beta}f\|_{L^{\infty}_{t}L^{r}_{x}L^{1}_{p}} \lesssim 1$, where $1\leq r \leq 2$ and $\beta >0$ is arbitrarily small, is a continuation criteria. The previously best known continuation criteria in the compact setting is $\|p_{0}^{\frac{4}{r} - 1+\beta}f\|_{L^{\infty}_{t}L^{r}_{x}L^{1}_{p}} \lesssim 1$, where $1\leq r < \infty$ and $\beta >0$ is arbitrarily small, is due to work by Kunze. Thus, our continuation criteria is an improvement in the $1\leq r \leq 2$ range. In addition, we consider also sufficient conditions for a global existence result to the three-dimensional relativistic Vlasov-Maxwell system without compact support in momentum space, building on previous work by Luk-Strain. In their work, it was shown that $\|p_{0}^{\theta}f\|_{L^{1}_{x}L^{1}_{p}} \lesssim 1$ is a continuation criteria for the relativistic Vlasov-Maxwell system without compact support in momentum space for $\theta > 5$. We improve this result to $\theta > 3$. We also build on another result by Luk-Strain, in which the authors proved the existence of a global classical solution in the compact momentum support setting given the condition that there exists a two-dimensional plane on which the momentum support of the particle density remains fixed. We prove well-posedness even if the plane varies continuously in time.