Strichartz estimates and moment bounds for the relativistic Vlasov-Maxwell system II. Continuation criteria in the 3D case

TL;DR

This paper establishes new criteria for continuing solutions to the 3D relativistic Vlasov-Maxwell system, extending previous results by relaxing support assumptions and providing bounds based on electromagnetic field integrals and weighted function norms.

Contribution

It introduces two novel continuation criteria for solutions with non-compact momentum support, generalizing classical and recent results in the field.

Findings

Solutions can be continued if electromagnetic field integrals are bounded.

Solutions can be extended if certain weighted norms of the distribution function are bounded.

Improves upon previous continuation criteria by relaxing assumptions.

Abstract

We consider the $3$-dimensional relativistic Vlasov-Maxwell system with data without compact support in momentum space. We prove two continuation criteria for solutions to this system. First, we show that a regular solution can be continued if the integral of the electromagnetic field along any characteristic is assumed to be bounded. This can be viewed as a generalization of the classical result of Glassey-Strauss (1986) to data with non-compact momentum support. Second, we extend the methods in our companion paper (Part I) to show that a regular solution can be extended as long as $\|\pel_0^{\th} f \|_{L^q_xL^1_{\pel}}$ remains bounded for $\th>\frac 2q$, $2<q\leq \infty$. This improves previous results of Pallard (2005).