# Onsager type conjecture and renormalized solutions for the relativistic Vlasov Maxwell system

## TL;DR

This paper proves an Onsager-type conjecture for the relativistic Vlasov-Maxwell system, identifying minimal regularity conditions for energy and entropy conservation in weak solutions, extending previous fluid model results.

## Contribution

It establishes new regularity thresholds ensuring conservation laws for weak solutions of the relativistic Vlasov-Maxwell equations, linking fractional Sobolev regularity to renormalization properties.

## Key findings

- Energy conservation holds if kinetic energy is in L^2.
- Renormalization property holds with fractional derivatives of electromagnetic fields.
- Onsager exponent for conservation is smaller than 1/3, unlike fluid models.

## Abstract

In this paper we give a proof of an Onsager type conjecture on conservation of energy and entropies of weak solutions to the relativistic Vlasov--Maxwell equations. As concerns the regularity of weak solutions, say in Sobolev spaces $W^{\alpha,p}$, we determine Onsager type exponents $\alpha$ that guarantee the conservation of all entropies. In particular, the Onsager exponent $\alpha$ is smaller than $\alpha = 1/3$ established for fluid models. Entropies conservation is equivalent to the renormalization property, which have been introduced by DiPerna--Lions for studying well-posedness of passive transport equations and collisionless kinetic equations. For smooth solutions renormalization property or entropies conservation are simply the consequence of the chain rule. For weak solutions the use of the chain rule is not always justified. Then arises the question about the minimal regularity needed for weak solutions to guarantee such properties. In the DiPerna--Lions and Bouchut--Ambrosio theories, renormalization property holds under sufficient conditions in terms of the regularity of the advection field, which are roughly speaking an entire derivative in some Lebesgue spaces (DiPerna--Lions) or an entire derivative in the space of measures with finite total variation (Bouchut--Ambrosio). In return there is no smoothness requirement for the advected density, except some natural a priori bounds. Here we show that the renormalization property holds for an electromagnetic field with only a fractional space derivative in some Lebesgue spaces. To compensate this loss of derivative for the electromagnetic field, the distribution function requires an additional smoothness, typically fractional Sobolev differentiability in phase-space. As concerns the conservation of total energy, if the macroscopic kinetic energy is in $L^2$, then total energy is preserved.