An endpoint case of the renormalization property for therelativistic Vlasov-Maxwell system

Recently C. Bardos et al. presented in their fine paper \cite{Bardos} a proof of an Onsager type conjecture on renormalization property and the entropy conservation laws for the relativistic Vlasov-Maxwell system. Particularly, authors proved that if the distribution function $u \in L^{\infty}(0,T;W^{\alpha,p}(\mathbb{R}^6))$ and the electromagnetic field $E,B \in L^{\infty}(0,T;W^{\beta,q}(\mathbb{R}^3))$, with $\alpha, \beta \in (0,1)$ such that $\alpha\beta + \beta + 3\alpha - 1>0$ and $1/p+1/q\le 1$, then the renormalization property and entropy conservation laws hold. To determine a complete proof of this work, in the present paper we improve their results under a weaker regularity assumptions for weak solution to the relativistic Vlasov-Maxwell equations. More precisely, we show that under the similar hypotheses, the renormalization property and entropy conservation laws for the weak solution to the relativistic Vlasov-Maxwell's system even hold for the end point case $\alpha\beta + \beta + 3\alpha - 1 = 0$. Our proof is based on the better estimations on regularization operators.