The aim of this paper is to improve the previous work on the relativistic Vlasov–Maxwell system, one of the most important equations in plasma physics. Recently, Bardos et al. [Q. Appl. Math. 78, 193–217 (2020)] presented a proof of an Onsager type conjecture on the renormalization property and the entropy conservation laws for the relativistic Vlasov–Maxwell system. Particularly, the authors proved that if the distribution function u∈L∞(0,T;Wθ,p(R6)) and the electromagnetic field E,B∈L∞(0,T;Wκ,q(R3)) with θ, κ ∈ (0, 1) such that θκ + κ + 3θ − 1 > 0 and 1/p + 1/q ≤ 1, then the renormalization property and entropy conservation laws hold. To determine a complete proof of this work, in this paper, we improve their results under weaker regularity assumptions for a weak solution to the relativistic Vlasov–Maxwell equations. More precisely, we show that under similar hypotheses, the renormalization property and entropy conservation laws for the weak solution to the relativistic Vlasov–Maxwell system even hold for the endpoint case θκ + κ + 3θ − 1 = 0. Our proof is based on better estimations on regularization operators.

中文翻译：

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相对论 Vlasov-Maxwell 系统重整化性质的一个端点案例

本文的目的是改进之前关于相对论 Vlasov-Maxwell 系统的工作，该系统是等离子体物理学中最重要的方程之一。最近，巴尔多斯等人。[问。应用程序 数学。78, 193–217 (2020)] 提出了关于相对论 Vlasov-Maxwell 系统的重整化性质和熵守恒定律的 Onsager 类型猜想的证明。特别地，作者证明了如果分布函数 u∈L∞(0,T;Wθ,p(R6)) 和电磁场 E,B∈L∞(0,T;Wκ,q(R3)) 与 θ , κ ∈ (0, 1) 使得 θκ + κ + 3θ − 1 > 0 且 1/p + 1/q ≤ 1，则重整化性质和熵守恒定律成立。为了确定这项工作的完整证明，在本文中，我们在较弱的正则性假设下改进了他们的结果，以获得相对论 Vlasov-Maxwell 方程的弱解。更确切地说，