An endpoint case of the renormalization property for the relativistic Vlasov–Maxwell system

• Published in: Journal of mathematical physics 2020-07, Vol.61 (7)
• Main Authors: Nguyen, Thanh-Nhan, Tran, Minh-Phuong
• Format: Article
• Language: English
• Subjects: Conservation laws; Distribution functions; Electromagnetic fields; Entropy; Mathematical analysis; Maxwell's equations; Physics; Plasma physics; Plasmas (physics); Regularization; Relativistic effects
• ISSN: 0022-2488; 1089-7658
• DOI: 10.1063/1.5144712

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.