{
  "id": "1905.05973",
  "version": "v1",
  "published": "2019-05-15T06:53:58.000Z",
  "updated": "2019-05-15T06:53:58.000Z",
  "title": "On the renormalization property and entropy conservation laws for the relativistic Vlasov-Maxwell system",
  "authors": [
    "Minh-Phuong Tran",
    "Thanh-Nhan Nguyen"
  ],
  "comment": "18 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-15T06:53:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "relativistic vlasov-maxwell system",
      "renormalization property",
      "entropy conservation laws hold",
      "weak solution",
      "onsager type conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}