{
  "id": "2203.02256",
  "version": "v1",
  "published": "2022-03-04T12:02:44.000Z",
  "updated": "2022-03-04T12:02:44.000Z",
  "title": "Global existence and analyticity of $L^p$ solutions to the compressible fluid model of Korteweg type",
  "authors": [
    "Zihao Song",
    "Jiang Xu"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We are concerned with a system of equations in $\\mathbb{R}^{d}(d\\geq2)$ governing the evolution of isothermal, viscous and compressible fluids of Korteweg type, that can be used as a phase transition model. In the case of zero sound speed $P'(\\rho^{\\ast})=0$, it is found that the linearized system admits the \\textit{purely} parabolic structure, which enables us to establish the global-in-time existence and Gevrey analyticity of strong solutions in hybrid Besov spaces of $L^p$-type. Precisely, if the full viscosity coefficient and capillary coefficient satisfy $\\bar{\\nu}^2\\geq4\\bar{\\kappa}$, then the acoustic waves are not available in compressible fluids. Consequently, the prior $L^2$ bounds on the low frequencies of density and velocity could be improved to the general $L^p$ version with $1\\leq p\\leq d$ if $d\\geq2$. The proof mainly relies on new nonlinear Besov (-Gevrey) estimates for product and composition of functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-04T12:02:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "compressible fluid model",
      "korteweg type",
      "global existence",
      "phase transition model",
      "hybrid besov spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}