Finite Energy Method for Compressible Fluids: The Navier-Stokes-Korteweg Model

Pierre Germain, Philippe Lefloch

Research output: Contribution to journalArticle

Abstract

This is the first of a series of papers devoted to the initial value problem for the one-dimensional Euler system of compressible fluids and augmented versions containing higher-order terms. In the present paper, we are interested in dispersive shock waves and analyze the zero viscosity-capillarity limit associated with the Navier-Stokes-Korteweg system. Specifically, we establish the existence of finite energy solutions as well as their convergence toward entropy solutions to the Euler system. Our method of proof combines energy and effective energy estimates, a nonlinear Sobolev inequality, high-integrability properties for the mass density and for the velocity, and compactness properties based on entropies.

Original languageEnglish (US)
Pages (from-to)3-61
Number of pages59
JournalCommunications on Pure and Applied Mathematics
Volume69
Issue number1
DOIs
StatePublished - Jan 2016

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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