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Finite Energy Method for Compressible Fluids: The Navier‐Stokes‐Korteweg Model
Author(s) -
Germain Pierre,
LeFloch Philippe
Publication year - 2016
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21622
Subject(s) - mathematics , nonlinear system , compressibility , euler system , sobolev space , euler equations , compact space , euler's formula , mathematical analysis , shock wave , compressible flow , energy method , limit (mathematics) , entropy (arrow of time) , viscosity , convergence (economics) , physics , mechanics , thermodynamics , economic growth , quantum mechanics , economics
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.© 2015 Wiley Periodicals, Inc.