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 1 2016

Fingerprint

Euler System
Compressible Fluid
Energy Method
Navier-Stokes
Entropy
Capillarity
Sobolev Inequality
Fluids
Entropy Solution
Initial value problems
Energy Estimates
One-dimensional System
Energy
Shock Waves
Shock waves
Integrability
Initial Value Problem
Compactness
Viscosity
Higher Order

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Finite Energy Method for Compressible Fluids : The Navier-Stokes-Korteweg Model. / Germain, Pierre; Lefloch, Philippe.

In: Communications on Pure and Applied Mathematics, Vol. 69, No. 1, 01.01.2016, p. 3-61.

Research output: Contribution to journalArticle

@article{8d77df52e2aa4ee8a0d54fd96c602ebb,
title = "Finite Energy Method for Compressible Fluids: The Navier-Stokes-Korteweg Model",
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.",
author = "Pierre Germain and Philippe Lefloch",
year = "2016",
month = "1",
day = "1",
doi = "10.1002/cpa.21622",
language = "English (US)",
volume = "69",
pages = "3--61",
journal = "Communications on Pure and Applied Mathematics",
issn = "0010-3640",
publisher = "Wiley-Liss Inc.",
number = "1",

}

TY - JOUR

T1 - Finite Energy Method for Compressible Fluids

T2 - The Navier-Stokes-Korteweg Model

AU - Germain, Pierre

AU - Lefloch, Philippe

PY - 2016/1/1

Y1 - 2016/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84947484025&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84947484025&partnerID=8YFLogxK

U2 - 10.1002/cpa.21622

DO - 10.1002/cpa.21622

M3 - Article

AN - SCOPUS:84947484025

VL - 69

SP - 3

EP - 61

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 1

ER -