Vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for compressible fluid flow

Author: 

Chen, G
Perepelitsa, M

Publication Date: 

1 November 2010

Journal: 

Communications on Pure and Applied Mathematics

Last Updated: 

2019-08-17T06:53:49.183+01:00

Issue: 

11

Volume: 

63

DOI: 

10.1002/cpa.20332

page: 

1469-1504

abstract: 

We establish the vanishing viscosity limit of the Navier-Stokes equations to the isentropic Euler equations for one-dimensional compressible fluid flow. For the Navier-Stokes equations, there exist no natural invariant regions for the equations with the real physical viscosity term so that the uniform sup-norm of solutions with respect to the physical viscosity coefficient may not be directly controllable. Furthermore, convex entropy-entropy flux pairs may not produce signed entropy dissipation measures. To overcome these difficulties, we first develop uniform energy-type estimates with respect to the viscosity coefficient for solutions of the Navier-Stokes equations and establish the existence of measure-valued solutions of the isentropic Euler equations generated by the Navier-Stokes equations. Based on the uniform energy-type estimates and the features of the isentropic Euler equations, we establish that the entropy dissipation measures of the solutions of the Navier-Stokes equations for weak entropy-entropy flux pairs, generated by compactly supported C2 test functions, are confined in a compact set in H-1, which leads to the existence of measure-valued solutions that are confined by the Tartar-Murat commutator relation. A careful characterization of the unbounded support of the measure-valued solution confined by the commutator relation yields the reduction of themeasurevalued solution to a Dirac mass, which leads to the convergence of solutions of the Navier-Stokes equations to a finite-energy entropy solution of the isentropic Euler equations with finite-energy initial data, relative to the different end-states at infinity. © 2010 Wiley Periodicals, Inc.

Symplectic id: 

203118

Submitted to ORA: 

Not Submitted

Publication Type: 

Journal Article