Vanishing Viscosity Limit of the Navier-Stokes Equations to the Euler Equations for Compressible Fluid Flow

Details Vanishing Viscosity Limit of the Navier-Stokes Equations to the Euler Equations for Compressible Fluid Flow Vanishing Viscosity Limit of the Navier-Stokes Equations to the Euler Equations for Compressible Fluid Flow

2009-10-13

arxiv.org/abs/0910.2360v1

Mathematics

Analysis of PDEs

30 pages

Scientific paper

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 and, 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 the 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 $C^2$ test functions, are confined in a compact set in $H^{-1}$, which lead 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 the measure-valued solution to a Delta mass, which leads to the convergence of solutions of the Navier-Stokes equations to a finite-energy entropy solution of the isentropic Euler equations.

Affiliated with

Also associated with

No associations

Rating

Vanishing Viscosity Limit of the Navier-Stokes Equations to the Euler Equations for Compressible Fluid Flow does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Vanishing Viscosity Limit of the Navier-Stokes Equations to the Euler Equations for Compressible Fluid Flow, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Vanishing Viscosity Limit of the Navier-Stokes Equations to the Euler Equations for Compressible Fluid Flow will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-640834