On the artificial compressibility method for the Navier Stokes Fourier system

TL;DR

This paper extends the artificial compressibility method to approximate weak solutions of the incompressible Navier Stokes Fourier system in bounded domains and the whole space, using dispersive estimates for convergence.

Contribution

It generalizes Temam's artificial compressibility approach to the Navier Stokes Fourier system in broader settings, employing Strichartz-type dispersive estimates for convergence.

Findings

Projection of velocity fields converges to a weak solution

Relatively compact divergence-free velocity projections

Effective approximation method for Navier Stokes Fourier system

Abstract

This paper deals with the approximation of the weak solutions of the incompressible Navier Stokes Fourier system. In particular it extends the artificial compressibility method for the Leray weak solutions of the Navier Stokes equation, used by Temam, in the case of a bounded domain and later in the case of the whole space. By exploiting the wave equation structure of the pressure of the approximating system the convergence of the approximating sequences is achieved by means of dispersive estimate of Strichartz type. It will be proved that the projection of the approximating velocity fields on the divergence free vectors is relatively compact and converges to a weak solution of the incompressible Navier Stokes Fourier system.