Analysis of semidiscretization of the compressible Navier-Stokes equations

TL;DR

This paper proves the existence of solutions for the two-dimensional compressible Navier-Stokes equations using time discretization, establishing a limit process as the time step approaches zero, based on a novel technique.

Contribution

It introduces a new method for proving existence of weak solutions for the compressible Navier-Stokes equations via time discretization and limit passage.

Findings

Existence of weak solutions for fixed time steps.

Successful limit passage as time step approaches zero.

Application of a new technique from steady Navier-Stokes analysis.

Abstract

The objective of this work is to present the existence result for the evolu- tionary compressible Navier-Stokes equations via time discretization. We consider the two-dimensional case with slip boundary conditions. First, the existence of weak solution for a fixed time step \Delta t > 0 is presented and then the limit passage as \Delta t \to 0+ is carried out. The proof is based on a new technique established for the stedy Navier-Stokes equations by Mucha P. and Pokorn\'y M. 2006 Nonlinearity 19 1747-1768.