Analysis of semidiscretization of the compressible Navier-Stokes equations

TL;DR

This paper proves the existence of solutions for the non-steady compressible Navier-Stokes equations using time discretization, focusing on the two-dimensional case with slip boundary conditions and building on previous steady-state techniques.

Contribution

It extends the existence results to the non-steady case by applying a novel time discretization approach and limit passage, based on techniques from steady Navier-Stokes analysis.

Findings

Existence of weak solutions for fixed time intervals.

Limit passage as time step approaches zero.

Application of new techniques from steady Navier-Stokes analysis.

Abstract

The objective of this work is to present the existence result of for the non- steady compressible Navier-Stokes equations via time discretization. We consider the two-dimensional case with a slip boundary conditions. First, the existence of weak solution for a fixed length of time interval \Delta t > 0 is presented and then the limit passage as \Delta t \rightarrow 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.