Semiflow selection for the compressible Navier-Stokes system

TL;DR

This paper develops a semiflow selection for the compressible Navier-Stokes system, aiming to address the non-uniqueness of weak solutions by establishing a solution selection with the semigroup property.

Contribution

It introduces a semiflow selection based on three state variables and a new selection depending only on initial density and momentum, with almost everywhere semigroup property.

Findings

Existence of a semiflow selection for the system.

A new solution selection depending only on initial density and momentum.

Semigroup property holds almost everywhere in time for the new selection.

Abstract

Although the existence of dissipative weak solutions for the compressible Navier-Stokes system has already been established for any finite energy initial data, uniqueness is still an open problem. The idea is then to select a solution satisfying the semigroup property, an important feature of systems with uniqueness. More precisely, we are going to prove the existence of a semiflow selection in terms of the three state variables: the density, the momentum and the energy. Finally, we will show that it is possible to introduce a new selection defined only in terms of the initial density and momentum; however, the price to pay is that the semigroup property will hold almost everywhere in time.