Globally bounded trajectories for the barotropic Navier-Stokes system with general boundary conditions

TL;DR

This paper proves that solutions to the barotropic Navier-Stokes system with general boundary conditions eventually stabilize within a bounded region, supporting stationary statistical solutions.

Contribution

It establishes the boundedness and long-term behavior of solutions for the Navier-Stokes system with general boundary conditions, including the existence of stationary statistical solutions.

Findings

All trajectories enter a bounded absorbing set.

Omega-limit sets are compact.

Stationary statistical solutions exist.

Abstract

We consider the barotropic Navier--Stokes system describing the motion of a viscous compressible fluid interacting with the outer world through general in/out flux boundary conditions. We consider a hard--sphere type pressure EOS and show that all trajectories eventually enter a bounded absorbing set. In particular, the associated omega-limit sets are compact and support a stationary statistical solution.