Blow-up criterion for the compressible Navier--Stokes system with inflow-outflow boundary conditions

TL;DR

This paper establishes a blow-up criterion for strong solutions of the 3D compressible Navier-Stokes equations with inflow-outflow boundary conditions, focusing on boundedness conditions that prevent solution singularities.

Contribution

It introduces a new blow-up criterion accommodating inhomogeneous boundary conditions by developing estimates on the density gradient.

Findings

Boundedness of $( ho^{-1}, u)$ prevents blow-up.

A new approach for $ abla_x ho$ estimates is developed.

Solution regularity is maintained under specified boundedness conditions.

Abstract

We consider the compressible Navier-Stokes system in three dimensions with general inflow-outflow boundary conditions, meaning that we prescribe a boundary velocity which has non-zero normal component and accordingly the density is prescribed on the inflow part of the boundary. We establish a blow-up criterion in a class of strong solutions in the $L^p-L^q$ framework. In particular assuming the boundedness of the quantities $(\varrho^{-1}, \bu)$ and of a suitable norm of $\nabla_x \varrho$ the solution remains regular and the blow-up does not occur. We develop the condition on $\nabla_x \varrho$ because we need a new approach in order to accommodate the inhomogeneous boundary conditions, as the standard estimates on the material time derivative works when the normal component of the boundary velocity is zero.