Dissipative Solutions to Compressible Navier–Stokes Equations with General Inflow–Outflow Data: Existence, Stability and Weak Strong Uniqueness

Abstract

So far existence of dissipative weak solutions for the compressible Navier–Stokes equations (i.e. weak solutions satisfying the relative energy inequality) is known only in the case of boundary conditions with non zero inflow/outflow (i.e., in particular, when the normal component of the velocity on the boundary of the flow domain is equal to zero). Most of physical applications (as flows in wind tunnels, pipes, reactors of jet engines) requires to consider non-zero inflow–outflow boundary condtions. We prove existence of dissipative weak solutions to the compressible Navier–Stokes equations in barotropic regime (adiabatic coefficient \(\gamma >3/2\), in three dimensions, \(\gamma >1\) in two dimensions) with large velocity prescribed at the boundary and large density prescribed at the inflow boundary of a bounded piecewise regular Lipschitz domain, without any restriction neither on the shape of the inflow/outflow boundaries nor on the shape of the domain. It is well known that the relative energy inequality has many applications, e.g., to investigation of incompressible or inviscid limits, to the dimension reduction of flows, to the error estimates of numerical schemes. In this paper we deal with one of its basic applications, namely weak–strong uniqueness principle.