Dynamical Stability of Non-Constant Equilibria for the Compressible Navier–Stokes Equations in Eulerian Coordinates

In this paper we establish global existence and uniqueness of strong solutions to the non-isothermal compressible Navier–Stokes equations in bounded domains. The initial data have to be near equilibria that may be non-constant due to considering large external forces. We are able to show exponential stability of equilibria in the phase space and, above all, to study the problem in Eulerian coordinates. The latter seems to be a novelty, since in works by other authors, global strong Lp-solutions have been investigated only in Lagrangian coordinates; Eulerian coordinates are even declared as impossible to deal with. The proof is based on a careful derivation and study of the associated linear problem.