Navier–Stokes–Fourier system with Dirichlet boundary conditions

Abstract

We consider the Navier–Stokes–Fourier system describing the motion of a compressible, viscous, and heat-conducting fluid in a bounded domain $\mathrm{\Omega }\subset R^d$, d = 2, 3, with general non-homogeneous Dirichlet boundary conditions for the velocity and the absolute temperature, with the associated boundary conditions for the density on the inflow part. We introduce a new concept of a weak solution based on the satisfaction of the entropy inequality together with a balance law for the ballistic energy. We show the weak–strong uniqueness principle as well as the existence of global-in-time solutions.

Keywords:

• Navier–Stokes–Fourier system
• Dirichlet boundary conditions
• weak solution
• weak–strong uniqueness

2020 Mathematics Subject Classification:

• 35Q30

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work of N.C. and E.F. was supported by the Czech Science Foundation (GAČR), Grant Agreement 18-05974S. The Institute of Mathematics of the Academy of Sciences of the Czech Republic is supported by RVO: 67985840.