Steady Navier- Stokes-Fourier system with slip boundary conditions
Abstract
We consider a problem modeling the steady flow of a compressible heat conducting Newtonian fluid subject to the slip boundary condition for the velocity. Assuming the pressure law of the form $p(\vr,\vt) \sim \vr^\gamma + \vr \vt$, we show (under additional assumptions on the heat conductivity and the viscosity) that for any $\gamma >1$ there exists a variational entropy solution to our problem (i.e. the weak formulation of the total energy balance is replaced by the entropy inequality and the global total energy balance). Moreover, if $\gamma > \frac 54$ (together with further restrictions on the heat conductivity), the solution is in fact a weak one. The results are obtained without any restriction on the size of the data.