## J. B\v rezina, A. Novotn\'y

*On weak solutions of steady Navier-Stokes equations for monatomic gas *

Comment.Math.Univ.Carolin. 49,4 (2008) 611-632. **Abstract:**We use $L^\infty $ estimates for the inverse Laplacian of the pressure introduced by Plotnikov, Sokolowski and Frehse, Goj, Steinhauer together with the nonlinear potential theory due to Adams, Hedberg, to get a priori estimates and to prove existence of weak solutions to steady isentropic Navier-Stokes equations with the adiabatic constant $\gamma >{1\over 3}(1+\sqrt {13})\approx 1.53$ for the flows powered by volume non-potential forces and with $\gamma >{1\over 8}(3+\sqrt {41}) \approx 1.175$ for the flows powered by potential forces and arbitrary non-volume forces. According to our knowledge, it is the first result that treats in three dimensions existence of weak solutions in the physically relevant case $\gamma \le {5\over 3}$ with arbitrary large external data. The solutions are constructed in a rectangular domain with periodic boundary conditions.

**Keywords:** steady compressible Navier-Stokes equations, periodic domain, isentropic flow, existence of the weak solution, potential theory

**AMS Subject Classification:** 35Q, 76N