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steady compressible Navier-Stokes equations; periodic domain; isentropic flow; existence of the weak solution; potential theory
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 $\gamma>{1\over3}(1+\sqrt{13})\approx 1.53$ for the flows powered by volume non-potential forces and with $\gamma>{1\over8}(3+\sqrt{41}) \approx1.175$ 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 $\gamma\le{5\over3}$ with arbitrary large external data. The solutions are constructed in a rectangular domain with periodic boundary conditions.
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