Geodesic
On weak solutions of steady Navier-Stokes equations for monatomic gas
Commentationes Mathematicae Universitatis Carolinae, Tome 49 (2008) no. 4, pp. 611-632

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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.
Classification : 35D05, 35Q30, 76D05, 76N15
Keywords: steady compressible Navier-Stokes equations; periodic domain; isentropic flow; existence of the weak solution; potential theory 
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     title = {On weak solutions of steady {Navier-Stokes} equations for monatomic gas},
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Březina, J.; Novotný, A. On weak solutions of steady Navier-Stokes equations for monatomic gas. Commentationes Mathematicae Universitatis Carolinae, Tome 49 (2008) no. 4, pp. 611-632. http://geodesic.mathdoc.fr/item/CMUC_2008__49_4_a6/