Existence of solutions of stationary compressible Navier-Stokes equations with large force
The authors show an existence result for the stationary Navier-Stokes equations for isoentropic compressible flows in a bounded domain, with no-slip boundary conditions. The body force is arbitrary: it is not a potential force and there is no smallness assumption on it. But there is a smallness assumption on the Mach number. The pressure is given by \[ p=C\rho^\gamma,\qquad \gamma\geq 1. \] First the authors obtain a dimensionless form of the equations. They split the density into a constant average term \(\overline\rho\) and a fluctuation \(\varepsilon^2\rho\), where \(\varepsilon\) is the Mach number. By means of this splitting, they also split the system in two new ones. The first one are the incompressible Navier-Stokes equations forced by \(\overline\rho f\), so that it contains most of the external force. The second one are the compressible Navier-Stokes equations, forced by \(\varepsilon^2F\), where \(F\) is a body force which depends on \(f\) and on the solution of the incompressible equations. At the end the Leray-Schauder fixed point theorem is applied, under the smallness assumption of the Mach number. Finally the authors show that the found solutions converge, up to a subsequence, to the solution of the incompressible equations, as the Mach number goes to zero.
scientific article; zbMATH DE number 1542414
scientific article