For the stationary compressible viscous Navier-Stokes equations with no-slip condition on a convex polygon
The compressible viscous Navier-Stokes equations are considered in a convex polygonal domain \(D\subset{\mathbb R}^2\) \[ -\mu\Delta v+\rho (v\cdot\nabla) v-(\mu+\nu)\nabla\operatorname{div}v+ \nabla p(\rho)= \rho f,\quad \operatorname{div}(\rho v)=g, \] \[ v=0,\quad \text{on }\partial D. \] Here \(v\) is the velocity field, \(\rho\) is the density and \(p=p\,(\rho)\) is the pressure, \(\mu\) and \(\nu\) are the viscous numbers with \(\mu>0\) and \(\mu+\nu>0\), \(f\) and \(g\) are given functions. In this paper \(p\,(\rho)=\rho\). It is shown that the solution to the problem exists in a small neighborhood of \((0,\tilde{\rho})\) for any convex polygon. The size of the neighborhood is determined by the data of the problem \((f,g)\). The singular behavior of the solution is described by the eigenvalue problem of the Stokes operator with zero Dirichlet condition, [see, \textit{V. A. Kozlov, V. G. Maz'ya} and \textit{J. Rossmann}, Spectral problems associated with corner singularities of solutions to elliptic equations. Providence, RI: American Mathematical Society (AMS) (2001; Zbl 0965.35003)].
scientific article; zbMATH DE number 5867207