On steady compressible Navier-Stokes equations in plane domains with corners
The following boundary value problem is considered in a bounded domain \(\Omega\subset \mathbb{R}^2\) with a corner point \[ -\mu_1 \Delta \overline v- (\mu_1+ \mu_2)\nabla\text{ div } \overline v+ \nabla \rho= \rho\overline f- \rho(\overline v\cdot \nabla) \overline v,\quad x\in \Omega, \] \[ \text{div}(\rho\overline v)= 0,\quad x\in \Omega,\quad \overline v|_{\partial\Omega}= 0. \] Here \(\overline v= (v_1, v_2)\) is the unknown velocity vector, \(\rho> 0\) is an unknown density, \(\overline f\) is a given vector, \(\mu_1\), \(\mu_2\) are constants satisfying the condition \(\mu_1> 0\), \(\mu_2\geq - \mu_1\). The problem is investigated in weighted Sobolev spaces. It is proved that if \(\overline f\) is sufficiently small then the problem has a unique solution in a certain ball of a suitable function space. The authors use the method of decomposition proposed in [the second author and \textit{M. Padula}, Arch. Ration. Mech. Anal. 126, No. 3, 243-297 (1994; Zbl 0809.76080)].
scientific article; zbMATH DE number 840217
scientific article