A singular expansion of solution for a regularized compressible Stokes system (Q1879222)

The elliptic system \[ -\varepsilon\Delta p+kU\cdot\nabla p+\lambda p +\text{div} u=g,\quad -\mu\Delta u+\nabla p=f \] is considered in the plane sector domain \[ \Omega =\{ (x,y)\in R^2:\theta_1 < \text{arccos}\, x/r < \theta_2 , \quad r^2 \leq x^2 +y^2 \}, \] with zero Dirichlet boundary conditions \(u|_{\partial\Omega}=0\), \(p|_{\partial\Omega}=0\). The corner singularity is studied for the solutions vanishing outside the ball \(r\leq 1\). The estimate \[ \mu \|u\|_s+ \varepsilon \|p\|_s \leq c (\|f\|_{s-2} +\|g\|_{s-2}) \tag{1} \] is proved valid only for \(1<s<s_1\), where \(\|\cdot\|_s\) is the norm in the Sobolev space \(H^s (\Omega)\). The constant \(s_1\) depends on the difference \(\theta_2 -\theta_1\) and on other data. For \(s>s_1\), the solution is represented through its regular and singular parts: \(u=\) \(u_R +\) \(u_s\), \(p=\) \(p_R +\) \(p_s\). The regular parts \(u_R\) and \(p_s\) obey the estimate (1). As for the singular parts, an algorithm is formulated for their construction.