A regularity result of solution to the compressible Stokes equations on a convex polygon
The author gives an abstract formulation of the incompressible Stokes problem to discuss its corner singularities and then state a higher singular expansion of the solution of the problem on a convex polygon. He also makes a study of the compressible Stokes system governed by equations \[ \begin{aligned} -\mu\Delta{\mathbf u}+ \nabla p & = {\mathbf f}\text{ n }D,\\ \text{div}\,{\mathbf u}+{\mathbf U}\cdot \nabla p+p\,\text{div}\,{\mathbf U} & =g \text{ in }D,\\ {\mathbf u} & ={\mathbf O} \text{ on }\partial D,\end{aligned} \] where \(D\) is a convex polygon in \(\mathbb{R}^2\), \({\mathbf f}\) and \(g\) are given function with \(\int_Dg\,d{\mathbf x}=0\), \(u=[u,v]\) is the unknown velocity vector, \(p\) is the unknown pressure field, \({\mathbf U}= [U,V]\) is a given smooth vector which is zero on the boundary \(\partial D\) of \(D\), and \(\mu\) is a viscosity parameter. A solution formula of this problem is proved and a regularity result of this problem is obtained in some detail. The author also discusses how to extract a singularity of the pressure function at the corner and proves the corresponding regularity result. It is shown that the lowest order of the regularity of the compressible Stokes system is the same as that of the incompressible Stokes problem. The entire paper is based on abstract functional analysis in a Sobolev space \[ H^{k,q}(D)=\left\{v\in L^q(D):\|v\|_{k, q,D}:=\left(\sum_{|\alpha| \leq k}\|\nabla^\alpha v\|^q_{q,D}\right)^{1/q}<\infty \right\} \] and the Sobolev imbedding theorems. In the opinion of the reviewer, this paper is interesting because a regularity of the solution of the compressible Stokes equations on a convex polygon is established by rigorous functional analysis.
scientific article; zbMATH DE number 2123172