The maximum regularity property of the steady Stokes problem associated with a flow through a profile cascade in $L^r$-framework

Abstract

The paper deals with the Stokes problem, associated with a flow of a viscous incompressible fluid through a spatially periodic profile cascade. We use results from [32] (the maximum regularity property in the $L^2$-framework) and [33] (the weak solvability in $W^{1,r}$), and extend the findings on the maximum regularity property to the general $L^r$-framework (for $1<r<\infty$). Using the reduction to one spatial period $\Omega$, the problem is formulated by means of boundary conditions of three types: the conditions of periodicity on curves ${\Gamma }_0$ and ${\Gamma }_1$, the Dirichlet boundary conditions on ${\Gamma }_{\mathrm{in}}$ and ${\Gamma }_P$ and an artificial "do nothing"-type boundary condition on ${\Gamma }_{\mathrm{out}}$ (see Fig. 1). We show that, although domain $\Omega$ is not smooth and different types of boundary conditions "meet" in the vertices of $\partial \Omega$, the considered problem has a strong solution with the maximum regularity property for "smooth" data. We explain the sense in which the "do nothing" boundary condition is satisfied for both weak and strong solutions.

arXiv admin note: text overlap with arXiv:2009.08234, arXiv:2006.15651