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

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 $1<r<\mathrm{\infty }$). Using the reduction to one spatial period $\mathrm{\Omega }$, the problem is formulated by means of boundary conditions of three types: the conditions of periodicity on curves ${\mathrm{\Gamma }}_0$ and ${\mathrm{\Gamma }}_1$, the Dirichlet boundary conditions on ${\mathrm{\Gamma }}_{\mathrm{i}\mathrm{n}}$ and ${\mathrm{\Gamma }}_P$ and an artificial “do nothing”-type boundary condition on ${\mathrm{\Gamma }}_{\mathrm{o}\mathrm{u}\mathrm{t}}$ (see Fig. 1). We show that, although domain $\mathrm{\Omega }$ is not smooth and different types of boundary conditions “meet” in the vertices of $\mathrm{\partial }\mathrm{\Omega }$, 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.