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 < ∞). Using the reduction to one spatial period Ω, the problem is formulated by means of boundary conditions of three types: the conditions of periodicity on curves Γ 0 and Γ 1 , the Dirichlet boundary conditions on Γ in and Γ P and an artificial "do nothing"-type boundary condition on Γ out (see Fig. 1). We show that, although domain Ω is not smooth and different types of boundary conditions "meet" in the vertices of ∂Ω, 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.