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

TL;DR

This paper extends the maximum regularity property of the steady Stokes problem to the general $L^r$-framework for flow through a periodic profile cascade, demonstrating strong solutions despite domain irregularities and mixed boundary conditions.

Contribution

It generalizes previous $L^2$ results to all $L^r$ spaces, handling complex boundary conditions and domain irregularities in the steady Stokes problem.

Findings

Established maximum regularity in $L^r$-framework for non-smooth domains

Proved existence of strong solutions with mixed boundary conditions

Clarified the interpretation of 'do nothing' boundary condition

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 $\Gamma_{\rm in}$ and $\Gamma_P$ and an artificial "do nothing"-type boundary condition on $\Gamma_{\rm out}$ (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.