The Maximum Regularity Property of the Steady Stokes Problem Associated with a Flow Through a Profile Cascade

Abstract

We deal with a steady Stokes-type problem, associated with a flow of a Newtonian incompressible fluid through a spatially periodic profile cascade. The used mathematical model is based on the reduction to one spatial period, represented by a bounded 2D domain \(\varOmega \). The corresponding Stokes–type problem is formulated by means of the Stokes equation, equation of continuity and three types of boundary conditions: the conditions of periodicity on the curves \(\varGamma _{\hspace {-1.1pt} 0}\) and \(\varGamma _{\hspace {-1.1pt} 1}\), the Dirichlet boundary conditions on \(\varGamma _{\hspace {-1.1pt} \mathrm {in}}\) and \(\varGamma _{\hspace {-1.1pt} p}\) and an artificial “do nothing”–type boundary condition on \(\varGamma _{\hspace {-1.1pt} \mathrm {out}}\). (See Fig. 1.) We explain on the level of weak solutions the sense in which the last condition is satisfied. We show that, although domain \(\varOmega \) is not smooth and different types of boundary conditions meet in the corners of \(\varOmega \), the considered problem has a strong solution with the so called maximum regularity property.