The maximum regularity property of the steady Stokes problem associated with a flow through a profile cascade in $L^r$-framework
Tomáš Neustupa · Applications of Mathematics · 2022
summary:We deal with the steady Stokes problem, associated with a flow of a viscous incompressible fluid through a spatially periodic profile cascade. Using the reduction to domain $\Omega $, which represents one spatial period, the problem is formulated by means of boundary conditions of three types: the conditions of periodicity on curves $\Gamma _{-}$ and $\Gamma _{+}$ (lower and upper parts of $\partial \Omega $), the Dirichlet boundary conditions on $\Gamma _{\rm in}$ (the inflow) and $\Gamma _{0}$ (boundary of the profile) and an artificial ``do nothing''-type boundary condition on $\Gamma _{\rm out}$ (the outflow). We show that the considered problem has a strong solution with the $L^r$-maximum regularity property for appropriately integrable given data. From this we deduce a series of properties of the corresponding strong Stokes operator.