Time‐periodic Stokes equations with inhomogeneous Dirichlet boundary conditions in a half‐space

Abstract

The time‐periodic Stokes problem in a half‐space with fully inhomogeneous right‐hand side is investigated. Maximal regularity in a time‐periodic Lp setting is established. A method based on Fourier multipliers is employed that leads to a decomposition of the solution into a steady‐state and a purely oscillatory part in order to identify the suitable function spaces.

Fluid-plate interaction under periodic forcing

“…Sobolev spaces defined on subdomains of T × T 2 0 × R, such as those appearing in Theorem 1.1 and 1.2, can be defined analogously to classical Sobolev spaces. We refer to [2] for a systematic approach.…”

“…T Z and ξ ∈ ( 2π L Z) 2 . At one point in the following, namely when we establish the L q estimates in Theorem 1.2, it is critical that a single Fourier transform F T×T 2 0 covering the whole time-space domain is employed instead of a sequential utilization of F T and F T 2 0 .…”

“…A pressure field corresponding to the weak solution obtained in Lemma 3.1 can be constructed and higher order regularity subsequently established. 2 (Ω; C) 3 ×W r,2 (T 2 0 ; C) and (u k , η k ) ∈ V k be the corresponding weak solution to (3.1) with g k = 0 constructed in Lemma 3.1. Then there exists a pressure field p k such that…”