Time periodic problem of the Navier-Stokes equations in an exterior domain with periodically moving boundary
In this paper we consider the Navier-Stokes equations in exterior domains of $\mathbb{R}^n$, $n\geq 3$, with a periodically in time moving boundary $\partial\Omega(t)$ and external force $f(t)$. For this case we prove the existence of a locally unique mild time periodic solution in weighted function spaces with radially symmetric Muckenhoupt weights. The solutions split into a stationary part controlled by potential theoretic estimates and a purely oscillatory part constructed as mild solution via analytic semigroup theory. To deal with perturbation terms of even second order - coming from a coordinate transform and the moving boundary - in weighted, homogeneous Sobolev spaces a maximal $L^1$ type regularity estimate will be used in weighted Lorentz spaces. To control the convective term an $\mathcal H^\infty$-calculus in weighted spaces of the Stokes operator, its $BIP$ property and embedding estimates of fractional powers are exploited, see a recent paper by the authors: The Stokes operator on exterior domains in homogeneous weighted function spaces: From weak theory to $\mathscr H^\infty$-calculus to fractional domains (2025).