Time periodic problem of the Navier-Stokes equations in an exterior domain with periodically moving boundary
Abstract
In this paper we consider the Navier-Stokes equations in exterior domains of ℝn, n≥ 3, with a periodically in time moving boundary ∂Ω(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 L1 type regularity estimate will be used in weighted Lorentz spaces. To control the convective term an ℋ∞-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 ℋ∞-calculus to fractional domains (2025).
Related
- Uniqueness results for the exterior Dirichlet and Neumann problem in two dimensions for the Euler-Lagrange equations of variational integrals with linear growth
- The exterior Dirichlet problem for special Lagrangian equations
- Uniform boundary observability for periodic wave equations with low frequency exponent 4/3
- The principal eigenvalue problem for time-periodic nonlocal equations with drift
- Nontrivial periodic solutions to the one-phase free boundary problem
BibTeX
@misc{farwig2026time,
title = {{Time periodic problem of the Navier-Stokes equations in an exterior domain with periodically moving boundary}},
author = {Reinhard Farwig and Kazuyuki Tsuda},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28999},
eprint = {2608.28999},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28999},
}
RIS
TY - UNPB
TI - Time periodic problem of the Navier-Stokes equations in an exterior domain with periodically moving boundary
AU - Reinhard Farwig
AU - Kazuyuki Tsuda
PY - 2026
DA - 2026/08/29
PB - arXiv
DO - 10.48550/arxiv.2608.28999
UR - https://arxiv.org/abs/2608.28999
AB - In this paper we consider the Navier-Stokes equations in exterior domains of ℝ^n, n≥ 3, with a periodically in time moving boundary ∂Ω(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 ℋ^∞-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 ℋ^∞-calculus to fractional domains (2025).
ER -