# On time-periodic Navier–Stokes flows with fast spatial decay in the whole space

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## Abstract

We investigate the pointwise behavior of time-periodic Navier–Stokes flows in the whole space. We show that if the time-periodic external force is sufficiently small in an appropriate sense, then there exists a unique time-periodic solution \(\{ u,p \}\) of the Navier–Stokes equation such that \(|u(t,x)|=O(|x|^{1-n})\), \(|\nabla u(t,x)|=O(|x|^{-n})\) and \(|p(t,x)|=O(|x|^{-n})\) uniformly in \(t \in {\mathbb {R}}\) as \(|x| \rightarrow \infty\). Our solution decays more rapidly than the time-periodic Stokes fundamental solution. The proof is based on the representation formula of a solution via the time-periodic Stokes fundamental solution and its properties.

## Keywords

Navier–Stokes equation Time-periodic solution Asymptotic property## Mathematics Subject Classification

35Q30 35B10 76D05 76D03## Notes

### Acknowledgements

The research was supported by the Academy of Sciences of the Czech Republic, Institute of Mathematics (RVO: 67985840). The author would like to thank Professor M. Kyed for useful comments.

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