Abstract
We consider the existence and uniqueness of time-periodic solutions to the Navier-Stokes equation in the whole space. We decompose periodic solutions into steady and purely periodic parts, and we analyze the equations they should satisfy. Based on the analysis of the purely periodic solutions represented by the Fourier transform to the Stokes equation, their additional regularity in time can be obtained and we use it to construct a time-periodic solution of the Navier-Stokes equation. Furthermore, we show that if the time-periodic solution is sufficiently small in an appropriate sense, then the Navier-Stokes equation admits no other solution in the same class.
Version Information
The current online version of this article, posted on 24 April 2024, supersedes the original version posted on 1 April 2024. The changes are as follows:
The display of $\frac{n}{3}$ appearing on pages 788, 789, 804, 805, 809, and 810 was corrected and replaced with $\frac{3}{2}$.
Citation
Tomoyuki Nakatsuka. "On unique solvability of the time-periodic problem for the Navier-Stokes equation." Adv. Differential Equations 29 (9/10) 783 - 814, September/October 2024. https://doi.org/10.57262/ade029-0910-783
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