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Time-periodic solutions to the Navier--Stokes equations on the whole space including the two-dimensional case

Mikihiro Fujii

TL;DR

This work resolves the time-periodic Navier–Stokes problem on the whole space by dimension: for $n\ge 3$ there exists a unique small $T$-periodic mild solution $u_{\rm per}$ under a small time-periodic force $f$ in a scaling-critical Besov-type framework, and this solution is stable to small perturbations. The approach exploits Chemin–Lerner spaces and maximal-regularity of the heat kernel to derive bilinear estimates and a fixed-point argument in a scaling-critical setting. In contrast, the 2D case is shown to be generically non-solvable for time-periodic forcing: there exist arbitrarily small $f_{\delta}$ for which no $T$-periodic solution exists in a natural class, demonstrated via a contradiction argument using a non-periodic construction and unconditional uniqueness. Together, these results delineate a sharp dimensional border for time-periodic solvability of the Navier–Stokes equations on $\mathbb{R}^n$ and contribute to the broader understanding of fluid dynamics in unbounded domains.

Abstract

Let us consider the incompressible Navier--Stokes equations with the time-periodic external forces in the whole space $\mathbb{R}^n$ with $n\geq 2$ and investigate the existence and non-existence of time-periodic solutions. In the higher dimensional case $n \geq 3$, we construct a unique small solution for given small time-periodic force in the scaling critical spaces of Besov type and prove its stability under small perturbations. In contrast, for the two-dimensional case $n=2$, the time-periodic solvability of the Navier--Stokes equations has been long standing open. It is the central work of this paper that we have now succeeded in solving this issue negatively by providing examples of small external forces such that each of them does not generate time-periodic solutions.

Time-periodic solutions to the Navier--Stokes equations on the whole space including the two-dimensional case

TL;DR

This work resolves the time-periodic Navier–Stokes problem on the whole space by dimension: for there exists a unique small -periodic mild solution under a small time-periodic force in a scaling-critical Besov-type framework, and this solution is stable to small perturbations. The approach exploits Chemin–Lerner spaces and maximal-regularity of the heat kernel to derive bilinear estimates and a fixed-point argument in a scaling-critical setting. In contrast, the 2D case is shown to be generically non-solvable for time-periodic forcing: there exist arbitrarily small for which no -periodic solution exists in a natural class, demonstrated via a contradiction argument using a non-periodic construction and unconditional uniqueness. Together, these results delineate a sharp dimensional border for time-periodic solvability of the Navier–Stokes equations on and contribute to the broader understanding of fluid dynamics in unbounded domains.

Abstract

Let us consider the incompressible Navier--Stokes equations with the time-periodic external forces in the whole space with and investigate the existence and non-existence of time-periodic solutions. In the higher dimensional case , we construct a unique small solution for given small time-periodic force in the scaling critical spaces of Besov type and prove its stability under small perturbations. In contrast, for the two-dimensional case , the time-periodic solvability of the Navier--Stokes equations has been long standing open. It is the central work of this paper that we have now succeeded in solving this issue negatively by providing examples of small external forces such that each of them does not generate time-periodic solutions.
Paper Structure (13 sections, 12 theorems, 161 equations)

This paper contains 13 sections, 12 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Known results and the position of our study
  3. Main results
  4. Higher dimensional case
  5. Two-dimensional case
  6. Organization and notations on this paper
  7. Preliminaries
  8. Higher dimensional analysis: Proofs of Theorems \ref{['thm:1']} and \ref{['thm:1-1']}
  9. Two-dimensional analysis: Proof of Theorem \ref{['thm:2']}
  10. Construction of non-periodic in time mild solutions
  11. Unconditional uniqueness
  12. Proof of Theorem \ref{['thm:2']}
  13. Remarks on the paradifferential calculus

Key Result

Theorem 1.1

Let $n \geqslant 3$ be an integer and let $1 \leqslant p < n$ and $1 \leqslant \sigma \leqslant \infty$. Then, there exists a positive constant $\delta_0=\delta_0(n,p,\sigma)$ and $\varepsilon_0=\varepsilon_0(n,p,\sigma)$ such that for any $T>0$ and $T$-periodic external force $f \in \widetilde{C}(\ the equation eq:NS_P possesses a unique $T$-periodic mild solution $u_{\rm per} \in U_{p,\sigma}(\m

Theorems & Definitions (31)

  • Theorem 1.1: Existence of time-periodic solutions on $\mathbb{R}^n$ with $n \geqslant 3$
  • Remark 1.2
  • Theorem 1.3: Time-periodic stability on $\mathbb{R}^n$ with $n \geqslant 3$
  • Remark 1.4
  • Theorem 1.5: Non-existence of the time-periodic solution on $\mathbb{R}^2$
  • Remark 1.6
  • Lemma 2.1
  • Remark 2.2
  • proof : Proof of Lemma \ref{['lemm:max-reg']}
  • Lemma 2.3
  • ...and 21 more