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Asymptotic instability for the forced Navier--Stokes equations in critical Besov spaces

Mikihiro Fujii, Hiroyuki Tsurumi

TL;DR

This work reveals sharp limits of asymptotic stability for forced Navier–Stokes flows in critical Besov spaces. By constructing forces with vanishing Besov norms in time, the authors show instability in high dimensions ($n\ge 3$) when $p\ge n$, driven by nonlinear interactions rather than linear dynamics, and they prove a corresponding instability in 2D for all $1\le p\le\infty$ in the Besov scale $\dot{B}_{p,1}^{2/p-1}$. The method hinges on intermittent, high-frequency forcing that induces a nonlinear $\text{high}\times\text{high}\to\text{low}$ cascade, yielding oscillatory tails in the solution that fail to decay to zero. The results delineate the precise regimes where asymptotic stability fails in critical spaces and establish a contrasting stability regime for $1\le p<n$. The work provides new insights into the nonlinear mechanisms that limit decay and has implications for the understanding of forced fluid flows in critical function spaces.

Abstract

The asymptotic stability is one of the classical problems in the field of mathematical analysis of fluid mechanics. In $\mathbb{R}^n$ with $n \geq 3$, it is easily proved by the standard argument that if the given small external force decays at temporal infinity, then the small forced Navier--Stokes flow also strongly converges to zero as time tends to infinity in the framework of the critical Besov spaces $\dot{B}_{p,q}^{n/p-1}(\mathbb{R}^n)$ with $1 \leq p < n$ and $1 \leq q < \infty$. In the present paper, we show that this asymptotic stability fails for $p \geq n$ with $n \geq 3$ in the sense that there exist arbitrary small external forces whose critical Besov norm decays in large time, whereas the corresponding Navier--Stokes flows oscillate and do not strongly converge as $t \to \infty$ in the framework of the critical Besov spaces $\dot{B}_{p,q}^{n/p-1}(\mathbb{R}^n)$. Moreover, we find that the situation is different in the two-dimensional case $n=2$ and show the forced Navier--Stokes flow is asymptotically unstable in $\dot{B}_{p,1}^{2/p-1}(\mathbb{R}^2)$ for all $1 \leq p \leq \infty$. Our instability does not appear in the linear level but is caused by the nonlinear interaction from external forces.

Asymptotic instability for the forced Navier--Stokes equations in critical Besov spaces

TL;DR

This work reveals sharp limits of asymptotic stability for forced Navier–Stokes flows in critical Besov spaces. By constructing forces with vanishing Besov norms in time, the authors show instability in high dimensions () when , driven by nonlinear interactions rather than linear dynamics, and they prove a corresponding instability in 2D for all in the Besov scale . The method hinges on intermittent, high-frequency forcing that induces a nonlinear cascade, yielding oscillatory tails in the solution that fail to decay to zero. The results delineate the precise regimes where asymptotic stability fails in critical spaces and establish a contrasting stability regime for . The work provides new insights into the nonlinear mechanisms that limit decay and has implications for the understanding of forced fluid flows in critical function spaces.

Abstract

The asymptotic stability is one of the classical problems in the field of mathematical analysis of fluid mechanics. In with , it is easily proved by the standard argument that if the given small external force decays at temporal infinity, then the small forced Navier--Stokes flow also strongly converges to zero as time tends to infinity in the framework of the critical Besov spaces with and . In the present paper, we show that this asymptotic stability fails for with in the sense that there exist arbitrary small external forces whose critical Besov norm decays in large time, whereas the corresponding Navier--Stokes flows oscillate and do not strongly converge as in the framework of the critical Besov spaces . Moreover, we find that the situation is different in the two-dimensional case and show the forced Navier--Stokes flow is asymptotically unstable in for all . Our instability does not appear in the linear level but is caused by the nonlinear interaction from external forces.

Paper Structure

This paper contains 16 sections, 16 theorems, 231 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The place of our study in the related literature
  3. Our results
  4. Idea of the proof
  5. Organization of the paper
  6. Preliminaries
  7. Function spaces
  8. Practical lemmas
  9. High-dimensional analysis
  10. Key propositions
  11. Proof of Theorem \ref{['thm:nD']}
  12. Two-dimensional analysis
  13. Key proposition
  14. Proof of Theorem \ref{['thm:2D']}
  15. Proof of Proposition \ref{['prop:stab']}
  16. ...and 1 more sections

Key Result

Proposition 1.1

Let $n$, $p$, and $q$ satisfy Then, there exist positive constants $\varepsilon=\varepsilon(n,p,q)$ and $C=C(n,p,q)$ such that if $a \in \dot{B}_{p,q}^{n/p-1}(\mathbb{R}^n)$ with $\operatorname{div} a = 0$ and $f \in \widetilde{C}((0,\infty);\dot{B}_{p,q}^{n/p-3}(\mathbb{R}^n))$ satisfy then eq:f-NS possesses a unique solution $u \in \widetilde{C}([0,\infty);\dot{B}_{p,q}^{n/p-1}(\mathbb{R}^n))$

Figures (3)

  • Figure 1: The behavior of $\|f_{\varepsilon}(t)\|_{\dot{B}_{p,q}^{n/p-3}}$
  • Figure 2: The behavior of $\|u_{\varepsilon}(t)\|_{\dot{B}_{p,q}^{n/p-1}}$
  • Figure 3: The Fourier support of the "high$\times$high$\to$low" cascade

Theorems & Definitions (29)

  • Proposition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Lemma 2.1: Bah-Che-Dan-11
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 3.1
  • ...and 19 more