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Sharp norm inflation for 3D Navier-Stokes equations in supercritical spaces

Xiaoyutao Luo

TL;DR

This work establishes norm inflation and strong ill-posedness for the 3D Navier–Stokes equations in almost all supercritical Besov spaces near the scaling-critical line $s=-1+\tfrac{3}{p}$ (except at $s=0$). The authors build an axisymmetric approximate solution by reducing the dynamics to a stationary radial vortex in the $rz$-plane and a transported swirl component, then induce inflation via two distinct mechanisms: $s>0$ uses mixing to drive a forward energy cascade, while $s<0$ leverages un-mixing to trigger growth in negative regularity norms. A careful stability analysis shows the approximate solution reflects in the full NS flow on a short time window $[0,t^{*}]$, yielding explicit lower bounds on Besov norms and, in particular, arbitrarily large finite-time growth of the vorticity. Together, these results imply strong ill-posedness in broad supercritical regimes and provide a rigorous link between norm inflation, small-scale formation, and rapid vorticity amplification, while highlighting open questions about sustained growth and potential blow-up.

Abstract

We prove that the incompressible Navier-Stokes equations exhibit norm inflation in $\dot B^{s}_{p,q}(\mathbb{R}^3)$ with smooth, compactly supported initial data. Such norm inflation is shown in all supercritical $\dot B^{s}_{p,q} $ near the scaling-critical line $s = -1+ \frac{3}{p}$ except at $s=0$. The growth mechanism differs depending on the sign of the regularity index $s$: forward energy cascade driven by mixing for $s>0$ and backward energy cascade caused by un-mixing for $s<0$. The construction also demonstrates arbitrarily large, finite-time growth of the vorticity, the first of such examples for the Navier-Stokes equations.

Sharp norm inflation for 3D Navier-Stokes equations in supercritical spaces

TL;DR

This work establishes norm inflation and strong ill-posedness for the 3D Navier–Stokes equations in almost all supercritical Besov spaces near the scaling-critical line (except at ). The authors build an axisymmetric approximate solution by reducing the dynamics to a stationary radial vortex in the -plane and a transported swirl component, then induce inflation via two distinct mechanisms: uses mixing to drive a forward energy cascade, while leverages un-mixing to trigger growth in negative regularity norms. A careful stability analysis shows the approximate solution reflects in the full NS flow on a short time window , yielding explicit lower bounds on Besov norms and, in particular, arbitrarily large finite-time growth of the vorticity. Together, these results imply strong ill-posedness in broad supercritical regimes and provide a rigorous link between norm inflation, small-scale formation, and rapid vorticity amplification, while highlighting open questions about sustained growth and potential blow-up.

Abstract

We prove that the incompressible Navier-Stokes equations exhibit norm inflation in with smooth, compactly supported initial data. Such norm inflation is shown in all supercritical near the scaling-critical line except at . The growth mechanism differs depending on the sign of the regularity index : forward energy cascade driven by mixing for and backward energy cascade caused by un-mixing for . The construction also demonstrates arbitrarily large, finite-time growth of the vorticity, the first of such examples for the Navier-Stokes equations.

Paper Structure

This paper contains 30 sections, 17 theorems, 187 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Application to small-scale formation
  3. Background and literature
  4. Outline of construction
  5. Notations
  6. Sobolev and Besov spaces
  7. Besov spaces
  8. The approximate solution
  9. Outline of construction
  10. Auxiliary coordinates
  11. The parameters and the setup
  12. Definition of ur and uz
  13. Definition of utheta
  14. Properties of the approximate solution
  15. Proof of Proposition \ref{['prop:approximate']}
  16. ...and 15 more sections

Key Result

Theorem 1.1

For any $s \neq 0$ and $1\leq p,q \leq \infty$ such that $-3 < s - \frac{3}{p} < -1$, the 3D Navier-Stokes equations eq:NS are strongly ill-posed in $\dot B^{s}_{p,q} (\mathbb{R}^3)$ in the following sense. For any $\epsilon>0$, there exists a time $0< t^* \leq \epsilon$ and a solution $u$ of eq:

Figures (2)

  • Figure 1: Left: Supercritical region in light blue. Right: Ill-posedness region of Theorem \ref{['thm:Besov']} in pink
  • Figure 2: Anisotropic setup in $rz$-plane with $\mu^{-1} \ll \nu^{-1}$.

Theorems & Definitions (32)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Lemma 1.4
  • Proposition 1.5
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 22 more