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Arbitrary norm growth in the 3D Navier-Stokes equations

Stan Palasek

Abstract

We construct a family of smooth initial data for the Navier-Stokes equations, bounded in $BMO^{-1}(\mathbb T^3)$, that gives rise to arbitrarily large global solutions. As a consequence, we rule out various hypothetical a priori estimates for strong solutions in terms of a critical norm of the initial data. To our knowledge, this is the first example of unbounded norm growth in the well-posed setting. The solutions exhibit an inverse cascade across an unbounded number of modes, with growth resulting from repeated squaring by the quadratic nonlinearity. This mechanism relies on largeness of the data in $B^{-1}_{\infty,\infty}$ and is fundamentally distinct from instantaneous norm inflation and ill-posedness phenomena.

Arbitrary norm growth in the 3D Navier-Stokes equations

Abstract

We construct a family of smooth initial data for the Navier-Stokes equations, bounded in , that gives rise to arbitrarily large global solutions. As a consequence, we rule out various hypothetical a priori estimates for strong solutions in terms of a critical norm of the initial data. To our knowledge, this is the first example of unbounded norm growth in the well-posed setting. The solutions exhibit an inverse cascade across an unbounded number of modes, with growth resulting from repeated squaring by the quadratic nonlinearity. This mechanism relies on largeness of the data in and is fundamentally distinct from instantaneous norm inflation and ill-posedness phenomena.

Paper Structure

This paper contains 12 sections, 9 theorems, 129 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Norm growth and ill-posedness in the Navier--Stokes equations
  3. Idea of the construction
  4. Notation, function spaces, and basic estimates
  5. Proof of Theorem \ref{['second-theorem']}
  6. Construction of the principal part
  7. The oscillatory part
  8. The coefficients and some estimates
  9. Construction of the full solution
  10. Final estimates
  11. Proof of Corollaries \ref{['a-priori-estimate-corollary']} and \ref{['kt-vmo-bmo-corollary']}
  12. Rank-one decomposition and Mikado flows

Key Result

Theorem 1.1

Let $\Theta_*\in{\mathbb R^3}$ and $\eta_*\in{\mathbb Z^3}$ be any non-zero vectors satisfying $\Theta_*\cdot\eta_*=0$. For any $\epsilon_*>0$ and $n_{max}\in\mathbb N$, there exists divergence-free initial data $u^0\in C^\infty({\mathbb T^3})$ with and such that the corresponding strong solution of nse is global and satisfies where for $n=0,1,2,\ldots,n_{max}$.

Figures (1)

  • Figure 1: We illustrate the conceptual distinction between the ill-posedness construction in bourgain2008ill (left) and the construction leading to Theorem \ref{['second-theorem']} (right). The axes represent time (horizontal) and frequency scale (vertical), while the thickness of the rectangles indicates the $B^{-1}_{\infty,\infty}$ norm of $u(t)$ projected to the shell. In both cases, a large concentration accumulates at the lowest frequency: in bourgain2008ill, by many single-step cascades; in the present work, via a single iterated cascade.

Theorems & Definitions (19)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Corollary 1.5
  • Remark 1.6
  • Remark 1.7
  • Corollary 1.8
  • Remark 2.1
  • Lemma 2.2: Nash lemma
  • ...and 9 more