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Quantitative classification of potential Navier-Stokes singularities beyond the blow-up time

Tobias Barker

TL;DR

This paper tackles the problem of classifying and quantifying potentially singular solutions to the 3D Navier–Stokes equations with initial data that are approximately axisymmetric. It develops a quantitative framework that avoids dependence on the exact blow-up time $T^*$ by leveraging improved energy bounds, an enhanced annulus of regularity, local-in-space smoothing, and a Carleman-based inductive backward-uniqueness scheme to propagate localized vorticity concentration forward in time. The main contributions are explicit lower bounds on localized energy in strategically chosen spatial regions near and beyond potential singular times, including a sharp exponential-type bound at the blow-up time and a double-exponential bound beyond blow-up on times in the irregular set. These results provide numerically testable criteria for viability of singular candidates and advance understanding of how singular behavior can manifest in axisymmetric-like NS flows, with potential extension to more general initial data under the stated bounds.

Abstract

In \cite{hou}, Hou gave a compelling numerical candidate for a singular solution of the 3D Navier-Stokes equations. We pioneer classifications of potentially singular solutions, motivated by the issue of investigating the viability of numerical candidates.For approximately axisymmetric initial data, we give the first quantitative classification of potentially singular solutions at \textit{any} given time in the region of potential blow-up times. Moreover, the quantitative bounds in the vicinity of any potential blow-up time are in principle amenable to numerical testing. To achieve this, we establish improved quantitative regions of regularity for approximately axisymmetric initial data, which may be of independent interest. Together with improved quantitative energy estimates from \cite{TB24}, this allows us to get a quantitative lower bound in the vicinity of a blow-up time by implementing the strategy of \cite{BP21}, which is a physical space analogue of Tao's strategy \cite{Ta21} for producing quantitative estimates for critically bounded solutions. To obtain a quantitative lower bound on the solution at any time in the region of potential blow-up times, we recursively apply quantitative Carleman inequality arguments from \cite{Ta21}. This necessitates careful bookkeeping to avoid exponential losses and to ensure that all forward-in-time iterations of (localized) vorticity concentration remain within the region of quantitative regularity of the solution.

Quantitative classification of potential Navier-Stokes singularities beyond the blow-up time

TL;DR

This paper tackles the problem of classifying and quantifying potentially singular solutions to the 3D Navier–Stokes equations with initial data that are approximately axisymmetric. It develops a quantitative framework that avoids dependence on the exact blow-up time by leveraging improved energy bounds, an enhanced annulus of regularity, local-in-space smoothing, and a Carleman-based inductive backward-uniqueness scheme to propagate localized vorticity concentration forward in time. The main contributions are explicit lower bounds on localized energy in strategically chosen spatial regions near and beyond potential singular times, including a sharp exponential-type bound at the blow-up time and a double-exponential bound beyond blow-up on times in the irregular set. These results provide numerically testable criteria for viability of singular candidates and advance understanding of how singular behavior can manifest in axisymmetric-like NS flows, with potential extension to more general initial data under the stated bounds.

Abstract

In \cite{hou}, Hou gave a compelling numerical candidate for a singular solution of the 3D Navier-Stokes equations. We pioneer classifications of potentially singular solutions, motivated by the issue of investigating the viability of numerical candidates.For approximately axisymmetric initial data, we give the first quantitative classification of potentially singular solutions at \textit{any} given time in the region of potential blow-up times. Moreover, the quantitative bounds in the vicinity of any potential blow-up time are in principle amenable to numerical testing. To achieve this, we establish improved quantitative regions of regularity for approximately axisymmetric initial data, which may be of independent interest. Together with improved quantitative energy estimates from \cite{TB24}, this allows us to get a quantitative lower bound in the vicinity of a blow-up time by implementing the strategy of \cite{BP21}, which is a physical space analogue of Tao's strategy \cite{Ta21} for producing quantitative estimates for critically bounded solutions. To obtain a quantitative lower bound on the solution at any time in the region of potential blow-up times, we recursively apply quantitative Carleman inequality arguments from \cite{Ta21}. This necessitates careful bookkeeping to avoid exponential losses and to ensure that all forward-in-time iterations of (localized) vorticity concentration remain within the region of quantitative regularity of the solution.
Paper Structure (22 sections, 12 theorems, 222 equations)

This paper contains 22 sections, 12 theorems, 222 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Comparison with previous literature and strategy
  4. Obtaining exponentially small lower bounds of the solution at a blow-up time
  5. Obtaining lower bounds of the solution beyond the blow-up time
  6. Remarks
  7. Optimality
  8. Vertical decay on the initial data
  9. Initial data that is not approximately axisymmetric
  10. Notation
  11. Universal constants, vectors and domains
  12. Function spaces and norms
  13. Approximate axisymmetric functions
  14. Notions of solutions and singularity
  15. Improved quantitative annulus of regularity
  16. ...and 7 more sections

Key Result

Theorem 1

Let $p\in (3,\infty]$. There exists a $0<c_{0}(p)<c_{1}(p)$, $c_{2}(p)$ and a positive universal constant $M_0>1$ such that the following statement holds true. Let $v:\mathbf{R}^3\times (0,\infty)\rightarrow \mathbf{R}^3$ be a suitable weak Leray-Hopf solution, with initial data $v_{0}:\mathbf{R}^3\ Moreover, suppose that $v$ possesses a singular point at $T^*$. Then the above assumptions imply th

Theorems & Definitions (24)

  • Theorem 1
  • Theorem 2
  • Definition 3: Approximately axisymmetric functions
  • Definition 4: Suitable weak solution
  • Definition 5: Singular point
  • Definition 6: Leray-Hopf solution
  • Definition 7: Suitable Leray-Hopf solution
  • Lemma 8
  • proof
  • Proposition 9
  • ...and 14 more