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On persistence of spatial analyticity in the hyper-dissipative Navier-Stokes models

Aseel Farhat, Zoran Grujic

TL;DR

This paper addresses finite-time blow-up for the 3D hyper-dissipative Navier–Stokes system with exponent $β>1$ by ruling out a class of analytic blow-up profiles. Leveraging Grujić’s sparse-regime regularity theory and an ascending-chain mechanism for the chain of derivatives, it demonstrates that the analytic radius of the flow can outpace the sparseness scale as derivatives grow, provided monotone growth in the blow-up profile. Under assumptions (A1)-(A2) on the analytic structure and focus of the singularity, the authors prove that no finite-time singularity occurs at $T^*$, and the solution extends analytically beyond $T^*$. The result strengthens the understanding of regularity in the turbulent regime and provides a self-contained argument showing that a broad class of analytic blow-up candidates cannot form when $β>1$.

Abstract

The goal of this note is to demonstrate that as soon as the hyper-diffusion exponent is greater than one, a class of finite time blow-up scenarios consistent with the analytic structure of the flow (prior to the possible blow-up time) can be ruled out. The argument is self-contained, in spirit of the regularity theory of the hyper-dissipative Navier-Stokes system in turbulent regime developed by Grujic and Xu.

On persistence of spatial analyticity in the hyper-dissipative Navier-Stokes models

TL;DR

This paper addresses finite-time blow-up for the 3D hyper-dissipative Navier–Stokes system with exponent by ruling out a class of analytic blow-up profiles. Leveraging Grujić’s sparse-regime regularity theory and an ascending-chain mechanism for the chain of derivatives, it demonstrates that the analytic radius of the flow can outpace the sparseness scale as derivatives grow, provided monotone growth in the blow-up profile. Under assumptions (A1)-(A2) on the analytic structure and focus of the singularity, the authors prove that no finite-time singularity occurs at , and the solution extends analytically beyond . The result strengthens the understanding of regularity in the turbulent regime and provides a self-contained argument showing that a broad class of analytic blow-up candidates cannot form when .

Abstract

The goal of this note is to demonstrate that as soon as the hyper-diffusion exponent is greater than one, a class of finite time blow-up scenarios consistent with the analytic structure of the flow (prior to the possible blow-up time) can be ruled out. The argument is self-contained, in spirit of the regularity theory of the hyper-dissipative Navier-Stokes system in turbulent regime developed by Grujic and Xu.
Paper Structure (5 sections, 7 theorems, 62 equations)

This paper contains 5 sections, 7 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The ascending chain condition and the improved local-in-time existence
  4. Proof of Theorem 1.1
  5. Acknowledgments

Key Result

Theorem 1.1

Let $\beta > 1$, $u_0 \in L^\infty \cap L^2$, and suppose that (A1) and (A2) hold. Then $T^*$ is not a singular time, and the solution $u$ can be continued analytically past $T^*$.

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • ...and 3 more