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Finite-time blow-up in an elementary model of the 3D Navier-Stokes equations

Stan Palasek

Abstract

We demonstrate finite-time blow-up in a simple, realistic shell model of the 3D Navier-Stokes equations, equipped with "smooth" (i.e., rapidly decaying in frequency) initial data and forcing. Previously studied models either exhibit a turbulent cascade that regularizes the three-dimensional viscous dynamics, or rely on highly artificial interactions not transparently realized in the true Euler nonlinearity. We also treat the inviscid, unforced case and obtain singularity formation just above the energy level. We conclude with a discussion of the prospects for embedding the behavior of the dyadic model into the full Euler and Navier-Stokes equations.

Finite-time blow-up in an elementary model of the 3D Navier-Stokes equations

Abstract

We demonstrate finite-time blow-up in a simple, realistic shell model of the 3D Navier-Stokes equations, equipped with "smooth" (i.e., rapidly decaying in frequency) initial data and forcing. Previously studied models either exhibit a turbulent cascade that regularizes the three-dimensional viscous dynamics, or rely on highly artificial interactions not transparently realized in the true Euler nonlinearity. We also treat the inviscid, unforced case and obtain singularity formation just above the energy level. We conclude with a discussion of the prospects for embedding the behavior of the dyadic model into the full Euler and Navier-Stokes equations.
Paper Structure (12 sections, 5 theorems, 90 equations, 2 figures)

This paper contains 12 sections, 5 theorems, 90 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Dyadic models and blow-up
  3. Statement of the model and main theorems
  4. Blow-up mechanism
  5. Inviscid case
  6. Viscous case
  7. Motivating the shell model
  8. Proof of blow-up theorems
  9. The trapping region
  10. Inviscid blow-up
  11. Viscous blow-up
  12. Prospects for transfer to the PDE setting

Key Result

Theorem 1.3

Let $\nu>0$, $\alpha>2$, and $N_k$ as in nk_choice. Then there exist positive initial data $X^0\in \mathcal{C}^\infty$ and force $f\in C_t^\infty([0,\infty);\mathcal{C}^\infty)$ that give rise to a finite-time blow-up of l2_obukhov. In particular, for any $s>0$, the data can be chosen so that $X$ be

Figures (2)

  • Figure 1: Comparison of several candidate models of blow-up: the Katz--Pavlović (KP) model MR2095627MR2038114, the Obukhov model with exponential or super-exponential frequency separation, and Tao's model from MR3486169. We illustrate each model using the quadratic gate notation from MR3486169. Filled arrows represent KP-type interactions or "pump gates"; open arrows represent Obukhov-type interactions or "amplifier gates"; and loops represent "rotor gates" whose oscillation rate is proportional to the mode to which the loop is connected.
  • Figure 2: A blow-up of the viscous Obukhov model at $t=0$. We depict snapshots of the solution $Y_k(t)=N_k^{\alpha-1}X_k(t)$, which models $\|P_ku(t)\|_{L^\infty}$, at various times $t\in[-T,0]$, including the blow-up profile $Y_k(0)=N_k^{1.4}$ (black). The system \ref{['l2_obukhov']} is simulated with $\nu=1$, $\alpha=\frac{5}{2}$, and $N_k=1.5^{1.15^k}$.

Theorems & Definitions (19)

  • Definition 1.1: Besov-type spaces
  • Definition 1.2
  • Theorem 1.3: Viscous blow-up
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8: Inviscid blow-up
  • Remark 1.9
  • Remark 1.10
  • ...and 9 more