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Novel Self-similar Finite-time Blowups with Singular Profiles of the 1D Hou-Luo Model and the 2D Boussinesq Equations: A Numerical Investigation

Bojin Chen, De Huang, Xiangyuan Li

Abstract

We present novel self-similar finite-time blowup scenarios for the 1D Hou--Luo model. We numerically demonstrate that solutions that initially satisfy certain derivative degeneracy condition can develop asymptotically self-similar finite-time blowups with singular self-similar profiles that are unbounded at some point. Moreover, this blowup phenomenon exhibits a two-stage feature: the solution first undergoes a local $L^{\infty}$ blowup at some time $\tilde{T}$, then continues in the weak sense beyond $\tilde{T}$ and develops a local $L^p$ blowup at a later time $T>\tilde{T}$ for some $p>0$. A further numerical investigation indicates that both stages are asymptotically self-similar. Finally, we extend our numerical study to the 2D Boussinesq equations and discover similar self-similar finite-time blowups with singular profiles that also exhibit a two-stage feature.

Novel Self-similar Finite-time Blowups with Singular Profiles of the 1D Hou-Luo Model and the 2D Boussinesq Equations: A Numerical Investigation

Abstract

We present novel self-similar finite-time blowup scenarios for the 1D Hou--Luo model. We numerically demonstrate that solutions that initially satisfy certain derivative degeneracy condition can develop asymptotically self-similar finite-time blowups with singular self-similar profiles that are unbounded at some point. Moreover, this blowup phenomenon exhibits a two-stage feature: the solution first undergoes a local blowup at some time , then continues in the weak sense beyond and develops a local blowup at a later time for some . A further numerical investigation indicates that both stages are asymptotically self-similar. Finally, we extend our numerical study to the 2D Boussinesq equations and discover similar self-similar finite-time blowups with singular profiles that also exhibit a two-stage feature.

Paper Structure

This paper contains 31 sections, 8 theorems, 123 equations, 40 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Dynamic Rescaling and Self-similar Blowups of 1-D Hou--Luo Model
  3. Dynamic rescaling formulation
  4. Finite-time singularity of the HL model: existing results
  5. Singular profiles and multi-scale self-similar blowups
  6. Scenario 1: novel asymptotically self-similar blowups with singular profiles.
  7. Scenario 2: novel self-similar finite-time blowups with positive regular profiles.
  8. Numerical Results of the HL Model: Scenario 1
  9. Case 1: odd data
  10. Case 2: one-sided data
  11. Numerical Results of the HL Model: Scenario 2
  12. Singular Steady Solutions of The HL Model
  13. Novel Self-similar Finite-time Blowup of the 2D Boussinesq Equations
  14. Scenario 1: novel asymptotically self-similar blowups with singular profiles.
  15. Scenario 2: novel self-similar finite-time blowups with positive regular profiles.
  16. ...and 16 more sections

Key Result

Lemma 2.1

Let $(\omega_0,\theta_0)$ be some suitable initial data, and let $x_1(t),x_2(t)$ be two continuous trajectory. Suppose that $(\omega_i,\theta_i,u_i)_{i=1,2}$ is the solution of eqt:1Dhouluo that satisfies $u_i(x_i(t))=0$. Let $c(t)$ be a time-dependent quantity that satisfies the following ODE: Then it holds that $\blacktriangleleft$$\blacktriangleleft$

Figures (40)

  • Figure 1.1: Evolution of $\omega(\cdot,t)$ with different settings of initial data.
  • Figure 1.2: Evolution of the profile $\Omega(\cdot,t)$. In the non-degenerate case (left), $\Omega$ converges to a regular profile, while in the degenerate case (right), $\Omega$ converges to a singular profile.
  • Figure 1.3: Self-similar profiles of the 2D Boussinesq equations in different settings. The left panel illustrates a regular profile arising from non-degenerate initial data, as studied in the existing literature. The right panel presents a novel singular profile evolved from degenerate initial data.
  • Figure 2.1: Spatial profiles of the initial data. Left: The odd-symmetric case. Right: The one-sided case.
  • Figure 3.1: Early evolution of the solution in the odd symmetry case in Scenario 1. In the top row, $(x,t)$ represents the physical space variables, while in the bottom row, $(x,t)$ represents the dynamic rescaling space variables. The two coordinate systems are related via the change of variables introduced in Section \ref{['sec:dynamic_rescaling_of_HL']}.
  • ...and 35 more figures

Theorems & Definitions (15)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • Conjecture 2.4
  • Lemma 5.1
  • Lemma 5.2
  • Theorem 5.3
  • Remark 5.4
  • proof : Proof of Theorem \ref{['thm:HL_steady_state']}
  • ...and 5 more