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Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data II: Rigorous Numerics

Jiajie Chen, Thomas Y. Hou

TL;DR

The paper proves finite-time blowup for the 2D Boussinesq and 3D axisymmetric Euler equations with smooth data by extending Part I's analytic framework with sharp, computer-assisted stability estimates. It develops space-time approximations to the linearized operator around an approximate self-similar profile, and constructs corrections to enforce boundary and normalization conditions while maintaining cubic vanishing near the singularity. A detailed a posteriori error analysis decomposes residuals into local and nonlocal parts, with rigorous bounds derived via rigorous numerics, adaptive meshes, and parallelized computations, ensuring the stability inequalities in the foundational lemma hold. The velocity field is estimated through a hierarchy of $L^ olinebreak{}^ olinebreak$ and $C^{1/2}$ Hölder bounds, aided by finite-rank approximations and kernel symmetrizations to manage nonlocal terms. Altogether, these results validate Part I’s stability framework and establish the finite-time singularity with near-self-similar blowup in a rigorously controlled setting, under smooth finite-energy initial data and boundary conditions.

Abstract

This is Part II of our paper in which we prove finite time blowup of the 2D Boussinesq and 3D axisymmetric Euler equations with smooth initial data of finite energy and boundary. In Part I of our paper [ChenHou2023a], we establish an analytic framework to prove stability of an approximate self-similar blowup profile by a combination of a weighted $L^\infty$ norm and a weighted $C^{1/2}$ norm. Under the assumption that the stability constants, which depend on the approximate steady state, satisfy certain inequalities stated in our stability lemma, we prove stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth initial data and boundary. In Part II of our paper, we provide sharp stability estimates of the linearized operator by constructing space-time solutions with rigorous error control. We also obtain sharp estimates of the velocity in the regular case using computer assistance. These results enable us to verify that the stability constants obtained in Part I [ChenHou2023a] indeed satisfy the inequalities in our stability lemma. This completes the analysis of the finite time singularity of the axisymmetric Euler equations with smooth initial data and boundary.

Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data II: Rigorous Numerics

TL;DR

The paper proves finite-time blowup for the 2D Boussinesq and 3D axisymmetric Euler equations with smooth data by extending Part I's analytic framework with sharp, computer-assisted stability estimates. It develops space-time approximations to the linearized operator around an approximate self-similar profile, and constructs corrections to enforce boundary and normalization conditions while maintaining cubic vanishing near the singularity. A detailed a posteriori error analysis decomposes residuals into local and nonlocal parts, with rigorous bounds derived via rigorous numerics, adaptive meshes, and parallelized computations, ensuring the stability inequalities in the foundational lemma hold. The velocity field is estimated through a hierarchy of and Hölder bounds, aided by finite-rank approximations and kernel symmetrizations to manage nonlocal terms. Altogether, these results validate Part I’s stability framework and establish the finite-time singularity with near-self-similar blowup in a rigorously controlled setting, under smooth finite-energy initial data and boundary conditions.

Abstract

This is Part II of our paper in which we prove finite time blowup of the 2D Boussinesq and 3D axisymmetric Euler equations with smooth initial data of finite energy and boundary. In Part I of our paper [ChenHou2023a], we establish an analytic framework to prove stability of an approximate self-similar blowup profile by a combination of a weighted norm and a weighted norm. Under the assumption that the stability constants, which depend on the approximate steady state, satisfy certain inequalities stated in our stability lemma, we prove stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth initial data and boundary. In Part II of our paper, we provide sharp stability estimates of the linearized operator by constructing space-time solutions with rigorous error control. We also obtain sharp estimates of the velocity in the regular case using computer assistance. These results enable us to verify that the stability constants obtained in Part I [ChenHou2023a] indeed satisfy the inequalities in our stability lemma. This completes the analysis of the finite time singularity of the axisymmetric Euler equations with smooth initial data and boundary.
Paper Structure (139 sections, 15 theorems, 563 equations, 4 figures)

This paper contains 139 sections, 15 theorems, 563 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Review of the analytic framework from Part I ChenHou2023a
  3. Dynamic rescaling formulation
  4. Nearly self-similar blowup and the blowup time
  5. The main steps in the proof of Theorem \ref{['thm:main']}
  6. Rigorous numerics
  7. Dependency tree
  8. Construct and estimate the approximate solution to the linearized equations
  9. Numerical methods
  10. A posteriori error estimates: decomposition of errors
  11. First correction and the construction of $\phi^N$
  12. The second correction
  13. Cubic interpolation in time
  14. Decomposing the time interval for parallel computing
  15. Compactly supported in time
  16. ...and 124 more sections

Key Result

Theorem 1

Let $$, $\mathbf{u}$ and $\omega$ be the density, velocity and vorticity in the 2D Boussinesq equations eq:bous1-eq:biot, respectively. There is a family of smooth initial data $( _0, \omega_0)$ with $_0(x,y)$ being even and $\omega_0(x,y)$ being odd in $x$, such that the solution of the 2D Boussine

Figures (4)

  • Figure 1: Left: The large box is $R(x, k)$ and the red box is $R_{s,1}(x, k)$. The small box containing $x$ has size $h \times h$. Right: The upper box is $R(x, k, N)$, and the shaded box is $R(x, k, S)$, the reflection of the region below the $y$-axis.
  • Figure 2: The largest box in the left and middle figure is $R(x, k)$. Left: The left and right blue regions are $X_{l, a} \times Y_{m, b}, X_{r, a} \times Y_{m, b}$. The four red regions correspond to $X_{\alpha, a } \times Y_{\beta, b}, \alpha = l, u, \beta = d, u$. Middle: Illustration of $R(x, k) R_s(x, k)$ and $R_s(x, k_2)$. $R(x, k) R_s(x, k)$ consists of the blue and the red regions. Right: different regions near the singularity for $u/x_1$. Blue, red, and white regions represent $S_{in, 1}, S_{in, 2}, S_{out}$, respectively.
  • Figure 3: Left: $R_{s,1}(x, k_3)$ and $R_{s,1}(z, k_3)$ with $x_2 = z_2$. The small square is a mesh grid containing $x$ or $z$. $x, z$ can have different locations relative to the grids. Right: The large rectangle is $R(k_2)$, the upper part is $R^+(k_2)$, and the lower part is $R^-(k_2)$. The blue region is $R^-(k_2) R^-(k_3)$. $\Gamma$ is part of its boundary.
  • Figure 4: Piecewise $L^{ }( _{elli})$ bound of the error $\bar{\varepsilon}_1, \hat{\varepsilon}_1$ in solving the Poisson equations. Left: error for the approximate steady steate. Right: error for the approximate space-time solution $\hat{W}_2$

Theorems & Definitions (29)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Remark 3.1
  • Remark 4.1
  • ...and 19 more