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Non-radial implosion for compressible Euler and Navier-Stokes in $\mathbb{T}^3$ and $\mathbb{R}^3$

Gonzalo Cao-Labora, Javier Gómez-Serrano, Jia Shi, Gigliola Staffilani

TL;DR

The paper advances the theory of singularity formation in compressible flows by constructing smooth, non-radial imploding solutions to the Euler and Navier–Stokes equations on both $ abla ext{T}^3$ and $ abla ext{R}^3$. It develops a robust linear stability framework around self-similar radial profiles, employing dissipativity, maximality, and a novel angular repulsivity mechanism via vector spherical harmonics to handle non-radial perturbations. The nonlinear analysis combines a truncated linear problem, a high-regularity energy method, and a finite-codimension topological argument to produce finite-time blow-up on a codimension-one manifold of initial data, with well-controlled self-similar profiles $(ar U,ar S)$. This work broadens the scope of imploding singularities beyond radial symmetry and provides a blueprint for extending stability analyses to angular perturbations in fluid PDEs, potentially impacting understanding of wave turbulence and norm inflation phenomena in periodic settings.

Abstract

In this paper we construct smooth, non-radial solutions of the compressible Euler and Navier-Stokes equation that develop an imploding finite time singularity. Our construction is motivated by the works [Merle, Raphaël, Rodnianski, and Szeftel, Ann. of Math., 196(2):567-778, 2022, Ann. of Math., 196(2):779-889, 2022], [Buckmaster, Cao-Labora, and Gómez-Serrano, arXiv:2208.09445, 2022], but is flexible enough to handle both periodic and non-radial initial data.

Non-radial implosion for compressible Euler and Navier-Stokes in $\mathbb{T}^3$ and $\mathbb{R}^3$

TL;DR

The paper advances the theory of singularity formation in compressible flows by constructing smooth, non-radial imploding solutions to the Euler and Navier–Stokes equations on both and . It develops a robust linear stability framework around self-similar radial profiles, employing dissipativity, maximality, and a novel angular repulsivity mechanism via vector spherical harmonics to handle non-radial perturbations. The nonlinear analysis combines a truncated linear problem, a high-regularity energy method, and a finite-codimension topological argument to produce finite-time blow-up on a codimension-one manifold of initial data, with well-controlled self-similar profiles . This work broadens the scope of imploding singularities beyond radial symmetry and provides a blueprint for extending stability analyses to angular perturbations in fluid PDEs, potentially impacting understanding of wave turbulence and norm inflation phenomena in periodic settings.

Abstract

In this paper we construct smooth, non-radial solutions of the compressible Euler and Navier-Stokes equation that develop an imploding finite time singularity. Our construction is motivated by the works [Merle, Raphaël, Rodnianski, and Szeftel, Ann. of Math., 196(2):567-778, 2022, Ann. of Math., 196(2):779-889, 2022], [Buckmaster, Cao-Labora, and Gómez-Serrano, arXiv:2208.09445, 2022], but is flexible enough to handle both periodic and non-radial initial data.
Paper Structure (26 sections, 37 theorems, 475 equations, 6 figures)

This paper contains 26 sections, 37 theorems, 475 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Historical Background
  3. Main result
  4. Setup of the problem
  5. Structure of the proof of Theorems \ref{['th:periodic']} and \ref{['th:euclidean']}
  6. Organization of the paper
  7. Notation
  8. Acknowledgements
  9. Study of the linearized operator
  10. Dissipativity
  11. Term $\mathcal{J}_{1, i}$
  12. Term $\mathcal{}{J}_3$
  13. Terms $\mathcal{} J_{2, i}$
  14. Term $\mathcal{} J_{4}$
  15. Conclusion
  16. ...and 11 more sections

Key Result

Theorem 1.2

Let $\nu = 1$. Let $\overline U, \overline S$ be self-similar profiles solving eq:ss_profiles and satisfying eq:profiles_positive--eq:angular_repulsivity, for some $r$ in the ranges eq:range_r, eq:condition_for_treating_dissipation. Let $T > 0$ sufficiently small, and $L > 0$ sufficiently large. The Moreover, there exists a finite codimension set of initial data satisfying the above conclusions (s

Figures (6)

  • Figure 1: The graph of $J(1-\chi_1)$ and $\chi_2$.
  • Figure 2: Phase portrait of the profile for $\gamma = 5/3$, $r=1.15$
  • Figure 3: Phase portrait to the left of $P_s$ for $\gamma = 5/3$ and $r = 1.05$. In green, the curve $(p_W(Z), Z)$ (that is, $N_W = 0$). The black curve represents $N_Z = 0$ and the red line represents $D_Z = 0$.
  • Figure 4: Phase portrait of the profile for $\gamma = 5/3$, $r=1.34$. The red triangle corresponds to the triangle where $D_Z > 0$, $W < W_0$, $W-Z > 0$. The black curve corresponds to $N_Z = 0$ and the brown curve to $\Xi_1 = 0$. The green line represents $U = U(\overline P_s)$. The quadrilateral $Q$ corresponds to the area of the picture that is located to the upper-right part of the green segment. Thus, it has three red sides and one green side, and $\overline P_s$, $P_s$ are two of its vertices.
  • Figure 5: Phase portrait of the profile for $\gamma = 5/2$, $r=1.20$. We are plotting the region corresponding to $W - Z > 0$ and $D_Z > 0$, with the boundary in red. The orange region corresponds to $U > U(P_s)$ and the blue to $U < U(P_s)$. The black curve corresponds to $N_Z = 0$. The brown curve corresponds to $\Xi_2 = 0$, and we want to show the solution stays in the region with $S$ below the brown curve (above the right branch of the brown curve). The green curves represent $N_W = 0$ and $U = U(\overline P_s)$.
  • ...and 1 more figures

Theorems & Definitions (77)

  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • proof
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Definition 1.7
  • Proposition 1.8
  • Lemma 2.1
  • ...and 67 more