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Self-similar blow-up solutions of incompressible Euler equations in $\mathbb R^d, d\geq 3$ with $C^{1,1-2/d-}$ velocity

Feng Shao, Dongyi Wei, Ping Zhang, Zhifei Zhang

Abstract

We investigate the axisymmetric incompressible Euler equations without swirl in $\mathbb R^d$ with $d\geq 3$. For any $α\in(0, α_d)$, where $α_d=1-2/d$, we construct a self-similar blow-up solution whose initial velocity fields satisfy $u_0\in C^{1,α}(\mathbb R^d)\cap C^\infty(\mathbb R^d\setminus\{0\})$. Our construction relies on a fixed-point framework formulated for the self-similar profile system, which takes the form of a coupled elliptic-transport system. Specifically, the transport equation recovers the vorticity profile from given data along characteristic curves, whereas the elliptic equation reconstructs the velocity field via Newtonian potentials defined in an auxiliary $(d+4)$-dimensional space. The main challenge lies in selecting suitable function spaces that remain invariant under such nonlinear compositions, while simultaneously capturing the exact singular behavior near the origin and symmetry axis.

Self-similar blow-up solutions of incompressible Euler equations in $\mathbb R^d, d\geq 3$ with $C^{1,1-2/d-}$ velocity

Abstract

We investigate the axisymmetric incompressible Euler equations without swirl in with . For any , where , we construct a self-similar blow-up solution whose initial velocity fields satisfy . Our construction relies on a fixed-point framework formulated for the self-similar profile system, which takes the form of a coupled elliptic-transport system. Specifically, the transport equation recovers the vorticity profile from given data along characteristic curves, whereas the elliptic equation reconstructs the velocity field via Newtonian potentials defined in an auxiliary -dimensional space. The main challenge lies in selecting suitable function spaces that remain invariant under such nonlinear compositions, while simultaneously capturing the exact singular behavior near the origin and symmetry axis.
Paper Structure (28 sections, 49 theorems, 446 equations, 1 figure)

This paper contains 28 sections, 49 theorems, 446 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Axisymmetric Euler without swirl
  3. Main result
  4. Organization of the paper.
  5. A road-map of the proof
  6. Self-similar formulation
  7. Functional spaces
  8. A fixed-point formulation
  9. Proof of Theorem \ref{['Thm.profile']}
  10. Existence of fixed-points
  11. Notations and conventions
  12. Analysis of the transport equation
  13. A coordinate transform and the characteristic geometry
  14. Upper bounds for $\Omega$
  15. Lower bound in the core region
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $d\in\mathbb N_{\geq 3}$ and $0<\alpha<\alpha_d:=1-2/d$. Then there is $\gamma_{*,0}>1$ such that for all $\gamma_*>\gamma_{*,0}$, the following statements hold: for each $T>0$, there exists a self-similar blow-up solution $u:[0, T)\times\mathbb R^d\to\mathbb R^d$ to Eq.Euler, and Moreover, for each $d,\alpha, \gamma_*$, there are uncountably many self-similar profiles $(\bm\omega_{\rm s}, u_

Figures (1)

  • Figure 1: The initial curve and characteristic curves for \ref{['Eq.Omega-rel-transport']}

Theorems & Definitions (99)

  • Theorem 1.1
  • Remark 2.1
  • Theorem 2.2: Existence of self-similar profiles
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5
  • Proposition 2.6
  • Lemma 2.7
  • Proposition 2.8
  • Definition 2.9
  • ...and 89 more