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Stationary self-similar profiles for the two-dimensional inviscid Boussinesq equations

Ken Abe, Daniel Ginsberg, In-Jee Jeong

Abstract

We consider ($-α$)-homogeneous solutions (stationary self-similar solutions of degree $-α$) to the two-dimensional inviscid Boussinesq equations in a half-plane. We show their non-existence and existence with both regular and singular profile functions.

Stationary self-similar profiles for the two-dimensional inviscid Boussinesq equations

Abstract

We consider ()-homogeneous solutions (stationary self-similar solutions of degree ) to the two-dimensional inviscid Boussinesq equations in a half-plane. We show their non-existence and existence with both regular and singular profile functions.

Paper Structure

This paper contains 55 sections, 44 theorems, 263 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Self-similar solutions
  3. The statement of the main results
  4. Self-similar Boussinesq solutions
  5. Self-similar Euler solutions
  6. Self-similar Pseudo-Beltrami solutions
  7. Self-similar irrotational solutions
  8. Desingularization of self-similar vortex sheets
  9. Relevant works on stratified flows: blow-up and stability
  10. The singularity formation
  11. Stability of static equilibrium
  12. Ideas of the proof
  13. The non-existence of homogeneous solutions with regular profiles
  14. The existence of homogeneous solutions with regular profiles
  15. The existence of homogeneous solutions with singular profiles
  16. ...and 40 more sections

Key Result

Theorem 1.2

The following holds for rotational ($-\alpha$)-homogeneous solutions to (1.1) and (1.4): (i) For $0\leq \alpha \leq 1$, no solutions $(u,p,\rho) \in C^{1}(\mathbb{R}^{2}\backslash \{0\})$ exist. (ii) For $-1/2\leq \alpha < 0$, no solutions $(u,p,\rho) \in C^{2}(\mathbb{R}^{2}\backslash \{0\})$ exist

Figures (11)

  • Figure 1: ($-\alpha$)-homogeneous solutions with regular profiles in Theorem 1.2
  • Figure 2: ($-\alpha$)-homogeneous solutions with singular profiles in Theorem 1.3 (the main result of this paper)
  • Figure 4: Forward self-similar weak solutions to 2D Euler and gSQG equations with scaling exponent $\alpha$
  • Figure 5: ($-\alpha$)-homogeneous Euler solutions with regular profiles in Theorem 1.5
  • Figure 6: ($-\alpha$)-homogeneous Euler solutions with singular profiles in Theorem 1.6
  • ...and 6 more figures

Theorems & Definitions (91)

  • Theorem 1.2: Rotational solutions with regular profiles
  • Theorem 1.3: Rotational solutions with singular profiles
  • Remark 1.4: $-1<\alpha<-1/2$
  • Theorem 1.5: 2D Euler solutions with regular profiles
  • Theorem 1.6: 2D Euler solutions with singular profiles
  • Theorem 1.7: Pseudo-Beltrami solutions with regular profiles
  • Theorem 1.8: Pseudo-Beltrami solutions with singular profiles
  • Theorem 1.9: Irrotational solutions with regular profiles
  • Theorem 1.10: Desingularization as $\alpha\to 1+0$
  • proof : Proof of Theorem 1.9
  • ...and 81 more