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Quadratic lifespan for the sublinear $α$-SQG sharp front problem

Riccardo Montalto, Federico Murgante, Stefano Scrobogna

Abstract

In this paper we consider the generalized surface quasi-geostrophic $α$-SQG equations, in the "sublinear regime" $α\in (0, 1)$ and we study the stability of vortex patches close to vortex discs. We shall prove that for regular, Sobolev initial vortex patches $\varepsilon$-close to a vortex disc, the solutions stay $\varepsilon$-close to a vortex disc for a time interval of order $O(\varepsilon^{- 2})$. The proof is based on a paradifferential Birkhoff normal form reduction, implemented in the case where the dispersion relation is sublinear.

Quadratic lifespan for the sublinear $α$-SQG sharp front problem

Abstract

In this paper we consider the generalized surface quasi-geostrophic -SQG equations, in the "sublinear regime" and we study the stability of vortex patches close to vortex discs. We shall prove that for regular, Sobolev initial vortex patches -close to a vortex disc, the solutions stay -close to a vortex disc for a time interval of order . The proof is based on a paradifferential Birkhoff normal form reduction, implemented in the case where the dispersion relation is sublinear.
Paper Structure (31 sections, 17 theorems, 127 equations)

This paper contains 31 sections, 17 theorems, 127 equations.

Table of Contents

  1. Presentation of the problem
  2. Results of the manuscript
  3. Ideas of the proof
  4. Energy estimate
  5. The Hamiltonian unknown
  6. The linearized equation
  7. Property of the dispersion relation
  8. Para-differential normal form
  9. Overview of the literature
  10. Foundational results
  11. Traveling waves
  12. Quasi-periodic solutions
  13. Normal forms for quasi-linear systems
  14. Further literature
  15. Background on the SQG patch problem as a contour equation
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $\alpha \in \left( 0, 1 \right)$. There exists $s_0 > 0$ such that for any $s \geq s_0$, there are $\varepsilon_0 > 0$, $c_{s,\alpha} > 0$, $C_{s,\alpha} > 0$ such that, for any $h_0$ in $H^s \left( \mathbb{T} ; \mathbb{R} \right)$ satisfying $\left\| h_0 \right\|_{H^s} \leq \varepsilon < \v satisfying $\left\| h\left( t \right) \right\|_{H^s} \leq C_{s, \alpha} \, \varepsilon$, for any $t

Theorems & Definitions (41)

  • Theorem 1.1: Quadratic life-span
  • Proposition 2.1
  • Remark 2.2: Symmetries and conservation laws
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6: Linearization of $\nabla H_{\alpha, \Omega}$ around zero
  • Definition 2.7
  • Lemma 2.8: Absence of three wave interactions
  • Definition 3.1: Symbols
  • ...and 31 more