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Strong illposedness for SQG in critical Sobolev spaces

In-Jee Jeong, Junha Kim

Abstract

We prove that the inviscid surface quasi-geostrophic (SQG) equations are strongly ill-posed in critical Sobolev spaces: there exists an initial data $H^{2}(\bbT^2)$ without any solutions in $L^\infty_{t}H^{2}$. Moreover, we prove strong critical norm inflation for $C^\infty$--smooth data. Our proof is robust and extends to give similar ill-posedness results for the family of modified SQG equations which interpolate the SQG with two-dimensional incompressible Euler equations.

Strong illposedness for SQG in critical Sobolev spaces

Abstract

We prove that the inviscid surface quasi-geostrophic (SQG) equations are strongly ill-posed in critical Sobolev spaces: there exists an initial data without any solutions in . Moreover, we prove strong critical norm inflation for --smooth data. Our proof is robust and extends to give similar ill-posedness results for the family of modified SQG equations which interpolate the SQG with two-dimensional incompressible Euler equations.

Paper Structure

This paper contains 12 sections, 8 theorems, 224 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Well-posedness theory for SQG
  4. Generalized SQG equations
  5. Critical space illposedness for Euler
  6. Ingredients of the proof
  7. Strategy in the Euler case
  8. Difficulties in the SQG case
  9. Organization of the paper
  10. The Key Lemma
  11. Nonexistence
  12. Norm inflation for smooth data

Key Result

Theorem A

For any $\epsilon, \delta, A> 0$, there exists $\theta_{0} \in C^\infty(\mathbb T^2)$ satisfying $\Vert\theta_0\Vert_{H^2 \cap W^{1,\infty}} < \epsilon$ such that the unique local-in-time smooth solution $\theta$ to eq:SQG with initial data $\theta_{0}$ exists on $[0,\delta^*]$ for some $0<\delta^*

Theorems & Definitions (18)

  • Theorem A: Strong norm inflation
  • Theorem B: Nonexistence
  • Remark 1.1
  • Lemma 3.1: Hardy's inequality
  • proof
  • Lemma 3.2
  • Remark 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 8 more