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Nonlinear instability for the surface quasi-geostrophic equation in the supercritical regime

Aynur Bulut, Hongjie Dong

Abstract

We consider the forced surface quasi-geostrophic equation with supercritical dissipation. We show that linear instability for steady state solutions leads to their nonlinear instability. When the dissipation is given by a fractional Laplacian, the nonlinear instability is expressed in terms of the scaling invariant norm, while we establish stronger instability claims in the setting of logarithmically supercritical dissipation. A key tool in treating the logarithmically supercritical setting is a global well-posedness result for the forced equation, which we prove by adapting and extending recent work related to nonlinear maximum principles. We believe that our proof of global well-posedness is of independent interest, to our knowledge giving the first large-data supercritical result with sharp regularity assumptions on the forcing term.

Nonlinear instability for the surface quasi-geostrophic equation in the supercritical regime

Abstract

We consider the forced surface quasi-geostrophic equation with supercritical dissipation. We show that linear instability for steady state solutions leads to their nonlinear instability. When the dissipation is given by a fractional Laplacian, the nonlinear instability is expressed in terms of the scaling invariant norm, while we establish stronger instability claims in the setting of logarithmically supercritical dissipation. A key tool in treating the logarithmically supercritical setting is a global well-posedness result for the forced equation, which we prove by adapting and extending recent work related to nonlinear maximum principles. We believe that our proof of global well-posedness is of independent interest, to our knowledge giving the first large-data supercritical result with sharp regularity assumptions on the forcing term.

Paper Structure

This paper contains 5 sections, 13 theorems, 175 equations.

Table of Contents

  1. Introduction and main results
  2. Local well-posedness and global perturbation for supercritical SQG
  3. Proof of Theorem \ref{['thm1']}: $(H^{2-\gamma},H^{2-\gamma})$ nonlinear instability for forced supercritical SQG
  4. Global well-posedness for log-supercritical SQG
  5. Proof of Theorem \ref{['thm2']}: $(H^\tau,L^2)$ nonlinear instability for forced log-supercritical SQG

Key Result

Theorem \oldthetheorem

Let $\gamma\in (0,1)$ be given and fix $f\in H^{3}(\mathbb{T}^2)$. Suppose that $\Theta_0\in H^{2-\gamma}(\mathbb{T}^2)$ is a solution to the steady-state supercritical SQG equation (eq-steady-supercritical) which is linearly unstable in the sense that the linearized operator $L_\gamma$ defined by (

Theorems & Definitions (24)

  • Definition \oldthetheorem: Lyapunov stability
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Proposition \oldthetheorem: Local well-posedness for supercritical SQG
  • Proposition \oldthetheorem: Local well-posedness for log-supercritical SQG
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • ...and 14 more