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

TL;DR

This paper demonstrates that linear instability in the supercritical regime of the surface quasi-geostrophic equation causes nonlinear instability, establishing new results on well-posedness and instability in supercritical dissipation settings.

Contribution

It provides the first large-data supercritical well-posedness result with sharp regularity assumptions and links linear instability to nonlinear instability in this context.

Findings

Linear instability implies nonlinear instability for steady states.

Established global well-posedness for the forced equation in supercritical regimes.

Proved stronger instability results in logarithmically supercritical dissipation.

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.