Norm Inflation For The Critical SQG Equation

TL;DR

This paper demonstrates that solutions to the critical SQG equation can exhibit norm inflation in certain Sobolev spaces despite the equation's global regularity, highlighting instability in the data-to-solution map.

Contribution

It constructs explicit solutions showing norm inflation at the critical regularity level and in supercritical Sobolev spaces, revealing instability phenomena in the SQG equation.

Findings

Solutions with large initial H^1 norm can experience norm inflation.

Small initial data in supercritical spaces can also lead to norm inflation.

The data-to-solution map is not uniformly bounded at the critical level.

Abstract

We consider the critical dissipative surface quasi-geostrophic (SQG) equation on $\mathbb{R}^2$ or $\mathbb{T}^2$. Despite global regularity of the equation, we show that the data-to-solution map at the critical level $H^1$ is not uniformly bounded. We construct solutions that experience $H^1$ norm inflation from smooth, compactly supported initial data with large $H^1$ norm. We also demonstrate small-data norm inflation in supercritical Sobolev spaces $W^{\beta,p}$ for $1<p<2$ and $1\le\beta<\tfrac{2}{p}$.