Non-existence and strong ill-posedness in $C^{k,\beta}$ for the generalized Surface Quasi-geostrophic equation

TL;DR

This paper demonstrates strong ill-posedness in certain Hölder spaces for the generalized Surface Quasi-geostrophic equation with highly singular velocity fields, and constructs solutions that lose regularity over time.

Contribution

It establishes strong ill-posedness results in $C^{k,eta}$ spaces for $ ext{gSQG}$ with $ heta$-dependent singular velocities and constructs solutions that degrade in regularity over time.

Findings

Proves strong ill-posedness in $C^{k,eta}$ for $ ext{gSQG}$ with $ heta$-more singular velocities.

Constructs solutions that start in $C^{k,eta}igcap L^2$ but exit $C^{k,eta}$ for $t>0$.

Solutions remain in a high Sobolev space for extended periods.

Abstract

We consider solutions to the generalized Surface Quasi-geostrophic equation ($\gamma$-SQG) when the velocity is more singular than the active scalar function (i.e. $\gamma\in(0,1)$). In this paper we establish strong ill-posedness in $C^{k,\beta}$ ($k\geq 1$, $\beta\in(0,1]$ and $k+\beta>1+\gamma$) and we also construct solutions in $\mathbb{R}^2$ that initially are in $C^{k,\beta}\cap L^2$ but are not in $C^{k,\beta}$ for $t>0$. Furthermore these solutions stay in $H^{k+\beta+1-2\delta}$ for some small $\delta$ and an arbitrarily long time.