Non existence and strong ill-posedness in $C^k$ and Sobolev spaces for SQG

TL;DR

This paper demonstrates the non-existence and strong ill-posedness of solutions to the surface quasi-geostrophic equations (SQG) in certain smooth and Sobolev spaces, highlighting limitations of well-posedness.

Contribution

It constructs solutions that lose regularity instantly and proves ill-posedness in the critical Sobolev space for SQG, advancing understanding of solution behavior.

Findings

Solutions in $C^k$ lose regularity immediately.

Ill-posedness established in $H^{2}$ space.

Finite energy solutions exhibit instant regularity loss.

Abstract

We construct solutions in $\mathbb{R}^2$ with finite energy of the surface quasi-geostrophic equations (SQG) that initially are in $C^k$ ($k\geq 2$) but that are not in $C^{k}$ for $t>0$. We prove a similar result also for $H^{s}$ in the range $s\in(\frac32,2)$. Moreover, we prove strong ill-posedness in the critical space $H^{2}$.