Non-existence and Strong lll-posedness in $$C^{k,\beta }$$ for the Generalized Surface Quasi-geostrophic Equation

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 \(C^{k,\beta }\) (\(k\ge 1\), \(\beta \in (0,1]\) 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.