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

Abstract

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