Non-existence and Strong lll-posedness in C^k, for the Generalized Surface Quasi-geostrophic Equation

Abstract We consider solutions to the generalized Surface Quasi-geostrophic equation (γ -SQG) when the velocity is more singular than the active scalar function (i. e. (0, 1) γ ∈ (0, 1) ). In this paper we establish strong ill-posedness in C^k, C k, β (k 1 k ≥ 1, (0, 1] β ∈ (0, 1 ] and k+ >1+ k + β > 1 + γ) and we also construct solutions in R² R 2 that initially are in C^k, L² C k, β ∩ L 2 but are not in C^k, C k, β for t>0 t > 0. Furthermore these solutions stay in H^k+ +1-2 H k + β + 1 - 2 δ for some small δ and an arbitrarily long time.