Non-existence and Strong lll-posedness in Ck,β for the Generalized Surface Quasi-geostrophic Equation

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