On well/ill-posedness for the generalized surface quasi-geostrophic equation in Hölder spaces

We establish the well/ill-posedness theories for the inviscid α -surface quasi-geostrophic ( α -SQG) equation in Hölder spaces, where α = 0 and α = 1 correspond to the two-dimensional Euler equation in the vorticity formulation and SQG equation of geophysical significance, respectively. We first prove the local-in-time well-posedness of α -SQG equation in L ∞ ( [ 0 , T ] ; C 0 , β ( R 2 ) ) with β ∈ ( α , 1 ) for some T > 0 . We then analyze the strong ill-posedness in C 0 , α ( R 2 ) constructing smooth solutions to the α -SQG equation that exhibit C 0 , α –norm growth in a short time. In particular, we develop the nonexistence theory for α -SQG equation in C 0 , α ( R 2 ) .