On well/ill-posedness for the generalized surface quasi-geostrophic equations in H\"older spaces

We establish the well/ill-posedness theories for the inviscid α-surface quasi-geostrophic (α-SQG) equations in H\"older 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 equations in C([0,T);C0,β(R²)) with β ∈ (α,1) for some $T>0$. We then analyze the strong ill-posedness in C0,α(R²) constructing smooth solutions to the α-SQG equations that exhibit C0,α--norm growth in a short time. In particular, we develop the nonexistence theory for α-SQG equations in C0,α(R²).