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

We establish the well/ill-posedness theories for the inviscid $\alpha$-surface quasi-geostrophic ($\alpha$-SQG) equations in Hölder spaces, where $\alpha =0$ and $\alpha =1$ 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 $\alpha$-SQG equations in $C([0,T);C^{0,\beta }({\mathbb{R}}^2))$ with $\beta \in (\alpha ,1)$ for some $T>0$. We then analyze the strong ill-posedness in $C^{0,\alpha }({\mathbb{R}}^2)$ constructing smooth solutions to the $\alpha$-SQG equations that exhibit $C^{0,\alpha }$–norm growth in a short time. In particular, we develop the nonexistence theory for $\alpha$-SQG equations in $C^{0,\alpha }({\mathbb{R}}^2)$.