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

Abstract

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];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 equation that exhibit C0,α–norm growth in a short time. In particular, we develop the nonexistence theory for α-SQG equation in C0,α(R2).