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

TL;DR

This paper investigates the well-posedness and ill-posedness of the inviscid α-surface quasi-geostrophic equations in Hölder spaces, establishing local existence in certain spaces and demonstrating norm growth and nonexistence in critical Hölder spaces.

Contribution

It provides the first local well-posedness results for α-SQG in Hölder spaces with exponent greater than α and proves strong ill-posedness and nonexistence in the critical space where the Hölder exponent equals α.

Findings

Proves local well-posedness in C([0,T);C^{0,β}) for β > α.

Shows solutions can exhibit rapid norm growth in C^{0,α} spaces.

Establishes nonexistence of solutions in the critical Hölder space C^{0,α}.

Abstract

We establish the well/ill-posedness theories for the inviscid $\alpha$-surface quasi-geostrophic ($\alpha$-SQG) equations in H\"older 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 $C^{0,\alpha}(\mathbb{R}^2)$.