On well-posedness of $\hat{I}\pm$-SQG equations in the half-plane

Abstract

We investigate the well-posedness of $\hat{I}\pm$-SQG equations in the half-plane, where $\hat{I}\pm =0$ and $\hat{I}\pm =1$ correspond to the 2D Euler and SQG equations respectively. For $0<\hat{I}\pm \le 1/2$, we prove local well-posedness in certain weighted anisotropic Hölder spaces. We also show that such a well-posedness result is sharp: for any $0<\hat{I}\pm \le 1$, we prove nonexistence of Hölder regular solutions (with the Hölder regularity depending on $\hat{I}\pm$) for initial data smooth up to the boundary.

20 pages, 1 figure