Local regularity and finite time singularity for the generalized SQG equation on the half-plane

We show that the generalized surface quasigeostrophic (SQG) equation with α∈(0,1 4] is locally well-posed on the half-plane in spaces of bounded integrable locally Lipschitz functions that are natural for its dynamic on domains with boundaries, and allow for some power growth of the derivative in the normal direction at the boundary. We also show existence of solutions with smooth initial data that exhibit finite time blowup in the whole local well-posedness parameter regime α∈(0,1 4]. These are the first general local well-posedness results on a domain with a boundary, and the first finite time singularity result, for equations of this type (as opposed to patch models). Moreover, we prove sharpness of both results by showing ill-posedness of the PDE in the above spaces when α>1 4.