Wellposedness of inviscid SQG in the half-plane

We consider the SQG equation without dissipation on the half-plane with Dirichlet boundary condition, and prove local wellposedness in the spaces $W^{3,p}$ and $C^{2,\beta }$ for any $1<p<\mathrm{\infty }$ and $0<\beta <1$. We complement this wellposedness by showing that for generic $C_0^{\mathrm{\infty }}$ initial data, the unique corresponding solution does not belong to $W^{3,\mathrm{\infty }}$.