Wellposedness of inviscid SQG in the half-plane

TL;DR

This paper establishes local well-posedness for the inviscid SQG equation on the half-plane in certain function spaces and demonstrates that solutions generally do not belong to a higher regularity space.

Contribution

It proves local well-posedness of the inviscid SQG equation in specific Sobolev and Hölder spaces and shows generic solutions lack higher regularity.

Findings

Well-posedness in $W^{3,p}$ and $C^{2,eta}$ spaces.

Generic solutions do not belong to $W^{3, abla}$ for higher regularity.

Provides insight into the regularity and uniqueness of solutions for the inviscid SQG equation.

Abstract

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 $1<p<\infty$ and $0<\beta<1$. We complement this wellposedness by showing that for generic $C^{\infty}_{0}$ initial data, the unique corresponding solution does not belong to $W^{3,\infty}$.