Local Well-posedness of Two Dimensional SQG Equation and Related Models

TL;DR

This paper provides a new, elementary proof of local existence and uniqueness of solutions for the 2D SQG equation and related transport equations using contraction mapping, simplifying previous approaches.

Contribution

It introduces a straightforward proof technique for local well-posedness of 2D SQG and related models, enhancing understanding and accessibility.

Findings

Established local existence and uniqueness of classical solutions for 2D SQG

Extended the method to certain transport equations with nonlocal velocity

Simplified the proof process using contraction mapping

Abstract

In this paper, we present a new and elementary proof of the local existence and uniqueness of the classical solution to the Cauchy problem of the two-dimensional generalized surface quasi-geostrophic (SQG) equation via the method of the contraction mapping principle. Also, same result holds true for a kind of transport equation with nonlocal velocity via the method of the contraction mapping principle.