Regularized and Approximate Equations for Sharp Fronts in the Surface Quasi-Geostrophic Equation and its Generalizations

TL;DR

This paper derives regularized and approximate contour dynamics equations for sharp fronts in the SQG and gSQG equations, providing well-posedness results for short times and extending understanding of front evolution in these fluid models.

Contribution

It introduces a cubic approximation of the contour dynamics equations and proves well-posedness, advancing the mathematical analysis of sharp front dynamics in SQG and gSQG models.

Findings

Derived regularized contour dynamics equations for sharp fronts.

Proved short-time well-posedness of the approximate equations.

Established weak well-posedness for SQG fronts.

Abstract

We derive regularized contour dynamics equations for the motion of infinite sharp fronts in the two-dimensional incompressible Euler, surface quasi-geostrophic (SQG), and generalized surface quasi-geostrophic (gSQG) equations. We derive a cubic approximation of the contour dynamics equation and prove the short-time well-posedness of the approximate equations for generalized surface quasi-geostrophic fronts and weak well-posedness for surface quasi-geostrophic fronts.