Two-Front Solutions of the SQG Equation and its Generalizations

TL;DR

This paper derives contour-dynamics equations for two-front solutions of the generalized SQG equations, analyzes their stability, and proves local existence and uniqueness of solutions depending on the parameter lpha.

Contribution

It introduces a new contour-dynamics framework for two-front solutions of GSQG and establishes stability and well-posedness results across different lpha regimes.

Findings

Derived contour-dynamics equations for two-front solutions.

Analyzed linearized stability of shear flows with two flat fronts.

Proved local existence and uniqueness of solutions for different lpha ranges.

Abstract

The generalized surface quasi-geostrophic (GSQG) equations are transport equations for an active scalar that depend on a parameter $0<\alpha \le 2$. Special cases are the two-dimensional incompressible Euler equations ($\alpha = 2$) and the surface quasi-geostrophic (SQG) equations ($\alpha = 1$). We derive contour-dynamics equations for a class of two-front solutions of the GSQG equations when the fronts are a graph. Scalar reductions of these equations include ones that describe a single front in the presence of a rigid, flat boundary. We use the contour dynamics equations to determine the linearized stability of the GSQG shear flows that correspond to two flat fronts. We also prove local-in-time existence and uniqueness for large, smooth solutions of the two-front equations in the parameter regime $1<\alpha\le 2$, and small, smooth solutions in the parameter regime $0<\alpha\le 1$.