Global Solutions of a Surface Quasi-Geostrophic Front Equation

TL;DR

This paper establishes local and global existence results for a nonlinear, nonlocal surface quasi-geostrophic front equation in one dimension, under certain smoothness and smallness conditions on initial data.

Contribution

It provides the first rigorous analysis of local and global solutions for a spatially nonlocal SQG front equation with specific convergence conditions.

Findings

Unique local smooth solutions under convergence conditions.

Global solutions for sufficiently smooth and small initial data.

Conditions for existence and uniqueness of solutions.

Abstract

We consider a nonlinear, spatially-nonlocal initial value problem in one space dimension on $\mathbb{R}$ that describes the motion of surface quasi-geostrophic (SQG) fronts. We prove that the initial value problem has a unique local smooth solution under a convergence condition on the multilinear expansion of the nonlinear term in the equation, and, for sufficiently smooth and small initial data, we prove that the solution is global.