Local wellposedness for the 2+1 dimensional monopole equation

TL;DR

This paper proves local well-posedness for the 2+1 dimensional monopole equation in Coulomb gauge using null form analysis and bilinear estimates, for small initial data in low regularity Sobolev spaces.

Contribution

It establishes local well-posedness for the monopole equation in 2+1 dimensions at low regularity, employing null form structures and Wave-Sobolev space techniques.

Findings

Proves local well-posedness for initial data in H^s with s > 1/4.

Identifies null forms in the nonlinearities of the monopole equation.

Employs optimal bilinear estimates in Wave-Sobolev spaces.

Abstract

The space-time monopole equation on $\R^{2+1}$ can be derived by a dimensional reduction of the anti-self-dual Yang Mills equations on $\R^{2+2}$. It can be also viewed as the hyperbolic analog of Bogomolny equations. We uncover null forms in the nonlinearities and employ optimal bilinear estimates in the framework of Wave-Sobolev spaces. As a result, we show the equation is locally wellposed in the Coulomb gauge for initial data sufficiently small in $H^s$ for $s>{1/4}$.