Global well-posedness of the Chern-Simons-Higgs equations with finite energy

TL;DR

This paper establishes the global well-posedness of the Chern-Simons-Higgs equations with finite energy initial data in (2+1)-dimensional Minkowski space-time, extending previous results to less regular data using null structure and bilinear estimates.

Contribution

It improves existing results by proving global well-posedness for less regular initial data and introduces techniques leveraging null structure and bilinear estimates.

Findings

Proved global well-posedness for finite energy data.

Extended well-posedness to data below energy regularity.

Utilized null structure and bilinear estimates in the analysis.

Abstract

We prove that the Cauchy problem for the Chern-Simons-Higgs equations on the (2+1)-dimensional Minkowski space-time is globally well posed for initial data with finite energy. This improves a result of Chae and Choe, who proved global well-posedness for more regular data. Moreover, we prove local well-posedness even below the energy regularity, using the the null structure of the system in Lorenz gauge and bilinear space-time estimates for wave-Sobolev norms.