On the global dynamics of Yang-Mills-Higgs equations

TL;DR

This paper analyzes the global behavior of solutions to Yang-Mills-Higgs equations with possibly infinite energy data, providing decay estimates and extending small data results to large data in the Maxwell-Klein-Gordon system.

Contribution

It introduces gauge-independent weighted energy estimates and extends small data results to large data for the Maxwell-Klein-Gordon system.

Findings

Derived pointwise decay estimates for Yang-Mills-Higgs fields.

Extended small data results to large data for Maxwell-Klein-Gordon.

Developed gauge-independent weighted energy estimates using backward light cones.

Abstract

We study solutions to the Yang-Mills-Higgs equations on the maximal Cauchy development of the data given on a ball of radius $R$ in $\mathbb{R}^3$. The energy of the data could be infinite and the solution grows at most inverse polynomially in $R-t$ as $t\rightarrow R$. As applications, we derive pointwise decay estimates for Yang-Mills-Higgs fields in the future of a hyperboloid or in the Minkowski space $\mathbb{R}^{1+3}$ for data bounded in the weighted energy space with weights $|x|^{1+\epsilon}$. Moreover, for the abelian case of Maxwell-Klein-Gordon system, we extend the small data result of Lindblad and Sterbenz to general large data (under same assumptions but without any smallness). The proof is gauge independent and it is based on the framework of Eardley and Moncrief together with the geometric Kirchhoff-Sobolev parametrix constructed by Klainerman and Rodnianski. The new ingredient is a class of weighted energy estimates through backward light cones adapted to the initial data.