On the Global Dynamics of Yang–Mills–Higgs Equations

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 R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^3$$\end{document}. The energy of the data could be infinite and the solution grows at most inverse polynomially in R-t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R-t$$\end{document} as t→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\rightarrow R$$\end{document}. As applications, we derive pointwise decay estimates for Yang–Mills–Higgs fields in the future of a hyperboloid or in the Minkowski space R1+3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^{1+3}$$\end{document} for data bounded in the weighted energy space with weights |x|1+ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|x|^{1+\epsilon }$$\end{document}. Moreover, for the abelian case of Maxwell–Klein–Gordon system, we extend the small data result of Lindblad and Sterbenz (IMRP Int Math Res Pap 109:1687-3017, 2006) 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 (Commun Math Phys 83(2):171–191, 1982a, 1982b) together with the geometric Kirchhoff–Sobolev parametrix constructed by Klainerman and Rodnianski (J Hyperbolic Differ Equ 4(3):401–433, 2007). The new ingredient is a class of weighted energy estimates through backward light cones adapted to the initial data.