Regularity of Wave-Maps in Dimension 2 + 1

Abstract

Abstract:In this article we prove a Sacks-Uhlenbeck/Struwe type global regularity result for wave-maps : R 2+1 → M into general compact target manifolds M.

Local Well-Posedness of the (4 + 1)-Dimensional Maxwell–Klein–Gordon Equation at Energy Regularity

“…This theorem is analogous to the threshold theorem for energy critical wave maps [18,29,30,[33][34][35][36][37]. The main result of this paper is the following local well-posedness theorem for (MKG) in the global Coulomb gauge at the energy regularity.…”

“…Taking the contrapositive, we see that any finite time blow up of a solution to (MKG) must be accompanied by energy concentration at a point. In [22,23], following the scheme successfully developed by one of the authors (D. Tataru) and J. Sterbenz in the context of energy critical wave maps [29,30], we establish global well-posedness of (MKG) for finite energy data by showing that such a phenomenon cannot occur. We refer to the last and the main paper of the sequence [23] for an overview of the entire series.…”

Scattering threshold for the focusing nonlinear Klein–Gordon equation

“…p C 2 D 2 ? in (1-2)); see [Akahori and Nawa 2010;Côte et al 2008;Duyckaerts et al 2008;Killip et al 2008;2009;Krieger and Schlag 2009;Sterbenz and Tataru 2010;Tao 2008a;2008b;2008c;2009a;2009b].…”

On Nash’s unique contribution to analysis in just three of his papers

“…In the context of Sobolev spaces these are the usual dot spacesẆ s,p spaces. 26 See [90,91], [58], and [94,95] for wave maps, [80,81] for Maxwell-Klein-Gordon, and [57] for Yang-Mills, as well as the references within.…”