Global solutions of nonlinear wave–Klein–Gordon system in one space dimension

Abstract

In this article we will develop some techniques aimed at the strong couplings in twodimensional wave-Klein-Gordon system. We distinguish the roles of different type of decay factors and develop a method which permits us to "exchange" one type of decay into the other. Then a global existence result of a model problem is established. We also give a sketch of the Klein-Gordon-Zakharov model system and establish the associate global existence result. * The present work belongs to a research project "Global stability of quasilinear wave-Klein-Gordon system in 2 + 1 space-time dimension" (11601414), supported by NSFC.

Stability of Quasilinear Waves in $$1+1$$ Dimension Under Null Condition

“…In Lemma 3.1 there is a log or polynomial growth in the bounds of the L 2 norms. As far as we understand, such growth also exists in [24,17,18].…”

“…is only true for n ≥ 3, and the L 2 norm of the potential can be bounded by the conformal energy only when n ≥ 3, see the remark in [24]. The L 2 estimates (possibly with weight) for waves in R 2+1 were obtained in [24,17,18], but the compact support assumption on the initial data is required.…”

“…This lemma is of vital importance in the proof of Theorem 1.1. Recall that the L 2 norm estimates on solutions to wave equations in R 2+1 have been obtained before [24,17,18], but the compactness assumption on the initial data is needed (although the compactness assumption can be removed in the theorem in [24]). To the best of our knowledge, the estimates in Lemma 3.1 is the first such result where no compactness assumptions are imposed on the initial data.…”

“…Later on Zha [25] had a thorough study on a large class of wave equations in R 2+1 . We also remind one the study of the system of coupled quasilinear wave and Klein-Gordon equations in R 2+1 by Ma [19,20,18], using the hyperboloidal foliation method [14] which dates back to [12]. But the compactness assumption on the initial data is needed in all of the above results.…”

Nonlinear stability of the totally geodesic wave maps in non-isotropic manifolds

“…See [27] for an analysis on the massive Maxwell-Klein-Gordon system in the complement of a fixed light cone, and more recently, [28] on a quasilinear wave system in the entire space time. Our approach relies on a generalized hyperboloidal foliation introduced in [29] and [30](which is called "Euclidean-hyperboloidal foliation" therein) . This is a smooth combination of hyperboloidal foliation in the interior of the light cone {r ≤ t − 1} and flat foliation outside of the light cone {r ≥ t}.…”