Asymptotic properties of solutions of the Maxwell Klein Gordon equation with small data

Abstract

We prove peeling estimates for the small data solutions of the Maxwell Klein Gordon equations with non-zero charge and with a non-compactly supported scalar field, in (3 + 1) dimensions. We obtain the same decay rates as in an earlier work by Lindblad and Sterbenz, but giving a simpler proof. In particular we dispense with the fractional Morawetz estimates for the electromagnetic field, as well as certain spacetime estimates. In the case that the scalar field is compactly supported we can avoid fractional Morawetz estimates for the scalar field as well. All of our estimates are carried out using the double null foliation and in a gauge invariant manner.

On global dynamics of the Maxwell–Klein–Gordon equations

“…To tackle the general case with nonzero charge, Shu in [24] proposed a frame work but without details. A complete proof towards this direction was contributed by Lindblad-Sterbenz in [18], also see a recent work [3] of Bieri-Miao-Shahshahani. However all these works are restricted to the small data regime or can be viewed as global stability problems of trivial solutions.…”

On the global behavior of solutions of the Maxwell–Klein–Gordon equations

“…Details for general case were not carried out. A complete proof towards this program was later contributed by , also see a more recent work [3].…”

Small Data Global Existence and Decay for Relativistic Chern–Simons Equations

“…where (6.48) is again the contributions of the terms with (A) d not falling on (ι Z ⋆) 3 , and (6.49)-(6.50) arise from applying Lemma 5.10 (three times).…”

“…We remark that a gauge covariant vector field method was employed in the study of massless Maxwell-Klein-Gordon equation in R 1+3 as well; see [34,3]. Furthermore, a suitable version of this method proved to be useful in the small data global existence problem for the closely related Chern-Simons-Schrödinger equation in R 1+2 [41].…”

“…In order to overcome this difficulty, we develop a gauge covariant vector field approach [46,47,34,3] for relativistic Chern-Simons theories, which allows us to avoid the gauge potential in the analysis. An issue for executing this strategy is the anomalous commutation property of the Chern-Simons coupled Klein-Gordon equation; see (1.16) for Let R 1+2 denote the (2+1)-dimensional Minkowski space equipped with the metric…”