Almost sharp nonlinear scattering in one-dimensional Born-Infeld equations arising in nonlinear Electrodynamics

摘要

We study decay of small solutions of the Born-Infeld equation in 1+1dimensions, a quasilinear scalar field equation modeling nonlinearelectromagnetism, as well as branes in String theory and minimal surfaces inMinkowski space-times. From the work of Whitham, it is well-known that there isno decay because of arbitrary solutions traveling to the speed of light just aslinear wave equation. However, even if there is no global decay in 1+1dimensions, we are able to show that all globally small $H^{s+1}\times H^s$,$s>\frac12$ solutions do decay to the zero background state in space, inside astrictly proper subset of the light cone. We prove this result by constructinga Virial identity related to a momentum law, in the spirit of works\cite{KMM,KMM1}, as well as a Lyapunov functional that controls the $\dot H^1\times L^2$ energy.