2017/07/09 by Miguel Á. Alejo, Miguel A. Alejo, Claudio Muñoz +2
Mathematics · Physics and Astronomy · #35Q53 #37K10 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #FOS: Mathematics #Primary 37K15 #Secondary 35Q51 #math.AP #msc:35Q51 #msc:35Q53 #msc:37K10 #msc:37K15
paper · pdf · doi:10.48550/arxiv.1707.02595
12 pages; This is version 2. Some typos corrected and sections organized differently for ease reading
openalex publication_date 2017/07/09 · arxiv created 2017/08/28 · arxiv updated 2017/08/30 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
We study decay of small solutions of the Born-Infeld equation in 1+1 dimensions, a quasilinear scalar field equation modeling nonlinear electromagnetism, as well as branes in String theory and minimal surfaces in Minkowski space-times. From the work of Whitham, it is well-known that there is no decay because of arbitrary solutions traveling to the speed of light just as linear wave equation. However, even if there is no global decay in 1+1 dimensions, we are able to show that all globally small Hs+1× Hs, s>\frac12 solutions do decay to the zero background state in space, inside a strictly proper subset of the light cone. We prove this result by constructing a Virial identity related to a momentum law, in the spirit of works \citeKMM,KMM1, as well as a Lyapunov functional that controls the H1 × L2 energy.