Global existence of smooth solution to relativistic membrane equation with large data

TL;DR

This paper proves the global existence of smooth solutions for the relativistic membrane equation with large initial data of short pulse type, revealing asymptotic geometric behavior and nonlinear effects at infinity.

Contribution

It establishes the global existence of smooth solutions for RME with large data using geometric multipliers and analyzes the asymptotic geometry at null infinity.

Findings

Global smooth solutions exist for large initial data.

Constructed multipliers adapted to membrane geometry.

Identified nonlinear expanding effects at infinity.

Abstract

This paper is concerned with the Cauchy problem for the relativistic membrane equation (RME) embedded in $\mathbb R^{1+(1+n)}$ with $n=2,3$. We show that the RME with a class of large (in energy norm) initial data admits a global, smooth solution. The initial data are given by the short pulse type, which is introduced by Christodoulou in his work on the formation of black holes [10]. Due to the quasilinear feature of RME, we construct two multipliers adapted to the geometry of membrane and present an efficient way for proving the global existence of smooth solution to the geometric wave equation with double null structure. We also derive the asymptotic geometry of the future null infinity and find out a nonlinear (expanding) effect at infinity.