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

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.

We also find out that the incoming null (and almost geodesic) congruences diverge near the null infinity of membrane (rather than encounter with a "shockwave" at the null infinity) and this nonlinear effect is due to the curved geometry of membrane. Comments are welcome!

References in corpus (2)

• Stable shock formation for nearly simple outgoing plane symmetric waves
• The weak null condition and global existence using the p-weighted energy method

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• Global nearly-plane-symmetric solutions to the membrane equation
• On the critical exponent $p_c$ of the 3D quasilinear wave equation $-(1+({\partial }_t\varphi {)}^p){\partial }_t^2\varphi +\Delta \varphi =0$ with short pulse initial data. I, global existence
• Global stability of the plane wave solutions to the relativistic string with non-small perturbations