Local well-posedness for membranes in the light cone gauge

TL;DR

This paper establishes the local well-posedness of the initial value problem for bosonic membranes in the light cone gauge by transforming the system into a hyperbolic form using the area-preserving constraint.

Contribution

It introduces an equivalent hyperbolic system for membrane evolution equations that leverages the area-preserving constraint, enabling well-posedness and regularity analysis.

Findings

Successfully reformulated membrane equations into a hyperbolic system

Proved local well-posedness of the initial value problem

Derived a blowup criterion and improved regularity estimates

Abstract

In this paper we consider the classical initial value problem for the bosonic membrane in light cone gauge. A Hamiltonian reduction gives a system with one constraint, the area preserving constraint. The Hamiltonian evolution equations corresponding to this system, however, fail to be hyperbolic. Making use of the area preserving constraint, an equivalent system of evolution equations is found, which is hyperbolic and has a well-posed initial value problem. We are thus able to solve the initial value problem for the Hamiltonian evolution equations by means of this equivalent system. We furthermore obtain a blowup criterion for the membrane evolution equations, and show, making use of the constraint, that one may achieve improved regularity estimates.