Global well-posedness for radial extremal hypersurface equation in $\left(1+3 \right)$-dimensional Minkowski space-time in critical Sobolev space

TL;DR

This paper establishes the global well-posedness of the radial extremal hypersurface equation in a critical Sobolev space within 1+3 dimensional Minkowski space-time, using novel analytical techniques.

Contribution

It introduces a new div-curl type lemma and combines it with energy and momentum balances to prove global well-posedness in the critical Sobolev space.

Findings

Proved global well-posedness for the radial extremal hypersurface equation.

Developed a new div-curl type lemma for this context.

Derived space-time estimates of the nonlinearity.

Abstract

In this article, we prove the global well-posedness in the critical Sobolev space $H_{rad}^2\left(\mathbb{R}^2\right) \times H_{rad}^1 \left(\mathbb{R}^2\right)$ for the radial time-like extremal hypersurface equation in $\left(1+3\right)$- dimensional Minkowski space-time. This is achieved by deriving a new div-curl type lemma and combined it with energy and ``momentum" balance law to get some space-time estimates of the nonlinearity.