Well-posedness of the equation for the three-form field in the eleven dimensional supergravity

TL;DR

This paper proves the global well-posedness of the three-form field equations in eleven-dimensional supergravity for small initial data, using advanced mathematical techniques to analyze their nonlinear wave structure.

Contribution

It establishes the first rigorous proof of global existence and uniqueness for these equations in a fixed gauge setting with small initial data.

Findings

Global well-posedness for small data

Decay estimates for solutions

Use of Klainerman vector fields in supergravity context

Abstract

We analyze the equations for the three-form field - a system of semi-linear gauge-invariant wave equations which arises in the theory of eleven dimensional supergravity. We prove that the Cauchy problem is well-posed globally in time for the fixed-gauge version of the equation for a small compactly supported smooth data. We employ the method of Klainerman vector fields along with a finer analysis of the nonlinearity to establish an integrable decay in the energy estimate.