Global Solutions of Evolutionary Faddeev Model With Small Initial Data

TL;DR

This paper proves global well-posedness for the evolutionary Faddeev model with small initial data, addressing complex nonlinear wave equations with null structure in high-dimensional Minkowski space.

Contribution

It establishes the first global existence result for the Faddeev model with small initial data in Sobolev spaces, handling semi-linear and quasi-linear nonlinearities.

Findings

Global well-posedness for small initial data

Handling of null structure in nonlinearities

Existence in Sobolev spaces

Abstract

We consider the Cauchy problem for evolutionary Faddeev model corresponding to maps from the Minkowski space $\mathbb{R}^{1 + n}$ to the unit sphere $\mathbb{S}^2$, which obey a system of non-linear wave equations. The nonlinearity enjoys the null structure and contains semi-linear terms, quasi-linear terms and unknowns themselves. We prove that the Cauchy problem is globally well-posed for sufficiently small initial data in Sobolev space.