Global existence of smooth solution to evolutionary Faddeev model with short-pulse data

TL;DR

This paper proves the global existence of smooth solutions for the evolutionary Faddeev model with large short-pulse initial data, using energy estimates and geometry-adapted multipliers.

Contribution

It establishes the global well-posedness of the Faddeev model for large initial data of short pulse type, a significant extension beyond small data results.

Findings

Global smooth solutions exist for large short-pulse initial data.

Energy estimates and geometric multipliers are effective in controlling the system.

The null structure of nonlinearities is crucial for the analysis.

Abstract

This paper is concerned with the Cauchy problem of the evolutionary Faddeev model, a system that maps from the Minkowski space $\mathbb{R}^{1+3}$ to the unit sphere $\mathbb{S}^2$. The model is a system of nonlinear wave equations whose nonlinearities exhibit a null structure and include semilinear terms, quasilinear terms, and the unknowns themselves. By considering a class of large initial data (in energy norm) of the short pulse type, we prove that the evolutionary Faddeev model admits a globally smooth solution via energy estimates. The main result is achieved through the selection of appropriate multipliers that are specially adapted to the geometry of the system.