Global existence of smooth solution to evolutionary Faddeev model with short-pulse data
Shaoying Luo, Jinhua Wang, Changhua Wei
TL;DR
The paper proves global smooth solutions for the evolutionary Faddeev model with short-pulse initial data by blending null-structure aware energy estimates with geometry-adapted vector-field multipliers. It introduces leading-part multipliers $L+(L\phi)^2\underline{L}$ and $\underline{L}+(\underline{L}\phi)^2L$, and constructs $\tilde{L}$ and $\tilde{\underline{L}}$ to cancel top-order cross terms, enabling robust control of high-order derivatives via a bootstrap argument in a region II setting. A two-stage strategy is then employed: first propagate large-data region II global existence, then, using small incoming energy on the short-pulse boundary slice $C_{\delta}$, solve a small-data region I Goursat problem to extend global regularity. The results demonstrate global well-posedness for a broad class of large short-pulse data, highlighting the role of geometry-adapted energy methods in quasilinear wave systems with null structures. The approach offers a framework applicable to other quasilinear wave equations with similar null and double-null structures. The findings have potential implications for understanding nonlinear wave interactions in constrained field theories and related geometric PDEs.
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.