Local and global well-posedness of wave maps on $\R^{1+1}$ for rough data

TL;DR

This paper establishes local and global well-posedness results for wave maps from 1+1 dimensional Minkowski space to an analytic manifold, even for large and rough initial data, extending previous work with new global existence results.

Contribution

It proves global existence for large, rough initial data in the scale-invariant norm and Sobolev spaces, advancing the understanding of wave maps in 1+1 dimensions.

Findings

Global existence for large data in the $ ext{L}^{1,1}$ norm.

Global existence in Sobolev spaces $H^s$ for $s > 3/4$.

Extension of previous results to rough initial data.

Abstract

We prove local and global existence from large, rough initial data for a wave map between 1+1 dimensional Minkowski space and an analytic manifold. Included here is global existence for large data in the scale-invariant norm $\dot L^{1,1}$, and in the Sobolev spaces $H^s$ for $s > 3/4$. This builds on previous work in 1+1 dimensions of Pohlmeyer, Gu, Ginibre-Velo and Shatah.