On the Division Problem for the Wave Maps Equation

TL;DR

This paper proves global well-posedness and scattering for Wave Maps into spheres with small initial data in critical Besov spaces across multiple dimensions, using a novel atomic space approach.

Contribution

It introduces a new proof method for the division problem in Wave Maps, leveraging atomic bilinear estimates and algebra properties of the iteration space.

Findings

Established small data global well-posedness for Wave Maps.

Developed an adaptable atomic space framework for analysis.

Provided a systematic approach to the division problem in dispersive PDEs.

Abstract

We consider Wave Maps into the sphere and give a new proof of small data global well-posedness and scattering in the critical Besov space, in any space dimension $n \geq 2$. We use an adapted version of the atomic space $U^2$ as the single building block for the iteration space. Our approach to the so-called division problem is modular as it systematically uses two ingredients: atomic bilinear (adjoint) Fourier restriction estimates and an algebra property of the iteration space, both of which can be adapted to other phase functions.