Global rough solutions for the Zakharov system in two spatial dimensions

TL;DR

This paper establishes an improved global well-posedness for the 2D Zakharov system with minimal regularity, allowing data with infinite energy, using a refined I-method and bilinear estimates.

Contribution

It extends global well-posedness results to rougher initial data for the Zakharov system in two dimensions, surpassing previous regularity limitations.

Findings

Global well-posedness with minimal regularity

Inclusion of data outside H^1 and L^2 spaces

Polynomial growth bound for solutions

Abstract

We show an improved global well-posedness result for the Zakharov system in two space dimensions with minimal regularity assumptions for the data. Especially we are able to allow Schroedinger and wave data, which do not belong to H^1 and L^2, respectively, thus with infinite energy. The proof uses a refined I-method originally initiated by Colliander, Keel, Staffilani, Takaoka and Tao and bilinear estimates by Bejenaru, Herr, Holmer and Tataru. A polynomial growth bound for the solution is also given.