On the 2d Zakharov system with L^2 Schr\"odinger data

TL;DR

This paper establishes local well-posedness for the 2D Zakharov system with large initial data in the optimal L^2-based Sobolev spaces, highlighting the natural setting related to the cubic nonlinear Schrödinger equation and the subsonic limit.

Contribution

It proves the well-posedness of the 2D Zakharov system at optimal regularity, with existence time depending solely on initial data norms, and demonstrates the sharpness of these results.

Findings

Well-posedness in L^2 x H^{-1/2} x H^{-3/2} spaces

Existence time depends only on initial data norms

Results are sharp due to blow-up solutions by Glangetas-Merle

Abstract

We prove local in time well-posedness for the Zakharov system in two space dimensions with large initial data in L^2 x H^{-1/2} x H^{-3/2}. This is the space of optimal regularity in the sense that the data-to-solution map fails to be smooth at the origin for any rougher pair of spaces in the L^2-based Sobolev scale. Moreover, it is a natural space for the Cauchy problem in view of the subsonic limit equation, namely the focusing cubic nonlinear Schroedinger equation. The existence time we obtain depends only upon the corresponding norms of the initial data - a result which is false for the cubic nonlinear Schroedinger equation in dimension two - and it is optimal because Glangetas-Merle's solutions blow up at that time.