On the 2d Zakharov system with L^2 Schrödinger data

Details On the 2d Zakharov system with L^2 Schrödinger data On the 2d Zakharov system with L^2 Schrödinger data

2008-11-19

arxiv.org/abs/0811.3047v2

Nonlinearity 22 (2009) 1063-1089

Mathematics

Analysis of PDEs

30 pages, 2 figures. Minor revision. Title has been changed

Scientific paper

10.1088/0951-7715/22/5/007

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.

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