Regularity Bounds on Zakharov System Evolutions

Spatial regularity properties of certain global-in-time solutions of the Zakharov system are established. In particular, the evolving solution $u(t)$ is shown to satisfy an estimate \Hsup s {u(t)} \leq C {{|t|}^{(s-1)+}}, where $H^s$ is the standard spatial Sobolev norm. The proof is an adaptation of earlier work on the nonlinear Schr\"odinger equation which reduces matters to bilinear estimates.Comment: 10 page

Global well-posedness for KdV in Sobolev Spaces of negative index

The initial value problem for the Korteweg-deVries equation on the line is shown to be globally well-posed for rough data. In particular, we show global well-posedness for initial data in H^s({\mathbb{R}), -3/10<s.Comment: 5 pages. Electronic Journal of Differential equations (submitted

Weighted Low-Regularity Solutions of the KP-I Initial Value Problem

In this paper we establish local well-posedness of the KP-I problem, with initial data small in the intersection of the natural energy space with the space of functions which are square integrable when multiplied by the weight y. The result is proved by the contraction mapping principle. A similar (but slightly weaker) result was the main Theorem in the paper " Low regularity solutions for the Kadomstev-Petviashvili I equation " by Colliander, Kenig and Staffilani (GAFA 13 (2003),737-794 and math.AP/0204244). Ionescu found a counterexample (included in the present paper) to the main estimate used in the GAFA paper, which renders incorrect the proof there. The present paper thus provides a correct proof of a strengthened version of the main result in the GAFA paper