Bounds on the growth of high Sobolev norms of solutions to 2D Hartree Equations

In this paper, we consider Hartree-type equations on the two-dimensional torus and on the plane. We prove polynomial bounds on the growth of high Sobolev norms of solutions to these equations. The proofs of our results are based on the adaptation to two dimensions of the techniques we previously used to study analogous problems on $S^1$, and on $\mathbb{R}$.Comment: 38 page

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