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.