Regularity Bounds on Zakharov System Evolutions

Abstract

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 ||u(t)||Hs ≤ C|t|(s - 1)+, where Hs is the standard spatial Sobolev norm. The proof is an adaptation of earlier work on the nonlinear Schrödinger equation which reduces matters to bilinear estimates.