Smoothing and global attractors for the Zakharov system on the torus

Abstract

In this paper we consider the Zakharov system with periodic boundary con- ditions in dimension one. In the rst part of the paper, it is shown that for xed initial data in a Sobolev space, the dierence of the nonlinear and the linear evolution is in a smoother space for all times the solution exists. The smoothing index depends on a parameter distinguishing the resonant and nonresonant cases. As a corollary, we obtain polynomial-in-time bounds for the Sobolev norms with regularity above the energy level. In the second part of the paper, we consider the forced and damped Zakharov system and obtain analogous smoothing estimates. As a corollary we prove the existence and smoothness of global attractors in the energy space. u(x; 0) = u0(x)2 H s0 (T); n(x; 0) = n0(x)2 H s1 (T); nt(x; 0) = n1(x)2 H s1 1 (T);