In this paper we consider the Zakharov system with periodic boundary conditions in dimension one. In the first part of the paper, it is shown that for fixed initial data in a Sobolev space, the difference 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.

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Let p a e be the Linnik density, that is, the probability density with the characteristic functionThe following problem is studied: Let (a, 0), (P,!?) be two points of PD. When is it possible to represent Pfi.t as a scale mixture of p ae l A subset of the admissible pairs (a, 0), (fi, #) is described.

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