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.
Keywords
Zakharov system, global attractor, smoothing, periodic
boundary conditions
