Smoothing and Global Attractors for the Zakharov System on the Torus

Details Smoothing and Global Attractors for the Zakharov System on the Torus Smoothing and Global Attractors for the Zakharov System on the Torus

2012-02-23

arxiv.org/abs/1202.5268v1

Mathematics

Analysis of PDEs

32 pages

Scientific paper

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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