Vol. 6, No. 3, 2013

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Smoothing and global attractors for the Zakharov system on the torus

Mehmet Burak Erdoğan and Nikolaos Tzirakis

Vol. 6 (2013), No. 3, 723–750

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.

Zakharov system, global attractor, smoothing, periodic boundary conditions
Mathematical Subject Classification 2010
Primary: 35Q55
Received: 23 February 2012
Revised: 19 June 2012
Accepted: 21 July 2012
Published: 11 July 2013
Mehmet Burak Erdoğan
Department of Mathematics
University of Illinois at Urbana-Champaign
Urbana, IL 61801
United States
Nikolaos Tzirakis
Department of Mathematics
University of Illinois at Urbana-Champaign
Urbana, IL 61801
United States