Smoothing and global attractors for the Zakharov system on the torus

Erdoğan, Burak; Tzirakis, Nikolaos

doi:10.2140/apde.2013.6.723

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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

DOI: 10.2140/apde.2013.6.723

Abstract

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

Mathematical Subject Classification 2010

Primary: 35Q55

Milestones

Received: 23 February 2012

Revised: 19 June 2012

Accepted: 21 July 2012

Published: 11 July 2013

Authors

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

Vol. 6, No. 3, 2013