Attractors for noncompact nonautonomous systems via energy equations

Ioana Moise; Ricardo Rosa; Xiaoming Wang

doi:10.3934/dcds.2004.10.473

Article Contents

2004, Volume 10, Issue 1&2: 473-496. Doi: 10.3934/dcds.2004.10.473

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Attractors for noncompact nonautonomous systems via energy equations

1.
Department of Mathematics, University of Texas at Austin, Austin, TX 78712
2.
Department of Applied Mathematics, Federal University of Rio de Janeiro, Caixa Postal 68530, Rio de Janeiro, RJ 21945–970
3.
Department of Mathematics, Iowa State University, Ames, IA 50011

Received: August 31, 2002

Revised: February 28, 2003

Published: December 2003

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Abstract

An extension to the nonautonomous case of the energy equation method for proving the existence of attractors for noncompact systems is presented. A suitable generalization of the asymptotic compactness property to the nonautonomous case, termed uniform asymptotic compactness, is given, and conditions on the energy equation associated with an abstract class of equations that assure the uniform asymptotic compactness are obtained. This general formulation is then applied to a nonautonomous Navier-Stokes system on an infinite channel past an obstacle, with time-dependent forcing and boundary conditions, and to a nonautonomous, weakly damped, forced Korteweg-de Vries equation on the real line.

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Mathematics Subject Classification: Primary: 35B40, 35Q30, 35Q53, 76D05.

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