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February  2004, 10(1&2): 473-496. doi: 10.3934/dcds.2004.10.473

Attractors for noncompact nonautonomous systems via energy equations

Ioana Moise 1, , Ricardo Rosa 2, and  Xiaoming Wang 3,

1. 

Department of Mathematics, University of Texas at Austin, Austin, TX 78712, United States
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, United States

Received  September 2002 Revised  March 2003 Published  October 2003

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.
Keywords: energy equation, uniform attractors, uniform asymptotic compactness, Korteweg-de-Vries equation., nonautonomous equations, Navier-Stokes equations.
Mathematics Subject Classification: Primary: 35B40, 35Q30, 35Q53, 76D0.
Citation: Ioana Moise, Ricardo Rosa, Xiaoming Wang. Attractors for noncompact nonautonomous systems via energy equations. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 473-496. doi: 10.3934/dcds.2004.10.473
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