Attractors for noncompact nonautonomous systems via energy equations

An extension to the nonautonomous case of the energy equation method for proving the existence of attractors for noncompact systems is presented. A suitable gen- eralization of the asymptotic compactness property to the nonautonomous case, termed uniform asymptotic compactness, is given, and conditions on the energy equation associ- ated 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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