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Attractors for non-compact semigroups via energy equations. (English) Zbl 0914.35023

Summary: The energy-equation approach used to prove the existence of the global attractor by establishing the so-called asymptotic compactness property of the semigroup is considered, and a general formulation that can handle a numher of weakly damped hyperbolic equations and parabolic equations on either bounded or unbounded spatial domains is presented. As examples, three specific and physically relevant problems we considered, namely the flows of a second-grade fluid, the flows of a Newtonian fluid in an infinite channel past an obstacle, and a weakly damped, forced Korteweg-de Vries equation on the whole line.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35Q30 Navier-Stokes equations
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
35Q53 KdV equations (Korteweg-de Vries equations)
76A05 Non-Newtonian fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
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