Attractors for Non-compact Autonomous and Non-autonomous Systems via Energy Equations

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 number of weakly damped hyperbolic equations and parabolic equations on either bounded or unbounded spatial domains is presented. A suitable extension of this approach to the existence of the uniform attractor for non-autonomous equations is also formulated. As examples, three specific and physically relevant problems are considered, namely the flows of a second-grade fluid, the flows of a Newtonian fluid in an infinite channel past an obstacle, and a non-autonomous weakly damped, forced Korteweg-deVries equation on the whole line.