On the Theory of Global Attractors and Lyapunov Functionals

From the point of view of longterm dynamics, we study multivalued and single-valued semigroups of operators acting on complete metric spaces. We provide necessary and sufficient conditions for the existence of the global attractor under minimal requirements in terms of continuity of the semigroup. In the case of single-valued semigroups possessing a Lyapunov functional, we exhibit a simple proof of the existence and the characterization of the attractor in terms of the unstable set of stationary points. As an application, we consider the multivalued semigroup generated by the equation ruling the evolution of the specific humidity in a system of moist air, and we prove the existence of a regular global attractor.