45 research outputs found
Minimality properties of set-valued processes and their pullback attractors
We discuss the existence of pullback attractors for multivalued dynamical
systems on metric spaces. Such attractors are shown to exist without any
assumptions in terms of continuity of the solution maps, based only on
minimality properties with respect to the notion of pullback attraction. When
invariance is required, a very weak closed graph condition on the solving
operators is assumed. The presentation is complemented with examples and
counterexamples to test the sharpness of the hypotheses involved, including a
reaction-diffusion equation, a discontinuous ordinary differential equation and
an irregular form of the heat equation.Comment: 33 pages. A few typos correcte
Uniformly attracting limit sets for the critically dissipative SQG equation
We consider the global attractor of the critical SQG semigroup on the
scale-invariant space . It was shown in~\cite{CTV13} that
this attractor is finite dimensional, and that it attracts uniformly bounded
sets in for any , leaving open the
question of uniform attraction in . In this paper we prove
the uniform attraction in , by combining ideas from DeGiorgi
iteration and nonlinear maximum principles.Comment: 17 page