Evolution Equations and Dynamical Systems (M. Conti, F. Dell'Oro, M. Grasselli, E. LaengE. MarchiniV. Pata)

The main focus is the longterm analysis of evolution equations modelling dissipative processes, mainly concerned with fluid dynamics, wave propagation, viscoelasticity, population dynamics, phase transitions, heat conduction and other reaction-diffusion phenomena. This research line essentially belongs to the theory of infinite-dimensional dynamical systems, whose peculiarity lies in the extension of the geometric approach to PDEs, viewed as ODEs on suitable infinite-dimensional phase spaces. This generalization, although introduces some technical difficulties as the lack of a simple characterization of compactness, has led to a better understanding of the evolution of complex phenomena, typically featuring a nontrivial and often chaotic asymptotic behavior. Among such phenomena, those exhibiting dissipation mechanisms are particularly relevant from the physical, chemical or biological viewpoint. Mathematically speaking, dissipation translates into the existence of small regions of the phase space capturing the trajectories of the corresponding dynamical system in the long time.

In the last years, the theory of dissipative dynamical systems has made remarkable progresses. Nowadays, sophisticated techniques allow to handle in full generality quite challenging issues. At the same time, by means of new theoretical tools, the analysis of attractors and their stability under perturbations has been extended to more and more complex models. In these recent achievements, our research unit has played a significant role, with original contributions covering a large variety of topics: abstract results on the existence and regularity of global attractors, finite-dimensional reduction of the longterm dynamics, robustness of exponential attractors with respect to (possibly singular) perturbations; convergence to equilibria of single trajectories. In this context, an important research line concerns with evolution equations with memory, describing many relevant physical phenomena whose evolutions are influenced by the past values of the variables.

Politecnico di Milano - Dipartimento di Matematica