Nonlinear evolution equations and applications

Nonlinear evolution models arise in several areas, like quantum Physics and phase-transition phenomena. The nonlinear feature of the equations, sometimes very strong, is due to the total or partial avoidance of linearization procedures. These, on the one hand, make the analytical problem more treatable while, on the other hand, decrease its efficacy since the model is a good approximation of the physical or chemical phenomenon only for a small range of data. Non-isothermal models became recently very interesting, in particular as far as existence, uniqueness and regularity of solutions are concerned. The physical consistency of the model may possibly lead to include singular potentials in the equations, like Dirac Delta or logarithmic free-energy potential, as well as dynamic boundary conditions. As a consequence, the mathematical analysis of the model is difficult from the very beginning, that is, from the local in time existence of a (weak) solution: this is the case in diffuse interface models, especially due to the coupling with the Navier-Stokes equations. Once this obstacle is overcome, the analysis of the longterm dynamics can be addressed. From this point of view, the knowledge of stationary states and bifurcation phenomena is relevant. In a more global perspective, when energy, heat or other type of dissipation occurs, one may try to devise a properly small set which at the same time represents the longterm dynamics, called global attractor.

 

[Ultimo aggiornamento: 04/02/2021 07:24:14]