Global variational methods for nonlinear evolution

Nonlinear evolution problems present a variety of challenging mathematical questions rang- ing from well-posedness, to qualitative behavior of solutions, to approximation. Variational methods bear a paramount importance in this context by providing novel formulations and efficient tools for modeling, analysis, and simulations. We propose a research project centered on a global variational approach, the Weighted Inertia-Dissipation-Energy (WIDE) principle, for investigating current issues in nonlinear evolution. The WIDE principle allows to reformulate a remarkable variety of evolution systems of applicative relevance in terms of a minimization plus a limit passage. This re- formulation paves the way for tackling evolution problems with techniques of the Calculus of Variations such as the Direct Method, Gamma-convergence, and relaxation. We aim to apply the WIDE principle to frontier themes in evolution PDEs. More precisely, we will address nonconvex evolution problems, focusing both on well-posedness and on approximability results. Moreover, we will exploit the WIDE principle for modeling and solving existence issues in dynamic problems in materials, with the specific aim of addressing dynamic fracture and damage.

The project has focused on the mathematical analysis of a range of time-dependent differential problems. The core of the project rests on a variational approach based on the minimization of global-in-time energies. This delivers a variational reformulation of the differential problem, paving the way to the use of the modern methods of the calculus of variations. In particular, it contributes a new approximation tool, with possible application to a variety of different systems. The outcome of the project is the proof of the viability of this variational approach in different contexts. On the more abstract side, we have successfully tackled classes of equations of so-called doubly nonlinear type, possibly also featuring nonconvex potentials. Apart from existence results, also in metric spaces, approximation and qualitative theories have been developed. On the more applied side, we have addressed problems from the continuum mechanics of solids and fluids. Most notably, the dynamic elastoplastic problem and the incompressible Navier-Stokes system have been solved by this variational method.