Analysis of evolutive partial differential equations

We are interested in the study of properties of solutions of partial derivative equations.
A first aspect is the study of the asymptotic behavior, for long times, of the solutions of some evolutionary PDEs with nonlinear diffusions. Particular attention is devoted to mean-curvature type diffusions, which find application, for example, in biophysics, chemical physics, population genetics and mathematics of ecology.
Another aspect concerns the study of the behavior of solutions of boundary problems for partial differential equations when, for example, the domain where the problem is defined, the boundary conditions, or the differential operator itself are perturbed.

We are interested in the study of partial differential equations of gradient flow type, with particular attention to equations that emerge in mathematical physics to describe the phenomenon of phase transitions (Allen-Cahn and Cahn-Hilliard equations). The study of such multiscale phenomena (fast-slow dynamics) is done mainly through the study of the energy associated to the system.

By means of potential theory, boundary problems for some differential operators (e.g., the Laplacian) can be reduced to integral equations by representing the solutions in terms of integral operators. We are interested in studying the properties of such integral operators and associated nonlinear operators. These results are then applied to the study of perturbation problems for boundary problems.

Numerical approximation of partial differential equations leads to a large system of ordinary differential equations which is computationally very expensive to approach. Therefore, we deal with algorithms that reduce the dimension of the problem by means of orthogonal projections. This reduction aims to reduce the computational cost and maintain a desired accuracy. One popular technique is e.g. Proper Orthogonal Decomposition. We also study algorithms to discover differential equations from experimental data.

We study the numerical approximation of optimal control problems for ordinary and partial differential equations. Specifically, we are interested in feedback control using dynamic programming equations (Hamilton-Jacobi-Bellman equations) and Model Predictive Control methods. The main focus is to develop efficient and accurate algorithms for large scale problems.