Alejandro Fernandez-Jimenez, University of Oxford, Aggregation-diffusion equations with saturation, Friday, January 31, 2025, time 11:00, Aula Seminari MOX - VI piano
Abstract:Abstract:
On this talk we will focus on the family of aggregation-diffusion equations $$ \frac{\partial \rho}{\partial t} = \mathrm{div}\left(\mathrm{m}(\rho) \nabla (U'(\rho) + V) \right). $$ Here, $\mathrm{m}(s)$ represents a continuous and compactly supported nonlinear mobility (saturation) not necessarily concave. $U$ corresponds to the diffusive potential and includes all the porous medium cases, i.e. $U(s) = \frac{1}{m-1} s^m$ for $m > 0$ or $U(s) = s \log (s)$ if $m = 1$. $V$ corresponds to the attractive potential and it is such that $V \geq 0$, $V \in W^{2, \infty}$. For this problem, we discuss: Existence using a suitable regularised approximation of the problem, we prove that the problem admits an $L^1$-contractive $C_0$-semigroup; $L^1$-local minimisers of the associated free-energy functional in the corresponding class of measures; and the long-time behaviour of the constructed solutions in view of its gradient flow structure. Furthermore, we observe saturation effects leading to ``freezing'' behavior, i.e. free boundaries at the saturation level. Finally, we explore the properties of a corresponding implicit finite volume scheme introduced by Bailo, Carrillo and Hu. The talk presents joint work with Prof. J.A. Carrillo and Prof. D. Gómez-Castro.

Giulia Meglioli, Bielefeld University, Gradient flow for a class of diffusion equations with Dirichlet boundary data, Wednesday, December 18, 2024, time 15:30, Aula Seminari - III Piano
Abstract:Abstract:
In the talk it will be presented a variational characterisation for a class of non-linear evolution equations with constant non-negative Dirichlet boundary conditions on a bounded domain as gradient flows in the space of non-negative measures. The relevant geometry is given by the modified Wasserstein distance introduced by Figalli and Gigli that allows for a change of mass by letting the boundary act as a reservoir. We give a dynamic formulation of this distance as an action minimisation problem for curves of non-negative measures satisfying a continuity equation in the spirit of Benamou-Brenier. Then we characterise solutions to non-linear diffusion equations with Dirichlet boundary conditions as metric gradient flows of internal energy functionals in the sense of curves of maximal slope. The topic has been addressed in a joint work with Matthias Erbar.

Francesca Gladiali, Università degli Studi di Sassari, Solutions of the torsion problem with many critical points in Riemannian manifolds, Friday, December 13, 2024, time 11:00, Aula Seminari - III Piano
Abstract:Abstract:
Given a complete Riemannian manifold (M,g) I will prove that, for any point p in M and for any integer n>2, there exists a family of sets containing p and a family of solutions to the torsion problem that admits at least n maximum points. Moreover the domains are close to be convex (in a suitable sense). The proof relies on similar results in R^d, for d>3. The talk is based on past and ongoing results involving M. Grossi and A. Enciso.

Filippo Dell'Oro, Politecnico di Milano, Vanishing viscosity limit for the compressible Navier-Stokes equations with non-linear density dependent viscosities, Friday, November 29, 2024, time 11:00, Aula seminari - III Piano
Abstract:Abstract:
In a three-dimensional bounded domain, we consider the compressible Navier-Stokes equations for a barotropic fluid with general non-linear density dependent viscosities and no-slip boundary conditions. A nonlinear drag term is added to the momentum equation. We establish two conditional Kato-type criteria for the convergence of the weak solutions to such a system towards the strong solution of the compressible Euler system when the viscosity coefficient and the drag term parameter tend to zero. Joint work with L. Bisconti and M. Caggio.

Marco Pozzetta, Politecnico di Milano, Isoperimetric inequalities on manifolds with curvature bounds, Friday, November 22, 2024, time 11:00, Aula Seminari - III Piano
Abstract:Abstract:
The celebrated Euclidean isoperimetric inequality provides a sharp estimate of the measure of the boundary of subsets of the Euclidean space in terms of their volume. This inequality is also rigid, as equality is achieved only by Euclidean balls. The study of sharp and rigid isoperimetric inequalities on Riemannian manifolds has been an active area of research over the past few decades and is closely connected to curvature bounds. In this introductory talk, we will review some classical and recent isoperimetric inequalities on classes of Riemannian manifolds with curvature bounded below. A possible unified approach to the proofs of such inequalities arises from the sharp concavity properties of the isoperimetric profile function. This latter result was first obtained on compact manifolds by C. Bavard--P. Pansu and S. Gallot in the '80s, and it has recently been extended to the noncompact setting in a joint work with G. Antonelli, E. Pasqualetto, and D. Semola.

Danica Basaric, Politecnico di Milano, On well-posedness of systems arising from fluid dynamics, Friday, November 15, 2024, time 10:00, Aula Seminari - III Piano
Abstract:Abstract:
Well-posedness of systems describing the motion of compressible fluids in the class of strong and weak solutions represents one of the most challenging problems of the modern theory of PDEs. In the first part of the talk, we are going to define and construct a larger class of solutions, the measure-valued and dissipative ones, which unable us to handle the problem of existence for large times and large initial data; we will also discuss the possible advantages of considering this weaker notion of solution when solving some related problems arising from fluid dynamics. In the second part of the talk, we are going to show that, by performing a suitable selection, it is possible to select one "good" solution satisfying the semigroup property, even in the context when the system lacks uniqueness. After showing this procedure in an abstract setting, we will apply it to specific systems, such as the compressible Euler and Navier-Stokes ones.