Enrico Laeng, Dipartimento di matematica, Politecnico di Milano, Successioni non nulle i cui infiniti momenti sono tutti nulli., Wednesday, March 01, 2023, Aula seminari, III piano, Dipartimento di Matematica, Politecnico di Milano
Abstract:Abstract:
Il teorema di Stone-Weierstrass implica che se tutti i momenti su un intervallo (integrali della funzione per una potenza intera della variabile) di una funzione continua sono nulli, allora anche tale funzione deve essere nulla. E` abbastanza ben noto che questo non e` piu` vero se prendiamo i momenti su una semiretta o su una retta. Noi affrontiamo il caso discreto, ovvero l'esistenza di successioni tali che le serie che hanno per addendi i termini della successione moltiplicati per una qualsiasi potenza intera dell'indice di convergono a zero grazie a un peculiare gioco di cancellazioni. E` un risultato generale che ha anche conseguenze per la teoria delle probabilita` perche` quantita` come media, varianza, curtosi, etc. sono legate ai momenti.

Bruno Volzone, Università degli Studi di Napoli 'Parthenope', Long-time behavior for local and nonlocal porous medium equations with small initial energy, Wednesday, Febraury 08, 2023, time 15:15, Aula Seminari III Piano
Abstract:Abstract:
In the first part of the talk, we will describe some aspects of a study developed in a joint paper with L. Brasco concerning the long-time behavior for the solution of the Porous Medium Equation in an open bounded connected set, with smooth boundary and sign-changing initial datum. Homogeneous Dirichlet boundary conditions are considered. We prove that if the initial datum has sufficiently small energy, then the solution converges to a nontrivial constant-sign solution of a sublinear Lane-Emden equation, once suitably rescaled. We also give a sufficient energetic criterion on the initial datum, which permits to decide whether convergence takes place towards the positive solution or to the negative one. The second part of the talk will be devoted to some new advances obtained in collaboration with G. Franzina, in the spirit of the ones explained above, for the study of the asymptotics of signed solutions for the Fractional Porous Medium Equation.

Simone Dovetta, Politecnico di Torino, Action versus energy ground states in nonlinear Schrödinger equations, Thursday, January 26, 2023, time 15:15, Aula Seminari III Piano
Abstract:Abstract:
The talk investigates the relation between normalized critical points of the nonlinear Schrödinger energy functional and critical points of the corresponding action functional on the associated Nehari manifold. First, we show that the ground state levels are strongly related by the following duality result: the (negative) energy ground state level is the Legendre–Fenchel transform of the action ground state level. Furthermore, whenever an energy ground state exists at a certain frequency, then all action ground states with that frequency have the same mass and are energy ground states too. We see that the converse is in general false and that the action ground state level may fail to be convex. Next we analyze the differentiability of the ground state action level and we provide an explicit expression involving the mass of action ground states. Finally we show that similar results hold also for local minimizers, and we exhibit examples of domains where our results apply. This is a joint work with Enrico Serra and Paolo Tilli.

Andrea Giorgini, Politecnico di Milano, Recent results for the Navier-Stokes-Cahn-Hilliard model with unmatched densities, Thursday, January 19, 2023, time 15:15, Aula Seminari III Piano
Abstract:Abstract:
We consider the initial-boundary value problem for the incompressible Navier-Stokes-Cahn-Hilliard system with non-constant density proposed by Abels, Garcke and Grün in 2012. This model arises in the diffuse interface theory for binary mixtures of viscous incompressible fluids. In particular, this system is a generalization of the well-known Model H in the case of fluids with unmatched densities. In this talk, I will present some recent results concerning the propagation of regularity of global weak solutions (for which uniqueness is not known) and their longtime convergence towards an equilibrium state in three dimensional bounded domains.

Andrea Signori, Politecnico di Milano, Chemotaxis model for tumour growth, Thursday, December 15, 2022, time 15:15, Aula Seminari III Piano
Abstract:Abstract:
We discuss analytic results for a new diffuse interface model describing the evolution of a tumour mass under the effects of a chemical substance (e.g., a nutrient). The process is described by utilising an order parameter representing the local proportion of tumour cells, and a variable describing the concentration of the chemical. The order parameter is assumed to satisfy a suitable form of the Cahn–Hilliard equation with mass source and logarithmic potential of Flory–Huggins type, whereas the chemical concentration satisfies a reaction-diffusion equation where the cross-diffusion term has the same expression as in the celebrated Keller–Segel model. Weak well-posedness, regularity, and continuous dependence results are presented. This is a joint work with E. Rocca (University of Pavia) and G. Schimperna (University of Pavia).

Andrea Giorgini, Politecnico di Milano, Recent results for the Navier-Stokes-Cahn-Hilliard model with unmatched densities, Thursday, December 01, 2022, time 15:15, Aula Seminari III Piano
Abstract:Abstract:
SI AVVISA CHE IL SEMINARIO IN OGGETTO È ANNULLATO PER INDISPONIBILITÀ DELLO SPEAKER