Cristopher Hermosilla, Universidad Técnica Federico Santa María, Valparaíso - CHILE, A Minimality Property of the Value Function in Optimal Control over the Wasserstein Space, Tuesday, March 05, 2024, time 15:00, Aula seminari - III piano
Abstract:Abstract:
In this talk we study an optimal control problem with (possibly) unbounded terminal cost in the space of Borel probability measures with finite second moment. We consider the metric geometry associated with the Wasserstein distance, and a suitable weak topology rendering this space locally compact. In this setting, we show that the value function of a control problem is the minimal viscosity supersolution of an appropriate Hamilton-Jacobi-Bellman equation. Additionally, if the terminal cost is bounded and continuous, we show that the value function is the unique viscosity solution of the HJB equation.

Filippo Giuliani, Politecnico di Milano, Arbitrarily large growth of Sobolev norms for a quantum Euler system, Thursday, Febraury 15, 2024, time 15:00, Aula seminari MOX VI piano
Abstract:Abstract:
In this talk we present a result of existence of solutions to the quantum hydrodynamic (QHD) system, under periodic boundary conditions, which undergo an arbitrarily large growth of higher order Sobolev norms in polynomial times. The proof is based on the connection between the QHD system and the cubic NLS equation, provided by the Madelung transform. We show that the cubic NLS equation on the two dimensional torus possesses solutions which starts close to plane waves and undergo an arbitrarily large growth of higher order Sobolev norms in polynomial times. This is an improvement of the result by Guardia-Hani-Haus-Maserp-Procesi (JEMS 2023) and it is achieved by a refined normal form approach. Then we show that the existence of such solutions to NLS implies the existence of solutions to the QHD system exhibiting a large growth in Sobolev norms.

Antonino De Martino, Politecnico di Milano, Spectral theories on the S-Spectrum, Tuesday, December 12, 2023, time 15:15, Aula Seminari - III Piano
Abstract:Abstract:
One of the deepest results in hypercomplex analysis is the Fueter extension theorem. It gives a two steps procedure to extend holomorphic functions to the hyperholomorphic setting. The first step gives the class of slice hyperholomorphic functions; their Cauchy formula allows to define the so-called S-functional calculus for noncommuting operators based on the S-spectrum. In the second step, this extension procedure generates monogenic functions; the related monogenic functional calculus, based on the monogenic spectrum, was widely studied by McIntosh and collaborators. In this talk, I will discuss the main notions of the S-spectrum and some concepts of the monogenic functional calculus. Moreover, I will also give some ideas on the new research direction of the fine structures.

Luca Gennaioli, Scuola Internazionale Superiore di Studi Avanzati (SISSA), Asymptotics as s -> 0+ of the fractional perimeter on Riemannian manifolds, Thursday, October 26, 2023, time 14:15, Aula seminari MOX - VI piano
Abstract:Abstract:
In this work we study the asymptotics of the fractional Laplacian as s -> 0+ on any complete Riemannian manifold (M, g), both of finite and infinite volume. Surprisingly enough, when M is not stochastically complete this asymptotics is related to the existence of bounded harmonic functions on M. As a corollary, we can find the asymptotics of the fractional s-perimeter on (essentially) every complete manifold, generalizing both the existing results: the classical result for Rn by Dipierro-Figalli-Palatucci-Valdinoci (2012) and the recent one for the Gaussian space by Carbotti-Cito-La Manna-Pallara (2021). In doing so, from many sets E contained in M we are able to produce a bounded harmonic function associated to E, which in general can be non-constant.

Ángel Castro, Instituto de Ciencias Matemáticas (Madrid), Traveling waves near shear flows, Monday, July 03, 2023, time 15:15, Aula Seminari III piano (nell'ambito delle iniziative del Dipartimento di Eccellenza)
Abstract:Abstract:
In this talk we will consider the existence of traveling waves arbitrarily close to shear flows for the 2D incompressible Euler equations. In particular we shall present some results concerning the existence of such solutions near the Couette, Taylor-Couette and the Poiseuille flows. In the first part of the talk we will introduce the problem and review some well known results on this topic. In the second one some of the ideas behind the construction of our traveling waves will be sketched.

Luigi Berselli, Università di Pisa, Energy conservation or anomalous dissipation for incompressible fluids, Thursday, June 29, 2023, time 15:15, Aula Seminari III piano (nell'ambito delle iniziative del Dipartimento di Eccellenza)
Abstract:Abstract:
We provide an overview and propose elementary proofs of energy conservation for weak solutions to the Euler and Navier-Stokes equations in the class of Holder continuous functions. Our focus is on exploring the interplay between space and time regularity. Additionally, we delve into the potential extension of these results to the Navier-Stokes equations in the presence of a solid boundary. Specifically, we consider the case of Dirichlet boundary conditions and our approach avoids any additional assumptions on the kinematic pressure.