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.

Alberto Ferrero, Università del Piemonte Orientale, Stabilità e altre proprietà qualitative delle soluzioni di equazioni ellittiche non lineari su varietà Riemanniane, Monday, May 29, 2023, time 16:15, Aula Seminari III piano
Abstract:Abstract:
Questo seminario prende spunto dai risultati ottenuti in due lavori, il primo del 2014 in collaborazione con Elvise Berchio e Gabriele Grillo ed il secondo del 2023 in collaborazione ancora con Elvise Berchio e con Debdip Ganguly e Prasun Roychowdhury. Essi trattano le proprietà di stabilità per le soluzioni di equazioni ellittiche con non linearità di tipo potenza o di tipo esponenziale su varietà Riemanniane con un polo di simmetria. Prima di entrare nello specifico dei problemi trattati nei due lavori sopra citati, si procederà con un'introduzione sul significato in ambito astrofisico di tali equazioni nello spazio euclideo tridimensionale. Si passerà poi alla presentazione di alcuni modelli nell'ambito della fluidodinamica che si riconducono allo studio di varianti delle suddette equazioni su sfere dello spazio euclideo. Infine si passerà a trattare le equazioni su varietà Riemanniane essenzialmente con curvatura definitivamente negativa per grandi distanze dal polo di simmetria, tra le quali rientra come caso particolare lo spazio iperbolico; seguirà poi la trattazione dei principali risultati presenti sull'argomento in letteratura ed in particolare nei due articoli del sottoscritto.

Hynek Kovarik, Università degli Studi di Brescia, Resolvent expansion and time decay of the wave functions of magnetic Hamiltonians in dimension two, Thursday, May 25, 2023, time 14:30 o'clock, Aula Seminari III piano
Abstract:Abstract:
I this talk I will present some results on the resolvent expansions of magnetic Hamiltonians at the threshold of the essential spectrum. I will show, in particular, that the nature of the expansion of a two-dimensional magnetic Hamiltonian is completely determined by the flux of the associated magnetic field. Applications to time decay of the wave functions will be discussed as well.

Edoardo Bocchi, Politecnico di Milano, Asymmetric equilibrium configurations of a body immersed in a 2D laminar flow, Thursday, May 18, 2023, time 15:15, Aula Seminari III piano
Abstract:Abstract:
We will study the equilibrium configurations of a fluid-structure interaction problem where a body is immersed in a fluid confined in a bounded planar channel and governed by the stationary Navier-Stokes equations with laminar inflow and outflow. The body is subject to both the lift force from the fluid and to some external elastic force. Motivated by an application to suspension bridges, asymmetry is taken into account. This requires the introduction of suitable assumptions to prevent collisions of the body with the boundary. We will present an existence and uniqueness result for sufficiently small inflow/outflow. This talk is based on a recent joint work with F. Gazzola.