A.A. 2022/23

Abstract

Perturbation of operators is a very rich branch of the modern analysis. The perturbation of Laplacian with inverse square potential arises, in particular, in the studies of non linear Schrödinger equation. So, in this talk, we want to give an explicit and easy characterization of the domain of the operator A=-Δ+b/|x|2. Such characterization is also usefull to determine the domain of A square in a similar way. The study of these operators gives to us a new proof of weighted perturbed Rellich inequalities. This talk is based on a joint work with V. Georgiev and H. Kubo.

Abstract

 Join remotely mercoledì 3 maggio 2023 alle ore 17:00:00 CEST Aula Seminari, Dipartimento di Matematica Masayuki Hayashi Università di Pisa Title: Travelling waves for a nonlinear Schrodinger system with quadratic interaction Abstract: Solitary waves (solitons, standing waves, traveling waves) are known to play a fundamental role in the analysis of nonlinear dispersive equations. First, we briefly explain nonlinear dispersive equations and solitary waves by using the nonlinear Schrodinger equation as an example. Next, as an introduction of recent results obtained by the speaker (jointly with N. Fukaya and T. Inui), we consider traveling waves of a nonlinear Schrodinger system with quadratic interaction. When the mass constants satisfy a certain condition, the system has no specific symmetry such as Galilean or pseudoconformal symmetry, which is of particular interest. We construct traveling wave solutions by variational methods, and also show global existence for oscillating large data as an application. We see that our results essentially come from the lack of Galilean invariance in the system. Finally, I will introduce a few open problems and interesting relevance to other types of nonlinear dispersive equations.

Abstract

The optimal matching problem is a popular random variational problem that received interest in the last 30 years. It was in 2014 that a PDE ansatz introduced by the statistical physics community allowed for explicit description of the cost and its minimizers. In this talk we review the recently introduced PDE approach to the matching problem and discuss some of its applications with a focus on geometrical properties of the optimizer.

Abstract

Let Δ be a finite set and, for each probability measure m on Δ, let G(m) be a transition kernel on Δ. Consider the sequence {X_n} of Δ-valued random variables such that, given X_0,...,X_n, the conditional distribution of X_{n+1} is G(L^{n+1})(X_n, . ), where L^{n+1} is the empirical measure at instant n. Under conditions on G, we establish a large deviation principle for the sequence {L^n}. As one application of this result, we obtain large deviation asymptotics for the Aldous-Flannery-Palacios (1988) approximation scheme for quasi-stationary distributions of finite state Markov chains. The conditions on G cover other models as well, including certain models with edge or vertex reinforcement.  

Arxiv link: https://arxiv.org/abs/2304.01384

Abstract

I will discuss non-linear stability results concerning the asymptotic behavior of solutions to classical models arising in kinetic theory. Firstly, I will present classical results on the stability of vacuum for the Vlasov--Poisson system. Later, I will speak about the stability of vacuum for the Vlasov--Poisson system with an unstable trapping potential (joint work with Anibal Velozo). If time permits, I will also present a stability result for the exterior of the Schwarzschild family of black holes for the spherically symmetric Einstein--massless Vlasov system

Abstract

In this talk I will discuss the Schrödinger operators with point interaction, which model the dynamics of a quantum particle interacting with a spatial defect. After introducing the adapted Sobolev spaces and the dispersive properties of the associated unitary evolution, I will consider the corresponding non-linear Schrödinger equation. Some recent results on the local and global well-posedness and the existence of standing waves solutions will be provided