Abstracts
The Phi^4_2 theory as limit of interacting inhomogeneous Bose gases – Cristina Caraci
ABSTRACT: Euclidean field theories have been extensively studied in the mathematical literature since the sixties, motivated by high-energy physics and statistical mechanics. Formally, they can be described by Gibbs measures associated with Euclidean action functionals over spaces of distributions. In the latest years it has been shown how such theories emerge as high-density limit of interacting Bose gases at positive temperature, giving a rigorous derivation from a realistic microscopic model of statistical mechanics. In this talk, I will present a result providing the derivation of such a field theory, in other words of the invariant Gibbs measure, with a quartic local interaction in two dimensions as a limit of an inhomogeneous interacting Bose gas, extending the previous work on the torus by Fröhlich-Knowles-Schlein-Sohinger. Based on a joint work with Antti Knowles, Alessio Ranallo and Pedro Torres Giesteira.
Fast reconnection for incompressible MHD equations – Gennaro Ciampa
ABSTRACT: In this talk, we study magnetic reconnection, understood as a change in the topology of magnetic field lines, for sufficiently regular solutions of the three-dimensional incompressible magnetohydrodynamic (MHD) equations. We construct a class of initial data for which reconnection occurs on time scales strictly shorter than the resistive diffusion time scale, despite the persistence of regularity. This separation of time scales is driven by enhanced dissipation induced by the advective dynamics, yielding the first rigorous examples in which advection, rather than resistivity alone, plays a genuinely active role in the reconnection process.
Instability of the 2D Taylor-Green vortex – Michele Dolce
ABSTRACT: The 2D Taylor-Green (TG) vortex is the prototypical example of an Euler steady state on T^2 possessing truly two-dimensional features, like elliptic and hyperbolic stagnation points. Its streamfunction, sin(x)sin(y), lives on the second Fourier shell, making it susceptible to large-scale destabilizing mechanisms. Despite the apparent simplicity of the steady state, a proof of its spectral instability has long remained elusive, and was only recently observed numerically. To solve this problem, I will introduce a new criterion to detect unstable eigenvalues for a wide class of linear Hamiltonian operators. We apply this to prove the stability of the TG vortex with respect to odd perturbations. In the subspace of functions even in both variables, we combine our criterion with a rigorous computer-assisted argument to locate two unstable eigenvalues. This fully characterizes the unstable spectrum of the TG vortex and implies nonlinear instability in velocity. This is a joint work with G. Cao-Labora, M. Colombo and P. Ventura.
KAM for pure gravity 3d traveling gravity water waves – Roberto Feola
ABSTRACT: Starting with the pioneering computations of Stokes in 1847, the search of traveling waves in fluid mechanics has always been a fundamental topic, since they can be seen as building blocks to determine the long time dynamics. In this talk we shall discuss the existence of time quasi-periodic traveling wave solutions for three-dimensional pure gravity water waves in finite depth, on flat tori, with an arbitrary number of speeds of propagation. These solutions are global in time, they do not reduce to stationary solutions in any moving reference frame and they are approximately given by finite sums of Stokes waves traveling with rationally independent speeds of propagation. The major difficulties arises from the fact that the 3D water waves system is a quasi-linear PDE in higher space dimension with “weak dispersion relation’’. As a consequence, in the search for quasi-periodic solutions one must deal with the presence of very strong resonance phenomena. We will focus on the spectral analysis of the linearized equations at any approximate traveling wave solutions. The strategy is suited for higher dimensional dispersive PDEs with sublinear dispersion.
Local Dynamics of Extended Fermi Gases at High Density – Luca Fresta
ABSTRACT: I will discuss the dynamics of pseudo-relativistic Fermi gases in a high-density semiclassical regime. I will present a result proving convergence of the many-body quantum evolution to the Hartree dynamics at the level of local observables, with estimates depending on the density only. The result applies to initial data describing equilibrium states confined in arbitrarily large domains, under the assumption of suitable local Weyl-type estimates. Time permitting, I will also discuss some open problems and future directions.
Based on joint work with Marcello Porta and Benjamin Schlein.
Commutator Estimates for Low-Temperature Fermi Gases with magnetic field – Laurent Lafleche
ABSTRACT: Due to the singularity of the interactions, the derivation of the Vlasov–Poisson equation with Coulomb or gravitational interaction remains an open problem. There have been recent advances in the study of singular potentials, which now allow the treatment of square-integrable interaction potentials. On the quantum side, a key ingredient in the strategy is the uniform-in-hbar propagation of Schatten norms of commutator, which are the quantum analogue of the Sobolev norms for classical phase space densities.
In this talk, I will discuss the size of commutators of typical states in various regimes, depending on the temperature and on the size of the magnetic field, and show their applications to semiclassical mean-field limits, as well as to the study of ground states.
The Lee-Huang-Yang formula for the hard-core dilute Bose gas: an upper bound – Alessandro Olgiati
ABSTRACT: The Lee-Huang-Yang formula captures the ground state energy of a dilute Bose gas in the thermodynamic limit. First derived heuristically in 1957, it predicts that, to second order in the dilute limit, the energy depends on the repulsive interaction only through the s-wave scattering length, independently of local integrability of the potential. This prediction has been largely confirmed by mathematical results over the past decades. Yet, while energy lower bounds are now known in considerable generality, all upper bounds known so far rely on local integrability assumptions. We construct a trial state for hard-core bosons whose energy matches the Lee-Huang-Yang asymptotics, thereby establishing the upper bound without requiring local integrability. Based on a joint work with Giulia Basti (Sapienza, University of Rome), Morris Brooks (University of Zurich), Serena Cenatiempo (GSSI L’Aquila), and Benjamin Schlein (University of Zurich).
Stability of quantum gases – Julien Sabin
ABSTRACT: The goal of this mini-course is to review some results of the last years concerning the large time stability of homogeneous quantum gases. We model the dynamics of these infinitely extended gases by a mean-field evolution equation on the one-body density ma-
trix of the system, which is a generalization of the nonlinear Schrödinger equation. The key concepts for the large time stability that we will be emphasized are: a criterion for the linearized stability, intensive use of the dispersion of free particles through Strichartz estimates. The plan of the course is the following:
1) Introduction to the model, linear stability
2) Strichartz estimates for density matrices
3) Nonlinear analysis
Formation and Dynamics of Ginzburg Landau Vortices for Vector Fields on Manifolds – Antonio Segatti
ABSTRACT: In this talk I will report on some joint works with Giacomo Canevari (Univ. Verona). I will discuss a parabolic Ginzburg–Landau equation for vector fields on a two-dimensional closed, oriented Riemannian manifold. The topology of the manifold induces energy concentration around a finite number of points, known as vortices. I will show that, in a suitable asymptotic regime, the evolution of these vortices is governed by the gradient flow of the so-called renormalized energy.
Periodic nonlinear Schrödinger equations and the evolution of its energy spectrum – Gigliola Staffilani
ABSTRACT: In this course we will investigate some questions related to weak turbulence theory by using as explicit example of wave interactions the solutions to the 2D periodic cubic Schrödinger equation. We will start by recalling Strichartz estimates and well-posedness. We will then explain how the evolution of the energy spectrum related to this equation can be studied in two different ways. We will first work on a method proposed by Bourgain and involving the growth of high Sobolev norms. Then we will present some recent results on energy cascade via the analysis of radial solutions to the wave kinetic equation, which is often used as the effective equation for the energy spectrum mentioned above.
Polynomial growth for the generalized DNLS – Nicola Visciglia
ABSTRACT: we present a joint work with M. Hayashi (Kyoto U.) and T. Ozawa (Waseda U.) concerning the polynomial growth of solutions to the generalized DNLS. The proof follows from a combination of the following main ingredients: the construction of suitable energies, the Zygmund $L^4$ estimate and the gauge transform.