Past Seminars: (see also GSSI Academic Calendar)



Speaker: Gennaro Ciampa (University of L'Aquila)

Title: Magnetic reconnection for incompressible MHD equations

Abstract: The goal of this talk is to provide examples of periodic smooth solutions of the incompressible Magnetohydrodynamics equations (MHD) for which the topology of the magnetic field lines changes under the evolution. This is known to be impossible in the non-resistive case by Alfvén's theorem. The reconnection of the magnetic field lines occurs instead in the resistive case, being responsible for many dynamic phenomena in astrophysics such as solar flares and the solar wind. Although numerical and experimental evidence exist, analytical examples of magnetic reconnection were not known. 



Speaker: Andrea Marchese (University of Trento)

Title: Closability of differential operators and structure of currents

Abstract: I will discuss recent results concerning the closability of certain directional derivative and Jacobian-type differential operators and their implications for the structure of flat chains and metric currents. Additionally, I will present a new, elementary proof of Ambrosio and Kirchhiem's flat chain conjecture, in the case of 1-dimensional currents. This conjecture asserts that metric currents in the Euclidean space correspond to Federer-Fleming flat chains. Our new proof sheds light on the obstructions that one needs to face towards a positive answer to the conjecture in full generality. This is based on joint works with G. Alberti, D. Bate, and A. Merlo. 



Speaker: Filippo Giuliani (Milan Politecnico)

Title: Sobolev norms inflation and transfers of energy in NLS equations under periodic boundary conditions

Abstract: In 2010 Colliander-Keel-Staffilani-Takaoka-Tao (Invent.Math.) proved the existence of solutions to the cubic defocusing nonlinear Schrödinger equation on the 2-dimensional torus undergoing an arbitrarily large (but finite) growth of high order Sobolev norms. Since in the defocusing case the H^1 norm of solutions is uniformly controlled in time, the study of Sobolev norms inflation can be used to detect energy transfers to high modes (energy cascades) and, in some sense, it could be seen as a deterministic approach to the study of weak wave turbulence. The result by Colliander-Keel-Staffilani-Takaoka-Tao is based on the analysis of the resonant dynamics of the NLS equation and it cannot be applied directly when the spatial domain is an irrational torus and/or in presence of convolution potentials, because the resonant structure may drastically change in these cases. In this talk I will show how to construct, in these situations, solutions exhibiting an arbitrarily large norm inflation by using normal form methods. If time permits, I will discuss also a recent result of this type for the quantum hydrodynamic system.



Special Event: School "Boundary Analysis for Dispersive and Viscous Fluids"

Information and registration



Speaker: Adriano Pisante (Sapienza University of Rome)

Title: Torus-like solutions for the Landau de Gennes model

Abstract: We report on some recent progress (in collaboration with F. Dipasquale and V. Millot) about the study of global minimizers of a continuum Landau-De Gennes energy functional for nematic liquid crystals in three-dimensional domains. First, we discuss absence of singularities for minimizing configurations under norm constraint, as well as absence of the isotropic phase for the unconstrained minimizers, together with the related biaxial escape phenomenon. Then, under suitable assumptions on the topology of the domain and on the Dirichlet boundary condition, we show that smoothness of energy minimizing configurations yields the emergence of nontrivial topological structure in their biaxiality level sets. Then, we discuss the previous properties under both the norm constraint and an axial symmetry constraint, showing that in this case only partial regularity is available, away from a finite set located on the symmetry axis. In addition, we show that singularities may appear due to energy efficiency and we describe precisely the asymptotic profile around singular points. Finally, in an appropriate class of domains and boundary data we obtain qualitative properties of the biaxial surfaces, showing that smooth minimizers exibit torus structure, as predicted in numerical simulations.



Speaker: Emanuela Radici (University of L'Aquila)

Title: Deterministic many particle limits for degenerate second order traffic models

Abstract: We study the well-posedness and the deterministic many particle limits for second order Follow-the-Leader type models involving challenging nonlinearities which naturally appear in the modelling of crowd dynamics. We aim to validate the second-order particle approach for a class of traffic models characterised by the intrinsic mechanism that the reaction-time of the drivers depends on both inertial and congestion terms. We also consider the many particle limit in the vanishing inertia regime to recover the expected transport equation with nonlinear mobility. This is a joint work with D. Mazzoleni and F. Riva.



Speaker: Roberto Feola (Roma Tre University)

Title: Long time dynamics of quasi-linear Hamiltonian Klein-Gordon equations

Abstract: We consider a class of Hamiltonian Klein-Gordon equations with a quasilinear, quadratic nonlinearity under periodic boundary conditions. We provide a precise description of the dynamics for an open set of small initial showing that the corresponding solutions remain close to oscillatory motions over a "large" time scale. The key ingredients of the proof are normal form methods, para-differential calculus and a modified energy approach.



Speaker: Federico Cacciafesta (University of Padua)

Title: Dispersive properties of the Dirac equation

Abstract: In this talk, I shall review the main dynamical properties of the linear Dirac equation, discussing in particular the validity of linear estimates (time decay, Strichartz, local smoothing) and comparing them with the ones for the Schroedinger equation. Time permitting, we shall see how these estimates are affected by the presence of external potentials (electric or magnetic).



Speaker: Birgit Schörkhuber (University of Innsbruck)

Title: Self-similar blowup for the 3d cubic NLS

Abstract: The cubic nonlinear Schrödinger equation arises in various physical applications and is one of the most fundamental models in dispersive PDEs. In the three dimensional focusing case the formation of singularities via self-similar solutions has been proposed in the 70s by Zakharov in the context of plasma physics.  Based on numerical experiments it is now widely believed that there is a self-similar “ground state”  solution which appears as an attractor in the time-evolution of generic large initial data. However, the existence of this (or any other finite-energy) self-similar solution has been a long-standing open problem. In this talk, I present recent joint work with Roland Donninger (University of Vienna), which proves, by using mildly computer assisted methods, the existence of a smooth finite-energy self-similar blowup solution to the 3d cubic NLS.



Speaker: Mickaël Latocca (University of Évry)

Title: Propagation of Regularity of the Free Boundary of an Inhomogeneous Fluid Governed by the Darcy Law

Abstract: We consider a 2d fluid in the domain defined by $b(x)<y<f(t,x)$, $x\in\mathbb{R}^d$ where $f(t,x)$ is the free interface, and $b(x)$ the fixed bottom. We assume that the fluid is incompressible and is governed by the Darcy law. In the case of constant densities, Nguyen--Pausader proved that the interface maintains Sobolev regularity for all time (provided the Sobolev regularity is large enough). Our long-term goal is to investigate stability of the constant-density case. The purpose of this talk is twofold: first I will explain how we can rewrite our problem as a quasilinear problem (a usual procedure in this context) and spend some time explaining why such problems pose non-trivial issues such as derivative losses. Then I will use the rest of my time to explain some specific difficulties linked to non-constant densities and explain how one can achieve the first part of our program, that is short-time propagation of the initial Sobolev regularity of the free boundary. This is based on a joint work with Huy Q. Nguyen (University of Maryland).



Speaker: Umberto Pappalettera (Bielefeld University)

Title: On anomalous regularity in Kraichnan’s model of turbulent transport

Abstract: In this talk I will present a new “anomalous regularisation” result for solutions of the stochastic transport equation \partial_t \rho + \circ \partial_t W \cdot \nabla \rho = 0 on \mathbb{R}^d, where W is a Gaussian, homogeneous, isotropic noise with \alpha-H\”older space regularity and compressibility ratio \wp < \frac{d}{4\alpha^2}. The proof is obtained by studying the local behaviour around the origin of solutions to a degenerate parabolic PDE in non-divergence form, which is of independent interest. Based on joint work with Theodore Drivas and Lucio Galeati.