Past Seminars: (see also GSSI Academic Calendar)



Speaker: Gianmarco Caldini (University of Trento)

Title: On regularity theory for generalized surfaces

Abstract: The natural question of how much smoother integral currents are with respect to their initial definition goes back to the late 1950s and to the origin of the theory with the seminal article of Federer and Fleming. In this seminar I will explain how closely one can approximate an integral current representing a given homology class by a smooth submanifold. Part of what will be discussed is derived from a joint study with William Browder and Camillo De Lellis, and builds on earlier preliminary work by the former author together with Frederick Almgren.



Speaker: Mahdi Haghshenas (Imperial College London)

Title: Boundedness and decay of waves on decelerated FLRW spacetimes

Abstract: After outlining the stability problem for Friedmann–Lemaître–Robertson–Walker (FLRW) spacetimes, we study the wave equation—as a proxy for the Einstein equations—on decelerated FLRW spacetimes with non-compact, flat spatial sections. We demonstrate how dispersion and expansion affect the long-time behaviour of waves. In particular, we present uniform energy bounds and integrated local energy decay estimates across the full decelerated expansion range. Furthermore, we describe a hierarchy of r-weighted energy estimates, in the spirit of the Dafermos–Rodnianski method, which lead to energy decay estimates.



Speaker: Gianluca Crippa (University of Basel)

Title: Anomalous dissipation in fluid dynamics

Abstract: The celebrated Kolmogorov's K41 theory of fully developed turbulence attempts to explain and quantify "wild, but typical" behaviors of "chaotic" fluids, most notably the lack of conservation of the total energy. The loss of energy is not due to friction between fluid molecules, but rather to the limited regularity of the flow. Kolmogorov's theory is numerically and experimentally validated to a very large extent, however, very little is known in rigorous mathematical terms. In my lecture, I will present some aspects of Kolmogorov's theory, and illustrate recent results on a related question for the linear advection equation.



Speaker: Dragos Iftimie (University of Lyon)

Title: Energy decay for 2D micropolar fluids

Abstract: We consider fluids where the particles possess a microstructure (micropolar flows) and are allowed to rotate in a two-dimensional setting. We find the asymptotic profile of the solution as the time goes to infinity. We deduce the remarkable fact that the large time behavior only depends on the kinematic viscosity, and not on the other parameters of the model (vortex-viscosity, spin viscosity and gyroviscosity). Our primary tool is a new enstrophy-like identity of independent interest, involving the difference between the fluid vorticity and the micro-angular velocity. This is joint work with L. Brandolese, V. Busuioc and C. F. Perusato.



Speaker: Luca Talamini (SISSA)

Title: Some results on L^∞ entropy solutions to systems of two conservation laws

Abstract: The pioneering work of Tartar and DiPerna established the existence of entropy solutions to systems of two conservation laws in the L^∞ setting by adapting the compensated compactness method. However, no regularity properties are known for these solutions. As a first step in this direction, we discuss a Liouville-type theorem for isentropic solutions of 2×2 genuinely nonlinear systems, together with some of its applications. The arguments are based on a kinetic formulation and on a Lagrangian representation at the kinetic level. Joint work with F. Ancona and E. Marconi.



Speaker: Gemma Hood (Leipzig University)

Title: A scattering construction for nonlinear wave equations on Kerr-Anti de Sitter spacetimes

Abstract: Given the sharp logarithmic decay of linear waves on the Kerr-AdS black hole (Holzegel, Smulevici, 2013), it is expected that the Kerr-AdS spacetime is unstable as a solution of the Einstein vacuum equations. However, the scattering construction presented here for exponentially decaying nonlinear waves on a fixed Kerr-AdS background serves as a first step to confronting the scattering problem for the full Einstein system. In this context, one may hope to derive a class of perturbations of Kerr-AdS which remain ‘close’ and dissipate sufficiently fast.



Speaker: Óscar Domínguez (CUNEF University)

Title: Uniqueness of transport equations via extrapolation

Abstract: We propose a novel extrapolation approach to uniqueness of weak solutions for a wide class of transport equations. In particular, this unifies and extends the classical works of Yudovich and Vishik on 2D Euler equations for incompressible fluid flows with associated nearly bounded and BMO vorticities, respectively.



Speaker: Federico Luigi Dipasquale (Scuola Superiore Meridionale)

Title: The formation of gradient-driven singular structures of codimension one and two in 2D: The case study of ferronematics

Abstract: We consider a variational model for ferronematics---composite materials formed by dispersing magnetic nanoparticles into a liquid crystal matrix. The model features two coupled order parameters: a Landau-de Gennes Q-tensor for the liquid crystal component and a magnetisation vector field M, both of them governed by a Ginzburg-Landau-type energy. The energy includes a singular coupling term favouring alignment between Q and M. We report on some recent results on the asymptotic behaviour of (not necessarily minimising) critical points as a small parameter $\eps$ tends to zero. Our main results show that the energy concentrates along distinct singular sets: the (rescaled) energy density for the Q-component concentrates, to leading order, on a finite number of singular points, while the energy density for the M-component concentrates along a one-dimensional rectifiable set. Moreover, we will see that the curvature of the singular set for the M-component (technically, the first variation of the associated varifold) is concentrated on a finite number of points, i.e. the singular set for the Q-component. Joint work with G. Canevari (University of Verona) and B. Stroffolini (University of Naples ``Federico II'').



Speaker: Ayman Said (CNRS/LMR)

Title: Generic small-scale creation in the two-dimensional Euler equation

Abstract: In this talk I will present a recent result in collaboration with Thomas Alazard (CNRS-École Polytechnique). While it is well known that the Cauchy problem for the two-dimensional incompressible Euler equation is globally well-posed for smooth  initial data, we show that for a dense $G_\delta$ set of initial data, the solutions lose regularity in infinite time, thereby confirming a long-standing conjecture of Yudovich in the smooth setting.



Speaker: Tim Heilman (TU Munich)

Title: Gamma-convergence of a nonlocal Modica-Mortola type energy

Abstract: In this talk we will present a Gamma-convergence result for a nonlocal critically scaled Modica-Mortola type energy. We first review some basic facts about Modica-Mortola type energy functionals and known results and then focus on explaining some of the ideas which allow to show Gamma-convergence with a geometric argument.