Mathematical Analysis Seminars @ DISIM

Title: Asymptotic behaviour of fractional Sobolev spaces on thin films

Abstract: We study the asymptotic behaviour of Gagliardo H^s seminorms on thin films of thickness tending to 0. In particular, we focus on to the "dimension-reduction" regimes (à la Le Dret-Raoult). We show that the behaviour is different in the low-integrability regime (s<1/2) and in the high-integrability regime (s>1/2). In both cases, there exists a critical scaling (depending on s) for which the limit is dimensionally reduced. In the case s>1/2 the scaling highlights a combined effect of the geometrical dimensions and the relevant range of interactions in the Gagliardo seminorms, and the limit is a H^1 seminorm. Moreover, in the light of the results by Bourgain, Brezis and Mironescu (BBM) and Maz'ya and Shaposhnikova (MS), we study the asymptotic behaviour as s tends to 1 and s tends to 0. In the case of s tending to 1 the asymptotic behaviour highlights a separation of scales and a “compatibility" with the BBM result. In the case of s tending to 0, the scaling is purely geometrical and leads to a H^{1/2} seminorm differently from the MS result. The analysis relies on two different approaches to compactness.

Work in collaboration with A. Pinamonti and M. Solci.

Title: A Li-Yau and Aronson-Bénilan approach for the Keller-Segel system

Abstract: In this talk we will focus on the Keller-Segel system, for d >= 2 and m = 2 - 2/d, i.e. the critical exponent. This system exhibits a rich behaviour and its dynamics depend on the initial mass. When the mass is below certain threshold (subcritical mass) there is global-in-time bounded solutions, if we are beyond this threshold (supercritical mass), one can construct solutions with finite time blow-up. Finally, if the mass is critical there exists global-in-time solutions but they are not bounded globally-in-time.

The main goal of the talk is to extend the classical Li-Yau and Aronson-Bénilan estimates in order to cover the Keller-Segel case. We are able to recover the estimate for subcritical and critical mass and, in particular, for a small (computable) mass we also obtain a regularising effect. We follow two strategies: for the small mass case we rely on concavity and harmonic analysis. For the general case of subcritical and critical mass our argument is based on a careful analysis of the subsolutions of the Liouville and the Lane--Emden equations combined with a contradiction argument. 

The talk presents joint work with C. Elbar and F. Santambrogio.

Title: Gradient flows on metric graphs with reservoirs: Microscopic derivation and multiscale limits

Abstract: We study evolution equations on metric graphs with reservoirs, that is graphs where a one-dimensional interval is associated to each edge and, in addition, the vertices are able to store and exchange mass with these intervals. Focusing on the case where the dynamics are driven by an entropy functional defined both on the metric edges and vertices, we provide a rigorous understanding of such systems of coupled ordinary and partial differential equations as (generalized) gradient flows in continuity equation format. Approximating the edges by a sequence of vertices, which yields a fully discrete system, we are able to establish existence of solutions in this formalism. Furthermore, we study several scaling limits using the recently developed framework of EDP convergence with embeddings to rigorously show convergence to gradient flows on reduced metric and combinatorial graphs.  Finally, numerical studies confirm our theoretical findings and provide additional insights into the dynamics under rescaling.

Title: The inelastic Lorentz gas: derivation, and long-time behaviour

Abstract: Understanding the Boltzmann equation for non-conservative particle systems, where kinetic energy is dissipated at each collision, is challenging and its derivation has remained an open problem. The main difficulty arises from the singularities that both the particle system and the associated kinetic equation may develop in finite time. The inelastic Lorentz gas, composed of light particles undergoing inelastic collisions with infinitely heavy scatterers and evolving according to the inelastic linear Boltzmann equation, provides a non-trivial yet tractable model in which many questions can be addressed.

In this talk, we present a rigorous derivation of the inelastic Lorentz gas from the deterministic dynamics of tagged particles colliding with inelastic scatterers distributed according to a Poisson point process. The proof relies on demonstrating the convergence of the series expansion of the solution in suitable weighted spaces, together with a weak-convergence approach that allows us to handle the singularities of the backward flow. We will also discuss the long-time behaviour of the inelastic Lorentz gas in the case of Maxwell molecules, and in the presence of a uniform gravitational field. In particular, we show the existence of an out-of-equilibrium steady state that attracts all solutions within the considered functional framework.

This is a joint work with Nicola Miele and Alessia Nota (GSSI) (arXiv:2504.02155, arXiv:2511.02934).

Title: H-compactness for nonlocal linear operators in fractional divergence form

Abstract: In this seminar we address the problem of the H-convergence of nonlocal linear operators in fractional divergence form, where the oscillations of the matrices are prescribed outside the reference domain. Our compactness argument bypasses the failure of the classical localisation techniques that mismatch with the nonlocal nature of the operators involved. 

If symmetry is also assumed, we extend the equivalence between the H-convergence of the operators and the $\Gamma$-convergence of the associated energies. If time allows we will talk about further generalizations. The seminar is based on a joint work with M. Caponi (Univaq) and A. Maione (CRM).

Title: On the topology of the magnetic lines of large solutions to the Magnetohydrodynamic equations in R^3 

Abstract: The purpose of this talk is to present the article [1], in which we establish two results: first, we introduce a new class of global strong solutions to the magnetohydrodynamic system in R^3 with initial data (u_0,b_0) of arbitrarily large size in any critical space. To do so, we impose a smallness condition on the difference u_0-b_0. Then we use this result to prove magnetic reconnection for a suitable class of (large) solutions. With this, we mean a change of topology of the integral lines of the magnetic field b under the evolution. The proof relies on counting the number of hyperbolic critical points of the solutions, and this instance is structurally stable.

[1] R. Lucà, C. Peña: "On the topology of the magnetic lines of large solutions to the Magnetohydrodynamic equations in R^3". Preprint: arXiv:2505.09340.

Title: Existence of gradient flow solutions for fractional thin film equations with aggregation on convex domains

Abstract: I will show a gradient flow structure for a family of fractional thin film equations with linear mobility and a reaction term. The problem is posed in a bounded convex domain with the homogeneous Neumann boundary condition. Existence and properties of weak solutions will be illustrated.

Title: Isoperimetric planar Tilings with unequal cells

Abstract: In this talk, we consider an isoperimetric problem for periodic planar Tilings allowing for unequal repeating cells. We discuss general existence and regularity results and we study classification results for double Tilings, i.e. Tilings with two repeating cells. In this case, we explicitly compute the associated energy profile and we give a complete description of the phase transitions. Based on joint works with M. Novaga and E. Paolini.

Title: Discrete and continuum models of robust biological transportation networks

Abstract: We study a discrete model for formation and adaptation of biological transport networks. The model consists of an energy consumption function constrained by a linear system on a graph. We discuss how structural properties of the optimal network patterns, like sparsity and (non)existence of loops, depend on the convexity/concavity of the metabolic part of the energy functional. We then introduce robustness of the network in terms of algebraic connectivity of the graph and explain its impact on the network structure. Passing to the continuum limit as the number of edges and nodes of the graph tends to infinity, we recover a nonlinear system of PDEs. This elliptic-parabolic system consists of a Darcy's type equation for the pressure field and a reaction-diffusion equation for the network conductance. We explain how the robustness property is reflected on the level of the PDE description. We give both analytical results and systematic numerical simulations for the PDE system, providing interesting insights into the mechanisms of network formation and adaption in biological context.