Abstract: We explore the dynamical behavior and energetic properties of a model of two species that interact nonlocally on finite graphs. The authors recently introduced the model in the context of non quadratic Finslerian gradient flows on generalized graphs featuring nonlinear mobilities. In a continuous and local setting, this class of systems exhibits a wide variety of patterns, including mixing of the two species, partial engulfment, or phase separation. This work showcases how this rich behavior carries over to the graph structure. We present analytical and numerical evidence thereof.

Abstract: Understanding the intermediate asymptotic and computing convergence rates towards equilibria are among the major problems in the study of parabolic equations. Convergence rates depend on the tail behaviour of solutions. This observation raised the following question: how can we understand the tail behaviour of solutions from the tail behaviour of the initial datum?

In this talk, I will discuss the asymptotic behaviour of solutions to the fast diffusion equation. It is well known that non-negative solutions behave for large times as the Barenblatt (or fundamental) solution, which has an explicit expression. In this setting, I will introduce the Global Harnack Principle (GHP), precise global pointwise upper and lower estimates of non-negative solutions in terms of the Barenblatt profile. I will characterize the maximal (hence optimal) class of initial data such that the GHP holds by means of an integral tail condition. As a consequence, I will provide rates of convergence towards the Barenblatt profile in entropy and in stronger norms such as the uniform relative error.

Abstract: The theory for variational evolutions—evolutions driven by one or more energies-or entropies—in spaces of measures has seen tremendous growth in the last decades, of which resulted in a rich framework for classical gradient systems in general metric spaces by Ambrosio, Gigli and Savaré, where the Wasserstein metric of optimal transport theory plays a fundamental role; and a theory for rate-independent systems. While these theories have allowed massive development of variational evolutions in a certain direction—gradient flows with homogeneous dissipation—physics and large-deviation theory suggest the study of generalized gradient flows—gradient flows with non-homogeneous dissipation—which are not covered in either of these established theories.

In this talk, I will discuss the motivation underlying the need for a generalized theory of gradient flows and how these structures can be used in practice.

Abstract: I will discuss a recent result in collaboration with Martin Burger (FAU Erlangen), where we connect the quadratic porous medium equation with a nonlocal interaction equation. We prove the convergence of solutions of a nonlocal interaction equation to the solution of the quadratic porous medium equation in the limit of a localising interaction kernel. The analysis is carried out at the level of the (nonlocal) partial differential equations and we use the gradient flow structure of the equations to derive bounds on energy, second order moments, and logarithmic entropy. The dissipation of the latter yields sufficient regularity to obtain compactness results and pass to the limit in the localised convolutions. The strategy we propose relies on a discretisation scheme which could be slightly modified in order to extend our result to PDEs with no gradient flow structure. Our analysis allows us to treat the case of limiting weak solutions of the non-viscous porous medium equation at relevant low regularity, assuming the initial value to have finite energy and entropy. However, the latter excludes particle solutions of the nonlocal interaction equation.

Abstract: In this talk, the evolution of a system of two species with nonlinear mobility on a graph with nonlocal interactions is discussed. We provide a rigorous interpretation of the interaction system as a gradient flow in a Finslerian setting. This not only extends the recent results of Esposito et. al. (2021) to systems of interacting species, but also translates the theory of gradient flows with concave, nonlinear mobilities to this setting. Weakening the notion of Minkowski norm and nonlocal gradient, in the spirit of Agueh (2011), the geometric interpretations and the analysis are carried over to p-Wasserstein-like distances. 

Abstract: In this talk I will discuss some recent results on the asymptotic behaviour of a family of weighted ultrafast diffusion PDEs. These equations are motivated by the gradient flow approach to the problem of quantization of measures, introduced in a series of joint papers with Emanuele Caglioti and François Golse. In this presentation I will focus on a recent result with Francesco Saverio Patacchini and Filippo Santambrogio where we the JKO scheme to obtain, existence, uniqueness, and exponential convergence to equilibrium under minimal assumptions on the data. 

Abstract:  We seek to establish qualitative convergence results to a general class of evolution PDEs described by gradient flows in optimal transportation distances. These qualitative convergence results come from dynamical systems under the general name of LaSalle Invariance Principle. By combining some of the basic notions of gradient flow theory and dynamical systems, we are able to reproduce this invariance principle in the setting of evolution PDEs under general assumptions. We apply this abstract theory to a non-exhaustive list of examples that recover, simplify, and even extend the results in their respective literatures.

Abstract: In this talk  We propose a nonlocal gradient structure approximating the aggregation equation motivated by the classical upwind scheme widely used for the numerical approximation of first order equations. We show that the nonlocal upwind metric is very useful very well suited for the variational formulation of first-order equations on graphs and graphons. The induced distance is a quasi-metric (non-symmetric) leading to a formal Finslerian structure. We show how it is possible in the variational framework to show unstability under suitable limits of the graph structure. 

Joint work with Antonio Esposito, Francesco Patacchini, Dejan Slepčev.

Abstract:  In more general diffusion equations the diffusivity depends on the solution itself yielding a nonlinear equation. In cross-diffusion systems we consider the simultaneous diffusion of several densities, where the diffusivity of one density depends on the other densities. These models appear, e.g., in mathematical biology for the movement of competing species where the most famous model is the SKT system. In many interesting cases the system admits a formal gradient flow strucutre with an entropy but little else is known as a maximum principle is lacking. In this talk I will present after an introduction and overview a recent work with Ayman Moussa (arXiv:2102.02893) about nonlocal versions which inherit the entropy structure and this can be used as a building block to construct solutions or to derive the model from particle systems.

Abstract:  In this talk I will discuss a class of interacting particle models with anisotripoc repulsive-attractive interaction forces. These models are motivated by the simulation of fingerprint databases, which are required in forensic science and biometric applications. In existing models, the forces are isotropic and particle models alead to non-local aggregation PDEs with radially symmetric potentials. The central novelty in the models I consider is an anisotropy induced by and underlying tensor field. This innovation does not only lead to the ability to describe real-world phenomena more accurately, but in contrast to their isotropic counterparts a gradient flow structure does not exist. I will discuss the role of anisotropic interaction in these models, present a stability analysis of line patterns, and show numerical results.

Abstract:  The aggregation-diffusion equation is a nonlocal PDE driven by two competing effects: local repulsion modeled by nonlinear diffusion, and long-range attraction modeled by nonlocal interaction. In this talk, I will discuss several qualitative properties of its steady states and dynamical solutions. Using continuous Steiner symmetrization techniques, we show that all steady states are radially symmetric up to a translation. (joint work with Carrillo, Hittmeir and Volzone). In a recent work, we further investigate whether they are unique within the radial class, and show that for general attractive potentials, uniqueness/non-uniqueness of steady states are determined by the power of the degenerate diffusion, with the critical power being m = 2. (joint work with Delgadino and Yan).

Abstract:  We propose fully-discrete, implicit-in-time finite-volume schemes for general non-linear non-local Fokker-Planck equations with a gradient flow structure. The schemes verify the positivity-preserving and energy-dissipating properties, done conditionally by the second order scheme and unconditionally by the first order counterpart. Dimensional splitting allows for the construction of these schemes with the same properties and a reduced computational cost in any dimension. We will showcase the handling of complicated phenomena: free boundaries, meta-stability, merging, and phase transitions.

Abstract: The Landau equation is an important PDE in kinetic theory modelling plasma particles in a gas. It can be derived as a limiting process from the famous Boltzmann equation. From the mathematical point of view, the Landau equation can be very challenging to study; many partial results require, for example, stochastic analysis as well as a delicate combination of kinetic and parabolic theory. The major open question is uniqueness in the physically relevant Coulomb case. I will present joint work with José Carrillo, Matias Delgadino, and Laurent Desvillettes where we cast the Landau equation as a generalized gradient flow from the optimal transportation perspective motivated by analogous results on the Boltzmann equation. A direct outcome of this is a numerical scheme for the Landau equation in the spirit of de Giorgi and Jordan, Kinderlehrer, and Otto. An extended area of investigation is to use the powerful gradient flow techniques to resolve some of the open problems and recover known results.

Abstract: Nonlocal interactions arise throughout the natural world, from collective dynamics in biological swarms to vortex motion in superconductors. Over the past fifteen years, there has been significant interest in aggregation diffusion equations, which consider the competing effects of nonlocal interactions and local diffusion, and constrained aggregation equations, which consider the competition between nonlocal interactions and a hard height constraint on the density. We will apply recent results on nonconvex gradient flows to prove that aggregation diffusion equations converge to the constrained aggregation equation in the slow diffusion limit. Then, we will use these dynamics to explore open conjectures in geometric shape optimization.

Abstract: I will discuss a result obtained in collaboration with M. Di Francesco and S. Fagioli, both from the University of L’Aquila. We investigate a class of systems of partial differential equations with nonlinear cross-diffusion and nonlocal interactions. Assuming a uniform “coerciveness” assumption on the diffusion part, which allows to consider a large class of systems with degenerate cross-diffusion (i.e. of porous medium type) and relaxes sets of assumptions previously considered in the literature, we prove global-in-time existence of weak solutions by means of a semi-implicit version of the Jordan–Kinderlehrer–Otto scheme. Our approach allows us to consider nonlocal interaction terms not necessarily yielding a formal gradient flow structure.

Abstract: I will present a systematic existence and uniqueness theory of weak measure solutions for systems of non-local interaction PDEs with two species, which are the PDE counterpart of systems of deterministic interacting particles with two species. The main motivations behind those models arise in cell biology, pedestrian movements, and opinion formation. In case of symmetrizable systems, we provide a complete existence and uniqueness theory within the Wasserstein gradient flow theory, which. In the general case, we provide an existence result of a semi-implicit version of the Jordan–Kinderlehrer–Otto (JKO) scheme. The results are also extended to systems with cross-diffusion with suitable coercivity properties. 

Abstract: One of the most fascinating phenomena observed in reaction-diffusion systems is the emergence of segregated solutions, i.e., population densities with disjoint supports. We analyse such a reaction cross-diffusion system. In order to prove the existence of weak solutions for a wide class of initial data without restriction about their supports or their positivity, we propose a variational splitting scheme combining ODEs with methods from optimal transport. In addition, this approach allows us to prove conservation of segregation for initially segregated data even in the presence of vacuum.

Abstract: Agregation-diffusion equations are used to model many phenomena, especially in asrophysics and in biology (chemotaxis), with the most well-known example being the Keller-Segel (KS) system. Here we consider a class of KS-type models with solutions which explode in the zero-diffusion limit. We give a sharp description of their behaviour (concentration, Lebesgue norms) in the small-diffusion regime for the radially symmetric setting.