# NCSU Differential Equations/Nonlinear Analysis Seminar Schedule Fall 2021

Wednesday, September 08, 15:00-16:00Zoom meeting:LinkSpeaker: Giulia Cavagnari, Politecnico di Milano, ItalyTitle:Evolution equations in Wasserstein spaces driven by dissipative probability vector fields: a variational approachAbstract:In this talk we present well posedness of Measure Differential Equations, i.e. evolution equations in the Wasserstein space of probability measures driven by dissipative probability vector fields. We take inspiration from the theory of \emph{dissipative operators} in Hilbert spaces and of Wasserstein gradient flows of geodesically convex functionals. Our approach is based on a measure-theoretic version of the Explicit Euler scheme, for which we prove convergence results with optimal error estimates. We finally characterize the limit solutions by a suitable Evolution Variational Inequality and compare this notion with the weaker barycentric/distributional one introduced by Piccoli. (This is a joint work with G. Savaré (Bocconi University) and G. E. Sodini (TUM-IAS).)

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Wednesday, September 15, 15:00-16:00Zoom meeting:LinkSpeaker: Saverio Salzo, Istituto Italiano di Tecnologia, ItalyTitle:The iterative Bregman projection method and applications to Optimal TransportAbstract:Iterative Bregman projections is a classical method to compute Bregman projections onto an intersection of affine sets. In statistics it was applied to the adjustment of distributions to a priori known marginals, and is best known as the Iterative proportional fitting procedures. In this talk I will present novel results concerning such classical method as well new applications in optimal transport.

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Wednesday, September 22, 15:00-16:00Zoom meeting:LinkSpeaker: Pierre Cardialaguet, Université Paris-Dauphine, FranceTitle:Microscopic derivation of a traffic flow model with a bifurcationAbstract: In this joint ongoing work with Nicolas Forcadel (INSA Rouen) we study traffic flows models with a bifurcation. The model consists in a single incoming road divided after a junction into several outgoing ones. There are basically two classes of models to describe this situation: microscopic models, which explain how each vehicle behaves in function of the vehicles in front; and macroscopic ones, taking the form of a conservation law on a junction (or, after integration, a Hamilton-Jacobi equation). Our aim is to derive the macroscopic models from the microscopic ones, thus providing a rigorous justification of the continuous models. The microscopic models being random (in order to take into account the fact that one knows only the distribution of cars taking a given road), the mathematical analysis requires the use of concentration inequalities as well as homogenization type arguments.

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Wednesday, September 29, 15:00-16:00Zoom meeting:LinkSpeaker: Riccardo Sacco, Politecnico di Milano, ItalyTitle:A Nonlinear Heterogeneous Transmission Model for Organic Polymer Retinal ProsthesesAbstract: In this talk we propose a model for the simulation of retinal prostheses based on the use of organic polymer nanoparticles (NP). The model consists of a nonlinearly coupled system of elliptic partial differential equations accounting for:

(1) light photoconversion into free charged carriers in the NP bulk;

(2) charge transport in the NP bulk due to drift and diffusion forces;

(3) net charge recombination in the NP bulk due to the balance between light

absorption and particle-particle recombination;

(4) electron-driven molecular oxygen reduction and capacitive coupling at the NP-solution interface;

(5) ion electrodiffusion in the solution bulk;

(6) capacitive and conductive coupling across the neuronal membrane.

Model dependent variables are represented by the electric potential, the number densities of photogenerated charge carriers in the NP bulk and the molar densities of moving ions in the aqueous solution surrounding the NP. The physical mechanisms (4) and (6) are mathematically expressed by nonlinear transmission conditions across interfaces which in turn give rise to the nonlinear coupling among the dependent variables throughout the whole computational domain. The proposed model is solved in stationary conditions and in one spatial dimension by resorting to a solution map which is a modification of the Gummel Decoupled algorithm conventionally used in inorganic semiconductor simulation. System discretization is conducted using the Finite Element Method, with stabilization terms to prevent spurious unphysical oscillations in the electric potential and ensure positivity of carrier and ion concentrations. Model predictions suggest that the combined effect of NP polarization and resistivity of the NP-neuron interface results in neuron depolarization and supports the efficacy of organic NPs in the design and development of retinal prostheses.

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Wednesday, October 06, 15:00-16:00Zoom meeting:LinkSpeaker: Gian Paolo Leonardi, University of Trento, ItalyTitle:A refined form of Cheeger’s inequalityAbstract:We improve Cheeger’s lower bound for the first nonzero eigenvalue

of the Laplacian on compact Riemannian manifolds with Ricci curvature bounded from below.

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Wednesday, October 13, 15:00-16:00Zoom meeting:LinkSpeaker: Yifeng Yu, University of California – Irvine, USATitle:High Degeneracy of Effective Hamiltonian in Two DimensionsAbstract:One of the major open problems in homogenization of Hamilton-Jacobi equations is to under deep properties of the effective Hamiltonian. In this talk, I will present some recent progress. In particular, consider the effective Hamiltonian associated with the mechanical Hamiltonian H(p,x)=(|p|^2)/2+V(x). We can show that for generic V, the effective Hamiltonian is piecewise 1d in a dense open set in two dimensions using Aubry-Mather theory. So the homogenization process has dramatically changed the local behavior of the original Hamiltonian.

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Wednesday, October 20, 15:00-16:00Zoom meeting:LinkSpeaker: Christian Seis, University of Münster, GermanTitle:Leading order asymptotics for fast diffusion on bounded domainsAbstract:On a smooth bounded Euclidean domain, Sobolev-subcritical fast diffusion with vanishing boundary trace leads to finite-time extinction, with a vanishing profile selected by the initial datum. In rescaled variables, we quantify the rate of convergence to this profile uniformly in relative error, showing the rate is either exponentially fast (with a rate constant predicted by the spectral gap) or algebraically slow (which is only possible in the presence of zero modes). In the first case, we identify the leading order asymptotics. Our results improve various results in the literature, while shortening their proofs. Joint work with Beomjun Choi and Robert J. McCann.

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Wednesday, October 27, 15:00-16:00Zoom meeting:LinkSpeaker: Radu Ioan Boţ, University of Vienna Oskar-Morgenstern-Platz 1, AustriaTitle:Primal-dual dynamical approaches to structured convex

minimization problemsAbstract:In this talk, we first propose a primal-dual dynamical approach to the minimization of a structured convex function consisting of a smooth term, a nonsmooth term, and the composition of another nonsmooth term with a linear continuous operator. To this end we introduce a dynamical system for which we prove that its trajectories asymptotically converge to a saddle point of the Lagrangian of the underlying convex minimization problem as time tends to infinity. In addition, we provide rates for both the violation of the feasibility condition by the ergodic trajectories and the convergence of the objective function along these ergodic trajectories to its minimal value. Explicit time discretization of the dynamical system results in a numerical algorithm which is a combination of the linearized proximal method of multipliers and the proximal

ADMM algorithm.

In the second part of the talk we give an outlook on a second order primal dual dynamical system with asymptotic vanishing term and on its fast convergence properties.

The talk relies on the papers (Bot ̧, Csetnek, La’szlo’, JDE, 2020) and (Bot ̧Nguyen, JDE, 2021).

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Wednesday, November 03, 15:00-16:00Sara Daneri, GSSI, ItalyZoom meeting:LinkSpeaker:On the sticky particle solutions to the pressureless Euler system in general dimensionTitle:Abstract: In this talk we consider the pressureless Euler system in dimension greater than or equal to two. Several works have been devoted to the search for solutions which satisfy the following adhesion or sticky particle principle: if two particles of the fluid do not interact, then they move freely keeping constant velocity, otherwise they join with velocity given by the balance of momentum. For initial data given by a finite number of particles pointing each in a given direction, in general dimension, it is easy to show that a global sticky particle solution always exists and is unique. In dimension one, sticky particle solutions have been proved to exist and be unique. In dimensions greater than or equal to two, it was shown that as soon as the initial data is not concentrated on a finite number of particles, it might lead to non-existence or non-uniqueness of sticky particle solutions.

In collaboration with S. Bianchini, we show that even though the sticky particle solutions are not well-posed for every measure-type initial data, there exists a comeager set of initial data in the weak topology giving rise to a unique sticky particle solution. Moreover, for any of these initial data the sticky particle solution is unique also in the larger class of dissipative solutions (where trajectories are allowed to cross) and is given by a trivial free flow concentrated on trajectories which do not intersect. In particular for such initial data there is only one dissipative solution and its dissipation is equal to zero. Thus, for a comeager set of initial data the problem of finding sticky particle solutions is well-posed, but the dynamics that one sees is trivial. Our notion of dissipative solution is lagrangian and therefore general enough to include weak and measure-valued solutions._____________________________________________________________________________________________

Wednesday, November 10, 15:00-16:00George Avalos, University of Nebraska-Lincoln, USAZoom meeting:LinkSpeaker:Stability Analysis of interactive fluid and multilayered structure PDE dynamicsTitle:Abstract:In this talk, we will discuss our recent work on a certain multilayered structure-fluid interaction (FSI) which arises in the modeling of vascular blood flow. The coupled PDE system under our consideration mathematically accounts for the fact that mammalian veins and arteries are typically composed of various layers of tissues: each layer will generally manifest its own intrinsic material properties, and will moreover be separated from the other layers by thin elastic laminae. Consequently, the resulting modeling FSI system will manifest an additional PDE, which evolves on the boundary interface, so as to account for the thin elastic layer. (This is in contrast to the FSI PDE’s which appear in the literature, wherein elastic dynamics are largely absent on the boundary interface.) As such, the PDE system will constitute a coupling of 3D fluid-2D elastic-3D elastic dynamics. For this multilayered FSI system, we will in particular present results of strong stability for finite energy solutions, and polynomial decay for sufficiently regular solutions. This is joint work with Pelin Güven Geredeli and Boris Muha.

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