# NCSU Differential Equations/Nonlinear Analysis Seminar Schedule 2023-2024

Wednesday, Sep 20, 15:00-16:00,Zoom meeting:LinkSpeaker: Liviu Ignat, Institute of Mathematics Simion Stoilow

of the Romanian Academy, RomaniaTitle:Asymptotic behavior of solutions for some diffusion problems on metric graphsAbstract:In this talk we present some recent result about the long time behavior of the solutions for some diffusion processes on a metric graph. We study evolution problems on a metric connected finite graph in which some of the edges have infinity length. We show that the asymptotic behaviour of the solutions of the heat equation (or even some nonlocal diffusion problems) is given by the solution of the heat equation, but on a star shaped graph in which there is only one node and as many infinite edges as in the original graph. In this way we obtain that the compact component that consists in all the vertices and all the edges of finite length can be reduced to a single point when looking at the asymptotic behaviour of the solutions. We prove that when time is large the solution behaves like a gaussian profile on the infinite edges. When the nonlinear convective part is present we obtain similar results but only on a star shaped tree.

This is a joint work with Cristian Cazacu (University of Bucharest), Ademir Pazoto (Federal University of Rio de Janeiro), Julio D. Rossi (University of Buenos Aires) and Angel San Antolin (University of Alicante).

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Wednesday, Sep 27, 15:00-16:00,SAS 4201Speaker: Shaoming Guo, University of Wisconsin MadisonTitle:Oscillatory integral operators on manifolds and related Kakeya and Nikodym problemsAbstract:The talk is about oscillatory integral operators on manifolds. Manifolds of constant sectional curvatures are particularly interesting, and we will see that very good estimates on these manifolds can be expected. We will also discuss Kakeya and Nikodym problems on general manifolds, in particular, manifolds satisfying Sogge’s chaotic curvatures.

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Wednesday, Oct 04, 15:00-16:00,Zoom meeting:LinkSpeaker: Weinan Wang, University of OklahomaTitle:Global well-posedness and the stabilization phenomenon for some two-dimensional fluid equationsAbstract:In this talk, I will talk about some recent well-posedness and stability results for several fluid models in 2D. More precisely, I will discuss the global well-posedness for the 2D Boussinesq equations with fractional dissipation. For the Oldroyd-B model, we show that small smooth data lead to global and stable solutions. When Navier-Stokes is coupled with the magnetic field in the magneto-hydrodynamics (MHD) system, solutions near a background magnetic field are shown to be always global in time. The magnetic field stabilizes the fluid. In all these examples the systems governing the perturbations can be converted to damped wave equations, which reveal the smoothing and stabilizing effect.

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Wednesday, Oct 11, 15:00-16:00,Zoom meeting:LinkSpeaker: Eduardo Casas Renteria, University of CantabriaTitle:Second Order Analysis for Optimal Control ProblemsAbstract:In this talk, we discuss second-order optimality conditions for optimal

control problems. This analysis is very important when we study the stability of the solution to the control problem with respect to small perturbations of the data. It is also crucial for proving superlinear or quadratic convergence of numerical algorithms for solving the problem, as well as for proving error estimates for numerical discretization of the control problem. We begin by establishing the analogies and differences between the optimality conditions for finite and infinite dimensional optimization problems. Then, an optimal control problem of a semilinear parabolic equation is considered. We highlight the difference between problems that do or do not involve the Tikhonov term in the cost

functional.

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Wednesday, Oct 18, 15:00-16:00,Zoom meeting:LinkSpeaker: Giuseppe Buttazzo, University of Pisa, ItalyTitle:Antagonistic cost functionals in shape optimization.Abstract:In several shape optimization problems one has to deal with cost functionals of the form ${\cal F}(\Omega)=F(\Omega)+kG(\Omega)$, where $F$ and $G$ are two shape functionals with a different monotonicity behavior and $\Omega$ varies in the class of domains with prescribed measure. In particular, the cost functional ${\cal F}(\Omega)$ is not monotone with respect to $\Omega$ and the existence of an optimal domain in general may fail. An interesting situation occurs when the functional $F(\Omega)$ is minimized by a ball, while the functional $G(\Omega)$ is maximized by a ball; several examples of this kind are present in the literature. We consider the particular case ${\cal F}(\Omega)=\lambda(\Omega)T^q(\Omega)$ where $\lambda(\Omega)$ is the first eigenvalue of the Dirichlet Laplacian, and $T(\Omega)$ is the so-called torsional rigidity; the interesting cases are $q$ small for the minimum problem and $q$ large for the maximum problem.

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Wednesday, Oct 25, 15:00-16:00,SAS 4201Speaker: Alex Dunlap, Duke UniversityTitle:Stochastic heat equations and Cauchy distributionsAbstract:I will describe how an invariant measure with Cauchy-distributed marginals arises from a supercritical stochastic heat equation with an additional, independent additive noise. Joint work with Chiranjib Mukherjee.

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Wednesday, Nov 01, 15:00-16:00,Zoom meeting:LinkSpeaker: Leon Bungert, University of WürzburgTitle:Adversarial robustness in machine learning: from worst-case to probabilisticAbstract:In this talk I will first review recent results which characterize adversarial training (AT) of binary classifiers as nonlocal perimeter regularization. Then I will speak about a probabilistic generalization of AT which also admits such a geometric interpretation, albeit with a different nonlocal perimeter. Using suitable relaxations one can prove existence of solutions for a large family of such probabilistically robust training problems. Furthermore, these problems have (mildly surprising) connections to the conditional value at risk (CVaR), a quantity from math finance. I will also discuss local asymptotics of the probabilistic perimeters and mention some open problems. This talk is based on joint work with N. García Trillos, R. Murray, M. Jacobs, D. McKenzie, D. Nikolic, and Q. Wang.

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Wednesday, Nov 08, 15:00-16:00,SAS 4201Speaker: Tu Nguyen Thai Son, Michigan State UniversityTitle:Generalized convergence of solutions for nonlinear Hamilton-Jacobi equationsAbstract:We examine the asymptotic behaviors of solutions to Hamilton-Jacobi equations while varying the underlying domains. We establish a connection between the convergence of these solutions and the regularity of the additive eigenvalues in relation to the domains. To accomplish this, we introduce a framework based on Mather measures that enables us to compute the one-sided derivative of these additive eigenvalues under different scenarios, including first-order, second-order, and contact first-order equations. Additionally, we provide examples of how this framework can be applied to other settings.

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Wednesday, Nov 15, 15:00-16:00,Zoom meeting:LinkSpeaker: Anna Doubova, University of SevilleTitle:Inverse problems connected with Burgers equation and some related systemsAbstract:We consider inverse problems concerning the one-dimensional viscous Burgers equation and some related nonlinear systems (involving heat effects, variable density, and fluid-solid interaction). We are dealing with inverse problems in which the goal is to find the size of the spatial interval from some appropriate boundary observations. Depending on the properties of the initial and boundary data, we prove uniqueness and non-uniqueness results. Moreover, we also solve these inverse problems numerically and computing approximations of the interval sizes.

This is a joint work with Jone Apraiz (University of Basque Country), Enrique Fernández-Cara (University of Sevilla and IMUS) and Masahiro Yamamoto (University of Tokyo).

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Wednesday, Nov 29, 15:00-16:00,SAS 4201Speaker: Benjamin Seeger, University of Texas at AustinTitle:Weak solutions of nonlinear, nonconservative transport systemsAbstract:I will discuss certain systems of transport type whose coefficients depend nonlinearly on the solution. Applications of such systems range from the modeling of pressure-less gases to the study of mean field games in a discrete state space. I will identify a notion of weak solution within the class of coordinate-wise decreasing functions, a condition which has particular relevance for applications arising in economics. I demonstrate the existence of a unique maximal and minimal solution, in an appropriate sense, and the discrepancy between the two can be precisely related to the formation of shocks. I will also present a selection principle for the family of solutions based on vanishing (nonlinear) viscosity. The analysis depends on new well-posedness results for linear transport equations with certain rough velocity fields, which are of independent interest. This is joint work with P.-L. Lions.

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Wednesday, Jan 17, 15:00-16:00,SAS 4201Speaker: Zane Li, NCSUTitle:An introduction to harmonic analysis over Q_pAbstract:The goal of this talk is to introduce the p-adics (from an analyst’s point of view) and harmonic analysis over Q_p. I will discuss not only how this serves as a good toy model for Euclidean harmonic analysis but is also strong enough to derive interesting exponential sum estimates. Some results mentioned in this talk are joint work with Shaoming Guo and Po-Lam Yung.

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Wednesday, Jan 24, 15:00-16:00,SAS 4201Speaker: Nicolás García Trillos, University of Wisconsin MadisonTitle:A tractable algorithm, based on optimal transport, for computing adversarial training lower boundsAbstract:Despite the success of deep learning-based algorithms, it is widely known that neural networks may fail to be robust to adversarial perturbations of data. In response to this, a popular paradigm that has been developed to enforce robustness of learning models is adversarial training (AT), but this paradigm introduces many computational and theoretical difficulties. Recent works have developed a connection between AT in the multiclass classification setting and multimarginal optimal transport (MOT), unlocking a new set of tools to study this problem. In this talk, I will leverage the MOT connection to discuss new computationally tractable numerical algorithms for computing universal lower bounds on the optimal adversarial risk. The key insight in the AT setting is that one can harmlessly truncate high order interactions between classes, preventing the combinatorial run times typically encountered in MOT problems. I’ll present a rigorous complexity analysis of the proposed algorithm and validate our theoretical results experimentally on the MNIST and CIFAR-10 datasets, demonstrating the tractability of our approach. This is joint work with Matt Jacobs (UCSB), Jakwang Kim (UBC), and Matt Werenski (Tufts).

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Wednesday, Feb 07, 15:00-16:00,Zoom meeting:LinkSpeaker: Thierry Champion, University of Toulon, FranceTitle:Relaxed multi-marginal costs in optimal transport and quantization effectsAbstract:In this talk, I shall present a relaxation formula and duality theory for the multi-marginal Coulomb cost that appears in optimal transport problems arising in Density Functional Theory. The related optimization problems involve probabilities on the entire space and, as minimizing sequences may lose mass at infinity, it is natural to expect relaxed solutions which are sub-probabilities. I will introduce the duality framework associated to this kind of cost, and apply it to a model type of minimization problems for which it appears a mass quantization effect for the optimal solutions. All the necessary material on optimal transport will be introduced during the talk.

This is a joint work with G. Bouchitté (Univ. Toulon), G. Buttazzo (Univ. Pisa) and L. De Pascale (Univ. Firenze)

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Wednesday, Feb 14, 15:00-16:00,Zoom meeting:LinkSpeaker: Russell Luke, Universität GöttingenTitle:Inconsistent Nonconvex Feasibility – Foundations and Application to Orbital TomographyAbstract:Feasibility models are a powerful approach to many real-world problems where simply finding a point that comes close enough to meeting many, sometimes contradictory demands is enough. In this talk I will outline the theoretical foundations for the convergence analysis of fixed point iterations of expansive mappings, and show how this specializes to fundamental algorithms for nonconvex, inconsistent feasibility problems. As a concrete instance, I will demonstrate the ideas and results on an orbital tomography problem of reconstructing 3 dimensional electron orbitals of small molecules directly from electron scattering measurements. Not only is this a nice demonstration of beautiful mathematics, but the resulting algorithms have enabled us to observe for first time electronic orbitals at several Angstrom resolutions directly from measurements.

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Wednesday, Feb 21, 15:00-16:00,Zoom meeting:LinkSpeaker: Ming Chen, University of PittsburghTitle:Global bifurcation for hollow vortex desingularizationAbstract:A hollow vortex is a region of constant pressure bounded by a vortex sheet and suspended inside a perfect fluid; we can think of it as a spinning bubble of air in water. In this talk, we present a general method for desingularizing non-degenerate steady point vortex configurations into collections of steady hollow vortices. The machinery simultaneously treats the translating, rotating, and stationary regimes. Through global bifurcation theory, we further obtain maximal curves of solutions that continue until the onset of a singularity. As specific examples, we obtain the first existence theory for co-rotating hollow vortex pairs and stationary hollow vortex tripoles, as well as a new construction of Pocklington’s classical co-translating hollow vortex pairs. All of these families extend into the non-perturbative regime, and we obtain a rather complete characterization of the limiting behavior along the global bifurcation curve. This is a joint work with Samuel Walsh (Missouri) and Miles Wheeler (Bath).

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Wednesday, Feb 28, 15:00-16:00,Zoom meeting:LinkSpeaker: Hakima Bessaih, FIUTitle:Various numerical scheme for stochastic hydrodynamic modelsAbstract:We will consider various models in hydrodynamic, including the 2d Navier-Stokes, Boussinesq equations, and a Brinkman-Forchheimer-Navier-Stokes equations in 3d. These models are driven by an external stochastic Brownian perturbation.

We will implement space-time numerical schemes and prove their convergence. We will show some rates of convergence as well. Furthermore, we will show the difference between the stochastic and the deterministic cases.

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Wednesday, Mar 06, 15:00-16:00,Zoom meeting:LinkSpeaker: Edouard Pauwels, Université de ToulouseTitle:Nonsmooth differentiation of parametric fixed points.Abstract:Recent developments in the practice of numerical programming require optimization problems not only to be solved numerically, but also to be differentiated. This allows to integrate the computational operation of evaluating a solution in larger models, which are themselves trained or optimized using gradient methods. Most well known applications include bilevel optimization and implicit input-output relations in deep network models. Fixed point of contraction mappings provide a natural high level description of these applications. We will briefly describe the interplay between implicit differentiation and derivatives of fixed point iterations in the smooth settings, dating back to early works in automatic differentiation. Motivated by the possibility to differentiate solutions of nonsmooth or constrained problems, we will describe recent extension of these results to the conservative gradient setting, a notion of generalized derivative which is compatible with calculus and gradient type optimization methods. We will also cover explicit examples related to the differentiation of solutions to monotone inclusions. Joint work with Jérôme Bolte, Tony Silvetti-Falls, Tam Le, Samuel Vaiter.

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Wednesday, March 20, 15:00-16:00, SAS 4201Speaker: Wojciech Ozanski, FSUTitle:Logarithmic spiral vortex sheetsAbstract:We will discuss a special family of 2D incompressible inviscid fluid flows in the form of logarithmic spiral vortex sheets. Such flows are determined by a vorticity distribution of a curve R^2, and they are notoriously hard to study analytically. In the talk we will discuss several results regarding logarithmic spiral vortex sheets: well-posedness of the spirals as solutions to the Euler equations (despite some recent evidence for the contrary), existence of nonsymmetric spirals, and linear instability of the symmetric spirals

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Wednesday, Mar 27, 15:00-16:00,SAS 4201Speaker: Hung Tran, University of Wisconsin MadisonTitle:Periodic homogenization of Hamilton-Jacobi equations: some recent progressAbstract:I first give a quick introduction to front propagations, Hamilton-Jacobi equations, level-set forced mean curvature flows, and homogenization theory. I will then show the optimal rates of convergence for homogenization of both first-order and second-order Hamilton-Jacobi equations. Based on joint works with J. Qian, T. Sprekeler, and Y. Yu.

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Wednesday, April 03, 15:00-16:00,Zoom meeting:LinkSpeaker: Jameson Graber, Baylor UniversityTitle:The Master Equation in Mean Field Game TheoryAbstract:Mean field game theory was developed to analyze Nash games with large numbers of players in the continuum limit. The master equation, which can be seen as the limit of an N-player Nash system of PDEs, is a nonlinear PDE equation over time, space, and measure variables that formally gives the Nash equilibrium for a given population distribution. In this talk I will emphasize the fact that the master equation can be seen as a nonlinear transport equation. In particular, the Nash equilibrium is unique if and only if the characteristics do not cross, and when they do cross, we are faced with the question of making a rational selection among multiple equilibria. I will provide some examples to show how subtle this problem is, and in particular I will show that the usual theory of entropy solutions is in general not sufficient for the purposes of equilibrium selection.

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Wednesday, April 10, 15:00-16:00,SAS 4201Speaker: Olivier GlassTitle:Small solids in Euler flowsAbstract:In this talk, I will discuss the evolution of rigid bodies in a perfect incompressible fluid, and the limit systems that can be obtained as the bodies shrink to points. The model is as follows: the fluid is driven by the incompressible Euler equation, while the solids evolve according Newton’s equations under the pressure force on their boundary. We investigate the question of the limit of the system as (some of) the solids converge to a point while keeping a constant velocity circulation on their boundary. We obtain a complete picture when each solid can belong to either of the three categories:

— solids that have a fixed size,

— solids that shrink to a point while keeping a fixed mass,

— solids that shrink to a point and having mass going to zero.

In the limit, we obtain a system coupling the Euler equation for the fluid, Newton’s equations for the non-shrinking bodies, and massive/non-massive point vortices for the remaining ones.

This is a joint work with Franck Sueur (Bordeaux, France), following works with Christophe Lacave (Grenoble, France), Alexandre Munnier (Nancy, France) and Franck Sueur.

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Wednesday, April 17, 15:00-16:00,Zoom meeting:LinkSpeaker: Aris Daniilidis, TU WienTitle:Slope determination: from convex to Lipschitz continuous functionsAbstract:A convex continuous function can be determined, up to a constant,

by its remoteness (distance of the subdifferential to zero). Based on this result, we discuss possible extensions in three directions: robustness (sensitivity analysis), slope determination (in the Lipschitz framework) and general determination theory.

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