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Differential Equations Seminar

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Today

Peter J. Olver, University of Minnesota, Fractalization and Quantization in Dispersive Systems and Lamb Oscillators

SAS 4201

The evolution, through spatially periodic linear dispersion, of rough initial data leads to surprising quantized structures at rational times, and fractal, non-differentiable profiles at irrational times.  Such phenomena, also known as the Talbot effect, have been observed in dispersive waves, optics, and quantum mechanics, and lead to intriguing connections with exponential sums arising in number…

Raymond Yat Tin, UCLA, An algorithm for overcoming the curse of dimensionality in Hamilton-Jacobi equations

SAS 4201

In this talk we discuss an algorithm to overcome the curse of dimensionality, in possibly non-convex/time/state-dependent Hamilton-Jacobi partial differential equations.  They may arise from optimal control and differential game problems, and are generally difficult to solve numerically in high dimensions. A major contribution of our works is to consider an optimization problem over a single…

Pierre-Emmanuel Jabin, University of Maryland, College Park, Quantitative estimates of propagation of chaos for many-particle systems with bounded stream functions

SAS 4201

We derive quantitative estimates proving the propagation of chaos for large stochastic systems of interacting particles. We obtain explicit bounds on the relative entropy between the joint law of the particles and the tensorized law at the limit. We have to develop for this new laws of large numbers at the exponential scale. But our…

Tom Witelski, Duke University, Dynamics of non-conservative fluid thin films

SAS 4201

The behavior of thin layers of viscous fluids coating solid surfaces is of importance in many industrial and natural applications. Interfacial instabilities and pattern formation can occur in films on water-repellent (also called hydrophobic or partially wetting) surfaces. Lubrication models asymptotically reduce the governing equations for the free-surface flow to a fourth-order nonlinear parabolic partial differential equation for the evolution…

John Harlim, Penn State University, Data-driven methods for estimating operator and parameters of dynamical systems

SAS 4201

I will discuss a nonparametric modeling approach for forecasting stochastic dynamical systems on smooth manifolds embedded in Euclidean space. This approach allows one to evolve the probability distribution of non-trivial dynamical systems with an equation-free modeling. In the second part of this talk, I will discuss a nonparametric estimation of likelihood functions using data-driven basis functions and the…

Daphne Klotsa, University of North Carolina at Chapel Hill, A touch of non-linearity at intermediate Reynolds numbers: where spheres “think” collectively and swim together

SAS 4201

From crawling cells to orca whales, swimming in nature occurs at different scales. The study of swimming across length scales can shed light onto the biological functions of natural swimmers or inspire the design of artificial swimmers with applications ranging from targeted drug delivery to deep-water explorations. In this talk, I will present experiments and simulations…

Roman Shvydkoy, University of Illinois at Chicago, Topological models of singular Cucker-Smale dynamics

SAS 4201

In this talk we will discuss new classes of models that seek to describe evolution of a congregation of agents based on laws of self-organization. These models appear in a broad range of applications -- from biological sciences to social behavior. We focus on two long time phenomena: flocking and alignment. It has been a mathematical challenge…

Giovanna Guidoboni, University of Missouri, Mathematical and computational properties of differential equations for fluid flows in deformable domains

SAS 4201

This talk focuses on differential problems describing the flow of a viscous fluid in deformable domains. Such problems include flow in compliant tubes, often adopted for the modeling of arterial blood flow, and flow through deformable porous media, often adopted for the modeling of tissue perfusion. The mixed hyperbolic-parabolic-elliptic nature of these systems guides the…

Khai Nguyen, NC State, Recent Results on Compactness Estimates for Nonlinear PDEs

SAS 4201

This talk will be devoted to a fundamental question on the compactness of sets of solutions. The key concept in this study is the Kolmogorov epsilon-entropy which is the logarithm of the minimum number of elements in an epsilon-covering of a given (totally bounded) set. I will use this concept to provide a sharp estimate…

Yaiza Canzani, UNC Chapel Hill, Understanding the growth of Laplace eigenfunctions

In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of L^2 mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical…

Malgorzata Peszynska, Modeling hysteresis using ODEs with constraints: Numerical stability and other properties

SAS 4201

In nonlinear conservation laws the flux function f(u) is usually single valued, but in many important applications it is hysteretic, i.e., it assigns different values depending on whether the input u(t) is increasing or decreasing in t. We present our recent results on a hysteresis model built with a collection of auxiliary ODEs under constraints. The model shares some similarities…

Marco Mazzola, Paris 6, On the controllability of a set valued evolution

SAS 4201

We consider a controllability problem for the evolution of a set in V(t). This problem was originally motivated by a model where a dog controls a flock of sheep. Here, V(t) is the region occupied by the sheep and the position of the dog is regarded as a control function. We will discuss necessary conditions…

Konstantina Trivisa, University of Maryland, College Park, On the dynamics of compressible flows: variational solutions, invariant measures, martingale solutions

SAS 4201

In this talk I’ll describe an overview of results on the Navier-Stokes equations for viscous compressible flows both in the deterministic and stochastic frame- work. A contrast will be drawn between the one-dimensional flows and the multidimensional case. In the case of 1d isentropic compressible flow, the existence of invariant mea- sures will be established…

Rossana Capuani, NC State, Mean field games with state constraints

SAS 4201

This talk will address deterministic mean field games for which agents are restricted in a closed domain of R^n with smooth boundary. In this case, the existence and uniqueness of Nash equilibria cannot be deduced as for unrestricted state space because, for a large set of initial conditions, the uniqueness of solutions to the minimization…

Pedro Aceves Sanchez, NC State, Fractional diffusion limit of a linear kinetic transport equation in a bounded domain

SAS 4201

In recent years, the study of evolution equations featuring a fractional Laplacian has received much attention due to the fact that they have been successfully applied into the modelling of a wide variety of phenomena, ranging from biology, physics to finance. The stochastic process behind fractional operators is linked, in the whole space, to an…

Charis Tsikkou, West Virginia University, Radial solutions to the Cauchy problem for the wave equation and compressible Euler system

SAS 4201

In the first part of this work, we consider the strategy of realizing the solution of the three-dimensional linear wave equation with radial Cauchy data as a limit of radial exterior solutions satisfying vanishing Neumann and Dirichlet conditions, on the exterior of vanishing balls centered at the origin. We insist on robust arguments based on energy methods and strong convergence. Our findings show that while one…

H.T. Banks, North Carolina State University, Population Models-The Prohorov Metric Framework and Aggregate Data Inverse Problems

SAS 4201

We consider nonparametric estimation of probability measures for parameters in problems where only aggregate (population level) data are available. We summarize an existing computational method for the estimation problem which has been developed over the past several decades. Theoretical results are presented which establish the existence and consistency of very general (ordinary, generalized and other)…

Alexander Kiselev, Duke University, Small scale formation in ideal fluids

SAS 4201

The incompressible Euler equation of fluid mechanics describes motion of ideal fluid, and was derived in 1755. In two dimensions, global regularity of solutions is known, and double exponential in time upper bound on growth of the derivatives of solution goes back to 1930s. I will describe a construction of example showing sharpness of this…

Wen Shen, Penn State University, Scalar Conservation Laws with Discontinuous and Regulated Flux

SAS 4201

Conservation laws with discontinuous flux functions arise in various models. In this talk we consider solutions to a class of conservation laws with discontinuous flux, where the flux function is discontinuous in both time and space, but regulated in the two variables. Convergence and the uniqueness of the vanishing viscosity limit for the viscous equation…