Events
Differential Equations Seminar
- Events
- Differential Equations Seminar
Weiwei Hu, Oklahoma State University, An Approximating Control Design for Optimal Mixing by Stokes Flows
SAS 4201We consider an approximating control design for optimal mixing of a non-dissipative scalar eld in unsteady Stokes ows. The objective of our approach is to achieve optimal mixing at a given nal time, via an active control of the ow velocity through boundary inputs. Due to the zero diusivity of the scalar eld, establishing the well-posedness of its…
Michael L. Overton, NYU Courant, Numerical Investigation of Crouzeix’s Conjecture
SAS 4201In many applications one wishes to minimize an objective function that is not convex and is not differentiable at its minimizers. We discuss two algorithms for minimization of nonsmooth, nonconvex functions. Gradient Sampling is a simple method that, although computationally intensive, has a nice convergence theory. The method is robust and the convergence theory has…
Constantine Dafermos, Brown University, Energy dissipation at maximal rate
SAS 4201The lecture will consider the case of evolution equations, endowed with a "free energy," for which the initial value problem possesses multiple solutions, with an eye to examining whether the particular solution that maximizes the rate of energy decay enjoys a special status.
Peter J. Olver, University of Minnesota, Fractalization and Quantization in Dispersive Systems and Lamb Oscillators
SAS 4201The 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 4201In 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 4201We 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 4201The 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 4201I 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 4201From 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 4201In 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 4201This 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 4201This 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 4201In 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 4201We 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 4201In 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 4201This 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 4201In 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 4201In 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…