Workshop on Nonlinear Differential Equations, Dynamical Systems and Applications

Plenary Speakers

Alberto Bressan, Pennsylvania State University

Multiple Solutions for the 2-Dimensional Euler Equations

For some time, it was expected that well-posedness results known for one- dimensional hyperbolic conservation laws could be extended to the multidimensional setting. However, fundamental work by De Lellis, Szekelyhidi, and collaborators, has shown that multidimensional hyperbolic Cauchy problems usually possess infinitely many weak solutions. Moreover, the known entropy criteria fail to select a single admissible one. In the first part of this talk I shall recall this approach based on a Baire category argument, yielding the existence of uncountably many weak solutions.

I will then discuss an alternative research program, aimed at constructing multiple solutions to some specific Cauchy problems. Starting with some numerical simulations, here the eventual goal is to achieve rigorous, computer-aided proofs of the existence of two distinct self-similar solutions with the same initial data. While solutions obtained via Baire category have turbulent nature, these self-similar solutions have Holder vorticity, and are smooth with the exception of one or two points of singularity. They are thus much easier to visualize and understand.

Chun Liu, Illinois Institute of Technology 

Charge Transport in Biological Environments

Almost all biological activities involve transport of charged ions. Ion channels play critical roles in biological systems including heart and nerves. The study of the dynamic properties of these systems provide formidable challenges and exciting opportunities for interdisciplinary researches and collaborations. Among them, the thermal effects play crucial role in the dynamics. In this talk, I will discuss a unified energetic variational approach developed specifically for these multiscale-multiphysics FFSI (field-fluid-structure interaction) problems. I will present some relevant theories, approaches and methods developed in the area.

Chongchun Zeng, Georgia Institute of Technology

Local Dynamics Near Traveling Waves of Hamiltonian PDEs: From the Linearization to Invariant Manifolds

Motivated by the stability/instability analysis of coherent states (standing waves, traveling waves, etc.) of nonlinear Hamiltonian PDEs such as BBM, GP, gKdV, and 2-D Euler equations, we first consider a general linear Hamiltonian system ut = JL u in a real Hilbert space X—the energy space. The main assumption is that the energy functional áLu, uñ has only finitely many negative dimensions—n-(L) < ¥, but little assumption on J other than J*=-J. After obtaining an L-orthogonal decomposition of X partially invariant under JL, our general results include an index theorem on the algebraic multiplicities of eigenvalues of JL, the linear exponential trichotomy and spectral mapping of the group tJLe , and the robustness of the stability/instability under small Hamiltonian perturbations. Based on the linear analysis, we study the nonlinear local dynamics and construct invariant manifolds of traveling wave manifolds in the energy space of the Gross-Pitaevskii equation in R . Unlike standing waves, the spatial translation involved in the study of local dynamics near traveling waves cause regularity issues, which we overcome by adopting a bundle coordinate system and applying certain space-time estimates.

Invited Speakers

Jiayin Jin, Georgia Institute of Technology

Dynamics Near the Solitary Waves of the Supercritical gKDV Equations

We construct smooth local center-stable, center-unstable and center manifolds near the manifold of solitary waves and give a detailed description of the local dynamics near solitary waves. In particular, the instability is characterized as following: any forward flow not starting from the center-stable manifold will leave a neighborhood of the manifold of solitary waves exponentially fast. Moreover, orbital stability is proved on the center manifold, which implies the uniqueness of the center manifold and the global existence of solutions on it. This is a joint work with Zhiwu Lin and Chongchun Zeng.

Pei Liu, Pennsylvania State University

Non-Isothermal Electrokinetics: Energetic Variational Approach

Fluid dynamics accompanies with the entropy production thus increases the local temperature, which plays an important role in charged systems such as the ion channel in biological environment and electrodiffusion in capacitors/batteries. In this article, we propose a general framework to derive the transport equations with heat flow through the Energetic Variational Approach. According to the rst law of thermodynamics, the total energy is conserved and we can use the Least Action Principle to derive the conservative forces. From the second law of thermodynamics, the entropy increases and the dissipative forces can be computed through the Maximum Dissipation Principle. Combining these two laws, we then conclude with the force balance equations and a temperature equation. To emphasis, our method provide a self consistent procedure to obtain the dynamical equations satisfying proper energy laws and it not only works for the charge systems but also for general systems.

Lina Ma, Trinity College

Coarse-Graining Langevin Dynamics using Reduction of Order Techniques

Langevin dynamics models arise from a wide variety of problems, especially where a mechanical system is subject to random forces that can be modeled by white noise. In this talk, I will present a derivation of a coarse-grained description, in the form of a generalized Langevin equation, from the Langevin dynamics model that describes the dynamics of bio-molecules. To facilitate the construction and implementation of the reduced models, the problem is then formulated as a reduced-order modeling problem. The reduced models can then be directly obtained from a Galerkin projection to appropriately defined Krylov subspaces.

Hongwei Mei, University of Kansas

Comparison Principle for a Hamilton-Jacobi Equation

In this talk, I will talk about the Comparison principle for a Hamilton-Jacobi equation which is from stochastic vortex dynamics.

Stefan Metzger, Illinois Institute of Technology 

Two-Phase Flow in Porous Media: An Upscaling Approach

Classical models for two-phase flow in porous media are based on a capillary pressure - saturation relationship. However, it is known for a long time that this approach has some shortcomings, as capillary pressure - saturation curves for drainage and imbibition show hysteresis effects. Applying homogenization techniques, we investigate the relation between saturation, droplet topology, and capillary pressure. Thereby, a phase-field description of the microscopic two-phase flow is the method of choice, as no artificial additional conditions are necessary to model the arising topological changes.

Tien Khai Nguyen, North Carolina State University

Kolmogorov Entropy Compactness Estimates for Hamilton-Jacobi Equations

Inspired by a question posed by Lax in 2002, in recent years it has received an increasing attention the study on the quantitative analysis of compactness for nonlinear PDEs. In this talk, I will present recent results on the sharp compactness estimates in terms of Kolmogorov epsilon entropy for Hamilton-Jacobi equations. Estimates of this type play a central role in various areas of information theory and statistics as well as of ergodic and learning theory. In the present setting, this concept could provide a measure of the order of “resolution” of a numerical method for the corresponding equation.

Ian Tice, Carnegie Mellon University 

The Stability of Contact Lines in Fluids

The contact line problem in interfacial fluid mechanics concerns the triple-junction between a fluid, a solid, and a vapor phase. Although the equilibrium configurations of contact lines have been well-understood since the work of Young, Laplace, and Gauss, the understanding of contact line dynamics remains incomplete and is a source of work in experimentation, modeling, and mathematical analysis. In this talk we consider a 2D model of contact point (the 2D analog of a contact line) dynamics for an incompressible, viscous, Stokes fluid evolving in an open-top vessel in a gravitational field. The model allows for fully dynamic contact angles and points. We show that small perturbations of the equilibrium configuration give rise to global-in-time solutions that decay to equilibrium exponentially fast. This is joint work with Yan Guo.

Charis Tsikkou, West Virginia University

On Similarity Flows for the Compressible Euler System

In this work we review the construction of globally defined radial similarity shock and cavity flows, and give a detailed description of their behavior following collapse. We then prove that similarity shock solutions provide bona fide weak solutions, of unbounded amplitude, to the multi-dimensional Euler system.

We also point out that both types of similarity flows involve regions of vanishing pressure prior to collapse (due to vanishing temperature and vacuum, respectively)— raising the possibility that Euler flows may remain bounded in the absence of such regions. This is joint work with Helge Kristian Jenssen (PSU).

Samuel Walsh, University of Missouri

Capillary-Gravity Water Waves with Exponentially Localized Vorticity

In this talk, we discuss recent success in establishing the existence of solutions to the water wave problem with exponentially decaying vorticity. These are two-dimensional stationary waves in a finite-depth body of water beneath vacuum. An external gravitational force acts in the bulk, and the effects of surface tension are felt on the air- sea interface. Our approach involves modeling the corresponding stream function as a spike solution to a singularly perturbed elliptic PDE. This is joint work with Mats Ehrnstrom (NTNU) and Chongchun Zeng (Georgia Tech).

Yao Yao, Georgia Institute of Technology 

Congested Aggregation via Newtonian Interaction

In this talk, we consider a congested aggregation model that describes the evolution of a density through the competing effects of nonlocal Newtonian attraction and a hard density constraint. It is formulated as the Wasserstein gradient flow of an interaction energy, with a penalization to enforce the density constraint. From this perspective, the problem can be seen as a slow diffusion limit of the Keller-Segel equation with degenerate diffusion. We focus on the patch dynamics where the initial data is a characteristic function, and show that the solution remains a patch for all time, and its boundary evolution is given by a Hele-Shaw type free boundary problem. In addition, in two dimensions, we show that all patch solutions will converge to a disk as the time goes to infinity with certain convergence rate. This is a joint work with Katy Craig and Inwon Kim.

Mingji Zhang, New Mexico Institute of Mining and Technology

Qualitative Properties of Ionic Flows via Poisson-Nernst-Planck Systems with Local Electrochemical Potentials

We study the Poisson-Nernst-Planck model for ionic flows through membrane channels for two ion species, one positively charged and one negatively charged. Of particular interest is the effects from finite ion sizes and boundary layers. More precisely, we first examine the flow properties of interest in terms of both individual fluxes and total flux of charge under the so-called electroneutrality boundary conditions. Interesting phenomena are observed. Furthermore, some important critical potentials are identified, which are crucial in characterizing the ion size effects on ionic flows. However, under the neutral conditions, two boundary layers disappears, while the effects from boundary layer could be important. This leads to our second part in this work, boundary layer effects on ionic flows. Compared to the first part, new features are observed, which further indicates the rich dynamics of ionic flows through membrane channels.

Qingtian Zhang, University of California-Davis 

Global Solution of SQG Front Equation

We consider a nonlinear, spatially-nonlocal initial value problem in one space dimension on R for the motion of surface quasi-geostrophic (SQG) fronts. We prove that the initial value problem has unique local smooth solutions and, under a smallness assumption on the initial data, that these solutions are global. This is a joint work with John Hunter and Jingyang Shu.