PDE and Applied Mathematics Seminar

Spring 2013

Organized by Dmitry Golovaty and Peter Gordon

Seminar is usually scheduled on Thursdays at 12:30 in CAS 436

January 22 (Leigh Hall 305, 3:30-4:30 pm). Vitaly Moroz (Swansea University) will speak on "Existence, nonexistence and optimal decay of solutions to nonlinear Choquard equations".

Abstract: Choquard (or Hartree, or nonlinear Schrodinger-Newton) equation is a stationary nonlinear Schrodinger type equation where the nonlinearity is coupled with a nonlocal convolution term (gravitational potential). We present sharp Liouville-type theorems on nonexistence of positive supersolutions of such equations in exterior domains. We also discuss existence and optimal decay properties of positive solutions under various assumptions on the decay of the external potential and the shape of the nonlinearity. This is joint work with Jean Van Schaftingen (Louvain-la-Neuve, Belgium).

January 31. George Chase (University of Akron) will speak on "Opportunities for Mathematical Models in Research in Filtration, Fluid-Droplet Separations, and Electrospinning"

Abstract: Brief descriptions of several current research projects will be presented. The presentation is informal to encourage discussion and exchange of ideas for mathematical modeling. Ideas to simplify the project descriptions or math models are welcome. The discussions may lead to future collaborations.
Projects that may be discussed include:
Droplet coalescence in thick and thin depth filters in gas-liquid and liquid-liquid systems,
Prediction of droplet motion and coalescence on surfaces of fibers in flow fields as individual fibers, thin fiber mats, and depth fiber media,
Motion and coalescence of droplets in electrowetting systems,
Calculation of contact lines between drops and curved surfaces (in particular, cylinders),
Calculation of electric fields in electrospinning processes in several geometries, with or without effects of the presence of the charged electrospinning jets,
Effect of a vibrating porous screen on the flow rate of a fluid moving through the screen,
Effects of screen vibrations on sand filter cake properties and motion of the filter cake across the screen,
Effects of vibrations of a fiber, fiber mat, or fiber filter medium on the motion of drops on the fibers.

February 7. Robert V. Kohn (Courant Institute, NYU) will speak on "A variational perspective on wrinkling patterns in thin elastic films."

Abstract: Thin sheets exhibit a daunting array of patterns. A key difficulty in their analysis is that while we have many examples, we have no classification of the possible "patterns." I have explored an alternative viewpoint in a series of recent projects with Peter Bella, Hoai-Minh Nguyen, and others. Our goal is to identify the scaling law of the minimum elastic energy (with respect to the sheet thickness, and the other parameters of the problem). Success requires proving upper bounds and lower bounds that scale the same way. The upper bounds are usually easier, since nature gives us a hint. The lower bounds are more subtle, since they must be ansatz-independent. In many cases, the arguments used to prove the lower bounds help explain "why" we see particular patterns. My talk will give an overview of this activity, and details of some examples.

February 14. Oleg D. Lavrentovich ( Kent State University) will speak on "Statics and Dynamics of Colloidal Particles in Liquid Crystals."

Abstract: Colloids and liquid crystals are two important classes of soft matter, usually explored independently of each other. The most studied colloids represent a dispersion of solid or liquid particles in an isotropic fluid such as water. The simplest liquid crystal, a nematic, is a fluid with long-range orientational order of molecules. This presentation reviews recent studies of liquid crystal colloids, i.e., dispersions of particles in a liquid crystal. The long-range orientational order imparts anisotropic elastic interactions of colloidal particles [1]. Elastic repulsion from the bounding walls opposes gravity and keeps the particles levitating in the liquid crystal bulk [2]. The levitating particles can be set into motion by applying an electric field. Liquid crystals enable a new mechanism of electrophoresis, in which the particle's velocity is proportional to the square of the electric field. The latter is dramatically different from the classic electrophoresis with a linear relationship. The nonlinear feature allows one to use an AC electric field driving to maintain steady flows, to move particles that are electrically neutral and to design three dimensional trajectories of their motion.
[1] P. Poulin, H. Stark, T.C. Lubensky and D.A. Weitz, Novel colloidal interactions in anisotropic fluids. Science 275, 1770 (1997).
[2] O.P. Pishnyak, S.V. Shiyanovskii, O.D. Lavrentovich, Inelastic collisions and anisotropic aggregation of particles in a nematic collider driven by backflow, Phys. Rev. Lett. 106, 047801 (2011).
[3] O.D. Lavrentovich, I. Lazo and O.P. Pishnyak, Nonlinear electrophoresis of dielectric and metal spheres in a nematic liquid crystal, Nature 467, 947 (2010).

February 21. Jonathan Selinger (Kent State University) will speak on "Curvature and defects in soft membranes with orientational order."

Abstract: Soft membranes have a coupling between curvature and in-plane orientational order: Defects in the orientational order can induce curvature, and conversely, curvature leads to an effective geometrical potential acting on defects. Recently, our group has done simulations which show that the interaction between curvature and defects depends on several important issues, including the baseline curvature of the membrane (flat, cylinder, sphere, torus), the phase of the defects (radial or tangential), and the relative contribution of in-plane (intrinsic) vs. out-of-plane (extrinsic) variations of the director. To understand the simulations, we develop a theoretical approach that can address those issues. Using this approach, we calculate the energy of defect structures in curved geometries, and determine how the energy varies as a function of the defect position and separation and the membrane distortions. The interaction energy depends on the relative magnitude of intrinsic vs. extrinsic couplings, and on the mechanical properties of the membrane. This approach provides opportunities to design membranes that will relax into selected shapes.

February 28. Adrian Tudorascu (West Virginia University ) will speak on "Weak Lagrangian solutions for the Semi-Geostrophic system in physical space"

Abstract: Proposed as a simplification for the Boussinesq system in a special regime, the Semi-Geostrophic (SG) system is used by metereologists to model how fronts arise in large scale weather patterns. We shall argue that weak (Eulerian) solutions for the Semi-Geostrophic system in physical space exhibiting some mild regularity in time cannot yield point masses in the dual space. However, such solutions are physically relevant to the model. Thus, we shall discuss a natural generalization of Cullen & Feldman's weak Lagrangian solutions in the physical space to include the possibility of singular measures in dual space. This presentation is based on joint work with M. Feldman (UW Madison).

March 7. Anna Ghazaryan (Miami University) will speak on "Fronts in a model for gasless combustion with heat loss."

Abstract: We consider a model of gasless combustion with heat loss, where the heat loss from the system to the environment is formulated according to Newton's law of cooling. The system of partial differential equations that describe evolution of the temperature and remaining fuel contains two small parameters, a diffusion coefficient for the fuel and a heat loss parameter. We use geometric singular perturbation theory to show existence of traveling waves in this system and then study their spectral and nonlinear stability.

March 21. Peter Sternberg (Indiana University, Bloomington) will speak on "Motion of Ginzburg-Landau and Gross-Pitaevskii Vortices on Surfaces".

Abstract: I will discuss some recent work regarding vortex behavior on surfaces in the dissipative Ginzburg-Landau setting and conservative Gross-Pitaevskii setting. This includes results generalizing the asymptotic vortex motion laws (systems of ODE's), previously obtained for the plane, to the setting of surfaces, as well as results on existence of special vortex solutions to GP on the two-sphere. These results were jointly obtained with Ko-Shin Chen and in part with Michael Gelantalis.

April 12 (11am in CAS 133) Alexei Novikov (Penn State) will speak on "Exit times of diffusions with incompressible drift."

Abstract: Consider a Brownian particle in a prescribed time-independent incompressible flow in a bounded domain. We investigate how the strength of the flow and its geometric properties affect the expected exit time of the particle. The two main questions we analyze in this talk are as follows. 1. Incompressible flows are known to enhance mixing in many contexts, but do they also always decrease the exit time? We prove that the answer is no, unless the domain is a disk. 2. Suppose the flow is cellular with amplitude A, and the domain is of size L. What could be said about the exit time when both L and A are large? We prove that there are two characteristic regimes: a) if L << A^4, then the exit time from the entire domain is compatible with the exit time from a single flow cell, and it can be determined from the Freidlin-Wentzell theory; b) if L>> A^4, then the problem `homogenizes' and the exit time is determined by the effective diffusivity of cellular

April 18. Ratnasingham Shivaji (UNC Greensboro) will speak on "Uniqueness results for classes of singular nonlinear eigenvalue problems"

Previous seminars:
Fall 2012