Ohio State University Partial Differential Equations Seminar    Year 2013-2014 Time/Location: Wednesdays 4:10-5:05pm/MW154 (unless otherwise noted)

Schedule of talks:

Abstracts

Chris Cosner (University of Miami) on October 2, 2013
Title: Evolutionary Stability of Ideal Free Dispersal Strategies: A Nonlocal Dispersal Model

Abstract: The dispersal of organisms has many significant ecological effects, and hence the evolution of dispersal has been a subject of considerable interest in evolutionary ecology. An important problem in the study of the evolution of dispersal is determining what kinds of dispersal strategies are evolutionarily stable in the sense that populations using them cannot be invaded by ecologically similar populations using other strategies. A class of strategies that have been shown to be evolutionarily stable in various contexts are those that produce an ideal free distribution of the population, that is, a spatial distribution where no individual can increase its fitness by moving to another location. This talk will present results on the evolutionary stability of ideal free dispersal strategies in the context of continuous time nonlocal dispersal models. These results partially extend some recent work on the evolutionary stability of ideal free dispersal for reaction-advection-diffusion equations and discrete diffusion models to nonlocal dispersal models. They also include an extension of an inequality from matrix theory to the case of nonlocal dispersal operators, which may be of independent interest.

Steve Cantrell (University of Miami) on October 9, 2013
Title: Avoidance Behavior in Intraguild Predation Commurnities: A Cross Diffusion Model

Abstract: A cross-diffusion model of an intraguild predation community, in which the intraguild prey species employs a fitness based avoidance strategy, is examined. The avoidance strategy employed is to increase motility in response to negative local fitness. Global existence of trajectories and the existence of a compact global attractor are proved. It is shown that if the intraguild prey has positive fitness at any point in the habitat when trying to invade, then it will be uniformly persistent in the system if its avoidance tendency is sufficiently strong. This type of movement strategy can lead to coexistence states where the intraguild prey is marginalized to areas with low resource productivity while the intraguild predator maintains high densities in regions with abundant resources, a pattern observed in many real world intraguild predation systems. This is joint work with Dan Ryan at NIMBIOS, University of Tennessee.

Marco Fontelos (Universidad Autonoma de Madrid) on November 6, 2013
Title: Mathematical modelling of Electrowetting phenomena

Abstract: The term electrowetting is commonly used for some techniques to change the shape and wetting behaviour of liquid droplets by the application of electric fields and charges. First, we describe the presence of symmetry breaking bifurcations and their physical role. We then develop and analyze a model for electrowetting that combines the Navier–Stokes system for fluid flow, a phase-field model of Cahn–Hilliard type for the movement of the interface, a charge transport equation, and the potential equation of electrostatics. A critical role in the deduction of suitable couplings between phase field and other physical fields is played by variational principles similar to those that apply for gradient flows. A consequence of such principle is the deduction of energy estimates that serve to prove existence and uniqueness of solutions for the resulting system of equations.

Mariana Smit Vega Garcia (Purdue University) on November 13, 2013
Title: New developments in the thin obstacle problem with Lipschitz coefficients

Abstract: We will describe the lower-dimensional obstacle problem for a uniformly elliptic, divergence form operator $L = div(A(x)\nabla)$ with Lipschitz continuous coefficients and discuss the optimal regularity of the solution. Our main result states that, similarly to what happens when $L = \Delta$, the variational solution has the optimal interior regularity. We achieve this by proving some new monotonicity formulas for an appropriate generalization of Almgren's frequency functional.

Bo Guan (Ohio State) on November 20, 2013
Title: The concavity and subsolution in estimates for fully nonlinear elliptic equations

Abstract: We report recent progresses in our effort of seeking methods to derive a priori second order estimates for fully nonlinear elliptic equations under general structure conditions. In this talk we shall focus on the Dirichlet problem in Euclidean space. The main topic will be the role of concavity and subsolutions.

Bo Guan (Ohio State) on Jan 22, 2014
Title: The Dirichlet Problem for Fully Nonlinear Elliptic Equations

Abstract: We report our recent work on fully nonlinear elliptic equations on Riemannian manifolds. We introduce some new techniques in deriving a priori estimates, which can be used to treat a variety of types of fully nonlinear elliptic and parabolic equations on real or complex manifolds. As a result we are able to solve the Dirichlet problem in a domain with no geometric restrictions under optimal structure conditions. The result is new and optimal in the Euclidean case.

Feride Tiglay (Ohio State) on Feb 12, 2014
Title: Integrable Evolution Equations on Spaces of Tensor Densities

Abstract: We present how two nonlinear partial differential equations arise naturally as Euler-Arnold equations on spaces of tensor densities. These equations, like several other equations from mathematical physics that fit in the same framework, possess some hallmarks of integrability. We discuss the well posedness of the Cauchy problem and break down of solutions.

Avner Friedman (Ohio State) on Feb 19, 2014
Title: The Dynamics of Ant Trails

Abstract: Armies of ants are known to move in trails. These trails are formed by a chemotactic force induced by pheromone secreted by the ants. In this talk I shall introduce a mathematical model consisting of two partial differential equations, which explain when and how these trails are formed. The first equation, for the ants, includes the chemotaxis effect of pheromone and the dispersion caused by overcrowding. The second equation is a reaction-diffusion equation for the pheromone concentration. The strength of the chemotactic force, χ, plays a critical role in the analysis. We prove that trails cannot be formed if χ is small, while many trails exist if χ is large.

Stephen Preston (Colorado) on Feb 26, 2014
Title: Higher-dimensional Hunter-Saxton and Camassa-Holm equations

Abstract: The Hunter-Saxton equation arises in a model of liquid crystals, while the Camassa-Holm equation arises in the study of water waves. Both equations are one-dimensional PDEs which are both completely integrable and are geodesic equations on infinite-dimensional manifolds. I will discuss some of their basic properties along with some natural generalizations of these equations to higher space dimensions.

Xiangwen Zhang (Columbia) on March 19, 2014
Title: A Proof of the Alexanderov's Uniqueness Theorem for Convex Surfaces in R^3

Abstract: A classical uniqueness theorem of Alexandrov says that: if M and M' are two closed strictly convex C^2 surface in R^3 and satisfy f(k1,k2)=f(k1',k2'), at pots of M, M' with parallel normals, for some C^1 function f(y1,y2) with (D_{y1}f)(D_{y2}f)>0, then M is equal to M' up to a translation. We will talk about a new PDE proof for this theorem by using the maximal principle and weak unique continuation theorem of Bers-Nirenberg. More generally, we prove a version of this theorem with certain weaker regularity assumption: the spherical hessians of the supporting functions for the corresponding convex bodies as Radon measures are nonsingular. This is a joint work with P. Guan and Z. Wang.

Yaping Wu (Capital Normal University, China) on March 20, 2014
Title: Steady States and Traveling Waves for SKT Competition Model with Cross-Diffusion

Abstract: In this talk we shall be focused on a quasilinear reaction diffusion system with cross diffusion, which was first proposed by Shigesada, Kawasaki and Teramoto in 1979 for investigating the spatial segregation of two competing species under inter- and intra-species population pressures. I shall talk about some recent research progress on the existence and stability of nontrivial steady states and travelling waves for the S-K-T competition model with cross diffusion, which may correspond to some new pattern formation and wave phenomena induced by cross diffusion.