# PDE Seminar

 Fridays 4:10pm, Stevenson Center 1307 Date: Friday, September 7, 2012 Speaker:  Jeremy LeCrone, Vanderbilt University Title: Stability and bifurcation of equilibria for the axisymmetric surface diffusion flow Abstract: The surface diffusion flow is a fourth-order quasilinear evolution law which models the motion of some surfaces in the presence of high temperatures. I will focus on the setting of two-dimensional surfaces which exhibit symmetry about a fixed axis of rotation and satisfy periodic boundary conditions.  I will discuss equilibria of the flow in this setting and analytic methods involved with stability/instability and bifurcation results. . Date: Friday, September 14, 2012 Speaker: Mary Ann Horn, NSF and Vanderbilt University Title: Mathematical challenges arising from the questions of controllability for linked elastic structures Abstract:  A tremendous challenge in the study of control and stabilization of dynamic elastic systems is the ability to rigorously address whether linked dynamic structures can be controlled using boundary feedback alone. When a structure is composed of a number of interconnected elastic elements or is modeled by a system of coupled partial differential equations, the behavior becomes much harder to both predict and to control. This talk focuses on a number of issues that arise when attempting to control these complex systems. Date: Monday, September 17, 4:10 pm, SC 1308  (jointly with the Symplectic and Differential Geometry Seminar) Speaker: Marcelo Disconzi, Vanderbilt University Title: Compactness results for the Yamabe problem, I Abstract: Abstract Date: Monday, September 24, 4:10 pm, SC 1308  (jointly with the Symplectic and Differential Geometry Seminar) Speaker: Marcelo Disconzi, Vanderbilt University Title: Compactness results for the Yamabe problem, II Abstract:  Abstract Date:  Monday, October 8, 4:10 pm, SC 1308  (jointly with the Symplectic and Differential Geometry Seminar) Speaker: Marcelo Disconzi, Vanderbilt University Title: Compactness results for the Yamabe problem, III Abstract:   Abstract Date: Friday, October 12, 2012 Speaker: Christoph Walker, Leibniz Unversität Hannover, Germany Title: Positive equilibrium solutions in structured population dynamics Abstract:  The talk focuses on positive equilibrium (i.e. time-independent) solutions to mathematical models for the dynamics of populations structured by age and spatial position. This leads to the study of quasilinear parabolic equations with nonlocal and possibly nonlinear initial conditions. We shall see in an abstract functional analytic framework how bifurcation techniques may be combined with optimal parabolic regularity theory to establish the existence of positive solutions. As an application of these results we give a description of the geometry of coexistence states in a two-parameter predator-prey model. Date: Friday, October 19, 2012 Speaker: Yuanzhen Shao, Vanderbilt University Title: Real analytic solutions of the surface diffusion flow Abstract: Date: Friday, November 2, 2012 Speaker: Xi Huo, Vanderbilt University Title: An age-structured model about contact tracing in the control of epidemic diseases Abstract: In this talk, I will present a deterministic population model in the spread of an epidemic disease with intervention of isolation and quarantine methods. I will provide the epidemic background, the PDE model, the existence and uniqueness theorem, the computation strategy, some simulation results and explanations. We are recently trying to use the model to fit the data of SARS spread in Taiwan, 2003. I will briefly present some of the recent fitting results as well. Date: Friday, November 9, 2012 Speaker: Marcelo Disconzi, Vanderbilt University Title: On the limit of strong surface tension for a fluid motion with free boundary Abstract: We study the free boundary Euler equations in two spatial dimensions. We prove that if the boundary is sufficiently regular, then solutions of the free boundary fluid motion converge to solutions of the Euler equations in a fixed domain when the coefficient of surface tension tends to infinity. This is  joint work with David G. Ebin. Date: Friday, November 16, 2012 Speaker: Naian Liao, Vanderbilt University Title:  On the local behavior of a logarithmically singular equation Abstract: The local properties of non-negative weak solutions to the singular parabolic equation $u_t-\Delta \ln u = 0$ are largely unclear though some research has been done for the Cauchy problem of such an equation.  In this talk, we address the local positivity of this equation in the form of a Harnack-type inequality. Under the assumption $\ln u$ is sufficiently integrable, we show if $u$ does not vanish identically in a space neighborhood of $x_0$ and on some time level $t_0$ then $u$ is positive in a neighborhood of $(x_0,t_0)$. Date: Friday, November 30, 2012 Speaker: Marcelo Disconzi, Vanderbilt University Title: On the Einstein equations for relativistic fluids Abstract: The Einstein equations have been a source of many interesting problems in Physics, Analysis and Geometry. Despite the great deal of work which has been devoted to them, with many success stories, several important questions remain open. One of the them is a satisfactory theory of isolated systems, such as stars, both from a perspective of the time development of the space-time, as well as from the point of view of the geometry induced on a space-like three surface. This talk will focus on the former situation. More specifically, we shall discuss relativistic fluids with and without viscosity, and prove a well-posedness result for the Cauchy problem. The viscous case, in particular, is of significant interest in light of recent developments in Astrophysics. Tuesday, January 8, 2013, 4:10 pm, SC1206. Special Colloquium Thursday, January 10, 2013, 4:30 pm,  SC5211. Special Colloquium  Tuesday, January 15, 2013, 4:10 pm, SC1206. Special Colloquium Friday, January 18, 2013, 4:10pm, SC1206. Special Colloquium Date: January-February, 2013 Speaker: Colin Klaus, Vanderbilt University Title:  Applications of Lie Groups to Differential Equations, Abstract: A series of informal talks, following the book by Peter J. Olver. Date: Friday, February 22, 12:10pm,  SC 1310 Speaker: Colin Klaus, Vanderbilt University Title:  Applications of Lie Groups to Differential Equations Abstract: A series of informal talks, following the book by Peter J. Olver. Date: Friday, February 22, 4:10-5:00 pm, SC 1307 Speaker: Zhongwei Tang, Beijing Normal University (visiting Vanderbilt University) Title: Multibump solutions of nonlinear Schrödinger equations with steep potential well and indefinite potential Abstract: abstract Date: Friday, March 1, 12:10pm,  SC 1313 Speaker: Colin Klaus, Vanderbilt University Title:  Applications of Lie Groups to Differential Equations Abstract: A series of informal talks, following the book by Peter J. Olver. Date: Friday, March 15, 2013. Time: 4:10 pm, SC 1307 Speaker: Glenn Webb, Vanderbilt University Title:  The ensemble interpretation of quantum mechanics and the two-slit experiment Abstract:  A partial differential equation model is provided for the two-slit experiment of quantum mechanics. The state variable of the equation is the probability density function of particle positions. The equation has a local diffusion term corresponding to stochastic variation of particles, and a nonlocal dispersion term corresponding to oscillation of particles in the transverse direction perpendicular to their forward motion. The model supports the ensemble interpretation of quantum mechanics and gives descriptive agreement with the Schrodinger equation model of the experiment. Date: Friday, March 22, 2013.  Time: 4:10 pm Speaker: Stefan Siegmund, TU Dresden Title:  Dynamical systems on graphs and chaotic monoid actions Abstract:  Boolean networks, neural networks and reaction-diffusion automata share a common structure which we identify as a special class of a new notion of dynamical systems on graphs for which we present a Lyapunov function type concept which implies phase-locking of the dynamics.  For dynamical systems with 'time' being a monoid instead of the integers or the reals, we define a notion of chaos which extends Devaney's classical chaos notion and we prove a theorem that sensitive dependence of initial conditions is a consequence of the two other properties in the definition. The common theme of the two topics is the intention to push the limits of dynamical systems theory in order to investigate how coupling or feedback motifs influence macroscopic behavior and discuss the role of time being a line. Date: Friday, March 29, 2013. Time: 4:10 pm, SC 1307 Speaker: Min Gao, Vanderbilt University Title:  Mathematical analysis of an age-structured population model applicable to early humans Abstract: The age structure of human populations is exceptional among animal species. Unlike most species, human juvenility is extremely extended and death is not coincident with the end of the reproductive period. Recently, a mathematical model was developed to examine the age structure of early humans, which reveals an extraordinary balance of human fertility and mortality. This model has two types of nonlinear mortalities, one term corresponding to the effects of crowding and the other term corresponding to the senescent burden on the juvenile population. We study this semilinear partial differential equation with a nonlinear boundary condition. We analyze the existence, uniqueness and regularity of solutions to the model equations. An intrinsic growth constant is obtained and linked to the existence and the stability of the trivial or the positive equilibrium.  The model supports the hypothesis that the age structure of early humans was robust in its balance of juvenile, reproductive, and senescent classes. Date: Friday, April 5, 2013. Time: 4:10 pm, SC 1307 Speaker: Gieri Simonett, Vanderbilt University Title:  On a thermodynamically consistent Stefan problem with variable surface energy Abstract:  A thermodynamically consistent two-phase Stefan problem with temperature-dependent surface tension and with or without kinetic undercooling is studied. It is shown that this problem generates a local semiflow on a well-defined state manifold. Moreover, stability and instability results of equilibrium configurations will be presented. It will be pointed out that surface heat capacity has a striking effect on the stability behavior of multiple equilibria. (Joint work with J. Prüss and M. Wilke). Date: Friday, April 12, 2013.  Time: 4:10 pm Speaker: Alan R. Parry, Duke University Title:  Mathematics meets astrophysics: an overview of general relativity and dark matter in galaxies Abstract: One of the most beautiful intersections of mathematics and physics is Einstein's theory of gravity, general relativity.  It has successfully resolved problems with Newton's notion of gravity and made incredible predictions about the universe that were later verified by observations.  Recently, mathematical motivations for general relativity have presented a possible and tantalizing description of dark matter, a mysterious and exotic form of matter that makes up approximately 23% of the energy density in the universe.  In this talk, I will introduce for general audiences the theory of general relativity and the mathematics behind it, discuss the major successes of general relativity, and describe this model for dark matter and the recent advances that have been made in its study. Date: Friday, April 19, 2013.   Time: 3:10pm, SC 1307  Speaker: Klaus Schmitt, University of Utah Title:   Nonlinear elliptic PDE: Some dimension dependent phenomena Abstract:  In this lecture I shall discuss a class of boundary value problems for nonlinear elliptic PDE's which have the property that the number of solutions depends on the space dimension. The class of equations considered has been studied extensively and a survey of the history of the results known is given. The tools used to study such problems are manifold and are a blend of continuation techniques, variational analysis, regularity theory, and dynamical systems methods. Date: Friday, April 19, 2013. Time: 4:30pm, SC 1307 Speaker:  Nemanja Kosovalic,  York University, Toronto Title:  Abstract evolution equation with state dependent delay and age structured population dynamics Abstract: Consider a population of individuals occupying some habitat, which is structured by age. Suppose that there are two distinct life stages, the immature stage and the mature stage. A natural question is What determines the age of maturity?". In many biological contexts, the age of maturity is determined by whether or not the quantity of food consumed by the immature population, reaches a prescribed threshold. Mathematically, this situation takes the form of a first order hyperbolic PDE coupled to an algebraic-delay term. We are led to consider the more general situation of the Lipschitz perturbation of a Hille-Yosida operator, coupled to an algebraic-delay term. We discuss the corresponding semiflow, and some of its dynamical properties. This is joint work with Dr. Felicia Magpantay and Dr. Jianhong Wu.