Schedule of talks:

Abstracts

Sze-Bi Hsu (Sep 17, 2014)
National Tsing-Hua University, Taiwan

In this talk we first introduce a system of reaction diffusion equations modeling two species competition in an unstirred chemostat with internal storage. Then we study the case of the singe population growth. Using the diffusion coefficient d as a bifurcation parameter , by the method of monotone dynamical system with cooperative order , we establish the global asymptotic stability of the washout steady state as d> d_0 and survival steady state as d

Shujun Shi (Sep 24, 2014)
Harbin Normal University and OSU

Title: On estimates for fully nonlinear parabolic equations on Riemannian manifolds. Abstract: In this talk, we will present some new ideas and methods to derive a priori second order estimates for a wide class of fully nonlinear parabolic equations. The methods can produce new existence results for the initial-boundary value problems in Euclidean space and also work in general Riemannian manifolds. The talk is about our recent paper (arXiv:1409.3633) jointed with Prof. Bo Guan and Zhenan Sui.

Thomas Hillen (Oct 8, 2014)
University of Alberta

Title: Chaotic patterns in chemotaxis Abstract: Chemotaxis describes the active orientation of moving individuals along chemical gradients. The Keller-Segel model for chemotaxis has become famous through the existence of finite-time blow up solutions. In this talk we add logistic growth to the chemotaxis model. Instead of blow-up we obtain global solutions which show interesting spatio-temporal chaotic patterns. I will motivate this model and analyse the resulting patterns. We find positive Lyapunov exponents and a period doubling sequence, both indications of chaotic behavior. Furthermore, I will present a discrete dynamical system, which can, surprisingly, reproduce most of these patterns. (joint work with K.J. Painter and J. Zielinski).

Junping Shi (Oct 15, 2014)
College of William and Mary

Title: New bifurcation theorems and their applications
Abstract: We review several old and new abstract bifurcation theorems, for both the local bifurcation based on implicit function theorem and global bifurcation based on degree theory. In particular we present three generalizations of the classical bifurcation from simple eigenvalue result of Crandall-Rabinowitz. Some novel examples are shown to demonstrate the applications of these results. The talk is partially based on work with Ping Liu and Yuwen Wang on local bifurcation theory, and work with Xuefeng Wang on global bifurcation theory.

Bei Hu (Oct 21, 2014)
University of Notre Dame and MBI

Title: A PDE Free Boundary Problem for Corporate Bond with Credit Rating Migration
Abstract: A free boundary model for pricing a corporate bond with credit rating migration is studied. Some interesting properties as well as numerical examples will be presented.

Alex Himonas (Oct 22, 2014)
University of Notre Dame

Title: Norm Inflation and ill-posedness for CH and related equations
Abstract: We shall consider the Cauchy problem for CH type equations and discuss the phenomenon of norm inflation in Sobolev spaces $H^s$ for $s$ less than the well-posednes critical index, which for these equations is equal to 3/2. This means that there exist solutions who are initially arbitrarily small and eventually arbitrarily large with respect to the $H^s$ norm, in an arbitrarily short time. When there is norm inflation, then we have ill-posedness since the data-to-solution map is not continuous.

Chongchun Zeng (Dec 3, 2014)
Georgia Tech

Title: Traveling water waves with compactly supported vorticity
Abstract: We consider the water wave problem -- the free boundary problem of the Euler equation -- and construct small traveling wave solutions with vorticity based on a bifurcation method. Unlike those perturbed from shear flows, the vorticity of these solutions are supported in a small domain away from the water surface. We also discuss the global bifurcation of such waves when the vorticity is a delta mess. This is a joint work with Jalal Shatah and Sam Walsh.

Yuriko Renardy (Jan 14, 2015)
Virginia Tech

Title: A viscoelastic constitutive model that predicts thixotropic yield stress behavior for large relaxation time.
AbstractThis talk will be on the predictions of a constitutive model which combines a viscoelastic constitutive model for an entangled microstructure, with a Newtonian solvent. An example of a thixotropic yield stress fluid is ketchup. Initial value problems started by an applied stress are examined with numerical computations, which show complicated transients for yielding to flow and unyielding back to a solid state. These transients are captured by asymptotic methods based on the multiple time scales, in the limit of large relaxation time. Two time scales are involved: a long one associated with the relaxation time of the microstructure, and a short one for the solvent. This is joint work with K. Maki (RIT) and H. Grant (Virginia Tech).

Huihui Zeng (Feb 11, 2015)
Math Sci Center, Tsinghua University & Center of Math Sci and Appl, Harvard University

Title: Global Solutions to the Gas-Vacuum interface Problem with Physical Singularity of Compressible Euler Equations with damping
Abstract: In this talk, I will present some recent results on the global solutions to the gas-vacuum interface problem of compressible Euler equations with physical singularity of the sound speed being $C^{1/2}$-Holder continuous near vacuum boundaries. For this problem, the global existence and convergence to the Barenblatt self-similar solution of the induced porous media equation as time goes to infinity is proved, with detailed convergence rate. The long time asymptotics of the gas-vacuum interfaces are also given. This talk is based on the Joint work with Tao Luo.

Yaobin Ou (Feb 11, 2015)
School of Information, Renmin University of China

Title: Global classical solutions to the vacuum free boundary problem of full compressible Navier-Stokes equations with large data
Abstract: In this talk, I'll present a recent result on the the free boundary value problem of one-dimensional Navier-Stokes equations for viscous polytropic and heat-conducting fluids, when the density connects to the vacuum continuously. The global existence of classical solution to the equations with large initial data was established in this work.

Min Tang (Feb 18, 2015)
Institute of Natural Sciences, Shanghai Jiaotong University

Title: The relaxation method for traveling wave and periodic traveling waves
Abstract: We present an original method that relies on the physical evolution to capture the traveling waves and periodic traveling waves. This method allows us to obtain the traveling wave profiles and their traveling speed simultaneously. We show the long time behavior of the scheme analytically for bistable reaction diffusion equations over the whole space R. The method is achieved by introducing an nonlocal advection in the original system, though it is a numerical way to find the traveling wave or periodic traveling wave, a lot of analytical question can be addressed.

Gabor Szekelyhidi (Mar 4, 2015)
University of Notre Dame

Title: Hessian quotient equations on compact Kahler manifolds
Abstract: I will discuss a priori estimates for Hessian type equations on compact Hermitian and Kahler manifolds, under the assumption that a suitable subsolution exists. This generalizes the approach of Hou-Ma-Wu and Dinew-Kolodziej for the complex Hessian equation, as well as results of Song-Weinkove on the J-flow.

Pierre-Emmanuel Jabin (Mar 11, 2015)
University of Maryland

Title: Weak solutions for compressible advection models
Abstract: We study the existence of weak solutions for some non linear advection models. Those include classical compressible fluid mechanics but also intricate models of transport of micro-organisms. In that last case, the organisms are assumed to interact through several chemical signals. This leads to possibly complex attraction-repulsion behaviors combined with possible anisotropy in the environment. The corresponding dynamics presents some unique difficulties requiring a completely new approach to handle them. This is a joint work with D. Bresch.

Feimin Huang (Mar 26, 2015) Note Date/Time/Location change
Academia Sinica

Title: Sonic-Subsonic Limit of Approximate Solutions to Multidimensional Steady Euler Equations
Abstract: A compactness framework is established for approximate solutions to sonic-subsonic flows governed by the steady full Euler equations for compressible fluids in arbitrary dimension. The existing compactness frameworks for the two-dimensional irrotational case do not directly apply for the steady full Euler equations in higher dimensions. The new compactness framework we develop in this paper applies for both non-isentropic and rotational flows. One of our main observations is that the compactness can be achieved by using only natural weak estimates for the mass balance and the vorticity, along with the Bernoulli law and the entropy relation, through a more delicate analysis on the phase space. As direct applications, we establish two existence theorems for multidimensional sonic-subsonic full Euler flows through infinitely long nozzles.

Wenxiong Chen (Apr 1, 2015)
Yeshiva University

Title: Direct methods of moving planes, moving spheres, and blowing-ups for the fractional Laplacian
Abstract: Many conventional approaches on partial differential operators do not work on the nonlocal fractional operator. To overcome this difficulty arising from non-localness, Caffarelli and Silvestre introduce the extension method to reduced the problem into a local one in one higher dimensions, which has become a powerful tool in studying such nonlocal problems and has yielded a series of fruitful results. However, due to technical restrictions, sometimes one needs to impose extra conditions when studying the extended problems in higher dimensions, and these conditions may not be necessary if we investigate the original nonlocal problems directly. In this talk, we will introduce direct methods of moving planes, moving spheres, and blowing-up and re-scaling arguments for the fractional Laplacian. By an elementary approach, we will first show the key ingredients needed in the method of moving planes either in a bounded domain or in the whole space, such as strong maximum principles for anti-symmetric functions, narrow region principles, and decay at infinity. Then, using simple examples, semi-linear equations involving the fractional Laplacian, we will illustrate how this new method of moving planes can be conveniently employed to obtain symmetry and non-existence of positive solutions, under much weaker conditions than in the previous literature. We firmly believe that these ideas and approaches can be effectively applied to a wide range of nonlinear problems involving fractional Laplacians or other nonlocal operators.

Avner Friedman (Apr 8, 2015)
Ohio State University

Title: A free boundary problem associated with the risk of high cholesterol
Abstract: Atherosclerosis is the leading cause of death in the United States and worldwide. The disease originates from a plaque that builds up on the artery, and may trigger heart attack or stroke. The growth of the plaque is initiated and maintained by LDL cholesterols which enter the plaque from the blood. In this talk I will describe a mathematical model, developed jointly with Wenrui Hao, of the growth of the plaque as a free boundary problem consisting a system of PDEs, with LDL and HDL cholesterol influxes from the free boundary. The risk of atherosclerosis will be visualized by a “risk map” in the (LDL, HDL)-plane. The existence, uniqueness, and asymptotic stability of small plaques have more recently been proved jointly with Wenrui Hao and Bei Hu.

Kun Zhao (Apr 15, 2015)
Tulane University

Title: Stability of Hydrostatic Equilibrium in 2D Boussinesq System
Abstract: The 2D Boussinesq equations, named after French mathematician and physicist Joseph Valentin Boussinesq, is a system of nonlinear PDEs describing the motion of natural convection. The model has been routinely used to model systems across a tremendous range of length and time scales from microfluidics and biophysics to meteorology, oceanography, and astrophysics. It plays an important role in the study of atmospheric and oceanographic turbulence, as well as other astrophysical situations where rotation and stratification play a dominant role. Rigorous mathematical analysis, including local/global well-posedness and finite-time blowup criteria, has been well documented in the literature. In contrast, the study of the long-time asymptotic behavior of the large-amplitude classical solutions to the model, which is physically relevant (related to the stability of hydrostatic equilibrium) and mathematically challenging, has been investigated relatively little. In this talk, I will discuss some recent result in this direction.

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