Ohio State University Partial Differential Equations Seminar 

  Year 2011-2012

Time/Location: Wednesdays 4:30 - 5:30pm/MW154 (unless otherwise noted)

Schedule of talks:


 
TIME  SPEAKER TITLE HOST

September 21 

No seminar 
() 		    
September 28   No seminar 
() 		    
Octobor 5  Avner Friedman 
Ohio State Unviersity 		   Conservations Laws in Mathematical Biology  
October 12 
No Seminar 
() 		    
October 19 
No Seminar 
() 		  
October 26 

No Seminar 
() 

    
November 2 

No Seminar 
() 

    
November 9
 Meijun Zhu 
University of Oklahoma		   Nonlinear Equations with Negative Exponents  Yuan Lou
November 16 

No Seminar 
() 

    
November 23 

No Seminar 
() 

    
November 30 
King-Yeung Lam (Adrian) 
Ohio State University 		   Diffusion and Directed Movements in Heterogeneous Environments  
December 7 
Zhongwei Shen  
University of Kentucky 		   Recent Progress on Elliptic Homogenization Problems  
January 4 
No Seminar 
() 		    
January 11 
No seminar 
() 		    
January 18 
No Seminar 
() 		    
January 25 
Junfang Li  
University of Alabama at Birmingham  		   A new mean curvature type of flow and its fully nonlinear version  
February 1 
Jiakun Liu  
Princeton 		   Light reflection problem and nonlinear optimization  
February 8 
No seminar 
() 		    
February 15 
No seminar 
() 		    
February 22 
Emmanuele DiBenedetto 
Vanderbilt  		   On the local behavior of solutions of logarithmically singular parabolic equations  
March 1 
No Seminar 
() 		    
March 7 
No Seminar 
() 		    
March 28
No Seminar 
 ()		    
April 5
No seminar 
() 		    
April 11   Dehua Wang 
Pittsburgh 		   Global solutions of a simplified Ericksen-Leslie system of nematic liquid crystals  
April 19 
No seminar 
() 		    
April 26 
No seminar 
() 		    
May 2 
Aihua Wood 
Air Force Institute of Technology 		   TBD  
May 9 
Alina Stancu 
Concordia 		   TBD  
May 16 
No seminar 
() 		    
May 23 
No seminar 
() 		    
May 30 
No seminar 
() 		    

Abstracts

Avner Friedman (Oct 5, 2011)

Many mathematical models in biology can be described by conservation laws. From a mathematical point of view one would like to establish, first of all, the existence and uniqueness of solutions under some prescribed initial (and possibly also boundary) conditions. However, the most interesting questions relate to establishing properties of solutions that are of biological interest. In this talk I shall give examples of biological processes whose mathematical models are represented by conservation laws. We described results and present open problems.

Meijun Zhu (Nov 9, 2011)

I will describe the motivation from geometric analysis view point for the study of nonlinear differential equations involving negative exponent. The associated geometric inequalities (Sobolev, Blaschke-Santalo inequalities, etc), geometric flow problems, as well as the existence result to equations with supcritical exponent will be discussed.

King-Yeung Lam (Nov 30, 2011)

In this talk I will talk about some recent results of the 2X2 Lotka-Volterra competition model in heterogeneous environment. Starting with the well known result of Dockery et. al. regarding the phenomena "Slower Diffuser Prevails", to the "Advection-Mediated Coexistence" proved by Cantrell et. al., we will try to understand how different dispersal strategies affect the dynamics/outcome of the competition.

Zhongwei Shen (Dec 7, 2011)

In this talk I will describe my recent work, joint with Carlos Kenig and Fanghua Lin, on homogenization of second-order elliptic equations and systems. In a series of papers we have obtained uniform Rellich estimates for L^2 solutions and boundary Lipschitz estimates for solutions with Neumann boundary conditions. These estimates are used to derive various convergence theorems and establish asymptotic expansions of Green's and Neumann functions.

Junfang Li (Jan 25, 2012)

(Joint with Pengfei Guan) In this talk, we will present a new type of mean curvature flow. For any closed star-shaped smooth hypersurface, this flow exists for all time t>0 and exponentially converges to a round sphere. Moreover, we will show that all the quermassintegrals evolve monotonically along this flow. Consequently, we prove a class of isoperimetric type of inequalities including the classical isoperimetric inequality on star-shaped domains. We will also present a fully non-linear parabolic equation of a function on the standard sphere and discuss its long-time existence and exponential convergence. As applications, we recover the well-known Alexandrov-Fenchel inequalities on bounded convex domains in Euclidean space.

Jiakun Liu (Feb 1, 2012)

The light reflection problem, due to its various applications, has been extensively studied by many mathematicians in recent years. In the special far field case, the light reflection is related to the reflector antenna design problem. Xu-Jia Wang showed that it is an optimal transportation, and so, is a linear optimization problem. In this talk, we study the general case of the light reflection problem, and show that it is related to a nonlinear optimization problem. This problem involves a fully nonlinear PDE of Monge-Ampere type, subject to a nonlinear boundary condition. If time permits, I will also briefly talk about some recent results on global regularity.

Emmanuele DiBenedetto (Feb 22, 2012)

The local positivity of solutions to logarithmically singular diffusion equations is investigated in some open space-time domain $E\times(0,T]$. It is shown that if at some time level $t_o\in(0,T]$ and some point $x_o\in E$ the solution $u(\cdot,t_o)$ is not identically zero in a neighborhood of $x_o$, in a measure-theoretical sense, then it is strictly positive in a neighborhood of $\pto$. The precise form of this statement is by an intrinsic Harnack-type inequality, which also determines the size of such a neighborhood.

Dehua Wang (Apr 11, 2012)

The multi-dimensional incompressible and compressible flows of nematic liquid crystals will be discussed. Some recent results of existence of global weak and strong solutions as well as the incompressible limits will be presented.

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