On Thomas-Fermi Theory and Extensions

Speaker: 

Gisele Goldstein

Institution: 

University of Memphis

Time: 

Thursday, April 28, 2016 - 3:00pm

Location: 

RH 440R

Of concern to quantum chemists and solid state physicists is the approximate numerical computation of the ground state wave function, and the ground state energy and density for molecular and other quantum mechanical systems. Since the number of molecules in bulk matter is of the order of 10e26 , direct computation is too cumbersome or impossible in many situations. In 1927, L. Thomas and E. Fermi proposed replacing the ground state wave function by the ground state density, which is a function of only three variables. Independently, each found an approximate expansion for the energy associated with a density. (The wave function uniquely determines the density, but not conversely.)

A computationally better approximate expansion was found in the 1960’s by W. Kohn and his collaborators; for this work Kohn got the Nobel Prize in Chemistry in 1998. A successful attempt to put Thomas-Fermi theory into a rigorous mathematical framework was begun in the 1970’s by E. Lieb and B. Simon and was continued and expanded by Ph. Benilan, H. Brezis and others. Very little rigorous mathematics supporting Kohn density functional theory is known. In this talk I will present a survey of rigorous Thomas-Fermi theory, including recent developments and open problems (in the calculus of variations and semilinear elliptic systems).

On nematic liquid crystal flows in dimensions two and three

Speaker: 

Changyou Wang

Institution: 

Purdue University

Time: 

Tuesday, February 2, 2016 - 3:00pm to 4:00pm

Host: 

Location: 

RH306

In this talk, I will discuss a simplified Ericksen-Leslie system modeling
the hydrodynamics of nematic liquid crystals, that is coupling between Navier-Stokes equations and harmonic map heat flows. I will describe some existence results of global weak solutions in dimensions two and three, and a finite time singularity result in dimension three. This is based on some joint works with Tao Huang, Junyu Lin, Fanghua Lin, and Chun Lin.

Line Defects in a Modified Ericksen Model of Nematic Liquid Crystals

Speaker: 

Robert M Hardt

Institution: 

Rice University

Time: 

Monday, October 12, 2015 - 3:00pm to 4:00pm

Host: 

Location: 

RH306

In 1985, J. Ericksen derived a model for uniaxial liquid crystals to allow for disclinations (i.e. line defects or curve singularities). It involved not only a unit orientation vectorfield on a region of R^3  but also a scalar order parmeter quantify- ing the expected inner product between this vector and the molecular orientation. FH.Lin, in several papers, related this model, for certain material constants, to harmonic maps to a metric cone over S^2. He showed that a minimizer would be continuous everywhere but would have higher regularity fail on the singular de- fect set s^{-1}(0). The optimal partial regularity result of R.Hardt-FH.Lin in 1993, for this model, led to regularity away from isolated points, which unfortunately still excluded line singularities. This paper accordingly also introduced a modified model involving maps to a metric cone over RP^2, the real projective plane. Here the nontrivial homotopy leads to the optimal estimate of the singular set being 1 dimensional. In 2010, J. Ball and A.Zarnescu discussed a derivation from the de Gennes Q tensor and interesting orientability questions using RP2. In recent ongo- ing work with FH.Lin and O. Alper, we see that the singular set with this model necessarily consists of Holder continuous curves. We will also survey some of the many more elaborate liquid crystal PDE’s involving a general director functional, the full Q tensor model, and possible coupling with fluid velocity. 

Coating flow of viscous Newtonian liquids on a rotating cylinder

Speaker: 

Marina Chugunova

Institution: 

Claremont Graduate University

Time: 

Thursday, October 22, 2015 - 3:00pm

Location: 

RH 440R

In this talk I discuss the different types of models which originate from a lubrication approximation of viscous coating flow dynamics on the outer surface of a rotating cylinder, that is in the presence of a gravitational field. Analytical and numerical results related to existence, uniqueness and stability of solutions will be presented.

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