Gibson 1st Floor SR

We discuss how various types of orientational and

translational ordering in different liquid crystal phases are

described by macroscopic tensor order parameters. In

particular, we consider a mean-field molecular-statistical

theory of the transition from the orthogonal uniaxial smectic

phase and the tilted biaxial phase composed of biaxial

molecules. The relationship between macroscopic order

parameters, molecular invariant tensors and the symmetry of

biaxial molecules is discussed in detail. Finally we use

microscopic and macroscopic symmetry arguments to consider the

mechanisms of the ferroelectric ordering in tilted smectic

phases determined by molecular chirality.

The question of local existence of a deformation of a simply connected body whose Left Cauchy Green Tensor matches a prescribed, symmetric, positive definite tensor field is considered. A sufficient condition is deduced after formulation as a problem in Riemannian Geometry. The compatibility condition ends up being surprisingly different from that of compatibility of a Right Cauchy Green Tensor field, a fact that becomes evident after the geometric formulation. The question involves determining conditions for the local existence of solutions to an overdetermined system of Pfaffian PDEs with algebraic constraints that is typically not completely integrable.

This talk is devoted to adaptivity in optimal control of PDEs with special emphasis on barrier methods for pointwise state constraints. The talk is divided into to major parts, first we will discuss the case of additional pointwise inequality constraints on the state variable, then we will transfer the results to constraints on the gradient of the state. Each part will start with a discussion of necessary optimality conditions and a brief overview about what is known and what is not known concerning a priori analysis. Then a posteriori error estimates for the discretization error as well as for the error from the barrier method will be presented. Finally we show some simple examples to illustrate the behavior of the estimators. 
The talk will be followed by an informal tea in the Gibson Building seminar room giving an opportunity to chat with Winnifried Wollner and Amit Acharya (our other current OxMOS visitor)

I will describe a topological approach to the motion planning problem of

robotics which leads to a new homotopy invariant of topological spaces

reflecting their "navigational complexity". Technically, this invariant is

defined as the genus (in the sense of A. Schwartz) of a specific fibration.

The talk will survey old and recent applications of variational techniques in studying the existence, stability and bifurcations of time harmonic, localized in space solutions of the nonlinear Schroedinger equation (NLS). Such solutions are called solitons, when the equation is space invariant, and bound-states, when it is not. Due to the Hamiltonian structure of NLS, solitons/bound-states can be characterized as critical points of the energy functional restricted to sets of functions with fixed $L^2$ norm.

In general, the energy functional is not convex, nor is the set of functions with fixed $L^2$ norm closed under weak convergence. Hence the standard variational arguments fail to imply existence of global minimizers. In addition for ``critical" and ``supercritical" nonlinearities the restricted energy functional is not bounded from below. I will first review the techniques used to overcome these drawbacks.

Then I will discuss recent results in which the characterizations of bound-states as critical points (not necessarily global minima) of the restricted energy functional is used to show their orbital stability/instability with respect to the nonlinear dynamics and symmetry breaking phenomena as the $L^2$ norm of the bound-state is varied.

In the lecture, I am going to explain a connection between

local regularity theory for the Navier-Stokes equations

and Liouville type theorems for bounded ancient solutions to

these equations.

This talk will review recent progress in understanding the effective

behavior of free boundaries in heterogeneous media.  Though motivated

by the pinning of martensitic phase boundaries, we shall explain

connections to other problems.  This talk is based on joint work with

Patrick Dondl.

In the past several years it has come to be appreciated that in low Reynolds number flow the nonlinearities provided by non-Newtonian stresses of a complex fluid can provide a richness of dynamical behaviors more commonly associated with high Reynolds number Newtonian flow. For example, experiments by V. Steinberg and collaborators have shown that dilute polymer suspensions being sheared in simple flow geometries can exhibit highly time dependent dynamics and show efficient mixing. The corresponding experiments using Newtonian fluids do not, and indeed cannot, show such nontrivial dynamics. To better understand these phenomena we study the Oldroyd-B viscoelastic model. We first explain the derivation of this system and its relation to more familiar systems of Newtonian fluids and solids and give some analytical results for small data perturbations. Next we study this and related models numerically for low-Reynolds number flows in two dimensions. For low Weissenberg number (an elasticity parameter), flows are "slaved" to the four-roll mill geometry of the fluid forcing. For sufficiently large Weissenberg number, such slaved solutions are unstable and under perturbation transit in time to a structurally dissimilar flow state dominated by a single large vortex, rather than four vortices of the four-roll mill state. The transition to this new state also leads to regions of well-mixed fluid and can show persistent oscillatory behavior with continued destruction and generation of smaller-scale vortices.