Forthcoming events in this series


Wed, 26 Nov 2008

13:30 - 14:30
Gibson 1st Floor SR

Variational Methods in Nonlinear Schroedinger Equations

Eduard Kirr
(University of Illinois at Urbana Champaign, USA)
Abstract

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.

Mon, 24 Nov 2008

13:30 - 14:30
Gibson 1st Floor SR

Fine structures arising in diblock copolymers and reaction-diffusion systems

Yoshihito Oshita
(Okayama University, Japan)
Abstract

We consider a class of energy functionals containing a small parameter ε and a long-range interaction. Such functionals arise from models for phase separation in diblock copolymers and from stationary solutions of FitzHugh–Nagumo type systems.

On an interval of arbitrary length, we show that every global minimizer is periodic, and provide asymptotic expansions for the periods.

In 2D, periodic hexagonal structures are observed in experiments in certain di-block

copolymer melts. Using the modular function and an heuristic reduction of a mathematical model, we present a mathematical account of a hexagonal pattern selection observed in di-block copolymer melts.

We also consider the sharp interface problem arising in the singular limit,

and prove the existence and the nondegeneracy of solutions whose interface is a distorted circle in a two-dimensional bounded domain without any assumption on the symmetry of the domain.

Thu, 20 Nov 2008
12:00
Gibson 1st Floor SR

Elliptic equations in the plane satisfying a Carleson measure condition

David Rule
(University of Edinburgh)
Abstract

We study the Neumann and regularity boundary value problems for a divergence form elliptic equation in the plane. We assume the gradient

of the coefficient matrix satisfies a Carleson measure condition and consider data in L^p, 1

Wed, 19 Nov 2008

14:00 - 15:00
Gibson 1st Floor SR

An approach to solvability of the generalised Navier-Stokes equation

Vasily V. Zhikov
(Moscow State University and Vladimir State University, Russia)
Abstract

The Navier-Stokes equation with a non-linear viscous term will be considered, p is the exponent of non-linearity.

An existence theorem is proved for the case when the convection term is not subordinate to the viscous

term, in particular for the previously open case p

Tue, 18 Nov 2008

11:00 - 12:00
Gibson 1st Floor SR

Dynamic fracture based on Griffith's criterion

Christopher Larsen
(Worcester Polytechnic Institute, USA)
Abstract

There has been much recent progress in extending Griffith's criterion for

crack growth into mathematical models for quasi-static crack evolution

that are well-posed, in the sense that there exist solutions that can be

numerically approximated. However, mathematical progress in dynamic

fracture (crack growth consistent with Griffith's criterion, together with

elastodynamics) has been meager. We describe some recent results on a

phase-field model of dynamic fracture, as well as some models based on a

"sharp interface" instead of a phase-field.

Some possible strategies for showing existence for these last models will

also be described.

Mon, 17 Nov 2008

12:30 - 13:30
Gibson 1st Floor SR

Order Parameters, Irreducible Tensors and the theory of Phase Transitions in Smectic Liquid Crystals

Mikhail Osipov
(Strathclyde)
Abstract

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.

Thu, 13 Nov 2008

13:30 - 14:30
Gibson 1st Floor SR

Asymptotic behaviour of the Stokes problem in cylinders

Sorin Mardare
(University of Rouen)
Abstract

We study the asymptotics of the Stokes problem in cylinders becoming unbounded in the direction of their axis. We consider

especially the case where the forces are independent of the axis coordinate and the case where they are periodic along the axis, but the same

techniques also work in a more general framework.

We present in detail the case of constant forces (in the axial direction) since it is probably the most interesting for applications and also

because it allows to present the main ideas in the simplest way. Then we briefly present the case of periodic forces on general periodic domains. Finally, we give a result under much more general assumptions on the applied forces.

Wed, 12 Nov 2008

10:45 - 11:45
Gibson 1st Floor SR

Compatibility conditions for the Left Cauchy Green Tensor field in 3-D

Amit Acharya
(Carnegie Mellon University)
Abstract

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.

Thu, 06 Nov 2008

12:30 - 13:30
Gibson 1st Floor SR

On the existence of extreme waves and the Stokes conjecture with vorticity

Eugen Varvaruca
(Imperial College)
Abstract

We present some recent results on singular solutions of the problem of travelling gravity water waves on flows with vorticity. We show that, for a certain class of vorticity functions, a sequence of regular waves converges to an extreme wave with stagnation points at its crests. We also show that, for any vorticity function, the profile of an extreme wave must have either a symmetric corner of 120 degrees or a horizontal tangent at any isolated stagnation point. Moreover, the profile necessarily has a symmetric corner of 120 degrees if the vorticity is nonnegative near the free surface.

Wed, 05 Nov 2008

13:30 - 14:30
Gibson 1st Floor SR

Propagation of free boundaries in heterogeneous materials

Kaushik Bhattacharya
(Caltech)
Abstract

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.

Thu, 23 Oct 2008

13:30 - 14:30
Gibson 1st Floor SR

Nonlinear stability of time-periodic viscous shocks

Margaret Beck
(Brown University, US)
Abstract

In order to understand the nonlinear stability of many types of time-periodic travelling waves on unbounded domains, one must overcome two main difficulties: the presence of embedded neutral eigenvalues and the time-dependence of the associated linear operator. This problem is studied in the context of time-periodic Lax shocks in systems of viscous conservation laws. Using spatial dynamics and a decomposition into separate Floquet eigenmodes, it is shown that the linear evolution for the time-dependent operator can be represented using a contour integral similar to that of the standard time-independent case. By decomposing the resulting Green's distribution, the leading order behavior associated with the embedded eigenvalues is extracted. Sharp pointwise bounds are then obtained, which are used to prove that the time-periodic Lax shocks are linearly and nonlinearly stable under the necessary conditions of spectral stability and minimal multiplicity of the translational eigenvalues. The latter conditions hold, for example, for small-oscillation time-periodic waves that emerge through a supercritical Hopf bifurcation from a family of time-independent Lax shocks of possibly large amplitude.

Fri, 17 Oct 2008

13:30 - 14:30
Gibson 1st Floor SR

Using global invariant manifolds to understand metastability in Burgers equation with small viscosity

Margaret Beck
(Brown University, US)
Abstract

The large-time behavior of solutions to Burgers equation with

small viscosity is described using invariant manifolds. In particular,

a geometric explanation is provided for a phenomenon known as

metastability,which in the present context means that

solutions spend a very long time near the family of solutions known as

diffusive N-waves before finally converging to a stable self-similar

diffusion wave. More precisely, it is shown that in terms of

similarity, or scaling, variables in an algebraically weighted $L^2$

space, the self-similar diffusion waves correspond to a one-dimensional

global center manifold of stationary solutions. Through each of these

fixed points there exists a one-dimensional, global, attractive,

invariant manifold corresponding to the diffusive N-waves. Thus,

metastability corresponds to a fast transient in which solutions

approach this ``metastable" manifold of diffusive N-waves, followed by

a slow decay along this manifold, and, finally, convergence to the

self-similar diffusion wave.

Thu, 16 Oct 2008

13:30 - 14:30
Gibson 1st Floor SR

One-homogeneous stationary points of elliptic systems in two dimensions.

Jon Bevan
(University of Surrey)
Abstract

A function $u: \mathbb{R}^{n} \to \mathbb{R}^{m}$ is one-homogeneous if $u(ax)=au(x)$ for any positive real number $a$ and all $x$ in $\R^{n}$. Phillips(2002) showed that in two dimensions such a function cannot solve an elliptic system in divergence form, in contrast to the situation in higher dimensions where various authors have constructed one-homogeneous minimizers of regular variational problems. This talk will discuss an extension of Phillips's 2002 result to $x-$dependent systems. Some specific one-homogeneous solutions will be constructed in order to show that certain of the hypotheses of the extension of the Phillips result can't be dropped. The method used in the construction is related to nonlinear elasticity in that it depends crucially on polyconvex functions $f$ with the property that $f(A) \to \infty$ as $\det A \to 0$.

Thu, 09 Oct 2008

13:30 - 14:30
Gibson 1st Floor SR

New Results on Travelling Waves in Hamiltonian Lattices

Michael Herrmann
(University of Oxford)
Abstract

Travelling waves are highly symmetric solutions to the Hamiltonian lattice equation and are determined by nonlinear advance-delay differential equations. They provide much insight into the microscopic dynamics and are moreover fundamental building blocks for macroscopic

lattice theories.

In this talk we concentrate on travelling waves in convex FPU chains and study both periodic waves (wave trains) and homoclinic waves (solitons). We present a new existence proof which combines variational and dynamical concepts.

In particular, we improve the known results by showing that the profile functions are unimodal and even.

Finally, we study the complete localization of wave trains and address additional complications that arise for heteroclinic waves (fronts).(joint work with Jens D.M. Rademacher, CWI Amsterdam)

Mon, 06 Oct 2008

11:30 - 12:30
Gibson 1st Floor SR

Decomposition Theorem, Orthogonality principle and Localization principle - the three components of the sufficiency proof (I)

Yury Grabovsky
(Temple University, US)
Abstract
The proof of quasiconvexity based sufficient conditions for strong local minima in vectorial variational problems consists of three major parts: the Decomposition Theorem, the Orthogonality principle and the Localization principle. The first and the last are the most technical.
In these talks I will explain the technical difficulties and the ways in which they were overcome.
Thu, 02 Oct 2008

13:30 - 14:30
Gibson 1st Floor SR

Mixing Transitions and Oscillations in Low-Reynolds Number Viscoelastic Fluids

Becca Thomases
(University of California, Davis)
Abstract

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.

Thu, 18 Sep 2008

13:30 - 14:30
Gibson 1st Floor SR

Characterization of generalized gradient Young measures in $W^{1,1}$ and $BV$

Filip Rindler
(Technical University of Berlin)
Abstract

This talk first introduces generalized Young measures (or DiPerna/Majda measures) in an $L^1$-setting. This extension to classical Young measures is able to quantitatively account for both oscillation and concentration phenomena in generating sequences.

We establish several fundamental properties like compactness and representation of nonlinear integral functionals and present some examples. Then, generalized Young measures generated by $W^{1,1}$- and $BV$-gradients are more closely examined and several tools to manipulate them (including averaging and approximation) are presented.

Finally, we address the question of characterizing the set of generalized Young measures generated by gradients in the spirit of the Kinderlehrer-Pedregal Theorem.

This is joint work with Jan Kristensen.

Tue, 16 Sep 2008

14:00 - 15:00
Gibson 1st Floor SR

Non-periodic Γ-convergence

Helia Serrano
(Universidad de Castilla-La Mancha)
Abstract

Γ-convergence is a variational convergence on functionals. The explicit characterization of the integrand of the Γ-limit of sequences of integral functionals with periodic integrands is by now well known. Here we focus on the explicit characterization of the limit energy density of a sequence of functionals with non-periodic integrands. Such characterization is achieved in terms of the Young measure associated with relevant sequences of functions. Interesting examples are considered.

Mon, 01 Sep 2008

13:30 - 14:30
Gibson 1st Floor SR

About yield surfaces of phase transformation for some shape memory alloys: duality and convexity. Application to fracture.

Christian Lexcellent
(University of Franche-Comte)
Abstract

Multiaxial mechanical proportional loadings on shape memory alloys undergoing phase transformation permit to determine the yield curve of phase transformation initiation in the stress space. We show how to transport this yield surface in the set of effective transformation strains of producted phase M. Two numerical applications are done concerning a Cu Al Be and a Ni Ti polycrystallines shape memory alloys. A special attention is devoted to establish a convexity criterium of these surfaces.

Moreover an application to the determination of the phase transformation surface around the crack tip for SMA fracture is performed.

At last some datas are given concerning the SMA damping behavior

AUTHORS

Christian Lexcellent, Rachid Laydi, Emmanuel Foltete, Manuel collet and Frédéric Thiebaud

FEMTO-ST Département de Mécanique Appliquée Université de Franche Comte Besançon France

Fri, 18 Jul 2008

13:30 - 14:30
Gibson 1st Floor SR

On Monge-Ampere type equations with supplementary ellipticity

Neil Trudinger
(Australian National University)
Abstract

We present a selection of recent results pertaining to Hessian

and Monge-Ampere equations, where the Hessian matrix is augmented by a

matrix valued lower order operator. Equations of this type arise in

conformal geometry, geometric optics and optimal transportation.In

particular we will discuss structure conditions, due to Ma,Wang and

myself, which imply the regularity of solutions.These conditions are a

refinement of a condition used originally by Pogorelev for general

equations of Monge-Ampere type in two variables and called strong

ellipticity by him.

Tue, 03 Jun 2008

13:30 - 14:30
Gibson 1st Floor SR

Non-conforming and conforming methods for minimization problems exhibiting the Lavrentiev phenomenon

Christoph Ortner
(University of Oxford)
Abstract

I will begin by talking briefly about the Lavrentiev phenomenon and its implications for computations. In short, if a minimization problem exhibits a Lavrentiev gap then `naive' numerical methods cannot be used to solve it. In the past, several regularization techniques have been used to overcome this difficulty. I will briefly mention them and discuss their strengths and weaknesses.

The main part of the talk will be concerned with a class of convex problems, and I will show that for this class, relatively simple numerical methods, namely (i) the Crouzeix--Raviart FEM and (ii) the P2-FEM with under-integration, can successfully overcome the Lavrentiev gap.