Past OxPDE Lunchtime Seminar

Motivated by the tensile experiments on titanium alloys of Petrinic et al

(2006), which show the formation of cracks through the formation and

coalescence of voids in ductile fracture, we consider the problem of

formulating a variational model in nonlinear elasticity compatible both

with cavitation and with the appearance of discontinuities across

two-dimensional surfaces. As in the model for cavitation of Müller and

Spector (1995) we address this problem, which is connected to the

sequential weak continuity of the determinant of the deformation gradient

in spaces of functions having low regularity, by means of adding an

appropriate surface energy term to the elastic energy. Based upon

considerations of invertibility we are led to an expression for the

surface energy that admits a physical and a geometrical interpretation,

and that allows for the formulation of a model with better analytical

properties. We obtain, in particular, important regularity properites of

the inverses of deformations, as well as the weak continuity of the

determinants and the existence of minimizers. We show further that the

creation of surface can be modelled by carefully analyzing the jump set of

the inverses, and we point out some connections between the analysis of

cavitation and fracture, the theory of SBV functions, and the theory of

cartesian currents of Giaquinta, Modica and Soucek. (Joint work with

Carlos Mora-Corral, Basque Center for Applied Mathematics).

Modelling the behaviour of light in photonic crystal fibres requires

solving 2nd-order elliptic eigenvalue problems with discontinuous

coefficients. The eigenfunctions of these problems have limited

regularity. Therefore, the planewave expansion method would appear to

be an unusual choice of method for such problems. In this talk I

examine the convergence properties of the planewave expansion method as

well as demonstrate that smoothing the coefficients in the problem (to

get more regularity) introduces another error and this cancels any

benefit that smoothing may have.

The relaxation of a free-energy functional which describes the

order-strain interaction in nematic liquid crystal elastomers is obtained

explicitly. We work in the regime of small strains (linearized

kinematics). Adopting the uniaxial order tensor theory or Frank

model to describe the liquid crystal order, we prove that the

minima of the relaxed functional exhibit an effective biaxial

microstructure, as in de Gennes tensor model. In particular, this

implies that the response of the material is soft even if the

order of the system is assumed to be fixed. The relaxed energy

density satisfies a solenoidal quasiconvexification formula.

I will discuss the investigatation of highly nonlinear solitary waves in heterogeneous one-dimensional granular crystals using numerical computations, asymptotics, and experiments. I will focus primarily on periodic arrangements of particles in experiments in which stiffer/heavier stainless stee are alternated with softer/lighter ones.

The governing model, which is reminiscent of the Fermi-Pasta-Ulam lattice, consists of a set of coupled ordinary differential equations that incorporate Hertzian interactions between adjacent particles. My collaborators and I find good agreement between experiments and numerics and gain additional insight by constructing an exact compaction solution to a nonlinear partial differential equation derived using long-wavelength asymptotics. This research encompasses previously-studied examples as special cases and provides key insights into the influence of heterogeneous, periodic lattice on the properties of the solitary waves.

I will briefly discuss more recent work on lattices consisting of randomized arrangements of particles, optical versus acoustic modes, and the incorporation of dissipation.

I will introduce the moving frame approach to the analysis of invariant variational problems and the evolution of differential invariants under invariant submanifold flows. Applications will include differential geometric flows, integrable systems, and image processing.

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.

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.

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

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

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.

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.

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.

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.

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.

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 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.

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.

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 $\det A \to 0$.

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)

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.

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.