Forthcoming Seminars › Seminar Series

OxPDE Lunchtime Seminar

Thu, 16/10/2008
13:30 		Jon Bevan (University of Surrey) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
One-homogeneous stationary points of elliptic systems in two dimensions.
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 $.
Fri, 17/10/2008
13:30 		Margaret Beck (Brown University, US) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Using global invariant manifolds to understand metastability in Burgers equation with small viscosity
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, 23/10/2008
13:30 		Margaret Beck (Brown University, US) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Nonlinear stability of time-periodic viscous shocks
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.
Wed, 05/11/2008
13:30 		Kaushik Bhattacharya (Caltech) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Propagation of free boundaries in heterogeneous materials
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, 06/11/2008
12:30 		Eugen Varvaruca (Imperial College) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
On the existence of extreme waves and the Stokes conjecture with vorticity
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, 12/11/2008
10:45 		Amit Acharya (Carnegie Mellon University) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Compatibility conditions for the Left Cauchy Green Tensor field in 3-D
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, 13/11/2008
13:30 		Sorin Mardare (University of Rouen) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Asymptotic behaviour of the Stokes problem in cylinders
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.
Mon, 17/11/2008
12:30 		Mikhail Osipov (Strathclyde) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Order Parameters, Irreducible Tensors and the theory of Phase Transitions in Smectic Liquid Crystals
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.
Tue, 18/11/2008
11:00 		Christopher Larsen (Worcester Polytechnic Institute, USA) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Dynamic fracture based on Griffith's criterion
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.
Wed, 19/11/2008
14:00 		Vasily V. Zhikov (Moscow State University and Vladimir State University, Russia) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
An approach to solvability of the generalised Navier-Stokes equation
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<2. The method of proof is based on ideas of the geometric measure theory and compensated compactness. A space-time measure is related to the solution whose existence is proved. This measure may have a singular component. A connection between this singularity and the known results from the partial regularity theory is discussed, in particular for the classical case p=2.
Thu, 20/11/2008
12:00 		David Rule (University of Edinburgh) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Elliptic equations in the plane satisfying a Carleson measure condition
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 < p \leq 2. We prove that if the norm of the Carleson measure is sufficiently small, we can solve both the Neumann and regularity problems with data in L^p. This is related to earlier work on the Dirichlet problem by other authors.
Mon, 24/11/2008
13:30 		Yoshihito Oshita (Okayama University, Japan) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Fine structures arising in diblock copolymers and reaction-diffusion systems
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.
Wed, 26/11/2008
13:30 		Eduard Kirr (University of Illinois at Urbana Champaign, USA) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Variational Methods in Nonlinear Schroedinger Equations
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.
Thu, 27/11/2008
13:30 		Michael Farber (University of Durham) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Topology of Robot Motion Planning
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.

For any corrections/updates to these listings, please contact seminars [-at-] maths [dot] ox [dot] ac [dot] uk.

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