Past OxPDE Lunchtime Seminar

20 January 2011
12:30
Thierry Goudon
Abstract
<p> Considering kinetic equations (Boltzmann, BGK, say...) in the small mean free path regime lead to conservation laws (the Euler system, typically) When the problem is set in a domain, boundary layers might occur due to the fact that incoming fluxes could be far from equilibrium states. We consider the problem from a numerical perspective and we propose a definition of numerical fluxes for the Euler system which is intended to account for the formation of these boundary layers. </p>
  • OxPDE Lunchtime Seminar
9 December 2010
12:30
Massimo Fornassier
Abstract
Free-discontinuity problems describe situations where the solution of interest is defined by a function and a lower dimensional set consisting of the discontinuities of the function. Hence, the derivative of the solution is assumed to be a "small function" almost everywhere except on sets where it concentrates as a singular measure. This is the case, for instance, in certain digital image segmentation problems and brittle fracture models. In the first part of this talk we show new preliminary results on the existence of minimizers for inverse free-discontinuity problems, by restricting the solutions to a class of functions with piecewise Lipschitz discontinuity set. If we discretize such situations for numerical purposes, the inverse free-discontinuity problem in the discrete setting can be re-formulated as that of finding a derivative vector with small components at all but a few entries that exceed a certain threshold. This problem is similar to those encountered in the field of "sparse recovery", where vectors with a small number of dominating components in absolute value are recovered from a few given linear measurements via the minimization of related energy functionals. As a second result, we show that the computation of global minimizers in the discrete setting is an NP-hard problem. With the aim of formulating efficient computational approaches in such a complicated situation, we address iterative thresholding algorithms that intertwine gradient-type iterations with thresholding steps which were designed to recover sparse solutions. It is natural to wonder how such algorithms can be used towards solving discrete free-discontinuity problems. This talk explores also this connection, and, by establishing an iterative thresholding algorithm for discrete inverse free-discontinuity problems, provides new insights on properties of minimizing solutions thereof.
  • OxPDE Lunchtime Seminar
26 November 2010
12:30
Richard Gratwick
Abstract

Tonelli gave the first rigorous treatment of one-dimensional variational problems, providing conditions for existence and regularity of minimizers over the space of absolutely continuous functions.  He also proved a partial regularity theorem, asserting that a minimizer is everywhere differentiable, possible with infinite derivative, and that this derivative is continuous as a map into the extended real line.  Some recent work has lowered the smoothness assumptions on the Lagrangian for this result to various Lispschitz and H\"older conditions.  In this talk we will discuss the partial regularity result, construct examples showing that mere continuity of the Lagrangian is an insufficient condition.

  • OxPDE Lunchtime Seminar
28 October 2010
12:30
Lisa Harris
Abstract
It has long been known that many materials are crystalline when in their energy-minimizing states. Two of the most common crystalline structures are the face-centred cubic (fcc) and hexagonal close-packed (hcp) crystal lattices. Here we introduce the problem of crystallization from a mathematical viewpoint and present an outline of a proof that the ground state of a large system of identical particles, interacting under a suitable potential, behaves asymptotically like fcc or hcp, as the number of particles tends to infinity. An interesting feature of this result is that it holds under no initial assumption on the particle positions. The talk is based upon a joint work in progress with Florian Theil.
  • OxPDE Lunchtime Seminar
18 June 2010
11:00
Abstract
It is well-known that Morrey's quasiconvexity is closely related to gradient Young measures, i.e., Young measures generated by sequences of gradients in $L^p(\Omega;\mathbb{R}^{m\times n})$. Concentration effects, however, cannot be treated by Young measures. One way how to describe both oscillation and concentration effects in a fair generality are the so-called DiPerna-Majda measures. DiPerna and Majda showed that having a sequence $\{y_k\}$ bounded in $L^p(\Omega;\mathbb{R}^{m\times n})$,$1\le p$ <$ +\infty$, and a complete separable subring ${\cal R}$ of continuous bounded functions on $\mathbb{R}^{m\times n}$ then there exists a subsequence of $\{y_k\}$ (not relabeled), a positive Radon measure $\sigma$ on $\bar\Omega$, and a family of probability measures on $\beta_{\cal R}\mathbb{R}^{m\times n}$ (the metrizable compactification of $\mathbb{R}^{m\times n}$ corresponding to ${\cal R}$), $\{\hat\nu_x\}_{x\in\bar\Omega}$, such that for all $g\in C(\bar\Omega)$ and all $v_0\in{\cal R}$ $$ \lim_{k\to\infty}\int_\Omega g(x)v(y_k(x))d x\ = \int_{\bar\Omega}\int_{\beta_{\cal R}\R^{m\times n}}g(x)v_0(s)\hat\nu_x(d s)\sigma(d x)\ , $$ where $v(s)=v_0(s)(1+|s|^p)$. Our talk will address the question: {\it What conditions must $(\sigma,\hat\nu)$ satisfy, so that $y_k=\nabla u_k$ for $\{u_k\}\subset W^{1,p}(\Omega;\mathbb{R}^m)$} We are going to state necessary and sufficient conditions. The notion of {\it quasiconvexity at the boundary} due to Ball and Marsden plays a crucial role in this characterization. Based on this result, we then find sufficient and necessary conditions ensuring sequential weak lower semicontinuity of $I:W^{1,p}(\Omega;\mathbb{R}^m)\to\mathbb{R}$, $$ I(u)=\int_\Omega v(\nabla u(x))\,\md x\ ,$$ where $v:\mathbb{R}^{m\times n}\to\mathbb{R}$ satisfies $|v|\le C(1+|\cdot|^p)$, $C$>$0$.
  • OxPDE Lunchtime Seminar
14 June 2010
12:30
Chuck Gartland
Abstract

We consider thin films of a cholesteric liquid-crystal material subject to an applied electric field.  In such materials, the liquid-crystal "director" (local average orientation of the long axis of the molecules) has an intrinsic tendency to rotate in space; while the substrates that confine the film tend to coerce a uniform orientation.

The electric field encourages certain preferred orientations of the director as well, and these competing influences give rise to several different stable equilibrium states of the director field, including spatially uniform, translation invariant (functions only of position across the cell gap) and periodic (with 1-D or 2-D periodicity in the plane of the film).  These structures depend on two principal control parameters: the ratio of the cell gap to the intrinsic "pitch" (spatial period of rotation) of the cholesteric and the magnitude of the applied voltage.

We report on numerical work (not complete) on the bifurcation and phase behavior of this system.  The study was motivated by potential applications involving switchable gratings and eyewear with tunable transparency. We compare our results with experiments conducted in the Liquid Crystal Institute at Kent State University.

  • OxPDE Lunchtime Seminar
11 June 2010
12:30
Abstract
Numerical homogenization/upscaling for problems with multiple scales have attracted increasing attention in recent years. In particular, problems with non-separable scales pose a great challenge to mathematical analysis and simulation. In this talk, we present some rigorous results on homogenization of divergence form scalar and vectorial elliptic equations with $L^\infty$ rough coefficients which allow for a continuum of scales. The first approach is based on a new type of compensation phenomena for scalar elliptic equations using the so-called ``harmonic coordinates''. The second approach, the so-called ``flux norm approach'' can be applied to finite dimensional homogenization approximations of both scalar and vectorial problems with non-separated scales. It can be shown that in the flux norm, the error associated with approximating the set of solutions of the PDEs with rough coefficients, in a properly defined finite-dimensional basis, is equal to the error associated with approximating the set of solutions of the same type of PDEs with smooth coefficients in a standard finite element space. We will also talk about the ongoing work on the localization of the basis in the flux norm approach.
  • OxPDE Lunchtime Seminar
10 June 2010
13:00
to
18:00
Walter Craig, Mikhail Feldman, John M. Ball, Apala Majumdar, Robert Pego
Abstract
{\bf Keble Workshop on Partial Differential Equations in Science and Engineering} \\ \\Place: Roy Griffiths Room in the ARCO Building, Keble College \\Time: 1:00pm-5:10pm, Thursday, June 10. \\ \\ Program:\\ \\ 1:00-1:20pm: Coffee and Tea \\ \\ 1:20-2:10pm: Prof. Walter Craig (Joint with OxPDE Lunchtime Seminar) \\ \\ 2:20-2:40pm Prof. Mikhail Feldman \\ \\ 2:50-3:10pm Prof. Paul Taylor \\ \\ 3:20-3:40pm Coffee and Biscuits \\ \\ 3:40-4:00pm: Prof. Sir John Ball \\ \\ 4:10-4:30pm: Dr. Apala Majumdar \\ \\ 4:40-5:00pm: Prof. Robert Pego \\ \\ 5:10-6:00pm: Free Discussion \\ \\{\bf Titles and Abstracts:} \\ 1.{\bf Title: On the singular set of the Navier-Stokes equations \\ Speaker: Prof. Walter Craig, McMaster University, Canada} \\ Abstract:\\ The Navier-Stokes equations are important in fluid dynamics, and a famous mathematics problem is the question as to whether solutions can form singularities. I will describe these equations and this problem, present three inequalities that have some implications as to the question of singularity formation, and finally, give a new result which is a lower bound on the size of the singular set, if indeed singularities exist. \\ \\{\bf 2. Title: Shock Analysis and Nonlinear Partial Differential Equations of Mixed Type. \\ Speaker: Prof. Mikhail Feldman, University of Wisconsin-Madison, USA} \\ \\ Abstract:\\ Shocks in gas or compressible fluid arise in various physical situations, and often exhibit complex structures. One example is reflection of shock by a wedge. The complexity of reflection-diffraction configurations was first described by Ernst Mach in 1878. In later works, experimental and computational studies and asymptotic analysis have shown that various patterns of reflected shocks may occur, including regular and Mach reflection. However, many fundamental issues related to shock reflection are not understood, including transition between different reflection patterns. For this reason it is important to establish mathematical theory of shock reflection, in particular existence and stability of regular reflection solutions for PDEs of gas dynamics. Some results in this direction were obtained recently. \\ In this talk we start by discussing examples of shocks in supersonic and transonic flows of gas. Then we introduce the basic equations of gas dynamics: steady and self-similar compressible Euler system and potential flow equation. These equations are of mixed elliptic-hyperbolic type. Subsonic and supersonic regions in the flow correspond to elliptic and hyperbolic regions of solutions. Shocks correspond to certain discontinuities in the solutions. We discuss some results on existence and stability of steady and self-similar shock solutions, in particular the recent work (joint with G.-Q. Chen) on global existence of regular reflection solutions for potential flow. We also discuss open problems in the area. \\ \\{\bf 3. Title: Shallow water waves - a rich source of interesting solitary wave solutions to PDEs \\ Speaker: Prof. Paul H. Taylor, Keble College and Department of Engineering Science, Oxford} \\ \\Abstract:\\ In shallow water, solitary waves are ubiquitous: even the wave crests in a train of regular waves can be modelled as a succession of solitary waves. When successive crests are of different size, they interact when the large ones catch up with the smaller. Then what happens? John Scott Russell knew by experiment in 1844, but answering this question mathematically took 120 years! This talk will examine solitary wave interactions in a range of PDEs, starting with the earliest from Korteweg and De Vries, then moving onto Peregrine's regularized long wave equation and finally the recently introduced Camassa-Holm equation, where solitary waves can be cartoon-like with sharp corners at the crests. For each case the interactions can be described using the conserved quantities, in two cases remarkably accurately and in the third exactly, without actually solving any of the PDEs. The methodology can be extended to other equations such as the various versions of the Boussinesq equations popular with coastal engineers, and perhaps even the full Euler equations. \\ {\bf 4. Title: Austenite-Martensite interfaces \\ Speaker: Prof. Sir John Ball, Queen's College and Mathematical Institute, Oxford} \\ \\Abstract:\\ Many alloys undergo martensitic phase transformations in which the underlying crystal lattice undergoes a change of shape at a critical temperature. Usually the high temperature phase (austenite) has higher symmetry than the low temperature phase (martensite). In order to nucleate the martensite it has to somehow fit geometrically to the austenite. The talk will describe different ways in which this occurs and how they may be studied using nonlinear elasticity and Young measures. \\ \\{\bf 5. Title: Partial Differential Equations in Liquid Crystal Science and Industrial Applications \\ Speaker: Dr. Apala Majumdar, Keble College and Mathematical Institute, Oxford} \\ \\Abstract:\\ Recent years have seen a growing demand for liquid crystals in modern science, industry and nanotechnology. Liquid crystals are mesophases or intermediate phases of matter between the solid and liquid phases of matter, with very interesting physical and optical properties. We briefly review the main mathematical theories for liquid crystals and discuss their analogies with mathematical theories for other soft-matter phases such as the Ginzburg-Landau theory for superconductors. The governing equations for the static and dynamic behaviour are typically given by systems of coupled elliptic and parabolic partial differential equations. We then use this mathematical framework to model liquid crystal devices and demonstrate how mathematical modelling can be used to make qualitative and quantitative predictions for practical applications in industry. \\ \\{\bf 6. Title: Bubble bath, shock waves, and random walks --- Mathematical models of clustering \\Speaker: Prof. Robert Pego, Carnegie Mellon University, USA} \\Abstract:\\ Mathematics is often about abstracting complicated phenomena into simple models. This talk is about equations that model aggregation or clustering phenomena --- think of how aerosols form soot particles in the atmosphere, or how interplanetary dust forms comets, planets and stars. Often in such complex systems one observes universal trend toward self-similar growth. I'll describe an explanation for this phenomenon in two simple models describing: (a) ``one-dimensional bubble bath,'' and (b) the clustering of random shock waves.
  • OxPDE Lunchtime Seminar
20 May 2010
12:30
Abstract
In this talk, we describe new profile decompositions for bounded sequences in Banach spaces of functions defined on $\mathbb{R}^d$. In particular, for "critical spaces" of initial data for the Navier-Stokes equations, we show how these can give rise to new proofs of recent regularity theorems such as those found in the works of Escauriaza-Seregin-Sverak and Rusin-Sverak. We give an update on the state of the former and a new proof plus new results in the spirit of the latter. The new profile decompositions are constructed using wavelet theory following a method of Jaffard.
  • OxPDE Lunchtime Seminar
13 May 2010
12:30
Abstract
In this talk we discuss the solution of the elastodynamic equations in a bounded domain with hereditary-type linear viscoelastic constitutive relation. Existence, uniqueness, and regularity of solutions to this problem is demonstrated for those viscoelastic relaxation tensors satisfying the condition of being completely monotone. We then consider the non-self-adjoint and non-linear eigenvalue problem associated with the frequency-domain form of the elastodynamic equations, and show how the time-domain solution of the equations can be expressed in terms of an eigenfunction expansion.
  • OxPDE Lunchtime Seminar

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