Forthcoming Seminars › Seminar Series

OxPDE Lunchtime Seminar

Mon, 26/04/2010
12:30 		Yong-Kum Cho (Chung-Ang University) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
A priori estimates for the weak solutions to the Boltzmann equation with grazing collisions
In this talk we consider the Boltzmann equation arising in gas dynamics with long-range interactions. Mathematically, it involves bilinear singular integral operators known as collision operators with non-cutoff collision kernels. As for the associated Cauchy problem, we develop a theory of weak solutions and present some of its a priori estimates related with physical quantities including the energy and moments.
Thu, 29/04/2010
12:30 		Dmitri Vassiliev (University College, London) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Rotational Elasticity
We consider a 3-dimensional elastic continuum whose material points can experience no displacements, only rotations. This framework is a special case of the Cosserat theory of elasticity. Rotations of material points of the continuum are described mathematically by attaching to each geometric point an orthonormal basis which gives a field of orthonormal bases called the coframe. As the dynamical variables (unknowns) of our theory we choose the coframe and a density. In the first part of the talk we write down the general dynamic variational functional of our problem. In doing this we follow the logic of classical linear elasticity with displacements replaced by rotations and strain replaced by torsion. The corresponding Euler-Lagrange equations turn out to be nonlinear, with the source of this nonlinearity being purely geometric: unlike displacements, rotations in 3D do not commute. In the second part of the talk we present a class of explicit solutions of our Euler-Lagrange equations. We call these solutions plane waves. We identify two types of plane waves and calculate their velocities. In the third part of the talk we consider a particular case of our theory when only one of the three rotational elastic moduli, that corresponding to axial torsion, is nonzero. We examine this case in detail and seek solutions which oscillate harmonically in time but depend on the space coordinates in an arbitrary manner (this is a far more general setting than with plane waves). We show [1] that our second order nonlinear Euler-Lagrange equations are equivalent to a pair of linear first order massless Dirac equations. The crucial element of the proof is the observation that our Lagrangian admits a factorisation. [1] Olga Chervova and Dmitri Vassiliev, "The stationary Weyl equation and Cosserat elasticity", preprint http://arxiv.org/abs/1001.4726
Thu, 13/05/2010
12:30 		David Al-Attar (Department of Earth Sciences, University of Oxford) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Eigenfunction Expansion Solutions of the Linear Viscoelastic Wave Equation
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.
Thu, 20/05/2010
12:30 		Gabriel Koch (OxPDE, University of Oxford) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Profile decompositions and applications to Navier-Stokes
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.
Thu, 10/06/2010
13:00 		Walter Craig, Mikhail Feldman, John M. Ball, Apala Majumdar, Robert Pego OxPDE Lunchtime Seminar Add to calendar
OxPDE lunchtime seminar and Keble Workshop on PDE
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

Titles and Abstracts:
1.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.

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.

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.
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.

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.

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.
Fri, 11/06/2010
12:30 		Lei Zhang (Hausdorff Center for Mathematics) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Homogenization approximation for PDEs with non-separated scales
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.
Mon, 14/06/2010
12:30 		Chuck Gartland (Kent State) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Numerical Investigations of Electric-Field-InducedTransitions in Cholesteric Liquid Crystal Films

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.
Fri, 18/06/2010
11:00 		Martin Kruzik (Academy of Sciences, Prague) OxPDE Lunchtime Seminar Add to calendar Gibson 1st Floor SR
Quasiconvexity at the boundary and weak lower semicontinuity of integral functionals
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: 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 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 $.

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

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