OxPDE lunchtime seminar and Keble Workshop on PDE
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Thu, 10/06/2010 13:00 |
Walter Craig, Mikhail Feldman, John M. Ball, Apala Majumdar, Robert Pego |
OxPDE Lunchtime Seminar  |
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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. |
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