Seminar Archive 2007/8

All dates/times are tentative. For more information contact Martin Dindos (Edinburgh), Lyonell Boulton or Wolfgang Staubach (Heriot-Watt) Fri 5 Sep 2008 Univeristy of Edinburgh JCMB 4312 3.00pm Barbara Keyfitz (Field's Institute, Toronto) Hyperbolic Conservation Laws - Past and Future Abstract. The field of conservation laws (quasilinear hyperbolic partial differential equations) has captured the attention of mathematics researchers, computational fluid dynamicists, and modelers of physical and engineering phenomena for over 70 years. This talk will survey some of the power, and some of the limitations, of the conservation law approach to modeling. It will also expound some of the major achievements in establishing a mathematical theory, and discuss why theoretical advances have been so slow, and why so much still seems to remain out of reach. Finally, I will describe recent work of a number of people on establishing mathematical properties of shock reflection phenomena. Thu 5 Jun 2008 University of Edinburgh JCMB 5215 3.00pm Alexander Volberg (Edinburgh) Belman function, harmonic analysis and GMT II Tue 27 May 2008 University of Edinburgh JCMB 5215 3.00pm Ian Doust (University of New South Wales) Enhanced negative type for finite metric trees Jose Rodrigo (Warwick) Singularities for aggregation equations with dissipation Thu 22 May 2008 University of Edinburgh JCMB 5215 3.00pm Alexander Volberg (Edinburgh) Belman function, harmonic analysis and GMT Thu 15 May 2008 University of Edinburgh JCMB 4312 3.00pm Denis Serre (Ecole Normale Superieur de Lyon) Shock profiles for finite difference schemes Mon 5 May 2008 University of Edinburgh JCMB 4312 3.00pm Ilja Bogaevsky (Moscow) Trajectories of viscosity solutions of Hamilton-Jacobi equation Abstract. In space of arbitrary dimension, we construct a physically natural motion of particles defined even at singularities of a viscosity solution of a Hamilton-Jacobi equation. For such a motion in plane, we describe all typical cases of appearance and interaction of clusters and draw figures showing how they participate in the well-known typical transitions of viscosity solution singularities. Fri 2 May 2008 ICMS, 14 India Street 4.00pm Andrej Zlatos (University of Chicago) Speed-up of reaction-diffusion fronts by flows Abstract. I will discuss some recent results on speed-up of traveling fronts by strong periodic flows in reaction-diffusion equations. I will present a characterization of the flows which can arbitrarily speed up fronts for general combustion-type reactions in two dimensions. The rate of this speed-up will also be determined. The problem turns out to be closely connected to the simpler question of effective diffusivity enhancement in the homogenization of the corresponding (linear) passive scalar equations. Wed 30 Apr - Fri 2 May 2008 ICMS, 14 India Street all day Classical and Modern Harmonic Analysis: from Theory to Numerical Computation Tue 29 Apr 2008 Heriot-Watt CM G.01 4.15pm Samir Bhowmik (Heriot-Watt) Numerical approximation of a nonlinear partial integro-differential equation Fri 25 Apr 2008 ICMS, 14 India Street 10.00am Harmonic Analysis / PDE Workshop Speakers: Jonathan Bennett (Birmingham) Alexander Mahalov (Arizona State) Gregory Seregin (Oxford) Paco Villarroya (Ediburgh) Tue 15 Apr 2008 Heriot-Watt CM G.01 4.15pm Volker Betz (Warwick) Exponentially small transitions in molecular quantum dynamics Abstract. Transitions of electronic states at closely avoided electronic energy levels  are an important topic in quantum chemistry. Mathematically, we are dealing with  a singularly perturbed system of partial differential equations, where the challenge is  to extract and characterise the exponentially small transitions between optimal superadiabatic subspaces. In a simplified version, where we reduce the PDE system to and ODE system,   the problem is now well understood. I will talk about the methods used to obtain the accurate exponentially small transition histories in the case of the ODE system,  and about the partial progress and main issues in the PDE case. Mon 14 Apr 2008 University of Edinburgh JCMB 6301 3.00pm Luis Vega (Universidad del Pais Vasco) On the stability of a singular vortex dynamics Fri 11 Apr 2008 University of Edinburgh JCMB 6206 4.00pm Juan Antonio Barcelo (Universidad Politecnica de Madrid) Bilinear Fourier mappings arising in Scattering Theory Tue 1 Apr 2008 Heriot-Watt CM G.01 4.15pm Bjorn Sandstede (Surrey) Stability of time-periodic viscous shocks Tue 18 Mar 2008 University of Edinburgh JCMB 5215 4.00pm Vilmos Komornik (University of Strasbourg) Ingham type theorems Abstract. We report on some joint works with C. Baiocchi and P. Loreti. In a paper of 1936, dedicated to Dirichlet series, Ingham established an elegant generalization of Parseval's equality. Later his theorem proved to be very useful in control theory. Motivated by various applications, we discuss several improvements and extensions of this result and we explain its connection to a classical variational problem. Thu 13 Mar 2008 ICMS, 14 India Street 3.00pm Mini-symposium on PDEs Tue 4 Mar 2008 Heriot-Watt CM G.01 4.15pm Sergey Zelik (Surrey) Interaction of solitons and Sinai-Bunimovich space-time chaos in dissipative PDEs Tue 5 Feb 2008 Heriot-Watt CM G.01 4.15pm Mathieu Lewin (Cergy-Pontoise) A variational model for relativistic electrons Abstract. I will present a variational model from relativistic quantum mechanics, involving the Dirac operator. The theory allows to describe the state of N electrons (for instance in an atom or a molecule). But uncommon effects like the polarization of the vacuum or the spontaneous creation of electron-positron pairs can also be described. Tue 29 Jan 2008 Heriot-Watt CM G.01 4.15pm David Dos Santos Ferreira (Paris 13) Anisotropic inverse problems and Carleman estimates Tue 22 Jan 2008 ICMS 14 India Street Mini-symposium in PDEs 3.00pm Pavel Plotnikov (Bath and Novosibirsk) Inhomogeneous boundary value problems for compressible Navier-Stokes and transport equations 4.30pm Nicholas Burq (University of Paris Sud) Random data Cauchy theory for nonlinear wave equations Tue 4 Dec 2007 Heriot-Watt CM G.01 4.15pm Jan Kristensen (University of Oxford) On the problem of regularity in the calculus of variations Mon 26 Nov 2007 Edinburgh University JCMB 5215 2.00pm Miroslav Chlebik (Sussex) Blowup behaviour for some nonlinear parabolic problems Tue 20 Nov 2007 Heriot-Watt CM G.01 4.15pm Igor Verbitsky (University of Birmingham) Form boundedness and global Green's function estimates Abstract. We intend to present a solution to the form boundedness problem for general second-order differential operators with distributional coefficients. This includes infinitesimal form boundedness, Trudinger's and Nash's inequalities. We will also give global bilateral estimates for Green's function and the conditional gauge associated with a class of differential and integral operators. Connections with some nonlinear elliptic PDE will be discussed. This talk is based on joint work with Michael Frazier and Vladimir Maz'ya. Wed 14 Nov 2007 ICMS 14 India Street Mini-symposium in PDEs 3.00pm Patrick Gerard (University of Paris Sud) The nonlinear Schrodinger equation on the sphere 4.30pm Ari Laptev (Imperial College) From Weyl type asymptotics to Lieb-Thirring inequalities Tue 13 Nov 2007 U. of Edinburgh JCMB 5325 2.00pm Michael G Cowling (Birmingham) A vector-valued version of Hardy's theorem and an uncertainty principle for operators Mon 12 Nov 2007 ICMS 14 India Street (Note there will be two talks) 3.00pm Nicolai V. Krylov (University of Minnesota) A new approach to parabolic and elliptic equations with VMO coefficients, Abstract. These two lectures consist of two parts. In the first part we present with proofs the main tools from Real Analysis and give a simple example of proving the solvability of elliptic equations with VMO coefficients. In the second part we present the results obtained by using the techniques explained in the first part. A wide class of elliptic and parabolic equations with measurable coefficients will be treated. As an immediate consequence we obtain weak uniqueness of diffusions for a class of operators much wider than those considered by Stroock and Varadhan. Tue 5 Nov 2007 Heriot-Watt CM Building, Room G.01 4.15pm Barbara Niethammer (University of Oxford) Effective theories for Ostwald Ripening Abstract. Ostwald Ripening denotes the late stage coarsening in a first order phase transition, where second phase particles interact by diffusional mass exchange to reduce their total surface area. In the low volume fraction regime the statistics of the ripening process can be described by the classical mean-field theory by Lifshitz, Slyozov and Wagner (LSW). However, due to several shortcomings of this theory, we are interested in higher order effects which are due to the finite volume fraction of particles. We compare the effects of screening induced fluctuations in particle densities with the ones induced by coalescence of particles. Tue 23 Oct 2007 Heriot-Watt CM Building, Room G.01 (Note unusual room) 4.15pm Martin Dindos (University of Edinburgh) L^p Dirichlet problem for elliptic operators with rough coefficients Abstract. We present of joint work with J. Pipher and S. Petermichl where we study the Dirichlet $L^p$ solvability of divergence type elliptic operators with (just) $L^\infty$ coefficients. Well know counterexamples show that boundedness and ellipticity is not sufficient for $L^p$ solvability, hence additional condition is required. Ussually, some kind of continuity or Dini-type condition is assumed. We instead present a much weaker Carleson type condition that is in some sence "sharp". In particular, we present result that for any $p>1$ if certain Carleson norm of coefficients of the operator is less than $C(p)$ the the $L^p$ problem is solvable. In addition, if coefficients satisfy vanishing Carleson condition, then the problem is solvable for all $p>1$. This can be used to show that the $L^p$ Dirichlet problem for the Laplace operator is solvable for all $p>1$ on Lipschitz domains with the property that $nabla phi$ is in the "vmo", where $\phi$ is the Lipschitz function that (locally) determines the boundary. "Vmo" is the space of functions of vanishing mean oscillations. Mon 15 Oct 2007 U. of Edinburgh JCMB 5215 4.15pm Sandra Pott (University of Glasgow) Logarithmic BMO on the bidisk Tue 9 Oct 2007 Heriot-Watt CM Building, Room T.01 4.15pm Lyonell Boulton (Heriot-Watt University) Basis properties of the eigenfunctions of the p-Laplacian in one dimension Abstract. Generalised sine functions are defined as the Dirichlet eigenfunctions associated to the first eigenvalue of the one-dimensional $p$-Laplacian equation on an interval of length $\pi_p:=2\pi/(p\sin(\pi/p))$. In the early eighties various properties of these functions were studied. Among others, $p\not=2$ versions of the Pythagorean relation and characterisations of the eigenfunctions associated to higher order eigenvalues. Despite this activity, it seems that analogues for $p\not=2$ of the standard completeness and expansion theorems for sine functions have not been considered previously. In this talk I will show that for $12/11\leq p < \infty$, the family of eigenfunctions of the $p$-Laplacian forms a Schauder basis of $L^2(0,\pi_p)$. Tue 8 Oct 2007 U. of Edinburgh JCMB 5215 4.15pm David Rule (University of Edinburgh) The Regularity and Neumann Problem for Non-Symmetric Elliptic Operators Mon 1 Oct 2007 U. of Edinburgh JCMB 5327 4.15pm Wolfgang Staubach (Heriot-Watt University) Pseudo-pseudodifferential operators and some of their applications in harmonic analysis Tue 25 Sep 2007 Heriot-Watt CM Building, Room T.01 4.15pm Alexander Belyaev (EECE) Nonlinear PDEs for Imaging and Graphics Abstract. In the past few years, the use of partial differential equations (PDEs) has become increasingly popular in the image processing and computer graphics communities. PDEs have brought new and powerful tools in imaging and graphics, the areas traditionally occupied by electrical engineers and computer scientists. In their turn, image processing and computer graphics bring to the PDE field many challenging new problems. Tue 20 Mar 2007 U. of Edinburgh JCMB, Room 5327 3.30pm Jose Rodrigo (Warwick) Contour dynamics for 2D active scalars 5.00pm Rafael Benguria (Pontificia Universidad Católica de Chile) On the speed of pulled fronts with a cutoff Abstract. We study the effect of a small cutoff ε on the velocity of a pulled front for the reaction diffusion equation in one dimension by means of a variational principle. We obtain a lower bound on the speed dependent on the cutoff, and for which the two leading order terms correspond to the Brunet Derrida expression. To do so we cast a known variational principle for the speed of propagation of fronts in new variables which makes it more suitable for applications. Tue 13 Mar 2007 Heriot-Watt CM Building, Room T.01 4.15pm Peter Kuchment (Texas A&M University) Thermoacoustic tomography, circular Radon transform, and the wave equation Abstract. The following problem has been emerging from several areas of mathematics and applications: can one recover a function on the plane or a higher dimensional space knowing its integrals over all circles (spheres) centered at the points of a given set S? One of its sources was approximation theory. Recent new types of tomography, e.g. photoacoustic and thermoacoustic tomography lead to similar problems. A major breakthrough was made in the middle of 90s in the work by M. Agranovsky and E. T. Quinto, albeit it did not completely resolve the issue. Significant progress has occurred recently. The talk will survey the status of this problem, known results, and open questions.