OxPDE Lunchtime Seminar
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Mon, 18/05/2009 15:30 |
Peter J. Olver (University of Minnesota) |
OxPDE Lunchtime Seminar  |
Gibson 1st Floor SR |
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Thu, 28/05/2009 12:30 |
Xanthippi Markenscoff (University of California, San Diego) |
OxPDE Lunchtime Seminar |
Gibson 1st Floor SR |
| The Cosserat brothers’ ingenuous and powerful idea (presented in several papers in the Comptes Rendus at the turn of the 20th century) of solving elasticity problems by considering the homogeneous Navier equations as an eigenvalue problem is presented. The theory was taken up by Mikhlin in the 1970’s who rigorously studied it in the context of spectral analysis of pde’s, and who also presented a representation theorem for the solution of the boundary-value problems of displacement and traction in elasticity as a convergent series of the ( orthogonal and complete in the Sobolev space H1) Cosserat eigenfunctions. The feature of this representation is that the dependence of the solution on geometry, material constants and loading is provided explicitly. Recent work by the author and co-workers obtains the set of eigenfunctions for the spherical shell and compares them with the Cosserat expressions, and further explores applications and a new variational principle. In cases that the loading is orthogonal to some of the eigenfunctions, the form of the solution can be greatly simplified. Applications, in addition to elasticity theory, include thermoelasticity, poroelesticity, thermo-viscoelasticity, and incompressible Stokes flow; several examples are presented, with comparisons to known solutions, or new solutions. | |||
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Thu, 04/06/2009 12:30 |
Elaine Crooks (University of Wales, Swansea) |
OxPDE Lunchtime Seminar |
Gibson 1st Floor SR |
Scalar balance laws with monostable reaction, possibly non-convex flux, and
viscosity  are known to admit so-called entropy travelling fronts for all velocities greater than or equal to an -dependent minimal value, both when is positive, when all fronts are smooth, and for  , when the possibly non-convex flux results in fronts of speed close to the minimal value typically having discontinuities where jump conditions hold.
I will discuss the vanishing-viscosity limit of these fronts. |
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Mon, 08/06/2009 12:30 |
Mitchell Luskin (University of Minnesota) |
OxPDE Lunchtime Seminar |
Gibson 1st Floor SR |
| The accuracy of the Nos-Hoover thermostat to sample the Gibbs measure depends on the dynamics being ergodic. It has been observed for a long time that this dynamics is actually not ergodic for some simple systems, such as the harmonic oscillator. In this talk, we rigourously prove the non-ergodicity of the Nos-Hoover thermostat, for the one-dimensional harmonic oscillator. We will also show that, for some multidimensional systems, the averaged dynamics for the limit of infinite thermostat "mass" has many invariants, thus giving theoretical support for either non-ergodicity or slow ergodization. Our numerical experiments for a two-dimensional central force problem and the one-dimensional pendulum problem give evidence for non-ergodicity. We also present numerical experiments for the Nose-Hoover chain with two thermostats applied to the one-dimensional harmonic oscillator. These experiments seem to support the non-ergodicity of the dynamics if the masses of the reservoirs are large enough and are consistent with ergodicity for more moderate masses. Joint work with Frederic Legoll and Richard Moeckel | |||
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Thu, 11/06/2009 12:30 |
João Lopes Costa (Lisbon and University of Oxford) |
OxPDE Lunchtime Seminar |
Gibson 1st Floor SR |
| I will address the celebrated and long standing “No-Hair” conjecture that aims for the classification of stationary, regular, electro-vacuum black hole space-times. Besides reviewing some of the necessary concepts from general relativity I will focus on the analysis of the singular harmonic map to which the source free Einstein-Maxwell equations reduce in the stationary and axisymmetric case. | |||
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Tue, 16/06/2009 12:30 |
Andre Sonnet (University of Strathclyde) |
OxPDE Lunchtime Seminar |
Gibson 1st Floor SR |
| The orientational order of a nematic liquid crystal in a spatially inhomogeneous flow situation is best described by a Q-tensor field because of the defects that inevitably occur. The evolution is determined by two equations. The flow is governed by a generalised Stokes equation in which the divergence of the stress tensor also depends on Q and its time derivative. The evolution of Q is governed by a convection-diffusion type equation that contains terms nonlinear in Q that stem from a Landau-de Gennes potential. In this talk, I will show how the most general evolution equations can be derived from a dissipation principle. Based on this, I will identify a specific model with three viscosity coefficients that allows the contribution of the orientation to the viscous stress to be cast in the form of a Q-dependent body force. This leads to a convenient time-discretised strategy for solving the flow-orientation problem using two alternating steps. First, the flow field of the Stokes flow is computed for a given orientation field. Second, with the given flow field, one time step of the orientation equation is carried out. The new orientation field is then used to compute a new body force which is again used in the Stokes equation and so forth. For some simple test applications at low Reynolds numbers, it is found that the non-homogeneous orientation of the nematic liquid crystal leads to non-linear flow effects similar to those known from Newtonian flow at high Reynolds numbers. | |||
