OxPDE Lunchtime Seminar
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Thu, 24/04/2008 13:00 |
I. Gallagher (Paris VII) |
OxPDE Lunchtime Seminar  |
Gibson 1st Floor SR |
| It is well known that the three dimensional, incompressible Navier-Stokes equations have a unique, global solution provided the initial data is small enough in a scale invariant space (say L3 for instance). We are interested in finding examples for which no smallness condition is imposed, but nevertheless the associate solution is global and unique. The examples we will present are due to collaborations with Jean-Yves Chemin, and with Marius Paicu. | |||
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Fri, 25/04/2008 13:30 |
Gui-Qiang Chen (USA) |
OxPDE Lunchtime Seminar |
Gibson 1st Floor SR |
| In this talk we will start with various shock reflection-diffraction phenomena, their fundamental scientific issues, and their theoretical roles in the mathematical theory of multidimensional hyperbolic systems of conservation laws. Then we will describe how the global shock reflection-diffraction problems can be formulated as free boundary problems for nonlinear conservation laws of mixed-composite hyperbolic-elliptic type.Finally we will discuss some recent developments in attacking the shock reflection-diffraction problems, including the existence, stability, and regularity of global regular configurations of shock reflection-diffraction by wedges. The approach includes techniques to handle free boundary problems, degenerate elliptic equations, and corner singularities, which is highly motivated by experimental, computational, and asymptotic results. Further trends and open problems in this direction will be also addressed. This talk will be mainly based on joint work with M. Feldman. | |||
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Wed, 21/05/2008 13:30 |
Nicolas Condette (Humboldt-Univ, Berlin) |
OxPDE Lunchtime Seminar |
Gibson 1st Floor SR |
| We propose and analyze a fully discrete Fourier collocation scheme to solve numerically a nonlinear equation in 2D space derived from a pattern forming gradient flow. We prove existence and uniqueness of the numerical solution and show that it converges to a solution of the initial continuous problem. We also derive some error estimates and perform numerical experiments to illustrate the theory. | |||
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Tue, 03/06/2008 13:30 |
Christoph Ortner (University of Oxford) |
OxPDE Lunchtime Seminar |
Gibson 1st Floor SR |
| I will begin by talking briefly about the Lavrentiev phenomenon and its implications for computations. In short, if a minimization problem exhibits a Lavrentiev gap then `naive' numerical methods cannot be used to solve it. In the past, several regularization techniques have been used to overcome this difficulty. I will briefly mention them and discuss their strengths and weaknesses. The main part of the talk will be concerned with a class of convex problems, and I will show that for this class, relatively simple numerical methods, namely (i) the Crouzeix–Raviart FEM and (ii) the P2-FEM with under-integration, can successfully overcome the Lavrentiev gap. | |||
