Past OxPDE Lunchtime Seminar

18 July 2008
13:30
Abstract
We present a selection of recent results pertaining to Hessian and Monge-Ampere equations, where the Hessian matrix is augmented by a matrix valued lower order operator. Equations of this type arise in conformal geometry, geometric optics and optimal transportation.In particular we will discuss structure conditions, due to Ma,Wang and myself, which imply the regularity of solutions.These conditions are a refinement of a condition used originally by Pogorelev for general equations of Monge-Ampere type in two variables and called strong ellipticity by him.
  • OxPDE Lunchtime Seminar
3 June 2008
13:30
Christoph Ortner
Abstract
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.
  • OxPDE Lunchtime Seminar
21 May 2008
13:30
Nicolas Condette
Abstract
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.
  • OxPDE Lunchtime Seminar
25 April 2008
13:30
Gui-Qiang Chen
Abstract
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.</p><p>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.
  • OxPDE Lunchtime Seminar
24 April 2008
13:00
I. Gallagher
Abstract
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 L<up>3</sup> 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.
  • OxPDE Lunchtime Seminar
6 March 2008
12:00
Sorin Mardare
Abstract
As all analysts know, solving a problem which contains some kind of nonlinearity is generally far from being obvious. This is of course the case when we deal with nonlinear PDEs, but also when we have linear systems with some nonlinear compatibility conditions. One example is constituted by the systems of first order linear partial differential equations. If the coefficients are regular, there is a classical way to solve this systems. However, if the coefficients are only of class L<sup>p</sup>, the classical methods cannot be applied. We will show how this problem can be solved by using a method of regularization which allows to preserve the nonlinear compatibility conditions. Then we will present some possible applications to the theory of elasticity. In the end some open problems related to similar aspects will be discussed. Example of such a problem: the rigidity of deformations of class H<sup>1</sup>.
  • OxPDE Lunchtime Seminar
28 February 2008
10:00
Patrizio Neff
Abstract
We are concerned with the derivation of the &Gamma;-limit to a three dimensional geometrically exact Cosserat model as the relative thickness h > 0 of a at domain tends to zero. The Cosserat bulk model involves already exact rotations as a second independent field and this model is meant to describe defective elastic crystals liable to fracture under shear. <BR> It is shown that the &Gamma;-limit based on a natural scaling assumption con- sists of a membrane like energy contribution and a homogenized transverse shear energy both scaling with h, augmented by an additional curvature stiffness due to the underlying Cosserat bulk formulation, also scaling with h. No specific bending term appears in the dimensional homogenization process. The formulation exhibits an internal length scale Lc which sur- vives the homogenization process. A major technical difficulty, which we encounter in applying the &Gamma;-convergence arguments, is to establish equi- coercivity of the sequence of functionals as the relative thickness <i>h</i> tends to zero. Usually, equi-coercivity follows from a local coerciveness assump- tion. While the three-dimensional problem is well-posed for the Cosserat couple modulus &mu;<sub>c</sub> &ge; 0, equi-coercivity forces us to assume a strictly pos- itive Cosserat couple modulus &mu;<sub>c</sub> > 0. The &Gamma;-limit model determines the midsurface deformation m &isin; H<sup>1,2</sup>(&omega;;R<sup>3</sup>). For the case of zero Cosserat couple modulus &mu;<sub>c</sub>= 0 we obtain an estimate of the &Gamma; - lim inf and &Gamma; - lim sup, without equi-coercivity which is then strenghtened to a &Gamma;- convergence result for zero Cosserat couple modulus. The classical linear Reissner-Mindlin model is "almost" the linearization of the &Gamma;-limit for &mu;<sub>c</sub> = 0 apart from a stabilizing shear energy term.
  • OxPDE Lunchtime Seminar
27 February 2008
12:00
Benson Muite
Abstract
We describe how to perform high resolution simulations of viscoelastic continuum mechanical models for martensitic transformations with diffuse interfaces. The computational methods described may also be of use in performing high resolution simulations of time dependent partial differential equations where solutions are sufficiently smooth.
  • OxPDE Lunchtime Seminar

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