Past OxPDE Lunchtime Seminar

22 September 2011
12:30
Gunther Uhlmann
Abstract
We will give a survey on some recent results on travel tomography which consists in determining the index of refraction of a medium by measuring the travel times of sound waves going through the medium. In differential geometry this is known as the boundary rigidity problem. We will also consider the related problem of tensor tomography which consists in determining a function, a vector field or tensors of higher rank from their integrals along geodesics.
  • OxPDE Lunchtime Seminar
20 September 2011
12:30
Gautam Iyer
Abstract
We consider an elliptic eigenvalue problem in the presence a fast cellular flow in a two-dimensional domain. It is well known that when the amplitude, <i>A</i>, is fixed, and the number of cells, $L^2$, increases to infinity, the problem `homogenizes' -- that is, can be approximated by the solution of an effective (homogeneous) problem. On the other hand, if the number of cells, $L^2$, is fixed and the amplitude $A$ increases to infinity, the solution ``averages''. In this case, the solution equilibrates along stream lines, and it's behaviour across stream lines is given by an averaged equation. <BR> In this talk we study what happens if we simultaneously send both the amplitude $A$, and the number of cells $L^2$ to infinity. It turns out that if $A \ll L^4$, the problem homogenizes, and if $A \gg L^4$, the problem averages. The transition at $A \approx L^4$ can quickly predicted by matching the effective diffusivity of the homogenized problem, to that of the averaged problem. However a rigorous proof is much harder, in part because the effective diffusion matrix is unbounded. I will provide the essential ingredients for the proofs in both the averaging and homogenization regimes. This is joint work with T. Komorowski, A. Novikov and L. Ryzhik.
  • OxPDE Lunchtime Seminar
7 July 2011
15:00
Abstract
<p>We consider an active scalar equation with singular drift velocity that is motivated by a model for the geodynamo. We show that the non-diffusive equation is ill-posed in the sense of Hadamard in Sobolev spaces. In contrast, the critically diffusive equation is globally well-posed. This work is joint with Vlad Vicol.</p>
  • OxPDE Lunchtime Seminar
23 June 2011
12:30
Lillian Pierce
Abstract
Discrete problems have a habit of being beautiful but difficult. This can be true even of discrete problems whose continuous analogues are easy. For example: computing the surface area of a sphere of radius N^{1/2} in k-dimensional Euclidean space (easy). Counting the number of representations of an integer N as a sum of k squares (historically hard). In this talk we'll survey a menagerie of discrete analogues of operators arising in harmonic analysis, including singular integral operators (such as the Hilbert transform), maximal functions, and fractional integral operators. In certain cases we can learn everything we want to know about the discrete operator immediately, from its continuous analogue. In other cases the discrete operator requires a completely new approach. We'll see what makes a discrete operator easy/hard to treat, and outline some of the methods that are breaking new ground, key aspects of which come from number theory. In particular, we will highlight the roles played by theta functions, exponential sums, Waring's problem, and the circle method of Hardy and Littlewood. No previous knowledge of singular integral operators or the circle method will be assumed.
  • OxPDE Lunchtime Seminar
26 May 2011
12:30
Fabrice Planchon
Abstract
Solutions which are time-bounded in L^3 up to time T can be continued past this time, by a landmark result of Escauriaza-Seregin-Sverak, extending Serrin's criterion. On the other hand, the local Cauchy theory holds up to solutions in BMO^-1; we aim at describing how one can obtain intermediate regularity results, assuming a priori bounds in negative regularity Besov spaces. This is joint work with J.-Y. Chemin, Isabelle Gallagher and Gabriel Koch.
  • OxPDE Lunchtime Seminar
19 May 2011
12:30
Abstract
We prove the existence of weak solutions to steady Navier Stokes equations $$\text{div}\, \sigma+f=\nabla\pi+(\nabla u)u.$$ Here $u:\mathbb{R}^2\supset \Omega\rightarrow \mathbb{R}^2$ denotes the velocity field satisfying $\text{div}\, u=0$, $f:\Omega\rightarrow\mathbb{R}^2$ and $\pi:\Omega\rightarrow\mathbb{R}$ are external volume force and pressure, respectively. In order to model the behavior of Prandtl-Eyring fluids we assume $$\sigma= DW(\varepsilon (u)),\quad W(\varepsilon)=|\varepsilon|\log (1+|\varepsilon|).$$ A crucial tool in our approach is a modified Lipschitz truncation preserving the divergence of a given function.
  • OxPDE Lunchtime Seminar
5 May 2011
12:30
Jose Rodrigo
Abstract
I will describe recent work with Charles Fefferman on a construction of families of analytic almost-sharp fronts for SQG. These are special solutions of SQG which have a very sharp transition in a very thin layer. One of the main difficulties of the construction is the fact that there is no formal limit for the family of equations. I will show how to overcome this difficulty, linking the result to joint work with C. Fefferman and Kevin Luli on the existence of a "spine" for almost-sharp fronts. This is a curve, defined for every time slice by a measure-theoretic construction, that describes the evolution of the almost-sharp front.
  • OxPDE Lunchtime Seminar
10 March 2011
12:30
Carolin Kreisbeck
Abstract
Modern mathematical approaches to plasticity result in non-convex variational problems for which the standard methods of the calculus of variations are not applicable. In this contribution we consider geometrically nonlinear crystal elasto-plasticity in two dimensions with one active slip system. In order to derive information about macroscopic material behavior the relaxation of the corresponding incremental problems is studied. We focus on the question if realistic systems with an elastic energy leading to large penalization of small elastic strains can be well-approximated by models based on the assumption of rigid elasticity. The interesting finding is that there are qualitatively different answers depending on whether hardening is included or not. In presence of hardening we obtain a positive result, which is mathematically backed up by Γ-convergence, while the material shows very soft macroscopic behavior in case of no hardening. The latter is due to the vanishing relaxation for a large class of applied loads. This is joint work with Sergio Conti and Georg Dolzmann.
  • OxPDE Lunchtime Seminar
24 February 2011
12:30
Kenneth H. Karlsen
Abstract

We propose a general framework for the study of $L^1$ contractive semigroups of solutions to conservation laws with discontinuous flux. Developing the ideas of a number of preceding works we claim that the whole admissibility issue is reduced to the selection of a family of "elementary solutions", which are certain piecewise constant stationary weak solutions. We refer to such a family as a "germ". It is well known that (CL) admits many different $L^1$ contractive semigroups, some of which reflects different physical applications. We revisit a number of the existing admissibility (or entropy) conditions and identify the germs that underly these conditions. We devote specific attention to the anishing viscosity" germ, which is a way to express the "$\Gamma$-condition" of Diehl. For any given germ, we formulate "germ-based" admissibility conditions in the form of a trace condition on the flux discontinuity line $x=0$ (in the spirit of Vol'pert) and in the form of a family of global entropy inequalities (following Kruzhkov and Carrillo). We characterize those germs that lead to the $L^1$-contraction property for the associated admissible solutions. Our approach offers a streamlined and unifying perspective on many of the known entropy conditions, making it possible to recover earlier uniqueness results under weaker conditions than before, and to provide new results for other less studied problems. Several strategies for proving the existence of admissible solutions are discussed, and existence results are given for fluxes satisfying some additional conditions. These are based on convergence results either for the vanishing viscosity method (with standard viscosity or with specific viscosities "adapted" to the choice of a germ), or for specific germ-adapted finite volume schemes.

This is joint work with Boris Andreianov and Nils Henrik Risebro.

  • OxPDE Lunchtime Seminar
17 February 2011
12:30
Yann Brenier
Abstract
The reconstruction of the early universe amounts to recovering the tiny density fluctuations of the early universe (shortly after the "big bang") from the current observation of the matter distribution in the universe. Following Zeldovich, Peebles and, more recently Frisch and collaboratoirs, we use a newtonian gravitational model with time dependent coefficients taking into accont general relativity effects. Due to the (remarkable) convexity of the corresponding action, the reconstruction problem apparently reduces to a straightforward convex minimization problem. Unfortunately, this approach completely ignores the mass concentration effects due to gravitational instabilities. In this lecture, we show a way of modifying the action in order to take concentrations into account. This is obtained through a (questionable) modification of the gravitation model, by substituting the fully nonlinear Monge-Amp`ere equation for the linear Poisson equation. (This is a reasonable approximation in the sense that it makes exact some approximate solutions advocated by Zeldovich for the original gravitational model.) Then the action can be written as a perfect square in which we can input mass concentration effects in a canonical way, based on the theory of gradient flows with convex potentials and somewhat related to the concept of self-dual Lagrangians developped by Ghoussoub. A fully discrete algorithm is introduced for the EUR problem in one space dimension.
  • OxPDE Lunchtime Seminar

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