Past OxPDE Lunchtime Seminar

24 January 2013
12:00
Abstract
{\bf This seminar is at ground floor!} \\ An important question in geometry and analysis is to know when two $k-$forms $f$ and $g$ are equivalent. The problem is therefore to find a map $\varphi$ such that% \[ \varphi^{\ast}\left( g\right) =f. \] We will mostly discuss the symplectic case $k=2$ and the case of volume forms $k=n.$ We will give some results when $3\leq k\leq n-2,$ the case $k=n-1$ will also be considered. \\ The results have been obtained in collaboration with S. Bandyopadhyay, G. Csato and O. Kneuss and can be found, in part, in the book below.\bigskip \\ \newline Csato G., Dacorogna B. et Kneuss O., \emph{The pullback equation for differential forms}, Birkha\"{u}ser, PNLDE Series, New York, \textbf{83} (2012).
  • OxPDE Lunchtime Seminar
17 January 2013
12:00
Parth Soneji
Abstract
We first provide a brief overview of some of the key properties of the space $\textrm{BV}(\Omega;\mathbb{R}^{N})$ of functions of Bounded Variation, and the motivation for its use in the Calculus of Variations. Now consider the variational integral \[ F(u;\Omega):=\int_{\Omega}f(Du(x))\,\textrm{d} x\,\textrm{,} \] where $\Omega\subset\mathbb{R}^{n}$ is open and bounded, and $f\colon\mathbb{R}^{N\times n}\rightarrow\mathbb{R}$ is a continuous function satisfying the growth condition $0\leq f(\xi)\leq L(1+|\xi|^{r})$ for some exponent $r$. When $u\in\textrm{BV}(\Omega;\mathbb{R}^{N})$, we extend the definition of $F(u;\Omega)$ by introducing the functional \[ \mathscr{F}(u,\Omega):= \inf_{(u_{j})}\bigg\{ \liminf_{j\rightarrow\infty}\int_{\Omega}f(Du_{j})\,\textrm{d} x\, \left| \!\!\begin{array}{r} (u_{j})\subset W_{\textrm{loc}}^{1,r}(\Omega, \mathbb{R}^{N}) \\ u_{j} \stackrel{\ast}{\rightharpoonup} u\,\,\textrm{in }\textrm{BV}(\Omega, \mathbb{R}^{N}) \end{array} \right. \bigg\} \,\textrm{.} \] \noindent For $r\in [1,\frac{n}{n-1})$, we prove that $\mathscr{F}$ satisfies the lower bound \[ \mathscr{F}(u,\Omega) \geq \int_{\Omega} f(\nabla u (x))\,\textrm{d} x + \int_{\Omega}f_{\infty} \bigg(\frac{D^{s}u}{|D^{s}u|}\bigg)\,|D^{s}u|\,\textrm{,} \] provided $f$ is quasiconvex, and the recession function $f_{\infty}$ ($:= \overline{\lim}_{t\rightarrow\infty}f(t\xi )/t$) is assumed to be finite in certain rank-one directions. This result is a natural extension of work by Ambrosio and Dal Maso, which deals with the case $r=1$; it involves combining work of Kristensen, Braides and Coscia with some new techniques, including a polyhedral approximation result and a blow-up argument that exploits fine properties of BV functions.
  • OxPDE Lunchtime Seminar
13 December 2012
12:00
Po Lam Yung
Abstract

In this talk, we will look at two non-linear wave equations in 2+1 dimensions, whose elliptic parts exhibit conformal invariance.

These equations have their origins in prescribing the Gaussian and mean curvatures respectively, and the goal is to understand well-posedness, blow-up and bubbling for these equations.

This is a joint work with Sagun Chanillo.

  • OxPDE Lunchtime Seminar
1 November 2012
12:30
Arghir D. Zarnescu
Abstract
H. Johnston and J.G. Liu proposed in 2004 a numerical scheme for approximating numerically solutions of the incompressible Navier-Stokes system. The scheme worked very well in practice but its analytic properties remained elusive.\newline In order to understand these analytical aspects they considered together with R. Pego a continuous version of it that appears as an extension of the incompressible Navier-Stokes to vector-fields that are not necessarily divergence-free. For divergence-free initial data one has precisely the incompressible Navier-Stokes, while for non-divergence free initial data, the divergence is damped exponentially.\newline We present analytical results concerning this extended system and discuss numerical implications. This is joint work with R. Pego, G. Iyer (Carnegie Mellon) and J. Kelliher, M. Ignatova (UC Riverside).
  • OxPDE Lunchtime Seminar
25 October 2012
12:00
Eylem Öztürk
Abstract
We investigate a mixed problem with Robin boundary conditions for a diffusion-reaction equation. We investigate the problem in the sublinear, linear and super linear cases, depending on the nonlinear part. We obtain relations between the parameters of the problem which are sufficient conditions for the existence of generalized solutions to the problem and, in a special case, for their uniqueness. The proof relies on a general existence theorem by Soltanov. Finally we investıgate the time-behaviour of solutions. We show that boundedness of solutions holds under some additional conditions as t is convergent to infinity. This study is joint work with Kamal Soltanov (Hacettepe University).
  • OxPDE Lunchtime Seminar
18 October 2012
12:00
Abstract
We establish the exact boundary controllability of nodal profile for general first order quasi linear hyperbolic systems in 1-D. And we apply the result in a tree-like network with general nonlinear boundary conditions and interface conditions. The basic principles of choosing the controls and getting the controllability are given.
  • OxPDE Lunchtime Seminar
11 October 2012
12:00
Virginia Agostiniani
Abstract
Nematic elastomers are rubbery elastic solids made of cross-linked polymeric chains with embedded nematic mesogens. Their mechanical behaviour results from the interaction of electro-optical effects typical of nematic liquid crystals with the elasticity of a rubbery matrix. We show that the geometrically linear counterpart of some compressible models for these materials can be justified via Gamma-convergence. A similar analysis on other compressible models leads to the question whether linearised elasticity can be derived from finite elasticity via Gamma-convergence under weak conditions of growth (from below) of the energy density. We answer to this question for the case of single well energy densities. We discuss Ogden-type extensions of the energy density currently used to model nematic elastomers, which provide a suitable framework to study the stiffening response at high imposed stretches. Finally, we present some results concerning the attainment of minimal energy for both the geometrically linear and the nonlinear model.
  • OxPDE Lunchtime Seminar

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