# Past OxPDE Special Seminar

1 August 2012
15:00
Abstract
$%\large We study nonnegative radial solutions to the problem \begin{equation*} \left\{ \begin{split} -\Delta u = \lambda K(\left|x \right|) f(u), \quad x \in \Omega \\u = 0 \quad \qquad \quad \qquad \mbox{if } \left|x \right| = r_0 \\u \rightarrow0 \quad \qquad \quad \qquad \mbox{as } \left|x \right|\rightarrow\infty, \end{split} \right. \end{equation*} where \lambda is a positive parameter, \Delta u=\mbox{div} \big(\nabla u\big) is the Laplacian of u, \Omega=\{x\in\ \mathbb{R}^{n}; n \textgreater 2, \left|x \right| \textgreater r_0\} and K belongs to a class of functions such that \lim_{r\rightarrow \infty}K(r)=0. For classes of nonlinearities f that are negative at the origin and sublinear at \infty we discuss existence and uniqueness results when \lambda is large.$
• OxPDE Special Seminar
5 June 2012
12:30
Dominic Breit
Abstract
<h1> We consider functions $u\in L^\infty(0,T;L^2({B}))\cap L^p(0,T;W^{1,p}({B}))$ where $p\in(1,\infty)$, $T$ is positive and ${B}\subset\mathbb R^d$ bounded. Solutions to non-linear evolutionary PDE's typically belong to these spaces. Many applications require an approximation $u_\lambda$ of $u$ which is Lipschitz-continous and coincides with $u$ on a large set. For problems arising in fluid mechanics one needs to work with functions which are divergence-free thus we construct a function $u_\lambda\in L^\infty(0,T;W^{1,\mathrm{BMO}}({B}))$ which is in addition to the properties from the known truncation methods solenoidal. As an application we revisit the existence proof for non-stationary generalized Newtonian fluids. Since $\mathrm{div}\,u_\lambda=0$ we can completely avoid the appearance of the pressure term and the proof can be heavily simplified. </h1>
• OxPDE Special Seminar
12 March 2012
12:30
Sanjay Govindjee
Abstract
<p>In the operation of high frequency resonators in micro electromechanical systems (MEMS)there is a strong need to be able to accurately determine the energy loss rates or alternativelythe quality of the resonance. The resonance quality is directly related to a designer’s abilityto assemble high fidelity system response for signal filtering, for example. This hasimplications on robustness and quality of electronic communication and also stronglyinfluences overall rates of power consumption in such devices – i.e. battery life. Pastdesign work was highly focused on the design of single resonators; this arena of work hasnow given way to active efforts at the design and construction of arrays of coupledresonators. The behavior of such systems in the laboratory shows un-necessarily largespread in operational characteristics, which are thought to be the result of manufacturingvariations. However, such statements are difficult to prove due to a lack of availablemethods for predicting resonator damping – even the single resonator problem is difficult.The physical problem requires the modeling of the behavior of a resonant structure (or setof structures) supported by an elastic half-space. The half-space (chip) serves as a physicalsupport for the structure but also as a path for energy loss. Other loss mechanisms can ofcourse be important but in the regime of interest for us, loss of energy through theanchoring support of the structure to the chip is the dominant effect.</p> <p>The construction of a basic discretized model of such a system leads to a system ofequations with complex-symmetric (not Hermitian) structure. The complex-symmetryarises from the introduction of a radiation boundary conditions to handle the semi-infinitecharacter of the half-space region. Requirements of physical accuracy dictate rather finediscretization and, thusly, large systems of equations. The core to the extraction of relevantphysical performance parameters is dependent upon the underlying modeling framework.In three dimensional settings of practical interest, such systems are too large to be handleddirectly and must be solved iteratively. In this talk, I will cover the physical background ofthe problem class of interest, how such systems can be modeled, and then solved. Particularinterest will be paid to the radiation boundary conditions (perfectly matched layers versushigher order absorbing boundary conditions), issues associated with frequency domainversus time domain methods, and how these choices interact with iterative solvertechnologies in sometimes unexpected ways. Time permitting I will also touch upon the issue of harmonic inversion methods of this class of problems.
• OxPDE Special Seminar
7 March 2012
12:30
Abstract
Let $M$ be the Cantor space or an $n$-dimensional manifold with $C(M,M)$ the set of continuous self-maps of $M$. We analyse the behaviour of the generic $f$ in $C(M,M)$ in terms of attractors and some notions of chaos.
• OxPDE Special Seminar
28 February 2012
16:00
Abstract

We show the solvability of a proposed Generalized Buckley-LeverettSystem, which is related to multidimensional Muskat Problem. More-over, we discuss some important questions concerning singular limitsof the proposed model.

• OxPDE Special Seminar
17 February 2012
16:00
to
17:15
Xue-Cheng Tai
Abstract
Image segmentation and a number of other problems from image processing and computer vision can be regarded as interface problems. Recently, diffusive and sharp interface techniques have been used for these problems. In this talk, we will first briefly explain these models and compare the advantages and disadvantages of these models. Numerically, these models can be solved through some PDEs. In the end, we will show some recent results on how to use graph cut to solve these interface problems. Moreover, the global minimizer can be guaranteed even the problem is nonconex and nonlinear. The use of max-flow in a network setting and also in an infinite dimensional setting will be explained.
• OxPDE Special Seminar
1 December 2011
15:00
to
17:00
Professor Charles A Stuart
Abstract

• Sufficient conditions for bifurcation from points that are not isolated eigenvalues of the linearisation.

• Odd potential operators.

• Defining min-max critical values using sets of finite genus.

• Formulating some necessary conditions for bifurcation.

• OxPDE Special Seminar
24 November 2011
15:00
to
17:00
Professor Charles A Stuart
Abstract

• Bifurcation from isolated eigenvalues of finite multiplicity of the linearisation.

• Pseudo-inverses and parametrices for paths of Fredholm operators of index zero.

• Detecting a change of orientation along such a path.

• Lyapunov-Schmidt reduction

• OxPDE Special Seminar
17 November 2011
15:00
to
17:00
Professor Charles A Stuart
Abstract
<p>• Review of the basic notions concerning bifurcation and asymptotic linearity.</p> <p>• Review of differentiability in the sense of Gˆateaux, Fréchet, Hadamard.</p> <p>• Examples which are Hadamard but not Fréchet differentiable.&nbsp; The Dirichlet problem for a degenerate elliptic equation on a bounded domain. The stationary nonlinear Schrödinger equation on RN</p>
• OxPDE Special Seminar
15 June 2011
13:30
Lawrence C Evans
Abstract
<p>I will discuss two of my papers that develop PDE methods for weak KAM theory, in the context of a singular variational problem that can be interpreted as a regularization of Mather's variational principle by an entropy term. This is, sort of, a statistical mechanics approach to the problem. I will show how the Euler-Lagrange PDE yield approximate changes to action-angle variables for the corresponding Hamiltonian dynamics.</p>
• OxPDE Special Seminar