# Past Partial Differential Equations Seminar

23 February 2009
17:00
Eric Cances
Abstract
Electronic structure calculations are commonly used to understand and predict the electronic, magnetic and optic properties of molecular systems and materials. They are also at the basis of ab initio molecular dynamics, the most reliable technique to investigate the atomic scale behavior of materials undergoing chemical reactions (oxidation, crack propagation, ...). In the first part of my talk, I will briefly review the foundations of the density functional theory for electronic structure calculations. In the second part, I will present some recent achievements in the field, as well as open problems. I will focus in particular on the mathematical modelling of defects in crystalline materials.
• Partial Differential Equations Seminar
16 February 2009
17:00
Reiner Sch&#228;tzle
Abstract
We prove the existence of a smooth minimizer of the Willmore energy in the class of conformal immersions of a given closed Riemann surface into $R^n$, $n = 3, 4$, if there is one conformal immersion with Willmore energy smaller than a certain bound $W_{n,p}$ depending on codimension and genus $p$ of the Riemann surface. For tori in codimension $1$, we know $W_{3,1} = 8\pi$ . Joint work with Enrst Kuwert.
• Partial Differential Equations Seminar
9 February 2009
17:00
Kenneth Falconer
Abstract
The talk will survey some recent and not so recent work on the Hausdorff and box dimension of self-affine sets and related attractors and repellers that arise in certain dynamical systems.
• Partial Differential Equations Seminar
2 February 2009
17:00
Andrea Malchiodi
Abstract
A classical problem in differential geometry is to deform the metric of a given manifold so that some of its curvatures become prescribed functions. Classical examples are the Uniformization problem for compact surfaces and the Yamabe problem for compact manifolds of dimension greater than two. <br /> We address a similar problem for the so-called Q-curvature, which plays an important role in conformal geometry and is a natural higher order analogue of the Gauss curvature. The problem is tackled using a variational and Morse theoretical approach.
• Partial Differential Equations Seminar
1 December 2008
13:00
Isaac Vikram Chenchiah
Abstract
he study of polycrystals of shape-memory alloys and rigid-perfectly plastic materials gives rise to problems of nonlinear homogenization involving degenerate energies. We present a characterisation of the strain and stress fields for some classes of problems in plane strain and also for some three-dimensional situations. Consequences for shape-memory alloys and rigid-perfectly plastic materials are discussed through model problems. In particular we explore connections to previous conjectures characterizing those shape-memory polycrystals with non-trivial recoverable strain.
• Partial Differential Equations Seminar
17 November 2008
17:00
Genevi&egrave;ve Raugel
Abstract
Y. Brenier, R. Natalini and M. Puel have considered a relaxation&quot; of the Euler equations in <b>R</b><sup>2</sup>. After an approriate scaling, they have obtained the following hyperbolic version of the Navier-Stokes equations, which is similar to the hyperbolic version of the heat equation introduced by Cattaneo, $$\varepsilon u_{tt}^\varepsilon + u_t^\varepsilon -\Delta u^\varepsilon +P (u^\varepsilon \nabla u^\varepsilon) \, = \, Pf~, \quad (u^\varepsilon(.,0), u_t^\varepsilon(.,0)) \, = \, (u_0(.),u_1(.))~, \quad (1)$$ where $P$ is the classical Leray projector and $\varepsilon$ is a small, positive number. Under adequate hypotheses on the forcing term $f$, we prove global existence and uniqueness of a mild solution $(u^\varepsilon,u_t^\varepsilon) \in C^0([0, +\infty), H^{1}({\bf R}^2) \times L^2({\bf R}^2))$ of (1), for large initial data $(u_0,u_1)$ in $H^{1}({\bf R}2) \times L^2({\bf R}2)$, provided that $\varepsilon&gt;0$ is small enough, thus improving the global existence results of Brenier, Natalini and Puel (actually, we can work in less regular Hilbert spaces). The proof uses appropriate Strichartz estimates, combined with energy estimates. We also show that $(u^\varepsilon,u_t^\varepsilon)$ converges to $(v,v_t)$ on finite intervals of time $[t_0,t_1]$, $0 <t_0>&lt;+ \infty$, when $\varepsilon$ goes to $0$, where $v$ is the solution of the corresponding Navier-Stokes equations $$v_t -\Delta v +P (v\nabla v) \, = \, Pf~, \quad v(.,0) \, = \, u_0~. \quad (2)$$ We also consider Equation (1) in the three-dimensional case. Here we expect global existence results for small data. Under appropriate assumptions on the forcing term, we prove global existence and uniqueness of a mild solution $(u^\varepsilon,u_t^\varepsilon) \in C^0([0, +\infty), H^{1+\delta}({\bf R}^3) \times H^{\delta}({\bf R}^3))$ of (1), for initial data $(u_0,u_1)$ in $H^{1 +\delta}({\bf R}^3) \times H^{\delta}({\bf R}^3)$ (where $\delta &gt;0$ is a small positive number), provided that $\varepsilon &gt; 0$ is small enough and that $u_0$ and $f$ satisfy a smallness condition. (Joint work with Marius Paicu) </t_0>
• Partial Differential Equations Seminar