# Past OxPDE Lunchtime Seminar

14 June 2012
12:30
Oliver Penrose
Abstract
A method of defining non-equilibrium entropy for a chaotic dynamical system is proposed which, unlike the usual method based on Boltzmann's principle $S = k\log W$, does not involve the concept of a macroscopic state. The idea is illustrated using an example based on Arnold's `cat' map. The example also demonstrates that it is possible to have irreversible behaviour, involving a large increase of entropy, in a chaotic system with only two degrees of freedom.
• OxPDE Lunchtime Seminar
7 June 2012
12:30
Leonid V. Berlyand
Abstract
We study minimizers of the Ginzburg-Landau (GL) functional $E_\epsilon(u):=\frac{1}{2}\int_A |\nabla u|^2 + \frac{1}{4\epsilon^2} \int_A(1-|u|^2)^2$ for a complex-valued order parameter $u$ (with no magnetic ﬁeld). This functional is of fundamental importance in the theory of superconductivity and superuidity; the development of these theories led to three Nobel prizes. For a $2D$ domain $A$ with holes we consider “semistiff” boundary conditions: a Dirichlet condition for the modulus $|u|$, and a homogeneous Neumann condition for the phase $\phi = \mathrm{arg}(u)$. The principal result of this work (with V. Rybalko) is a proof of the existence of stable local minimizers with vortices (global minimizers do not exist). These vortices are novel in that they approach the boundary and have bounded energy as $\epsilon\to0$. In contrast, in the well-studied Dirichlet (“stiff”) problem for the GL PDE, the vortices remain distant from the boundary and their energy blows up as $\epsilon\to 0$. Also, there are no stable minimizers to the homogeneous Neumann (“soft”) problem with vortices. \\ Next, we discuss more recent results (with V. Rybalko and O. Misiats) on global minimizers of the full GL functional (with magnetic ﬁeld) subject to semi-stiff boundary conditions. Here, we show the existence of global minimizers with vortices for both simply and doubly connected domains and describe the location of their vortices.
• OxPDE Lunchtime Seminar
31 May 2012
12:30
Isaac Vikram Chenchiah
Abstract
We present a variational model for the quasi-static evolution of brutal brittle damage for geometrically-linear elastic materials. We allow for multiple damaged states. Moreover, unlike current formulations, the materials are allowed to be anisotropic and the deformations are not restricted to anti-plane shear. The model can be formulated either energetically or through a strain threshold. We explore the relationship between these formulations. This is joint work with Christopher Larsen, Worcester Polytechnic Institute.
• OxPDE Lunchtime Seminar
24 May 2012
12:30
Mikhail Feldman
Abstract
We discuss shock reflection problem for compressible gas dynamics, and von Neumann conjectures on transition between regular and Mach reflections. Then we will talk about some recent results on existence, regularity and geometric properties of regular reflection solutions for potential flow equation. In particular, we discuss optimal regularity of solutions near sonic curve, and stability of the normal reflection soluiton. Open problems will also be discussed. The talk will be based on the joint work with Gui-Qiang Chen, and with Myoungjean Bae.
• OxPDE Lunchtime Seminar
17 May 2012
12:30
Gianluca Crippa
Abstract
In this seminar I will present two results regarding the uniqueness (and further properties) for the two-dimensional continuity equation and the ordinary differential equation in the case when the vector field is bounded, divergence free and satisfies additional conditions on its distributional curl. Such settings appear in a very natural way in various situations, for instance when considering two-dimensional incompressible fluids. I will in particular describe the following two cases:\\ (1) The vector field is time-independent and its curl is a (locally finite) measure (without any sign condition).\\ (2) The vector field is time-dependent and its curl belongs to L^1.\\ Based on joint works with: Giovanni Alberti (Universita' di Pisa), Stefano Bianchini (SISSA Trieste), Francois Bouchut (CNRS & Universite' Paris-Est-Marne-la-Vallee) and Camillo De Lellis (Universitaet Zuerich).
• OxPDE Lunchtime Seminar
9 May 2012
12:30
Apala Majumdar
Abstract
In this talk, we make quantitative comparisons between two widely-used liquid crystal modelling approaches - the continuum Landau-de Gennes theory and mesoscopic mean-field theories, such as the Maier-Saupe and Onsager theories. We use maximum principle arguments for elliptic partial differential equations to compute explicit bounds for the norm of static equilibria within the Landau-de Gennes framework. These bounds yield an explicit prescription of the temperature regime within which the LdG and the mean-field predictions are consistent, for both spatially homogeneous and inhomogeneous systems. We find that the Landau-de Gennes theory can make physically unrealistic predictions in the low-temperature regime. In my joint work with John Ball, we formulate a new theory that interpolates between mean-field and continuum approaches and remedies the deficiencies of the Landau-de Gennes theory in the low-temperature regime. In particular, we define a new thermotropic potential that blows up whenever the mean-field constraints are violated. The main novelty of this work is the incorporation of spatial inhomogeneities (outside the scope of mean-field theory) along with retention of mean-field level information.
• OxPDE Lunchtime Seminar
3 May 2012
12:30
Beatrice Pelloni
Abstract
In this talk I will survey the results on the existence of solutions of the semigeostrophic system, a fully nonlinear reduction of the Navier-Stokes equation that constitute a valid model when the effect of rotation dominate the atmospheric flow. I will give an account of the theory developed since the pioneering work of Brenier in the early 90's, to more recent results obtained in a joint work with Mike Cullen and David Gilbert.
• OxPDE Lunchtime Seminar
18 April 2012
12:30
to
25 April 2012
13:30
Abstract
In this talk I will discuss the refraction of shocks on the interface for 2-d steady compressible flow. Particularly, the class of E-H type regular refraction is defined and its global stability of the wave structure is verified. The 2-d steady potential flow equations is employed to describe the motion of the fluid. The stability problem of the E-H type regular refraction can be reduced to a free boundary problem of nonlinear mixed type equations in an unbounded domain. The corresponding linearized problem has similarities to a generalized Tricomi problem of the linear Lavrentiev-Bitsadze mixed type equation, and it can be reduced to a nonlocal boundary value problem of an elliptic system. The later is finally solved by establishing the bijection of the corresponding nonlocal operator in a weighted H\"older space via careful harmonic analysis. This is a joint work with CHEN Shuxing and HU Dian.
• OxPDE Lunchtime Seminar
8 March 2012
12:30
Yoshihito Oshita
Abstract

We study the mean-field models describing the evolution of distributions of particle radii obtained by taking the small volume fraction limit of the free boundary problem describing the micro phase separation of diblock copolymer melts, where micro phase separation consists of an ensemble of small balls of one component. In the dilute case, we identify all the steady states and show the convergence of solutions.

Next we study the dynamics for a free boundary problem in two dimension, obtained as a gradient flow of Ohta- Kawasaki free energy, in the case that one component is a distorted disk with a small volume fraction. We show the existence of solutions that a small, almost circular interface moves along a curve determined via a Green’s function of the domain. This talk is partly based on a joint work with Xiaofeng Ren.

• OxPDE Lunchtime Seminar
1 March 2012
12:30
François Murat
Abstract
<p>In this lecture I will report on joint work with J. Casado-Díaz, T. Chacáon Rebollo, V. Girault and M.~Gómez Marmol which was published in Numerische Mathematik, vol. 105, (2007), pp. 337-510.</p> We consider, in dimension $d\ge 2$, the standard $P^1$ finite elements approximation of the second order linear elliptic equation in divergence form with coefficients in $L^\infty(\Omega)$ which generalizes Laplace's equation. We assume that the family of triangulations is regular and that it satisfies an hypothesis close to the classical hypothesis which implies the discrete maximum principle. When the right-hand side belongs to $L^1(\Omega)$, we prove that the unique solution of the discrete problem converges in $W^{1,q}_0(\Omega)$ (for every $q$ with $1 \leq q$ &lt; ${d \over d-1}$) to the unique renormalized solution of the problem. We obtain a weaker result when the right-hand side is a bounded Radon measure. In the case where the dimension is $d=2$ or $d=3$ and where the coefficients are smooth, we give an error estimate in $W^{1,q}_0(\Omega)$ when the right-hand side belongs to $L^r(\Omega)$ for some $r$ &gt; $1$.
• OxPDE Lunchtime Seminar