# Past OxPDE Lunchtime Seminar

12 November 2013
12:00
Prof. Jose Francisco Rodrigues
Abstract
We prove existence of solution for evolutionary variational and quasivariational inequalities defined by a first order quasilinear operator and a variable convex set, characterized by a constraint on the absolute value of the gradient (which, in the quasi-variational case, depends on the solution itself). The only required assumption on the nonlinearity of this constraint is its continuity and positivity. The method relies on an appropriate parabolic regularization and suitable a priori estimates. Uniqueness of solution is proved for the variational inequality. We also obtain existence of stationary solutions, by studying the asymptotic behaviour in time. We shall illustrate a simple “sand pile” example in the variational case for the transport operator were the problem is equivalent to a two-obstacles problem and the solution stabilizes in finite time. Further remarks about these properties of the solution will be presented.This is a joint work with Lisa Santos. If times allows, using similar techniques, I shall also present the existence, uniqueness and continuous dependence of solutions of a new class of evolution variational inequalities for incompressible thick fluids. These non-Newtonian fluids with a maximum admissible shear rate may be considered as a limit class of shear-thickening or dilatant fluids, in particular, as the power limit of Ostwald-deWaele fluids.
• OxPDE Lunchtime Seminar
7 November 2013
12:00
Abstract
Dislocations are line defects in crystals, and were first posited as the carriers of plastic flow in crystals in the 1934 papers of Orowan, Polanyi and Taylor. Their hypothesis has since been experimentally verified, but many details of their behaviour remain unknown. In this talk, I present joint work with Christoph Ortner on an infinite lattice model in which screw dislocations are free to be created and annihilated. We show that configurations containing single geometrically necessary dislocations exist as global minimisers of a variational problem, and hence are globally stable equilibria amongst all finite energy perturbations.
• OxPDE Lunchtime Seminar
24 October 2013
12:00
Dr. Shiwu Yang
Abstract
<p><span>We study the nonlinear wave equations on a class of asymptotically flat Lorentzian manifolds $(\mathbb{R}^{3+1}, g)$ with time dependent inhomogeneous metric g. Based on a new approach for proving the decay of solutions of linear wave equations, we give several applications to nonlinear problems. In particular, we show the small data global existence result for quasilinear wave equations satisfying the null condition on a class of time dependent inhomogeneous backgrounds which do not settle to any particular stationary metric.</span></p>
• OxPDE Lunchtime Seminar
17 October 2013
12:00
Abstract
<p><span>Penrose’s Weyl Curvature Hypothesis, which dates from the late 70s, is a hypothesis, motivated by observation, about the nature of the Big Bang as a singularity of the space-time manifold. His Conformal Cyclic Cosmology is a remarkable suggestion, made a few years ago and still being explored, about the nature of the universe, in the light of the current consensus among cosmologists that there is a positive cosmological constant.&nbsp; I shall review both sets of ideas within the framework of general relativity, and emphasise how the second set solves a problem posed by the first.</span></p>
• OxPDE Lunchtime Seminar
20 June 2013
12:00
Abstract
The Bayesian approach to inverse problems is of paramount importance in quantifying uncertainty about the input to and the state of a system of interest given noisy observations. Herein we consider the forward problem of the forced 2D Navier Stokes equation. The inverse problem is inference of the forcing, and possibly the initial condition, given noisy observations of the velocity field. We place a prior on the forcing which is in the form of a spatially correlated temporally white Gaussian process, and formulate the inverse problem for the posterior distribution. Given appropriate spatial regularity conditions, we show that the solution is a continuous function of the forcing. Hence, for appropriately chosen spatial regularity in the prior, the posterior distribution on the forcing is absolutely continuous with respect to the prior and is hence well-defined. Furthermore, the posterior distribution is a continuous function of the data. \\ \\ This is a joint work with Andrew Stuart and Kody Law (Warwick)
• OxPDE Lunchtime Seminar
6 June 2013
12:00
Mayte Pérez-Llanos
Abstract
<ul>In this talk we study numerical approximations of continuous solutions to a nonlocal $p$-Laplacian type diffusion equation, $u_t (t, x) = \int_\Omega J(x − y)|u(t, y) − u(t, x)|^{p-2} (u(t, y) − u(t, x)) dy.$</ul> <ul> First, we find that a semidiscretization in space of this problem gives rise to an ODE system whose solutions converge uniformly to the continuous one, as the mesh size goes to zero. Moreover, the semidiscrete approximation shares some properties with the continuous problem: it preserves the total mass and the solution converges to the mean value of the initial condition, as $t$ goes to infinity.</ul> <ul> Next, we discretize also the time variable and present a totally discrete method which also enjoys the above mentioned properties.</ul> <ul> In addition, we investigate the limit as $p$ goes to infinity in these approximations and obtain a discrete model for the evolution of a sandpile.</ul> <ul>Finally, we present some numerical experiments that illustrate our results. </ul> <ul>This is a joint work with J. D. Rossi.</ul>
• OxPDE Lunchtime Seminar
30 May 2013
12:00
James Robinson
Abstract
<ul>In 1985 Moffatt suggested that stationary flows of the 3D Euler equations with non-trivial topology could be obtained as the time-asymptotic limits of certain solutions of the equations of magnetohydrodynamics. Heuristic arguments also suggest that the same is true of the system<br /> $-\Delta u+\nabla p=(B\cdot\nabla)B\qquad\nabla\cdot u=0\qquad$ <br /> $B_t-\eta\Delta B+(u\cdot\nabla)B=(B\cdot\nabla)u$ when $\eta=0$.<br /> <br /> In this talk I will discuss well posedness of this coupled elliptic-parabolic equation in the two-dimensional case when $B(0)\in L^2$ and $\eta$ is positive. </ul> <ul>Crucial to the analysis is a strengthened version of the 2D Ladyzhenskaya inequality: $\|f\|_{L^4}\le c\|f\|_{L^{2,\infty}}^{1/2}\|\nabla f\|_{L^2}^{1/2}$, where $L^{2,\infty}$ is the weak $L^2$ space. I will also discuss the problems that arise in the case $\eta=0$. </ul> <br /> <br /> <ul> This is joint work with David McCormick and Jose Rodrigo.</ul>
• OxPDE Lunchtime Seminar
23 May 2013
12:00
Francesco Solombrino
Abstract
Inspired by some recents developments in the theory of small-strain elastoplasticity, we both revisit and generalize the formulation of the quasistatic evolutionary problem in perfect plasticity for heterogeneous materials recently given by Francfort and Giacomini. We show that their definition of the plastic dissipation measure is equivalent to an abstract one, where it is defined as the supremum of the dualities between the deviatoric parts of admissible stress fields and the plastic strains. By means of this abstract definition, a viscoplastic approximation and variational techniques from the theory of rate-independent processes give the existence of an evolution statisfying an energy- dissipation balance and consequently Hill's maximum plastic work principle for an abstract and very large class of yield conditions.
• OxPDE Lunchtime Seminar
16 May 2013
12:00
Paolo Secchi
Abstract
<ul>We consider the free boundary problem for the plasma-vacuum interface in ideal compressible magnetohydrodynamics (MHD). In the plasma region the flow is governed by the usual compressible MHD equations, while in the vacuum region we consider the pre-Maxwell dynamics for the magnetic field. At the free-interface, driven by the plasma velocity, the total pressure is continuous and the magnetic field on both sides is tangent to the boundary. The plasma density does not go to zero continuously at the interface, but has a jump, meaning that it is bounded away from zero in the plasma region and it is identically zero in the vacuum region. The plasma-vacuum system is not isolated from the outside world, because of a given surface current on the fixed boundary that forces oscillations.</ul> <ul>Under a suitable stability condition satisfied at each point of the initial interface, stating that the magnetic fields on either side of the interface are not collinear, we show the existence and uniqueness of the solution to the nonlinear plasma-vacuum interface problem in suitable anisotropic Sobolev spaces.</ul> <ul>The proof follows from the well-posedness of the homogeneous linearized problem and a basic a priori energy estimate, the analysis of the elliptic system for the vacuum magnetic field, a suitable tame estimate in Sobolev spaces for the full linearized equations, and a Nash-Moser iteration.</ul> <ul>This is a joint work with Y. Trakhinin (Novosibirsk).</ul>
• OxPDE Lunchtime Seminar