Forthcoming events in this series


Thu, 27 Feb 2014

12:00 - 13:00
L6

The rigidity problem for symmetrization inequalities

Dr. Filippo Cagnetti
(University of Sussex)
Abstract

Steiner symmetrization is a very useful tool in the study of isoperimetric inequality. This is also due to the fact that the perimeter of a set is less or equal than the perimeter of its Steiner symmetral. In the same way, in the Gaussian setting,

it is well known that Ehrhard symmetrization does not increase the Gaussian perimeter. We will show characterization results for equality cases in both Steiner and Ehrhard perimeter inequalities. We will also characterize rigidity of equality cases. By rigidity, we mean the situation when all equality cases are trivially obtained by a translation of the Steiner symmetral (or, in the Gaussian setting, by a reflection of the Ehrhard symmetral). We will achieve this through the introduction of a suitable measure-theoretic notion of connectedness, and through a fine analysis of the barycenter function

for a special class of sets. These results are obtained in collaboration with Maria Colombo, Guido De Philippis, and Francesco Maggi.

Thu, 20 Feb 2014

13:00 - 14:00
L6

On extremizers for Fourier restriction inequalities

Diogo Oliveira e Silva
(Universitat Bonn)
Abstract

This talk will focus on extremizers for

a family of Fourier restriction inequalities on planar curves. It turns

out that, depending on whether or not a certain geometric condition

related to the curvature is satisfied, extremizing sequences of

nonnegative functions may or may not have a subsequence which converges

to an extremizer. We hope to describe the method of proof, which is of

concentration compactness flavor, in some detail. Tools include bilinear

estimates, a variational calculation, a modification of the usual

method of stationary phase and several explicit computations.

Thu, 13 Feb 2014

12:00 - 13:00
L6

Modelling collective motion in biology

Prof. Philip Maini
(University of Oxford)
Abstract

We will present three different recent applications of cell motion in biology: (i) Movement of epithelial sheets and rosette formation, (ii) neural crest cell migrations, (iii) acid-mediated cancer cell invasion. While the talk will focus primarily on the biological application, it will be shown that all of these processes can be represented by reaction-diffusion equations with nonlinear diffusion term.

Fri, 07 Feb 2014

12:00 - 13:00
L6

Transonic shocks in steady compressible Euler flows

Prof. Hairong Yuan
(East China Normal University)
Abstract

I will introduce the physical phenomena of transonic shocks, and review the progresses on related boundary value problems of the steady compressible Euler equations. Some Ideas/methods involved in the studies will be presented through specific examples. The talk is based upon joint works with my collaborators.

Thu, 23 Jan 2014

12:00 - 13:00
L6

On Stability of Steady Transonic Shocks in Supersonic Flow around a Wedge

Prof. Beixiang Fang
(Shanghai JiaoTong University)
Abstract

In this talk we are concerned with the stability of steady transonic shocks in supersonic flow around a wedge. 2-D and M-D potential stability will be presented.

This talk is based on the joint works with Prof. G.-Q. Chen, and Prof. S.X. Chen.

Thu, 05 Dec 2013

12:00 - 13:00
L5

An analysis of crystal cleavage in the passage from atomistic models to continuum theory

Manuel Friedrich
(Universität Augsburg)
Abstract

We study the behavior of atomistic models under uniaxial tension and investigate the system for critical fracture loads. We rigorously prove that in the discrete-to- continuum limit the minimal energy satisfies a particular cleavage law with quadratic response to small boundary displacements followed by a sharp constant cut-off beyond some critical value. Moreover, we show that the minimal energy is attained by homogeneous elastic configurations in the subcritical case and that beyond critical loading cleavage along specific crystallographic hyperplanes is energetically favorable. We present examples of mass spring models with full nearest and next-to-nearest pair interactions and provide the limiting minimal energy and minimal configurations.

Thu, 28 Nov 2013

12:00 - 13:00
L6

Contact Solutions for fully nonlinear PDE systems and applications to vector-valued Calculus of Variations in $L^{\infty}$

Dr. Nicholas Katzourakis
(University of Reading)
Abstract

Calculus of Variations for $L^{\infty}$ functionals has a successful history of 50 years, but until recently was restricted to the scalar case. Motivated by these developments, we have recently initiated the vector-valued case. In order to handle the complicated non-divergence PDE systems which arise as the analogue of the Euler-Lagrange equations, we have introduced a theory of "weak solutions" for general fully nonlinear PDE systems. This theory extends Viscosity Solutions of Crandall-Ishii-Lions to the general vector case. A central ingredient is the discovery of a vectorial notion of extremum for maps which is a vectorial substitute of the "Maximum Principle Calculus" and allows to "pass derivatives to test maps" in a duality-free fashion. In this talk we will discuss some rudimentary aspects of these recent developments.

Tue, 12 Nov 2013

12:00 - 13:00
L6

Variational and Quasi-variational Solutions to Nonlinear Equations with Gradient Constraint

Prof. Jose Francisco Rodrigues
(Portugal)
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.

Thu, 07 Nov 2013

12:00 - 13:00
L6

Existence and stability of screw dislocations in an anti-plane lattice model

Thomas Hudson
(OxPDE, University of Oxford)
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.

Thu, 24 Oct 2013

12:00 - 13:00
L6

Nonlinear wave equations on time dependent inhomogeneous backgrounds

Dr. Shiwu Yang
(University of Cambridge)
Abstract

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.

Thu, 17 Oct 2013

12:00 - 13:00
L6

Penrose’s Weyl Curvature Hypothesis and his Conformal Cyclic Cosmology

Prof. Paul Tod
(OxPDE, University of Oxford)
Abstract

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.  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.

Thu, 20 Jun 2013
12:00
Gibson 1st Floor SR

Determining White Noise Forcing From Eulerian Observations in the Navier Stokes Equation

Hoang Viet Ha
(Nanyang Technological University)
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.

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This is a joint work with Andrew Stuart and Kody Law (Warwick)

Thu, 06 Jun 2013

12:00 - 13:00
Gibson 1st Floor SR

Numerical approximations for a nonloncal model for sandpiles

Mayte Pérez-Llanos
(Universidad Autonoma de Madrid)
Abstract
    In this talk we study numerical approximations of continuous solutions to a nonlocal $p$-Laplacian type diffusion equation, \[ u_t (t, x) = \int_\Omega J(x − y)|u(t, y) − u(t, x)|^{p-2} (u(t, y) − u(t, x)) dy. \]
    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.
    Next, we discretize also the time variable and present a totally discrete method which also enjoys the above mentioned properties.
    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.
    Finally, we present some numerical experiments that illustrate our results.
    This is a joint work with J. D. Rossi.
Thu, 30 May 2013
12:00
Gibson 1st Floor SR

A coupled parabolic-elliptic system arising in the theory of magnetic relaxation

James Robinson
(University of Warwick)
Abstract
    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
    \[ -\Delta u+\nabla p=(B\cdot\nabla)B\qquad\nabla\cdot u=0\qquad \]
    \[ B_t-\eta\Delta B+(u\cdot\nabla)B=(B\cdot\nabla)u \] when $\eta=0$.

    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.
    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 $\eta=0$.


    This is joint work with David McCormick and Jose Rodrigo.
Thu, 23 May 2013
12:00
Gibson 1st Floor SR

Quasistatic evolution problems in perfect plasticity for generalized multiphase materials

Francesco Solombrino
(Technical University of Munich)
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.

Thu, 16 May 2013
12:00
Gibson 1st Floor SR

The plasma-vacuum interface problem with external excitation

Paolo Secchi
(University of Brescia)
Abstract
    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.
    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.
    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.
    This is a joint work with Y. Trakhinin (Novosibirsk).
Wed, 15 May 2013
12:00
Gibson 1st Floor SR

Decay of positive waves to hyperbolic systems of balance laws

Cleopatra Christoforou
(University of Cyprus)
Abstract

Historically, decay rates have been used to provide quantitative and qualitative information on the solutions to hyperbolic conservation laws. Quantitative results include the establishment of convergence rates for approximating procedures and numerical schemes. Qualitative results include the establishment of results on uniqueness and regularity as well as the ability to visualize the waves and their evolution in time.

In this talk, I will present two decay estimates on the positive waves for systems of hyperbolic and genuinely nonlinear balance laws satisfying a dissipative mechanism. The result is obtained by employing the continuity of Glimm-type functionals and the method of generalized characteristics. Using this result on the spreading of rarefaction waves, the rate of convergence for vanishing viscosity approximations to hyperbolic balance laws will also be established. The proof relies on error estimates that measure the interaction of waves using suitable Lyapunov functionals. If time allows, a further application of the recent developments in the theory of balance laws to differential geometry will be addressed.

Thu, 09 May 2013
12:01
Gibson 1st Floor SR

Weak solutions to the barotropic Navier-Stokes system with slip boundary conditions in time dependent domains and incompressible limits

Šárka Nečasová
(Academy of Sciences of the Czech Republic)
Abstract
We consider the compressible (barotropic) Navier-Stokes system on time-dependent domains, supplemented with slip boundary conditions. Our approach is based on penalization of the boundary behaviour, viscosity, and the pressure in the weak formulation. Global-in-time weak solutions are obtained. Secondly, we suppose that the characteristic speed of the fluid is domi- nated by the speed of sound and perform the low Mach number limit in the framework of weak solutions. The standard incompressible Navier-Stokes system is identified as the target problem. References:
    [1] E. Feireisl, O. Kreml, S. Nečasová, J. Neustupa, and J. Stebel. Weak solutions to the barotropic NavierStokes system with slip boundary conditions in time dependent domains. J. Differential Equations, 254:125–140, 2013.
    [2] E. Feireisl, O. Kreml, S. Nečasová, J. Neustupa, and J. Stebel. Incompressible limits of fluids excited by moving boundaries. Submitted
Thu, 02 May 2013
12:00
Gibson 1st Floor SR

Partial Regularity for constrained minimisers of quasi convex functionals with $p$-growth

Christopher Hopper
(OxPDE, University of Oxford)
Abstract

We consider minimisers of integral functionals $F$ over a ‘constrained’ class of $W^{1,p}$-mappings from a bounded domain into a compact Riemannian manifold $M$, i.e. minimisers of $F$ subject to holonomic constraints. Integrands of the form $f(Du)$ and the general $f(x,u,Du)$ are considered under natural strict $p$-quasiconvexity and $p$-growth assumptions for any exponent $1 < p <+\infty$. Unlike the harmonic and $p$-harmonic map case, the quasiconvexity condition requires one to linearise the map at the level of the gradient. In a bid to give a direct proof of partial $C^{1,α}-regularity for such minimisers, we developing an appropriate notion of a tangential harmonic approximation together with a discussion on the difficulties in establishing Caccioppoli-type inequalities. The need in the latter problem to construct suitable competitors to the minimiser via the so-called Luckhaus Lemma presents difficulties quite separate to that of the unconstrained case. We will give a proof of this lemma together with a discussion on the implications for higher integrability.

Thu, 25 Apr 2013
12:00
Gibson 1st Floor SR

From nonlinear to linearized elasticity via $\Gamma$-convergence: the case of multi-well energies satisfying weak coercivity conditions

Konstantinos Koumatos
(OxPDE, University of Oxford)
Abstract
We derive geometrically linear elasticity theories as $\Gamma$-limits of rescaled nonlinear multi-well energies satisfying a weak coercivity condition, in the sense that the standard quadratic growth from below of the energy density $W$ is replaced by the weaker p-growth far from the energy wells, where $1

Thu, 07 Mar 2013

12:00 - 13:00
Gibson 1st Floor SR

Characterisation of electric fields in periodic composites

Marc Briane
(Université de Rennes)
Abstract
This is work done in collaboration with G.W. Milton and A. Treibergs (University of Utah). Our purpose is to characterise, among all the regular periodic gradient fields, the ones which are isotropically realisable electric fields, namely solutions of a conduction equation with a suitable isotropic conductivity. In any dimension a sufficient condition of realisability is that the gradient field does not vanish. This condition is also necessary in dimension two but not in dimension three. However, when the conductivity also needs to be periodic, the previous condition is shown to be not sufficient. Then, using the associated gradient flow a necessary and sufficient condition for the isotropic realisability in the torus is established and illustrated by several examples. The realisability of the matrix gradient fields and the less regular laminate fields is also investigated.
Thu, 28 Feb 2013
12:00
Gibson 1st Floor SR

Quadratic interaction functional and structure of solutions to hyperbolic conservation laws

Stefano Bianchini
(SISSA-ISAS)
Abstract

The proof of several properties of solutions of hyperbolic systems of conservation laws in one space dimension (existence, stability, regularity) depends on the existence of a decreasing functional, controlling the nonlinear interactions of waves. In a special case (genuinely nonlinear systems) the interaction functional is quadratic, while in the general case it is cubic. Several attempts to prove the existence of a a quadratic functional also in the most general case have been done. I will present the approach we follow in order to prove this result, an some of its implication we hope to exploit.

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Work in collaboration with Stefano Modena.

Thu, 21 Feb 2013
12:00
Gibson 1st Floor SR

1D Burgers Turbulence as a model case for the Kolmogorov Theory

Alexandre Boritchev
(Ecole Polytechnique)
Abstract

The Kolmogorov 1941 theory (K41) is, in a way, the starting point for all

models of turbulence. In particular, K41 and corrections to it provide

estimates of small-scale quantities such as increments and energy spectrum

for a 3D turbulent flow. However, because of the well-known difficulties

involved in studying 3D turbulent flow, there are no rigorous results

confirming or infirming those predictions. Here, we consider a well-known

simplified model for 3D turbulence: Burgulence, or turbulence for the 1D

Burgers equation. In the space-periodic case with a stochastic white in

time and smooth in space forcing term, we give sharp estimates for

small-scale quantities such as increments and energy spectrum.