Thu, 04 Jun 2015

12:00 - 13:00
L6

Higher gradient integrability for σ -harmonic maps in dimension two

Mariapia Palombaro
(University of Sussex)
Abstract

I will present some recent results concerning the higher gradient integrability of

σ-harmonic functions u with discontinuous coefficients σ, i.e. weak solutions of

div(σ∇u) = 0. When σ is assumed to be symmetric, then the optimal integrability

exponent of the gradient field is known thanks to the work of Astala and Leonetti

& Nesi. I will discuss the case when only the ellipticity is fixed and σ is otherwise

unconstrained and show that the optimal exponent is attained on the class of

two-phase conductivities σ: Ω⊂R27→ {σ1,σ2} ⊂M2×2. The optimal exponent

is established, in the strongest possible way of the existence of so-called

exact solutions, via the exhibition of optimal microgeometries.

(Joint work with V. Nesi and M. Ponsiglione.)

Fri, 28 Nov 2014
14:00
L2

An optimal control approach for modelling Neutrophil cell migration

Dr Anotida Madzvamuse
(University of Sussex)
Abstract

Cell migration is of vital importance in many biological studies, hence robust cell tracking algorithms are needed for inference of dynamic features from (static) in vivo and in vitro experimental imaging data of cells migrating. In recent years much attention has been focused on the modelling of cell motility from physical principles and the development of state-of-the art numerical methods for the simulation of the model equations. Despite this, the vast majority of cell tracking algorithms proposed to date focus solely on the imaging data itself and do not attempt to incorporate any physical knowledge on cell migration into the tracking procedure. In this study, we present a mathematical approach for cell tracking, in which we formulate the cell tracking problem as an inverse problem for fitting a mathematical model for cell motility to experimental imaging data. The novelty of this approach is that the physics underlying the model for cell migration is encoded in the tracking algorithm. To illustrate this we focus on an example of Zebrafish (Danio rerio's larvae} Neutrophil migration and contrast an ad-hoc approach to cell tracking based on interpolation with the model fitting approach we propose in this talk.

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, 16 Jan 2014

16:00 - 17:30
L3

Topology of Sobolev spaces and Local minimizers

Ali Taheri
(University of Sussex)
Abstract

Attempting to extend the methods of critical point theory (e.g., those of Morse theory and Lusternik-Schnirelman theory) to the study of strong local minimizers of integral functionals of the calculus of variations I will describe how the obstruction method of algebraic topology can be successfully used to tackle the enumeration problem for various homotopy classes of maps in Sobolev spaces and that how this will result in precise lower bounds on the number of such local minimizers in terms of convenient topological invariants of the underlying spaces. I will then move on to dicussing variants as well as applications of the result to some classes of geometric nonlinear PDEs in particular problems in nonlinear elasticity.

Mon, 18 Nov 2013

17:00 - 18:00
L6

Blow-up of arbitrarily rough critical Besov norms at any Navier-Stokes singularity

Gabriel Koch
(University of Sussex)
Abstract

We show that the spatial norm in any critical homogeneous Besov

space in which local existence of strong solutions to the 3-d

Navier-Stokes equations is known must become unbounded near a singularity.

In particular, the regularity of these spaces can be arbitrarily close to

-1, which is the lowest regularity of any Navier-Stokes critical space.

This extends a well-known result of Escauriaza-Seregin-Sverak (2003)

concerning the Lebesgue space $L^3$, a critical space with regularity 0

which is continuously embedded into the spaces we consider. We follow the

``critical element'' reductio ad absurdum method of Kenig-Merle based on

profile decompositions, but due to the low regularity of the spaces

considered we rely on an iterative algorithm to improve low-regularity

bounds on solutions to bounds on a part of the solution in spaces with

positive regularity. This is joint work with I. Gallagher (Paris 7) and

F. Planchon (Nice).

Thu, 01 Nov 2012

12:30 - 13:30
Gibson 1st Floor SR

Analytical and numerical aspects of an extended Navier-Stokes system

Arghir D. Zarnescu
(University of Sussex)
Abstract

H. Johnston and J.G. Liu proposed in 2004 a numerical scheme for approximating numerically solutions of the incompressible Navier-Stokes system. The scheme worked very well in practice but its analytic properties remained elusive.\newline

In order to understand these analytical aspects they considered together with R. Pego a continuous version of it that appears as an extension of the incompressible Navier-Stokes to vector-fields that are not necessarily divergence-free. For divergence-free initial data one has precisely the incompressible Navier-Stokes, while for non-divergence free initial data, the divergence is damped exponentially.\newline

We present analytical results concerning this extended system and discuss numerical implications. This is joint work with R. Pego, G. Iyer (Carnegie Mellon) and J. Kelliher, M. Ignatova (UC Riverside).

Thu, 11 Oct 2007

14:00 - 15:00
Comlab

Explicit A Posteriori Error Analysis for Evolution Equation's Finite Element Approximation

Dr Omar Lakkis
(University of Sussex)
Abstract

I will address the usage of the elliptic reconstruction technique (ERT) in a posteriori error analysis for fully discrete schemes for parabolic partial differential equations. A posteriori error estimates are effective tools in error control and adaptivity and a mathematical rigorous derivation justifies and improves their use in practical implementations.

The flexibility of the ERT allows a virtually indiscriminate use of various parabolic PDE techniques such as energy methods, duality methods and heat-kernel estimates, as opposed to direct approaches which leave less maneuver room. Thanks to ERT parabolic stability techniques can be combined with different elliptic a posteriori error analysis techniques, such as residual or recovery estimators, to derive a posteriori error bounds. The method has the merit of unifying previously known approaches, as well as providing new ones and providing us with novel error bounds (e.g., pointwise norm error bounds for the heat equation). [These results are based on joint work with Ch. Makridakis and A. Demlow.]

Another feature, which I would like to highlight, of the ERT is its simplifying power. It allows us to derive estimates where the analysis would be very complicated otherwise. As an example, I will illustrate its use in the context of non-conforming methods, with a special eye on discontinuous Galerkin methods. [These are recent results obtained jointly with E. Georgoulis.]

Thu, 16 Feb 2006

14:00 - 15:00
Comlab

The finite element method for Cahn-Hilliard-Navier-Stokes equations

Dr David Kay
(University of Sussex)
Abstract

The Cahn-Hilliard equations provides a model of phase transitions when two or more immiscible fluids interact. When coupled with the Navier-Stokes equations we obtain a model fro the dynamics of multiphase flow. This model takes into account the viscosity and densities of the various fluids present.

A finite element discretisation of the variable density Cahn-Hilliard-Navier-Stokes equations is presented. An analysis of the discretisation and a reliable efficient numerical solution method are presented.

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