All posts by Paul Jewell

Generalized junction conditions for degenerate parabolic equations

Abstracts

Vinh Nguyen (Cardiff, UK)

With C. IMBERT, we study degenerate parabolic equations in multi-domains whose coefficients are discontinuous along interfaces.  We observe that the approach proposed by IMBERT and Monneau (2014) for Hamilton-Jacobi equations can be further developed to handle generalized junction conditions (such as the generalized Kirchoff ones) and second order terms. We first prove that generalized junction conditions reduce to flux-limited ones.  We then use then vertex test function (Imbert, Monneau — 2014) to prove a comparison principle.  We  then determine the vanishing viscosity limit associated with Hamilton-Jacobi equations posed on multi-domains and networks.  In the two-domain and convex case, the maximal Ishii solution identified by Barles, Briani and Chasseigne (2012) is selected. Finally, we give a short and simple PDE proof for the large deviation result of Boue, Dupuis and Ellis (2000).

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Models of charged drops

Abstracts

Matteo Novaga (Pisa, Italy)

Electrified liquids are well known to be prone to a variety of interfacial instabilities that result in the onset of apparent interfacial singularities and liquid fragmentation. In the case of electrically conducting liquids, one of the basic models describing the equilibrium interfacial configurations and the onset of instability assumes the liquid to be equipotential and interprets those configurations as local minimizers of the energy consisting of the sum of the surface energy and the electrostatic energy. Surprisingly, this classical geometric variational model is mathematically ill-posed irrespectively of the degree to which the liquid is electrified.

Specifically, an isolated spherical droplet is never a local minimizer, no matter how small is the total charge on the droplet, since the energy can always be lowered by a smooth, arbitrarily small distortion of the droplet’s surface. This is in sharp contrast with the experimental observations that a critical amount of charge is needed in order to destabilize a spherical droplet. We discuss some possible regularization mechanisms for the considered free boundary problem

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Scalar conservation laws with nonlinear multiplicative rough signal

Abstracts

Panagiotis E. Souganidis (Chicago, USA)

I will present a recently developed theory for scalar conservation laws with nonlinear multiplicative rough signal dependence. I will describe the difficulties, introduce the notion of pathwise entropy/kinetic solution and its well-posedness. I will also talk about the long time behavior of the solutions as well as some regularization by noise type results.

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A tour of Lipschitz truncations

Abstracts

Bianca Stroffolini (Napoli, Italy)

The purpose of the Lipschitz truncation is to regularize a given function by a Lipschitz continuous one by changing it only on a small  bad set. It is crucial for the applications that the function is not changed globally, which rules out the possibility of convolutions.

The Lipschitz truncation technique was introduced by Acerbi-Fusco to show lower semicontinuity of certain variational integrals.

Since then this technique has been successfully applied in many different areas: biting lemmas, existence theory  and regularity results of non-linear elliptic PDE . It was also successfully applied in the framework of non-Newtonian fluids of power law type and even in the context of numerical analysis.

I will try to present some Lipschitz truncations Lemmas.
As an application, existence/ regularity of solutions of PDEs will be discussed.

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Round table “Women in Maths” with the participation of Pro-VC Prof. Karen Holford

Abstracts

This round table session is addressed at all the female participants but it is of course open to interested male participants as well (please contact Federica if you are interested in participating).

The aim is to discuss the problem of gender imbalance in mathematical academic positions in particular in UK departments. We will focus on the recruiting problem, in particular from the post-doc level on. This seems to be the academic level where female applicants “disappear”. The contribution to the discussion of the many overseas female participants will be very valuable in understanding if this problem is UK-specific or a general trend across Europe, and in sharing best practise for increasing female representation in maths departments.

Location: Room M/1.25

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About numerically extracting the metric for entropy-driven diffusive systems

Posters

Peter Embacher (Cardiff)

We consider a diffusive system that is microscopically driven by an underlying stochastic process. For some of these systems it has been shown that their macroscopical evolution can be described by a gradient flow along its entropy, as long as the corresponding thermodynamical metric is chosen conveniently. We propose a numerical method to extract this metric from given experimental data. The method was applied to zero-range-processes and proved successfull there.

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Star-shaped and convex sets in the Heisenberg group

Posters

Doaa Filali (Cardiff, UK)

Star-shaped sets play a signifcant role in analysis and PDEs. In particular convex sets are star-shaped with respect to the all interior points. In the Euclidean case there are several equivalent definitions for star-shaped sets. This is not true in more degenerate geometries. For example in the Heisenberg group we generalised several Euclidean definitions for starshapedness and they turned to be not equivalent. For example we looked at the definition of strong star-shaped sets (related to the dilations) and the definition of weak star-shaped sets (related to the horizontal line segments). We construct counterexamples showing that the two definitions are not equivalent. While the first definition is important for some PDE application, the second weaker definition is key for convexity. In fact weak star-shaped with respect to all interior points is equivalent to the horizontal convexity (and all other equivalent notions known in the Heisenberg group). Beside that we study the relation between many different notions of star-shaped sets and their relations with convexity and convex functions. These results have been applied in the study of geometrical properties for level sets of nonlinear subelliptic PDEs.

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Well-posedness and scattering of the Chern-Simons-Schrödinger system

Posters

Zhuo Min Harold Lim (Cambridge, UK)

The Chern-Simons-Schrödinger system is a gauge-covariant version of the cubic nonlinear Schrödinger equation in two space dimensions. It describes the effective dynamics of a large system of nonrelativistic

charged quantum particles in the plane, interacting with each other and also with a self-generated long-range electromagnetic field. I will present my recent work establishing well-posedness in the energy space and scattering for the defocusing system, which describes a repulsive binary interaction. The scattering result is surprising from a physical point of view, and reflects subtle cancellations in the long-range electromagnetic interactions

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Minimal cost for the macroscopic motion of an interface

Posters

Panagiota Birmpa (Sussex, UK)

We will discuss the power needed to force a motion of a interface between two different phases of a given ferromagnetic sample with a prescribed speed V. In this model, the interface is the non-homogeneous stationary solution of a non local evolution equation. Considering a stochastic microscopic system of Ising spins with Kac interaction evolving in time according to Glauber dynamics, we assign the cost functional which penalizes deviations from the solutions of the mesoscopic evolution equation by considering the underlying microscopic process. Then, we study the optimal way to displace the interface.

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Stochastic Filtering for Rotating Shallow Water Equations

Posters

Oana Lang (Imperial, UK)

The aim of the talk/poster is to present a stochastic filtering problem consisting of a signal that models the motion of an incompressible fluid below a free surface whenthe vertical length scale is much smaller than the horizontal one. The evolution of the two-dimensional rotating system is represented by an infinite dimensional stochastic PDE and observed via a finite dimensional observation process. The deterministic

part of the SPDE consists of a classical shallow water equation (with an added viscosity term) and a new type of noise, namely the one introduced in [2]. Although this is a single layer model, therefore it does not completely reflect the complex stratification of the real atmosphere, it allows for important geophysical phenomena such as gravity and Rossby waves, eddy formation and geophysical turbulence.

References:

[1] A. Bain, D. Crisan, Fundamentals of Stochastic Filtering, Springer, 2009

[2] D. Holm , Variational Principles for Stochastic Fluid Dynamics, 2015

[3] G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics, 2005

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