Forthcoming events in this series


Wed, 07 Feb 2024
12:00
L6

Pressure jump in the Cahn-Hilliard equation

Charles Elbar
(Laboratoire Jacques Louis Lions, Sorbonne Université)
Abstract

We model a tumor as an incompressible flow considering two antagonistic effects: repulsion of cells when the tumor grows (they push each other when they divide) and cell-cell adhesion which creates surface tension. To take into account these two effects, we use a 4th-order parabolic equation: the Cahn-Hilliard equation. The combination of these two effects creates a discontinuity at the boundary of the tumor that we call the pressure jump.  To compute this pressure jump, we include an external force and consider stationary radial solutions of the Cahn-Hilliard equation. We also characterize completely the stationary solutions in the incompressible case, prove the incompressible limit and prove convergence of the parabolic problems to stationary states.

Wed, 17 Jan 2024
12:00
L6

A new understanding of the grazing limit

Prof Tong Yang
(Department of Applied Mathematics, The Hong Kong Polytechnic University)
Abstract

The grazing limit of the Boltzmann equation to Landau equation is well-known and has been justified by using cutoff near the grazing angle with some suitable scaling. In this talk, we will present a new approach by applying a natural scaling on the Boltzmann equation. The proof is based on an improved well-posedness theory for the Boltzmann equation without angular cutoff in the regime with an optimal range of parameters so that the grazing limit can be justified directly that includes the Coulomb potential. With this new understanding, the scaled Boltzmann operator in fact can be decomposed into two parts. The first one converges to the Landau operator when the parameter of deviation angle tends to its singular value and the second one vanishes in the limit. Hence, the scaling and limiting process exactly capture the grazing collisions. The talk is based on a recent joint work with Yu-Long Zhou.

Thu, 30 Nov 2023

12:00 - 13:00
L3

Gravitational Landau Damping

Matthew Schrecker
(University of Bath)
Abstract

In the 1960s, Lynden-Bell, studying the dynamics of galaxies around steady states of the gravitational Vlasov-Poisson equation, described a phenomenon he called "violent relaxation," a convergence to equilibrium through phase mixing analogous in some respects to Landau damping in plasma physics. In this talk, I will discuss recent work on this gravitational Landau damping for the linearised Vlasov-Poisson equation and, in particular, the critical role of regularity of the steady states in distinguishing damping from oscillatory behaviour in the perturbations. This is based on joint work with Mahir Hadzic, Gerhard Rein, and Christopher Straub.

Thu, 23 Nov 2023

12:00 - 13:00
L3

Recent developments in fully nonlinear degenerate free boundary problems

Edgard Pimentel
(University of Coimbra)
Abstract

We consider degenerate fully nonlinear equations, whose degeneracy rate depends on the gradient of solutions. We work under a Dini-continuity condition on the degeneracy term and prove that solutions are continuously differentiable. Then we frame this class of equations in the context of a free transmission problem. Here, we discuss the existence of solutions and establish a result on interior regularity. We conclude the talk by discussing a boundary regularity estimate; of particular interest is the case of point-wise regularity at the intersection of the fixed and the free boundaries. This is based on joint work with David Stolnicki.

Thu, 02 Nov 2023

12:00 - 13:00
L3

Coarsening of thin films with weak condensation

Hangjie Ji
(North Carolina State University)
Abstract

A lubrication model can be used to describe the dynamics of a weakly volatile viscous fluid layer on a hydrophobic substrate. Thin layers of the fluid are unstable to perturbations and break up into slowly evolving interacting droplets. In this talk, we will present a reduced-order dynamical system derived from the lubrication model based on the nearest-neighbour droplet interactions in the weak condensation limit. Dynamics for periodic arrays of identical drops and pairwise droplet interactions are investigated which provide insights to the coarsening dynamics of a large droplet system. Weak condensation is shown to be a singular perturbation, fundamentally changing the long-time coarsening dynamics for the droplets and the overall mass of the fluid in two additional regimes of long-time dynamics. This is joint work with Thomas Witelski.

Thu, 19 Oct 2023

12:00 - 13:00
L3

Extrinsic flows on convex hypersurfaces of graph type.

Hyunsuk Kang
(Gwangju Institute of Science and Technology and University of Oxford)
Abstract

Extrinsic flows are evolution equations whose speeds are determined by the extrinsic curvature of submanifolds in ambient spaces.  Some of the well-known ones are mean curvature flow, Gauss curvature flow, and Lagrangian mean curvature flow.

We focus on the special case in which the speed of a flow is given by powers of mean curvature for smooth convex hypersurfaces of graph type, i.e., ones that can be represented as the graph of a function.  Convergence and long-time existence of such flow will be discussed. Furthermore, C^2 estimates which are independent of height of the graph will be derived to see that the boundary of the domain of the graph is also a smooth solution for the same flow as a submanifold with codimension two in the classical sense.  Some of the main ideas, notably a priori estimates via the maximum principle, come from the work of Huisken and Ecker on mean curvature evolution of entire graphs in 1989.  This is a joint work with Ki-ahm Lee and Taehun Lee.

Thu, 09 Mar 2023

12:00 - 13:00
L4

TBA

Vincent Calvez
(Institut Camille Jordan, Université Claude Bernard)
Abstract

TBA

Thu, 23 Feb 2023

13:00 - 14:00
L4

Failure of the CD condition in sub-Riemannian and sub-Finsler geometry

Mattia Magnabosco
(Hausdorff Center for Mathematics)
Abstract

The Lott-Sturm-Villani curvature-dimension condition CD(K,N) provides a synthetic notion for a metric measure space to have curvature bounded from below by K and dimension bounded from above by N. It was proved by Juillet that the CD(K,N) condition is not satisfied in a large class of sub-Riemannian manifolds, for every choice of the parameters K and N. In a joint work with Tommaso Rossi, we extended this result to the setting of almost-Riemannian manifolds and finally it was proved in full generality by Rizzi and Stefani. In this talk I present the ideas behind the different strategies, discussing in particular their possible adaptation to the sub-Finsler setting. Lastly I show how studying the validity of the CD condition in sub-Finsler Carnot groups could help in proving rectifiability of CD spaces.

Thu, 23 Feb 2023

12:00 - 13:00
L4

Ocean Modelling at the Met Office

Mike Bell
(Met Office Fellow in Ocean Dynamics)
Abstract

Mike will briefly describe the scope and shape of science within the Met Office and of his career there. He will also outline the coordination of the development of the NEMO ocean model, which he leads, and work to ensure the marine systems at the Met Office work efficiently on modern High Performance Computers (HPCs).  In the second half of the talk, Mike will focus on two of his current scientific interests: accurate calculation of horizontal pressure forces in models with steeply sloping coordinates; and dynamical interpretations of meridional overturning circulations and ocean heat uptake.

Thu, 19 Jan 2023

12:00 - 13:00
L6

On the Incompressible Limit for a Tumour Growth Model Incorporating Convective Effects

Markus Schmidtchen
(TU Dresden)
Abstract

In this seminar, we study a tissue growth model with applications to tumour growth. The model is based on that of Perthame, Quirós, and Vázquez proposed in 2014 but incorporated the advective effects caused, for instance, by the presence of nutrients, oxygen, or, possibly, as a result of self-propulsion. The main result of this work is the incompressible limit of this model, which builds a bridge between the density-based model and a geometry free-boundary problem by passing to a singular limit in the pressure law. The limiting objects are then proven to be unique.

Thu, 09 Jun 2022

11:30 - 15:00
Linbury Building, Worcester College, University of Oxford

Research Working Lunch TT22

Further Information

Details including speakers, tiles and abstracts coming soon ...

Registration is required, please CLICK HERE or scan the below QR code.

QR Code for Research Working Lunch TT22

Organisers: 

Dr Benjamin Fehrman

Eliana Fausti

 

Administrator:

Kerri Louise Howard FInstAM

Abstract

CDT PDE Research Working Lunch Poster

11:30 Refreshments (tea, coffee and homemade biscuits)

12:00 Talks (main room)

13:15 Buffet Style Lunch (incl. tea, coffee and homemade cakes)

15:00 End

Fri, 20 Jun 2014

12:00 - 13:00
L6

Deformations of Axially Symmetric Initial Data and the Angular Momentum-Mass Inequality

Dr. Ye Sle Cha
(State University of New York at Stony Brook)
Abstract

We show how to reduce the general formulation of the mass-angular momentum inequality, for axisymmetric initial data of the Einstein equations, to the known maximal case whenever a geometrically motivated system of equations admits a solution. This procedure is based on a certain deformation of the initial data which preserves the relevant geometry, while achieving the maximal condition and its implied inequality (in a weak sense) for the scalar curvature; this answers a question posed by R. Schoen. The primary equation involved, bears a strong resemblance to the Jang-type equations studied in the context of the positive mass theorem and the Penrose inequality. Each equation in the system is analyzed in detail individually, and it is shown that appropriate existence/uniqueness results hold with the solution satisfying desired asymptotics. Lastly, it is shown that the same reduction argument applies to the basic inequality yielding a lower bound for the area of black holes in terms of mass and angular momentum.

Mon, 16 Jun 2014

14:00 - 15:00
L4

Weighted norms and decay properties for solutions of the Boltzmann equation

Prof. Irene M. Gamba
(University of Texas at Austin)
Abstract

We will discuss recent results regarding generation and propagation of summability of moments to solution of the Boltzmann equation for variable hard potentials.
These estimates are in direct connection to the understanding of high energy tails and decay rates to equilibrium.

Fri, 13 Jun 2014

12:00 - 13:00
L6

Shock Reflection, von Neumann conjectures, and free boundary problems

Prof. Mikhail Feldman
(University of Wisconsin-Madison)
Abstract

We discuss shock reflection problem for compressible gas dynamics, various patterns of reflected shocks, and von Neumann conjectures on transition between regular and Mach reflections. Then

we will talk about recent results on existence of regular reflection solutions for potential flow equation up to the detachment angle, and discuss some techniques. The approach is to reduce the shock

reflection problem to a free boundary problem for a nonlinear equation of mixed elliptic-hyperbolic type. Open problems will also be discussed. The talk is based on the joint work with Gui-Qiang Chen.

Fri, 13 Jun 2014

10:30 - 11:30
L6

Fluid-Composite Structure Interaction Problems

Prof. Suncica Canic
(University of Houston)
Abstract

Fluid-structure interaction (FSI) problems arise in many applications. The widely known examples are aeroelasticity and biofluids.

In biofluidic applications, such as, e.g., the study of interaction between blood flow and cardiovascular tissue, the coupling between the fluid and the

relatively light structure is {highly nonlinear} because the density of the structure and the density of the fluid are roughly the same.

In such problems, the geometric nonlinearities of the fluid-structure interface

and the significant exchange in the energy between a moving fluid and a structure

require sophisticated ideas for the study of their solutions.

In the blood flow application, the problems are further exacerbated by the fact that the walls of major arteries are composed of several layers, each with

different mechanical characteristics.

No results exist so far that analyze solutions to fluid-structure interaction problems in which the structure is composed of several different layers.

In this talk we make a first step in this direction by presenting a program to study the {\bf existence and numerical simulation} of solutions

for a class of problems

describing the interaction between a multi-layered, composite structure, and the flow of an incompressible, viscous fluid,

giving rise to a fully coupled, {\bf nonlinear moving boundary, fluid-multi-structure interaction problem.}

A stable, modular, loosely coupled scheme will be presented, and an existence proof

showing the convergence of the numerical scheme to a weak solution to the fully nonlinear FSI problem will be discussed.

Our results reveal a new physical regularizing mechanism in

FSI problems: the inertia of the fluid-structure interface regularizes the evolution of the FSI solution.

All theoretical results will be illustrated with numerical examples.

This is a joint work with Boris Muha (University of Zagreb, Croatia, and with Martina Bukac, University of Pittsburgh and Notre Dame University).

Thu, 05 Jun 2014

12:00 - 13:00
L6

A nonlinear model for nematic elastomers

Dr. Marco Barchiesi
(Universita di napoli)
Abstract

I will discuss the well-posedness of a new nonlinear model for nematic

elastomers. The main novelty is that the Frank energy penalizes

spatial variations of the nematic director in the deformed, rather

than in the reference configuration, as it is natural in the case of

large deformations.

Fri, 30 May 2014

12:00 - 13:00
L6

Weak universality of the stochastic Allen-Cahn equation

Dr. Weijun Xu
(University of Warwick)
Abstract

We consider a large class of three dimensional continuous dynamic fluctuation models, and show that they all rescale and converge to the stochastic Allen-Cahn equation, whose solution should be interpreted after a suitable renormalization procedure. The interesting feature is that, the coefficient of the limiting equation is different from one's naive guess, and the renormalization required to get the correct limit is also different from what one would naturally expect. I will also briefly explain how the recent theory of regularity structures enables one to prove such results. Joint work with Martin Hairer.

Fri, 23 May 2014

12:00 - 13:00
C6

Analysis of variational model for nematic shells

Dr. Antonio Segatti
Abstract

In this talk, I will introduce and analyse an elastic

surface energy recently introduced by G. Napoli and

L. Vergori to model thin films of nematic liquid crystals.

As it will be clear, the topology and the geometry of

the surface will play a fundamental role in understanding

the behavior of thin films of liquid crystals.

In particular, our results regards the existence of

minimizers, the existence of the gradient flow

of the energy and, in the case of an axisymmetric

toroidal particle, a detailed characterization of global and local minimizers.

This last item is supplemented with numerical experiments.

This is a joint work with M. Snarski (Brown) and M. Veneroni (Pavia).

Fri, 09 May 2014

12:00 - 13:00
L6

On Local Existence of Shallow Water Equations with Vacuum

Prof. Yachun Li
(Shanghai JiaoTong University)
Abstract

In this talk, I will present our new local existence result to the shallow water equations describing the motions of vertically averaged flows, which are closely related to the $2$-D isentropic Navier-Stokes equations for compressible fluids with density-dependent viscosity coefficients. Via introducing the notion of regular solutions, the local existence of classical solutions is established for the case that the viscosity coefficients are degenerate and the initial data are arbitrarily large with vacuum appearing in the far field.

Fri, 09 May 2014

11:00 - 12:00
L6

Study of the Prandtl boundary layer theory

Prof. Ya-Guang Wang
(Shanghai JiaoTong University)
Abstract

We shall talk our recent works on the well-posedness of the Prandtl boundary layer equations both in two and three space variables. For the two-dimensional problem, we obtain the well-posedness in the Sobolev spaces by using an energy method under the monotonicity assumption of tangential velocity, and for the three-dimensional Prandtl equations, we construct a special solution by using the Corocco transformation, and obtain it is linearly stable with respect to any three-dimensional perturbation. These works are collaborated with R. Alexandre, C. J. Liu, C. Xu and T. Yang.

Fri, 02 May 2014

12:00 - 13:00
C6

Using multiple frequencies to satisfy local constraints in PDE and applications to hybrid inverse problems

Giovanni Alberti
(University of Oxford)
Abstract

In this talk I will describe a multiple frequency approach to the boundary control of Helmholtz and Maxwell equations. We give boundary conditions and a finite number of frequencies such that the corresponding solutions satisfy certain non-zero constraints inside the domain. The suitable boundary conditions and frequencies are explicitly constructed and do not depend on the coefficients, in contrast to the illuminations given as traces of complex geometric optics solutions. This theory finds applications in several hybrid imaging modalities. Some examples will be discussed.

Thu, 13 Mar 2014

12:00 - 13:00
L6

Stochastic homogenization of nonconvex integral functionals with non-standard convex growth conditions

Prof. Antoine Gloria
(Université Libre de Bruxelles and Inria)
Abstract

One of the main unsolved problems in the field of homogenization of multiple integrals concerns integrands which are not bounded polynomially from above. This is typically the case when incompressible (or quasi-incompressible) materials are considered, although this is still currently a major open problem.
In this talk I will present recent progress on the stochastic homogenization of nonconvex integral functionals in view of the derivation of nonlinear elasticity from polymer physics, and consider integrands which satisfy very mild convex growth conditions from above.
I will first treat convex integrands and prove homogenization by combining approximation arguments in physical space with the Fenchel duality theory in probability. In a second part I will generalize this homogenization result to the case of nonconvex integrands which can be written in the form of a convex part (with mild growth condition from above) and a nonconvex part (that satisfies a standard polynomial growth condition). This decomposition is particularly relevant for the derivation of nonlinear elasticity from polymer physics.
This is joint work with Mitia Duerinckx (ULB).
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.