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Workshop Programme

for period 23 - 27 June 2014

International Conference on Free Boundary Problems: Theory and Applications

23 - 27 June 2014

Timetable

Monday 23 June
09:00-09:55 Registration
09:55-10:00 Welcome from John Toland (INI Director)
Chair: G-Q Chen
10:00-10:45 Goldstein, R (University of Cambridge)
  Plenary Lecture 1: How a Volvox embryo turns itself inside out Sem 1
 

During the growth of daughter colonies of the multicellular alga Volvox the spherical embryos must turn themselves inside out to complete their development. This process of 'inversion' has many features in common with gastrulation, the process by which an initially convex spherical shell of animal cells develops an invagination, leading to the formation of a gastric system. In both cases it is understood that cell shape changes play a major role in guiding the process, but quantification of the dynamics, and formulation of a mathematical description of the process, have been lacking. In this talk I will describe advances my group has made recently on both fronts. Using the technique of SPIM (selective plane illumination microscopy) we have obtained the first real-time three-dimensional time-lapse movies of inversion in Volvox, using several species displaying distinct morphological events. The beginnings of an elastic theory of these processes will also be descri bed.

 
11:00-11:30 Morning Coffee
11:30-12:15 Wagner, B (Technische Universität Berlin)
  Plenary Lecture 2: Unstaedy non-uniform base states and their stability Sem 1
 

In this talk we consider several pattern forming systems, ranging from phase separation of polymer blends, self-assembly of crystalline films to dewetting of polymer films. These systems all have unstaedy non-uniform base states. We develop asymptotic techniques to analyse their associated linear stability problems and derive expressions for predicting the dominant wave-length of the pattern.

 
12:30-13:30 Lunch at Wolfson Court
Chair: JL Vázquez
13:30-14:15 Glimm, J (Stony Brook University)
  Plenary Lecture 3: Free Boundaries and Fluid Mixing at the Micro Level Sem 1
 

Turbulent mixing often occurs with immiscible fluids or with miscible fluids over rapid time scales, so that the flow is locally inhomogeneous at a micro level for periods of interest. We start with a review of problems in which such flows arise.

The flow regions in which the mixing occurs can generally be identified reliably; examples will be given. The challenge for current research is to describe the microscopic and inhomogeneous mixture in a statistical sense.

Full resolution of the flows is generally out of the question and will remain so for decades. Thus we are interested in statistical properties of the flow that are stable and appear to converge under mesh refinement, with sufficient detail in the statistical description (for example a pdf or cdf (cumulative distribution function) to support reaction processes in flows of engineering interest. A first step, generally insufficient, is to compute means and variances of fluctuating processes.

This goal is still in the future. Partial results leading in this direction will be presented. We formulate a notion of stochastic convergence and present numerical algorithms which appear to be convergent in this metric. We introduce theoretical ideas related to convergence based on the renormalization group, including the important notion that the solution, at the LES level of resolution of necessity considered here, is not unique. In other words, the usual standard of convergence under mesh refinement is not sufficient to guarantee a simulation in agreement with experimental data. We discuss methods to mitigate this serious obstacle to scientific progress. Basically, experiments are essential to select the correct non-unique solution and the algorithm and its adjustable parameters to reach this goal. While flows of interest are commonly at high Reynolds numbers outside the regime of relevant experiments, the expansion parameter is 1/Reynolds number. In terms of this parameter, the perturbation from experiment to applications is small, within normally accepted ranges for perturbative extensions of validation regimes. "cambridge.14.abs" 37L, 2201C

 
14:20-14:50 Afternoon Tea
Session: Focus Session Fluids (parallel session)
Chair: A Lacey
14:50-15:20 Ockendon, J (University of Oxford)
  Some Free Boundary Problems for Flows at high Re Sem 1
 

This talk will describe some models and open problems for (i) disturbance waves in annular 2-phase flow and (ii) bubble collapse.

 
15:20-15:50 Fontelos, M (ICMAT)
  Shapes of charged drops in an electric field and Rayleigh jets Sem 1
 

When a viscous drop is electrically charged or subject to an external electric field, it may undergo deformations and, occasionally produce thin jets (called Rayleigh jets) that are of practical interest. We study the drop's shapes and related bifurcation problems, as well as the possible mechanisms for jet formation.

 
16:00-16:30 Oliver, J (University of Oxford)
  On contact-line dynamics with mass transfer Sem 1
 

We investigate the effect of mass transfer on the evolution of a thin (two-dimensional) partially wetting drop. While the effects of viscous dissipation, capillarity, slip and uniform mass transfer are taken into account, other effects, such as gravity, surface tension gradients, vapour transport and heat transport, are neglected in favour of mathematical tractability. Our matched-asymptotic analysis reveals that the leading-order outer formulation and contact-line law that is selected in the small-slip limit depends delicately on both the sign and size of the mass transfer flux, leading in particular to novel generalisations of Tanner's law. We analyse the resulting evolution of the drop and report good agreement with numerical simulations. Co-Authors: Matthew Saxton (University of Oxford), Jonathan Whiteley (University of Oxford), Dominic Vella (University of Oxford); John King (University of Nottingham).

 
16:30-17:00 Lam, A (University of Warwick)
  Surfactants in two-phase flow Sem 1
 

Co-authors: Harald Garcke (Universitat Regensburg), Bjorn Stinner (University of Warwick)

Emulsification is an important industrial process that involves mixing two or more fluids that normally are unmixable. More precisely, in the process of emulsification, it is desirable to have stable dispersions of one fluid in another. Common examples of emulsions are milk, fire extinguishers and hand cream. The mixture is thermodynamically unstable and will progressively revert back to their unmixed states over time. Surface active agents (or surfactants) are often added to increase the stability of the mixture and hence there is great interest, especially in industrial applications, to understand the influence of surfactants on the dynamics of multi-fluid systems.

In this talk, I will outline the derivation of two new models on soluble surfactants in two phase flow. The first is a sharp interface model that describes the interfaces with moving hypersurfaces, and the second is a phase field model. Both models are thermodynamically consistent and generalise existing models in the literature. I will then discuss the relationship between two models, which is supported with some numerical simulations.

 
Session: Focus Session Recent Developments 1 (parallel session)
Chair: D Bucur
14:50-15:20 Paoli, LA (Université Jean Monnet)
  Dynamics of an Euler beam with unilateral constraints Sem 2
 

We study the vibrations of an elastic beam between rigid obstacles. The non penetrability condition leads to a description of the dynamics as a hyperbolic fourth order variational inequality. For this free boundary problem we construct a sequence of approximate solutions by combining some classical space discretizations with time-stepping schemes especially suited to unilateral contact problems for discrete mechanical systems. We prove the stability and the convergence of these numerical methods and we obtain an existence result for our original problem under very general assumptions on the geometry of the obstacles.

 
15:20-15:50 Yeressian Negarchi, K (University of Zurich)
  Nondegeneracy in the Obstacle Problem with a Degenerate Force Term Sem 2
 

In this talk I present the proof of the optimal nondegeneracy of the solution $u$ of the obstacle problem $\triangle u=f\chi_{\{u>0\}}$ in a bounded domain $D\subset\mathbb{R}^{n}$, where we only require $f$ to have a nondegeneracy of the type $f(x)\geq\lambda\vert (x_1,\cdots,x_p)\vert^{\alpha}$ for some $\lambda>0$, $1\leq p\leq n$ (an integer) and $\alpha>0$. We prove optimal uniform $(2+\alpha)$-th order and nonuniform quadratic nondegeneracy, more precisely we prove that there exists $C>0$ (depending only on $n$, $p$ and $\alpha$) such that for $x$ a free boundary point and $r>0$ small enough we have $\sup_{\partial B_r(x)}u\geq C\lambda (r^{2+\alpha}+\vert(x_1,\cdots,x_p)\vert^{\alpha}r^{2})$. I also present the proof of the optimal growth with the assumption $\vert f(x)\vert\leq\Lambda\vert (x_1,\cdots,x_p)\vert^{\alpha}$ for some $\Lambda\geq 0$ and the porosity of the free boundary.

Preprint: http://www.newton.ac.uk/preprints/NI14045.pdf

 
16:00-16:30 Du, Y (Univ of New England, Australia)
  Spreading profile and nonlinear Stefan problems Sem 2
 

I will report some recent progresses (in joint works with Z. Guo, B. Lou, H. Matsuzawa, H. Matano, K. Wang, M. Zhou etc.) on the study of a general nonlinear Stefan problem, used as a model for the understanding of a variety of spreading phenomena, where the unknown function u(t,x) represents the density of concentration of a certain (chemical or biological) species at time t and space location x, with the free boundary standing for the spreading front. Such spreading problems are usually modeled by the corresponding Cauchy problem, which has attracted extensive research starting from the well-known 1937 paper of Kolmogorov-Petrovski-Piskunov. We will discuss the similarity and differences of the long-time behavior of these two types of mathematical models by closely examining their spreading profiles.

 
16:30-17:00 Toader, R (Università degli Studi di Udine)
  A model for the quasi-static crack growth in hydraulic fracture Sem 2
 

We present a variational model for the quasi-static crack growth in hydraulic fracture in the framework of the energy formulation of rate-independent processes. The cracks are assumed to lie on a prescribed plane and to satisfy a very weak regularity assumption.

 
17:00-18:00 Welcome Wine Reception and Poster Session
Tuesday 24 June
Chair: J King
09:00-09:45 Sethian, J (University of California, Berkeley)
  Plenary Lecture 4 Sem 1
09:50-10:20 Morning Coffee
Session: Focus Session FBPs in Biology (parallel session)
Chair: R Kornhuber
10:20-10:50 O'Dea, R (University of Nottingham)
  Multiscale analyses of tissue growth and front propagation Sem 1
 

The derivation of continuum models which represent underlying discrete or microscale phenomena is emerging as an important part of mathematical biology: integration between subcellular, cellular and tissue-level behaviour is crucial to understanding tissue growth and mechanics. I will consider the application of a multiscale method to two problems on this theme.

Firstly a new macroscale description of nutrient-limited tissue growth, which is formulated as a microscale moving-boundary problem within a porous medium, is introduced. A multiscale homogenisation method is employed to enable explicit accommodation of the influence of the underlying microscale tissue structure, and its evolution, on the macroscale dynamics.

A challenging consideration in continuum models of tissue is the accommodation of (spatially-discrete) cell-signalling events, a feature of which being the progression of moving fronts of cell-signalling activity across a lattice. New (continuum) analyses of monotone waves in a discrete diffusion equation are presented, and extended to modulated fronts exhibited in cell signalling models.

 
10:50-11:20 Li, B (University of California, San Diego)
  Dielectric Boundary in Biomolecular Solvation Sem 1
 

A dielectric boundary in a biomolecular system is a solute-solvent (e.g., protein-water) interface that defines the dielectric coefficient to be one value in the solute region and another in solvent. The inhomogeneous dielectric medium gives rise to an effective dielectric boundary force that is crucial to the biomolecular conformation and dynamics. This lecture presents a precise definition and explicit formula of the dielectric boundary force based on the classical Poisson-Boltzmann theory of electrostatic interactions. These results are used to study the motion of a cylindrical dielectric boundary driven by the competition between the surface tension, dielectric boundary force, and solvent viscous force. Implications of the mathematical findings to biomolecular conformational stabilities are discussed.

 
11:30-12:00 Bozorgnia, F (Instituto Superior Técnico, Lisboa)
  Numerical investigations of long range segregation systems Sem 1
 

We investigate numerical approximation for a class of elliptic and parabolic competition-diffusion systems of long range segregation models for two and more competing species. Also we study the long term behavior for parabolic system. We prove that as the competition rate goes to infinity the solution converges, along with suitable sequences, to a spatially long range segregated state satisfying some differential inequalities. Moreover, we provide numerical simulations for parabolic and elliptic cases.

 
12:00-12:30 Venkataraman, C (University of Sussex)
  Moving boundary problems in cell biology Sem 1
 

We discuss the mathematical formulation and numerical solution of free and moving boundary problems that arise in the study of cell motility. We consider the dynamics of the cell membrane and species that reside on the cell membrane. We also present novel finite element methods for the simulation of the model equations and report on numerical results.

 
Session: Focus Session Mixed FBPs (parallel session)
Chair: M Feldman
10:20-10:50 Hunter, JK (University of California, Davis)
  Weak shock diffraction Sem 2
 

Co-author: Allen Tesdall (CUNY)

We study the diffraction of a weak, self-similar shock in two space dimensions near a point where its shock strength approaches zero and the shock turns continuously into an expansion wavefront. For example, this happened when a weak shock hits a semi-infinite screen. The local asymptotic solution satisfies the unsteady transonic small disturbance equation. We also consider a related half-space problem where a shock whose strength approaches zero reflects off a ``soft'' boundary. Numerical solutions show a complex reflection pattern similar to one that occurs in the Guderley Mach reflection of weak shocks.

 
10:50-11:20 Xiang, W (University of Oxford)
  Shock diffraction by convex cornered wedges Sem 2
 

Co-author: Prof. Gui-Qiang G. Chen (University of Oxford)

In this talk, I would like to present one of our current research projects, that is on the mathematical analysis of shock diffraction by convex cornered wedges. The existence of the regular configuration is established up to the critical wedge angle, which should be the criterion of the transition between the regular configuration and the Mach configuration. This work is joint with Prof. G.-Q. Chen.

 
11:30-12:00 Stinga, PR (University of Texas at Austin)
  The Signorini problem, fractional Laplacians and the language of semigroups Sem 2
 

The Signorini problem can be equivalently formulated as a thin obstacle problem for an elastic membrane. The resulting free boundary problem turns out to be equivalent to the obstacle problem for the fractional Laplacian on the whole space. We will show how to understand this problem under the light of the language of semigroups that I developed in my PhD thesis (2010). In particular, we are able to consider different kinds of Signorini problems that are equivalent to obstacle problems for fractional powers of operators different than the Laplacian on the whole space. Boundary conditions of different kinds (Dirichlet, Neumann, periodic) and radial solutions can also be treated with this unified language. Another advantage is that this language avoids the use of the Fourier transform. The basic regularity results (Harnack inequalities, Schauder estimates) for these fractional nonlocal operators can be studied by means of the generalization of the Caffarelli--Silvestre extensio n problem that I proved in my PhD thesis. It turns out that the solution for the extension problem can be written in terms of the heat semigroup.

 
12:00-12:30 Fang, B (Shanghai Jiao Tong University)
  On Stability of Steady Transonic Shocks in Supersonic Flow around a Wedge Sem 2
 

In this talk we are concerned with the stability of steady transonic shocks in supersonic flow around a wedge. When a uniform supersonic flow comes against a straight wedge with small vertex angle, a shock-front attached to the wedge appears. It is indicated in the book "Supersonic Flow and Shock Waves" by Courant and Friedrichs that there are two admissible shock solutions satisfying both Rankine-Hugoniot conditions and the entropy condition. The weaker shock solution may be transonic, while the stronger one must be transonic. In this talk, we shall present stability results for 2-D and M-D flows.

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

 
12:30-13:30 Sandwich lunch at INI
Chair: C Elliot
13:30-13:35 King, J; Elliot, C
  Welcome and Introduction Sem 1
13:35-13:45 King, J (University of Nottingham)
  Free Boundary Problems in Biology and Medicine Sem 1
13:45-14:10 Bretschneider, T (University of Warwick)
  How the Geometry of the Cell Boundary Couples Cellular Blebbing and Actin Based Protrusions Sem 1
14:10-14:35 O'Dea, R (University of Nottingham)
  Mathematical Modelling of Tissue Growth in a Perfusion Bioreactor Sem 1
14:35-15:00 Lowengrub, J (University of California, Irvine)
  Feedback, Lineages and Cancer Therapy Sem 1
15:00-15:20 Tea and Coffee Break
15:20-15:45 Ward, J
  Free Boundary Problems in Models for Bacterial Biofilms Sem 1
15:45-16:10 Ladds, G
  Modelling the Spatial Temporal Dynamics of Small Monomeric G Proteins Sem 1
16:10-17:00
  Open Discussion Sem 1
17:00-18:00 Drinks Reception and Networking
Wednesday 25 June
Chair: H Shahgholian
09:00-09:45 Novaga, M (Università di Pisa)
  Plenary Lecture 5: Obstacle problems in geometric evolutions Sem 1
 

I will consider geometric evolutions of fronts in the presence of obstacles. I will discuss in particular the mean curvature flow and the Willmore flow, presenting some new results and some open questions.

 
09:50-10:20 Morning Coffee
10:20-11:05 Markowich, P (KAUST)
  Plenary Lecture 6: On Boltzmann-type and Free Boundary Models for Price Formation Sem 1
 

We present a theory for a price formation free boundary model formulated by Lasry and Lions. Also we introduce a mesoscopic Boltzmann-type model for price formation in economic markets of single goods, which gives the Lasry-Lions model in the limit of large trading frequency.

 
Session: Focus Session Recent Developments 2A (parallel session)
Chair: N Wolanski
11:15-11:45 Cherniha, R (University of Nottingham)
  Symmetries of Boundary Value Problems: Definitions, Algorithms and Applications to Physically Motivated Problems Sem 1
 

One may note that the symmetry-based methods were not widely used for solving boundary-value problems (BVPs). To the best of our knowledge, the first rigorous definition of Lie's invariance for BVPs was formulated by George Bluman in early 1970s and applied to some classical BVPs. However, Bluman's definition cannot be directly applied to BVPs of more general form, for example, to those involving boundary conditions on the moving surfaces, which are described by unknown functions. In our recent papers, a new definition of Lie's invariance of BVP with a wide range of boundary conditions (including those at infinity and moving surfaces) was formulated. Moreover, an algorithm of the group classification for the given class of BVPs was worked out. The definition and algorithm were applied to some classes of nonlinear two-dimensional and multidimensional BVPs of Stefan type with the aim to show their efficiency. In particular, the group classification problem for these classes of BVPs was solved, reductions to BVPs of lower dimensionality were constructed and examples of exact solutions with physical meaning were found. Very recently, the definition and algorithms were extended on the case of conditional invariance for BVPs and applied to some nonlinear BVPs. This research was supported by a Marie Curie International Incoming Fellowship within the 7th European Community Framework Programme.

 
11:45-12:15 Gwinner, J (Universität der Bundeswehr München)
  Free Boundary Value Problems in Contact Mechanics Sem 1
 

In this talk we report on recent progress in the solution of various non-smooth free boundary value problems that arise in contact mechanics, including unilateral contact with Tresca friction and non-monotone adhesion/delamination problems. We analyse these problems and employ for their numerical treatment optimization and regularization techniques combined with fi nite element discretisation. This presentation is based on the recent papers [1, 2] and the recent PhD Thesis [3] of N. Ovcharova. [1] J.Gwinner, hp-FEM convergence for unilateral contact problems with Tresca friction in plane linear elastostatics, J. Comput. Appl. Math. 254 (2013) 175 -184 [2] N. Ovcharova and J.Gwinner, A study of regularization techniques of nondi erentiable optimization in view of application to hemivariational inequalities, J. Optim Theory Appl., DOI 10.1007/s10957-014-0521-y (2014) [3] N. Ovcharova, Regularization Methods and Finite Element Approximation of Some Hemivariational Inequalities with Applications to Nonmonotone Contact Problems, Ph.D. Thesis, Universitaet der Bundeswehr Muenchen, 2012.

 
Session: Focus Session Recent Developments 2B (parallel session)
Chair: K Yeressian Negarchi
11:15-11:45 Tarzia, D (Universidad Austral)
  Generalized Neumann solutions for the two-phase fractional Lam\'e-Clapeyron-Stefan problems Sem 2
 

A generalized Neumann solution for the two-phase fractional Lam\'e-Clapeyron-Stefan problem for a semi-infinite material with constant boundary and initial conditions is obtained. In this problem, the two governing diffusion equations and a governing condition for the free boundary include a fractional time derivative in the Caputo sense of order 0

Moreover, we also obtain a new generalized Neumann solutions for the two-phase fractional Lam\'e-Clapeyron-Stefan problems when a heat flux or a convective boundary condition is imposed on the fixed face x=0. These solutions are obtained when an inequality for the coefficient which characterizes the heat flux or the convective boundary condition is satisfied. In these cases, when a? 1 we also recover the explicit solution given in Tarzia, Quart. Appl. Math., 39 (1981), 491-499 and in MAT-Serie A, 8 (2004), 21-27.

The first part of this presentation is a joint work with S.D. Roscani (Univ. Rosario, Argentina).

 
11:45-12:15 Boukrouche, M (Institut Camille Jordan)
  Convergence of optimal control problems governed by second kind parabolic variational inequalities Sem 2
 

We consider a family of optimal control problems where the control variable is given by a free boundary condition of Neumann type. This family is governed by parabolic variational inequalities of the second kind. We prove the strong convergence of the optimal controls and state systems associated to this family to a similar optimal control problem.

 
12:30-13:30 Lunch at Wolfson Court
Chair: J Lowengrub
13:30-14:15 Deckelnick, K (Otto-von-Guericke-Universität Magdeburg)
  Plenary Lecture 7: Double obstacle phase field approach for an elliptic inverse problem with discontinuous coefficients Sem 1
 

We consider the inverse problem of recovering interfaces where the diffusion coefficient in an elliptic PDE has jump discontinuities. We employ a least squares approach together with a perimeter regularization. A suitable relaxation of the perimeter leads to a sequence of Cahn--Hilliard type functionals for which we obtain a $\Gamma$--convergence result. Using a finite element discretization of the elliptic PDE and a suitable adjoint problem we derive an iterative method in order to approximate discrete critical points. We prove convergence of the iteration and present results of numerical tests. This is joint work with C.M. Elliott (Warwick) and V. Styles (Sussex).

 
14:20-14:50 Afternoon Tea
Session: Focus Session Numerics for FBPS (parallel session)
Chair: C Elliott
14:50-15:20 Ranner, T (University of Leeds)
  Computational methods for an optimal partition problem on surfaces Sem 1
 

In this talk I will explore computational techniques for solving a free boundary problem posed on a surface. The problem is to divide a surface into regions such that the sum of the first eigenvalue of the Dirichlet Laplace-Beltrami operator over each region is minimized. Different relaxations of this problem will be explored. Each takes the form of large system of partial differential equations which will be solved using algorithms designed for high performance computing techniques.

 
15:20-15:50 Styles, V (University of Sussex)
  Multi-material structural topology optimization using phase field methods Sem 1
 

Co-authors: Luise Blank (University of Regensburg), Hassan Farshbaf-Shaker (WIAS Berlin), Harald Garcke (University of Regensburg)

A phase field approach for structural topology optimization which allows for topology changes and multiple materials is analyzed. First order optimality conditions are rigorously derived and several numerical results for mean compliance problems are presented.

 
16:00-16:30 Kahle, C (Universität Hamburg)
  A stable discretization for the simulation of two-phase flow Sem 1
 

Co-authors: Harald Garcke (University of Regensburg), Michael Hinze (University of Hamburg)

In [Abels, Garcke, Gruen, M3AS, 22, 2012] a model for the simulation of variable density two-phase flow using the diffuse interface approach is presented. Due to an additional transport term in the Navier-Stokes system this model enjoys consistency with thermodynamics. To preserve this important property in the numerical simulation we propose a time discretization that uses a carefully designed weak formulation and delivers a time discrete variant of the continuous energy equality. The spatial discretization is enriched by a suitable post processing step to ensure the energy inequality to hold in the fully discrete setting. An a-posteriori adaptive procedure is proposed that deals with the errors both in the Navier-Stokes and the Cahn-Hilliard system.

 
16:30-17:00 Banas, L (University of Bielefeld)
  Numerical approximation of phase-field models for multiphase flow Sem 1
 

Flow of mixtures of incompressible fluids with complex fluid interactions (surface tensions, contact angles) can be described by a system of (multicomponent) Cahn-Hilliard-Navier-Stokes equations. We propose finite element based numerical approximations of some non-smooth phase-field models for mixtures of incompressible fluids with variable densities and viscosities. We discuss theoretical and practical issues related to the proposed numerical approximations and present computational experiments to demonstrate main features of the considered models.

 
Session: Focus Session Analysis 1 (parallel session)
Chair: H Shahgholian
14:50-15:20 Danielli, D (Purdue University)
  Optimal Regularity and the Free Boundary in the Parabolic Signorini Problem Sem 2
 

Co-authors: Nicola Garofalo (University of Padova), Arshak Petrosyan (Purdue University), Tung To (Purdue University)

In this talk we will give an overview of our comprehensive treatment of the parabolic Signorini problem, based on a generalization of Almgren's monotonicity of the frequency. In particular, we will discuss the optimal regularity of solutions, the classi fication of free boundary points, the regularity of the regular set, and the structure of the singular set.

 
15:20-15:50 Bucur, D (Université de Savoie)
  Isoperimetric inequalities and free discontinuity problems Sem 2
 

The Faber-Krahn inequality for the first eigenvalue of the Laplace operator with Robin boundary conditions was recently proved by D. Daners in the context of Lipschitz sets. This talk introduces a new approach to deal with this type of isoperimetric inequalities and is based on recent advances on the regularity theory for general free discontinuity problems, in particular on a new monotonicity formula jointly obtained with S. Luckhaus. As a consequence, we not only extend isoperimetric inequalities to arbitrary sets, but also obtain a range of new ones (joint work with A. Giacomini).

 
16:00-16:30 Lee, KA (Seoul National University)
  Curvature Flows with Free Boundaries Sem 2
 

In this talk, we are going to discuss alpha- Gauss Curvature Flows with free boundaries. For a=1n, a flat spot on convex manifold will persists for some time, which is similar as the waiting time in Porous Medium Equations. Naturally, the interface between strictly convex region and flat spot will form a free boundary that will moves along the time. We will show the relation between optimal regularity and bounded nondegenerate speed of free boundaries. The optimal regularity and the smoothness of the free boundary will be discussed. We will also consider the dynamics and regularity of free boundary of the manifold with a concave region.

 
16:30-17:00 Karakhanyan, A (University of Edinburgh)
  The Proximity of Free Boundary to Wedges and Planes Sem 2
 

The talk is about the geometry of free boundary striking the centre of the half ball B_1^+ where a wedge like boundary data is prescribed near the origin. It will be seen that the effect of the wedge-like data dooms the asymptotic shape of the free boundary and the homogeneous global solutions turn out to have wedge-like free boundary too. A delicate analysis involving monotonicity formula techniques is employed to furnish the main results, which are from a joint paper with H. Shahgholian (to appear in the Transactions of the AMS).

 
17:20-18:00 FRBW04 organiser's meeting
19:30-22:00 Conference Dinner at Christ's College
Thursday 26 June
Chair: J Toland
09:00-09:45 Ball, J (University of Oxford)
  Plenary Lecture 8: Interfaces arising from solid phase transformations Sem 1
 

Solid phase transformations give rise to a variety of unusual kinds of interfaces between different phases, some only observed in recent experiments. The lecture will discuss ways of describing and predicting these, and related questions of nonlinear analysis.

 
09:50-10:20 Morning Coffee
Session: Focus Session Material Science (parallel session)
Chair: B Wagner
10:20-10:50 Majumdar, A (University of Bath)
  Free Boundary-Value Problems for Liquid Crystals Sem 1
 

We discuss modelling strategies for free boundary-value problems arising in liquid crystal applications. We have recently modelled stable liquid crystal patterns in shallow cuboid-shaped nematic wells and have computed phase diagrams as a function of geometrical aspect ratio and anchoring strength on well boundaries. Recent work by Telo da Gama group (CFTC, Lisbon) shows that our computed patterns are also relevant to wedge-shaped nematic-filled wells at the isotropic-nematic transition temperature. At the isotropic-nematic transition temperature, the isotropic and nematic phases are separated by a free boundary, depending on the geometry and the surface effects at play. We discuss modelling approaches to this free-boundary problem and the role of such free boundaries in applications. This is joint work with Chong Luo, Samo Kralj and Samo Kralj.

 
10:50-11:20 Stewart, P (University of Glasgow)
  On the production of high-porosity metallic solids Sem 1
 

Co-author: Stephen Davis (Northwestern University)

High-porosity metallic solids are useful as lightweight materials in many engineering disciplines. However, batch processing techniques for producing such materials by solidification of a molten metallic foam are problematic, and it is not currently possible to control the porosity distribution of the final product {\it a priori}. The molten metallic foam is inherently unstable; the thin liquid bridges between bubbles drain rapidly and rupture due to intermolecular forces, leading to bubble coalescence and large-scale topological rearrangement. In this talk we will consider the competition between coalesecence and freezing in these dynamically evolving foams, examining in particular the coupling between the microscale hydrodynamics in the molten liquid films and the progression of a solidification front. The foam is modelled as a coupled system of free boundary problems involving both liquid/gas and liquid/solid interfaces, which we solve using a combination of asymptotic and numerical techniques. An understanding of these microscale processes motivates other protocols for solid foam production where the porosity distribution of the final product can be more effectively controlled.

 
11:30-12:00 Hecht, C (Universität Regensburg)
  A phase field approach for shape optimization in fluids Sem 1
 

Co-author: Harald Garcke (University of Regensburg)

We consider the problem of shape optimization with a general objective functional using the Stokes equations as state constraints. By combining a porous medium and a phase field approach we obtain a well-posed problem in a diffuse interface setting. We discuss well-posedness and optimality conditions and relate the phase field problem to a perimeter penalized sharp interface problem.

 
12:00-12:30 Glicksman, M (University of Florida)
  Deterministic Pattern Formation in Diffusion-Limited Systems Sem 1
 

An interface evolving under local equilibrium develops gradients in the Gibbs-Thomson-Herring interface temperature distribution that provide tangential energy fluxes. The Leibniz-Reynolds transport theorem exposes a 4th-order, net-zero energy field (the 'Bias' field) that autonomously deposits and removes capillary-mediated thermal energy. Where energy is released locally, the freezing rate is persistently retarded, and where energy is removed, the rate is enhanced. These contravening dynamic field responses balance at points (roots) where the surface Laplacian of the chemical potential vanishes, inducing an inflection, or ‘curling’, of the interface. Interfacial inflection couples to the main transport fields producing pattern branching, folding, and complexity.

Precision noise-free numerical schemes, including integral equation sharp-interface solvers (J. Lowengrub, S. Li) and, recently, three noise-free phase-field simulations (A. Mullis, M. Zaeem, K. Reuther) independently confirm that pattern branching initiates at locations predicted using analytical methods for smooth, noise-free starting shapes in 2-D. A limit cycle may develop as the interface and its energy field co-evolve, synchronizing the inflection points to produce classical dendritic structures. Noise and stochastics play no direct role in the proposed deterministic mechanism of branching and pattern morphogenesis induced by persistent 'perturbations'.

 
Session: Focus Session Recent Developments 3 (parallel session)
Chair: D Danielli
10:20-10:50 Dallaston, M (University of Oxford)
  Free boundary problems in glacial hydrology Sem 2
 

Many aspects of a glacier’s dynamics depend on the evolution of an interface between slowly creeping ice and fast flowing water. As ice can be modelled as a very viscous fluid, variations on free boundary Stokes flow arise in the modelling of such interfaces. In this talk I will focus on the shape of a channel that is cut through or under the ice by viscous dissipation of meltwater. Networks of these channels form a vital part of a glacier’s hydrological system, which carries meltwater from the surface to the glacier margin, where mixing with dense ocean water has a strong effect on melting at the ice face. The evolving cross section of the channel is related to the problem of a contracting or expanding bubble in two dimensional Stokes flow, which allows us to derive analytic results. I will also discuss the impact of a meltwater source, such as a channel, on the spatial distribution of melting at the ice face.

 
10:50-11:20 Feehan, P (Rutgers, The State University of New Jersey)
  A classical Perron method for existence of smooth solutions to boundary value and obstacle problems for boundary-degenerate elliptic operators via holomorphic m Sem 2
 

We prove existence of solutions to boundary value problems and obstacle problems for degenerate-elliptic, linear, second-order partial differential operators with partial Dirichlet boundary conditions using a new version of the Perron method. The elliptic operators considered have a degeneracy along a portion of the domain boundary which is similar to the degeneracy of a model linear operator identified by Daskalopoulos and Hamilton (1998) in their study of the porous medium equation or the degeneracy of the Heston operator (Heston, 1993) in mathematical fi nance. Our Perron method relies on weak and strong maximum principles for degenerate-elliptic operators, concepts of continuous subsolutions and supersolutions for boundary value and obstacle problems for degenerate-elliptic operators, and maximum and comparison principle estimates previously developed by us.

 
11:30-12:00 Moroz, V (Swansea University)
  Existence and Qualitative Properties of Grounds States to the Choquard-Type Equations Sem 2
 

Co-author: Jean Van Schaftingen (Louvain-la-Neuve, Belgium)

The Choquard equation, also known as the Hartree equation or nonlinear Schrodinger-Newton equation is a stationary nonlinear Schrodinger type equation where the nonlinearity is coupled with a nonlocal convolution term given by an attractive gravitational potential. We present sharp Liouville-type theorems on nonexistence of positive supersolutions of such equations in exterior domains and consider existence, positivity, symmetry and optimal decay properties of ground state solutions under various assumptions on the decay of the external potential and the shape of the nonlinearity. We also discuss the existence of semiclassical solutions to the equation.

 
12:00-12:30 Cheng, C-H (National Central University)
  Some moving boundary value problems consisting of viscous incompressible fluids moving and interacting with nonlinear elastic shells Sem 2
 

In this talk, I will briefly talk about my past works on the interaction of incompressible Navier-Stokes with either the biofluid shells or the Koiter shells. The biofluid shells can be used to model the cell membranes, while the Koiter shells are used to model general elastic thin shells that have equilibrium state. The dynamics of both kinds of shells are described by nonlinear PDEs derived from energy functionals depending on geometric structures of the shells. The well-posedness of the problems concerning the coupling of the fluids and the shells are non-trivial, and we established the existence and uniqueness of solutions in Sobolev spaces.

 
12:30-13:30 Lunch at Wolfson Court
Chair: JF Rodrigues
13:30-14:15 Bertozzi, AL (University of California, Los Angeles)
  Plenary Lecture 9: Curvature flow on graphs for large data classification Sem 1
 

In the continuum, close connections exist between mean curvature ow, the Allen-Cahn (AC) partial di erential equation, and the Merriman-Bence-Osher (MBO) threshold dynamics scheme. Graph analogues of these processes have recently seen a rise in popularity as relaxations of NP-complete combinatorial problems, which demands deeper theoretical underpinnings of the graph processes. We discuss several applications including supervised and unsupervised machine learning and community detection in social networks. We discuss connections to spectral graph theory and fast algorithms and some recent results for curvature flow on graphs and open problems.

 
14:20-14:50 Afternoon Tea
Session: Focus Session Analysis 2 (parallel session)
Chair: JL Vázquez
14:50-15:20 Chipot, M (University of Zurich)
  Obstacle problems in unbounded domains and related issues Sem 1
 

We will present a formulation of obstacle problems in unbounded domains when the energy method does not work i.e. when the force does not belong to H-1(O). We will report also on some new results for the pure Neumann problem in unbounded cylinders.

 
15:20-15:50 Lindgren, EK (KTH - Royal Institute of Technology)
  Pointwise estimates and regularity for the parabolic obstacle problem Sem 1
 

Co-author: Régis Monneau (ENPC)

I will discuss pointwise regularity properties for parabolic equations. In particular, I will talk about a method that can be used to obtain a second order Taylor expansion. I will also discuss how the same method can be applied to the parabolic obstacle problem in order to study the regularity of the free boundary.

 
16:00-16:30 Quirós, F (U. Autónoma de Madrid)
  Free boundaries in fractional filtration equations Sem 1
 

Co-authors: Arturo de Pablo (U. Carlos III de Madrid), Ana Rodríguez (U. Politécnica de Madrid), Juan Luis Vázquez (U. Autónoma de Madrid)

In this talk we will review some recent results on the Cauchy problem for the fractional filtration equation $\partial_t u+(-\Delta)^{\sigma/2}\varphi(u)$, $\sigma\in(0,2)$, where the nonlinearity $\varphi$ satisfies some not very restrictive conditions.

Solutions to these problems become instantaneously positive if the initial data are nonnegative. However, a free boundary emerges in certain cases in which there is a distinguished value, as in the fractional Stefan problem, $\varphi(u)=(u-1)_+$, or in the ‘mesa’ limit, $m\to\infty$, for the fractional porous media equation, $\varphi(u)=u^m$.

 
16:30-17:00 Simonett, G (Vanderbilt University)
  On a thermodynamically consistent Stefan problem with variable surface energy Sem 1
 

A thermodynamically consistent two-phase Stefan problem with temperature dependent surface tension is studied. It is shown that this problem generates a local semiflow on a well-defined state manifold. Moreover, stability and instability results of equilibrium configurations will be presented. It will be pointed out that surface heat capacity has a striking effect on the stability behavior of multiple equilibria.

 
Session: Focus Session Recent Developments 4 (parallel session)
Chair: A Karakhanyan
14:50-15:20 Matusik, M (University of Gdansk)
  Oscillating facets Sem 2
 

Co-author: Piotr Rybka (The University of Warsaw)

We study a very singular one-dimensional parabolic problem with initial data in the energy space. We study facet creation and extinction of solutions caused by the evolution of facets.

 
15:20-15:50 Heinemann, C (Weierstraß-Institut für Angewandte Analysis und Stochastik)
  Analysis of coupled phase-field models describing damage phenomena Sem 2
 

Co-authors: Christiane Kraus (WIAS Berlin), Elisabetta Rocca (WIAS Berlin / University of Pavia)

In this talk we are going to investigate evolution inclusions describing damage processes in elastic media which are coupled with further physical phenomena such as heat conduction or phase separation. The main difficulty in the analysis of these PDE systems is the irreversibility condition for the damage progression. To handle this highly nonlinear behavior we develop an appropriate notion of weak solution consisting of a variational inequality and a full energy inequality. Results ensuring the existence of weak solutions will be presented. The talk is based on joint works with C. Kraus (WIAS) and E. Rocca (WIAS/University of Pavia).

 
16:00-16:30 Mitchell, S (University of Limerick)
  Numerical and asymptotic solutions of vertical continuous casting with and without superheat Sem 2
 

Co-author: Michael Vynnycky (Royal Institute of Technology (Sweden))

In a continuous casting process, such as the strip casting of copper, molten metal first passes through a water-cooled mould region, before being subjected to a high cooling rate further downstream. Consequently, the molten metal solidifies and the solidified metal is withdrawn at a uniform casting speed. Industrialists need to understand the factors influencing product quality and process productivity. Of key significance is the heat transfer that occurs during solidification, particularly the location of the interface between molten metal and solid.

The modelling of the continuous casting of metals is known to involve the complex interaction of non-isothermal fluid and solid mechanics. Typically, the flow in the molten metal is turbulent, and it is generally believed that a computational fluid dynamics (CFD) approach is necessary in order to correctly capture the heat transfer characteristics. However, we can show that an asymptotically reduced version of the CFD-based model, which neglects this turbulence, gives predictions for the pool depth, local temperature profiles and mould wall heat flux that agree very well with results of the original CFD model.

This reduced model can be described as a steady state 2D heat flow Stefan problem, with a degenerate initial condition and non-standard Neumann-type boundary condition. If we assume the incoming metal is at the melt temperature then we obtain a one-phase model but with potentially two stages, depending whether the metal is fully solidified before leaving the mould. However, in reality, the incoming temperature is greater than the melt temperature, termed as including superheat, and this leads to a two-phase model with a pre-solidification stage, where the second phase only first appears after a finite delay.

In this work we highlight some numerical challenges in solving the systems with and without superheat. The Keller box finite-difference scheme is used, along with a boundary immobilisation method.

 
16:30-17:00 Mitrou, G (University of Southampton)
  Higher order convergent trial methods for Bernoulli's free boundary problem Sem 2
 

Co-author: Helmut Harbrecht (University of Basel)

Free boundary problem is a partial differential equation to be solved in a domain, a part of whose boundary is unknown – the so-called free boundary. Beside the standard boundary conditions that are needed in order to solve the partial differential equation, an additional boundary condition is imposed at the free boundary. One aims thus to determine both, the free boundary and the solution of the partial differential equation.

This work is dedicated to the solution of the generalized exterior Bernoulli free boundary problem which is an important model problem for developing algorithms in a broad band of applications such as optimal design, fluid dynamics, electromagnentic shaping etc. For its solution the trial method, which is a fixed-point type iteration method, has been chosen.

The iterative scheme starts with an initial guess of the free boundary. Given one boundary condition at the free boundary, the boundary element method is applied to compute an approximation of the violated boundary data. The free boundary is then updated such that the violated boundary condition is satisfied at the new boundary. Taylor's expansion of the violated boundary data around the actual boundary yields the underlying equation, which is formulated as an optimization problem for the sought update function. When a target tolerance is achieved, the iterative procedure stops and the approximate solution of the free boundary problem is detected.

The efficiency of the trial method as well as its speed of convergence depends significantly on the update rule for the free boundary, and thus on the violated boundary condition. This talk focuses on the trial method with violated Dirichlet boundary data and on the development of higher order convergent versions of the trial method with the help of shape sensitivity analysis.

 
Friday 27 June
Chair: M Feldman
09:00-09:45 Shkoller, S (University of Oxford)
  Plenary Lecture 10: Absence of the interface splash singularity for the two-fluid Euler equations Sem 1
 

An interface splash singularity occurs when a locally smooth fluid interface self-intersects. Such interface singularities occur for one-fluid interfaces in the Euler equations and other fluids models.

By means of elementary arguments in Lagrangian coordinates, we prove that such a singularity cannot occur in finite-time for a two-fluid interface evolved by either the incompressible Euler equations (with surface tension) or the Muskat equations. By assuming that such a singularity can occur, we find a sharp blow-up rate for the vorticity, and characterize the geometry of the evolving interface. This leads to a contradiction, showing that such a singularity can occur. This is joint work with D. Coutand.

 
10:00-10:45 Lowengrub, J (University of California, Irvine)
  Plenary Lecture 11: Numerical simulation of endocytosis Sem 1
 

Co-authors: Sebastian Aland (UC Irvine/TU Dresden), Jun Allard (UC Irvine)

Many cell processes involve the formation of membrane vesicles from a larger membrane, including endocytosis, inter-organelle transport and virus entry. These events are typically orchestrated by curvature-inducing molecules attached to the membrane, such as clathrin and bar-domain proteins. Recent reports demonstrate that in some circumstances vesicles can form de novo in a few milliseconds, e.g., ultrafast endocytosis at the neurological synapse. Membrane dynamics at these scales (millisecond, nanometer) are dominated by hydrodynamic interactions, as the membrane pushes the intracellular and extracellular fluids around to accommodate curvature. To study this problem, we develop new diffuse interface models for the dynamics of inextensible vesicles in a viscous fluid with stiff, curvature-inducing molecules. A new feature of this work is the implementation of the local inextensibility condition by using a local Lagrange multiplier harmonically extended off the interface. To make the method even more robust, we develop a local relaxation scheme that dynamically corrects local stretching/compression errors thereby preventing their accumulation. This is critical to accurately capturing hydrodynamic effects during endocytosis. By varying the membrane coverage of curvature-inducing molecules, we find that there is a cri tical (smallest) neck radius and a critical (fastest) budding time.

 
11:00-11:30 Morning Coffee
Chair: J King
11:30-12:15 Jensen, O (University of Manchester)
  Plenary Lecture 12: Slamming in flexible-channel flows Sem 1
 

Co-author: Feng Xu (University of Nottingham)

Large-amplitude self-excited oscillations of high-Reynolds-number flow in a long flexible-walled channel can exhibit vigorous slamming motion, whereby the channel is almost completely occluded over a very short interval in space and time. Treating the flexible channel wall as an inertialess elastic membrane, this near-singular behaviour is exhibited in two-dimensional Navier-Stokes simulations and can be captured in a reduced one-dimensional PDE model (Stewart et al. J. Fluid Mech. 662:288, 2010). The properties of the rigid parts of the system, upstream and downstream of the membrane, play a major role in determining the onset of oscillations. In order to investigate the extreme flow structure that arises during a brief slamming event, we systematically reduce the PDE model to a third-order nonlinear algebraic-differential system, which identifies the likely dominant physical balances.

 
12:30-13:30 Lunch at Wolfson Court
Chair: C Elliott
13:30-14:15 Feldman, M (University of Wisconsin-Madison)
  Plenary Lecture 13: Shock Reflection, von Neumann conjectures, and free boundary problems Sem 1
 

Co-author: Gui-Qiang Chen (Oxford)

We discuss shock reflection problem for compressible gas dynamics, 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.

 
14:30-15:00 Afternoon Tea
15:00-15:45 Vázquez, JL (Universidad Autonoma de Madrid)
  Plenary Lecture 14: Free boundary problems for mechanical models of tumor growth Sem 1
 

Mathematical models of tumor growth, now commonly used, present several levels of complexity, both in terms of the biomedical ingredients and the mathematical description. The simplest ones contain competition for space using purely fluid mechanical concepts. Another possible ingredient is the supply of nutrients. The models can describe the tissue either at the level of cell densities, or at the scale of the solid tumor, in this latter case by means of a free boundary problem. We first formulate a free boundary model of Hele-Shaw type, a variant including growth terms, starting from the description at the cell level and passing to a certain singular limit which leads to a Hele-Shaw type problem. A detailed mathematical analysis of this purely mechanical model is performed. Indeed, we are able to prove strong convergence in passing to the limit, with various uniform gradient estimates; we also prove uniqueness for the limit problem. At variance with the classical Hele-Shaw problem, here the geometric motion governed by the pressure is not sufficient to completely describe the dynamics. Using this theory as a basis, we go on to consider a more complex model including nutrients. Here, technical difficulties appear, that reduce the generality and detail of the description. We prove uniqueness for the system, a main mathematical difficulty. Joint work with Benoit Perthame, Paris, and Fernando Quiros, Madrid

 
15:45-16:00
  Closing Remarks Sem 1

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