FBP2012 --

Speakers 2008

Plenary Lecturers
Focus Sessions
Communications

Plenary Lecturers 2008

Martin Burger (Westfälische Wilhelms-Universität Münster)
"Free Boundaries in Biological Aggregation Models"
This talk will discuss the appearance and simulation of free boundary motion in biological aggregation models with nonlocal attractive forces. Due to the attractive part clusters will form, which however remain at finite size due to a local repulsive force (nonlinear diffusion). Hence, a similar coarsening behavior as in phase-change models happens, which we analyze and numerically compute based on appropriate gradient flow structures.
Luis Caffarelli (University of Texas at Austin)
"Integral Minimal Surfaces"
I will talk about Integral Minimal Surfaces. We will discuss movement by "integral curvature" and integral minimal surfaces. This type of problems occur when long range interactions determine the local diffusion speed along a surface.
Nicolas Dirr (University of Bath)
"Interfaces in Heterogeneous Media"
We consider energies which consist of an interfacial energy and a heterogeneous (periodic or random) "bulk" or "volume" term, with heterogeneities that change on a "small" scale. This competition of the interfacial energy and the heterogenities creates a complex energy landscape. We present some results for the qualitative behavior on large scales, both for the energies and for the associated gradient flows.
Ute Ebert (Centrum voor Wiskunde en Informatica)
"Streamer Ionization Fronts As a Moving Boundary Problem"
Streamer discharges are the earliest stages of sparks and lightning; they are growing cold plasma channels that propagate due to the nonlinear coupling of electron drift, impact ionization reaction and space charge effects, which leads to a strong field enhancement at the propagating tips of the streamer fingers. Tens of kilometers large sprite discharges above thunderclouds are driven by the same physical mechanisms. As streamers and sprites propagate with velocities of 10⁵ to 10⁷ m/s, experiments and observations only recently allow to characterize them properly.
The minimal model for a streamer discharge (e.g., in pure gases like nitrogen or argon) consists of two reaction-advection-diffusion equations for electron or ion densities, coupled to the Poisson equation of electrostatics. Numerical solutions show the formation of a thin space charge layer around the ionized streamer interior. A moving boundary approximation for this thin layer is of viscous fingering type. As a regularization of the boundary dynamics, we have suggested and approximately derived a mixed Dirichlet-Neumann boundary condition, similar to kinetic undercooling.
We have thoroughly analyzed the stability of convecting circles in this boundary problem, as these are the simplest uniformly translating solution. For a particular ratio ε of regularization length over radius, the linear parturbation theory about the convecting circle can be solved exactly, and universal asymptotes for time to ∞ as well as the discrete spectrum can be determined. The extension of linear perturbation theory to arbitrary ε> 0, nonlinear solutions as well as the dynamics of finger solutions will be treated by Fabian Brau in his talk.
The moving boundary analysis is joint work with Fabian Brau, Chiu-Yen Kao, Bernard Meulenbroek, Lothar Schaer and Saleh Tanveer.
Antonio Fasano (Universitá degli Studi di Firenze)
"Recent Trends in Modelling Cancer Invasion"
The literature on cancer modelling has increased at an impressive pace during the last decade, addressing a number of different topics. Several mathematical models have been formulated for tumors with a prescribed geometry (namely the multicellular spheroids and the so-called tumor cords), dealing with various aspects of their evolution (growth, metabolism, mechanical behavior, angiogenesis, treatments etc.), both in vitro and in vivo. The possibility that such simple shapes can be unstable, giving rise to more complicated structure (even of fractal type) has also been analyzed in a series of papers. Recently much attention has been devoted to the critical problem of tumor invasiveness, which in turn involves the study of cells motility (chemotaxis, haptotaxis, intra- and extra-vasation), as well as the ability of tumor cells to exert an aggressive action on the host tissue. All these topics seem to be relevant to the phenomenon of metastasis. Here we want to concentrate on the question of the acid-mediated tumor invasion. Such a process is based on the fact that the host tissue is more sensitive to pH lowering than the tumor cells. Thus the tumor invasion can be favored by the production of an acidic environment. The latter is a consequence of the products of a metabolism shifted towards glycolysis, which is typical of hypoxic tumors. Two classes of papers will be considered. The first class studies the progression of the tumor by assuming some simple law of production of an excess of H₊-ions and contains papers dealing with one-dimensional traveling waves or with spherical tumors (vascular or not). The second class is concerned more specifically with the metabolism of spheroids responsible for the pH lowering. The two classes present interesting features and are in a sense complementary, suggesting that much progress can result from combining the two approaches.
Marco Fontelos (Consejo Superior de Investigaciones Cientificas)
"Free Boundary Problems Involving Electrically"
Free boundary problems involving electrically charged fluids and the presence of surface tension forces are attracting the attention of broad communities of physicists and engineers. The main reason is the possibility of controlling the behaviour of fluids at micro and nanometer length scales by means of electric and magnetic fields with all the potential applications that should provide. Mathematically, these problems involve solving Stokes equations in the fluid domain subject to boundary conditions that balance viscous stresses with surface tension forces and electrostatic repulsion of charges. If the fluid is partly in contact with a solid, then no-slip boundary conditions are imposed at the solid-fluid interface. The presence of repulsive, and hence destabilizing, forces at the interface that oppose the stabilizing surface tension forces gives rise to various interesting phenomena. These include: 1) the existence of nonspherical equilibrium configurations for levitating drops that can be characterized as symmetry-breaking bifurcations of spherical configurations, 2) the existence of instabilities of the interface leading to the formation of finite-time geometrical singularities in the form of cones and emission of jets, 3) the phenomenon of electrowetting, consisting in the control of the wetting properties of fluids by means of electric fields. Mathematically, problem 1) is addressed via Crandall-Rabinowitz's theorems suitably adapted to the study of free boundary problems, problem 2) is studied via boundary integral formulations and local analysis near the singularities and problem 3) is essentially variational. We will review recent contributions as well as some open problems.
Sam Howison (University of Oxford)
"From Splish to Splash: Aspects of Water Impact Problems"
I shall describe asymptotic approaches to a range of "violent impact" problems involving collisions between liquid and solid or liquid bodies. Applications include impact of a blunt body on water, early stages of droplet collison, and high Froude number shallow-water flows on a sloping base. Mathematical connections range from variational inequalities to delta shocks.
Bei Hu (University of Notre Dame)
"PDE free boundary problems in tumor models"
We shall discuss the recent progress (joint work with Avner Friedman) on the PDE tumor models, the linear stability of the tumor, the nonlinear stability of the tumor, and the comparison of the stability between different models. These comparisons have implications on the physical problems.
Arshak Petrosyan (Purdue University)
"Monotonicity Formulas and the Singular Set in the Thin Obstacle Problem"
We construct two new one-parameter families of monotonicity formulas to study the free boundary points in the lower dimensional obstacle problem. The first one is a family of Weiss type formulas geared for points of any given homogeneity and the second one is a family of Monneau type formulas suited for the study of singular points. We show the uniqueness and continuous dependence of the blowups at singular points of given homogeneity. This allows to prove a structural theorem for the singular set.
Our approach works both for zero and smooth non-zero lower dimensional obstacles. The study in the latter case is based on a generalization of Almgren's frequency formula, first established by Caffarelli, Salsa, and Silvestre.
This is a joint work with Nicola Garofalo.
Mario Primicerio (Universitá degli Studi di Firenze)
"A Non-Standard Free Boundary Problem in Pasta Cooking"
The talk will describe a work in progress in which a free boundary problem for the diffusion equation is studied modelling the cooking of pasta. Water infiltration and gelatinization seem to be the most relevant processes occurring during the cooking; their interplay originates a problem which is non-standard and presents some interest in itself.
José Francisco Rodrigues (Universidade de Lisboa)
"Remarks on a Class of Two Phase Free Boundary Problems"
We present some properties of the solution to a class of free boundary problems with two phases. Under a natural nondegeneracy assumption on the interface, for which a sufficient conditon is given, we prove a continuous dependence result for the characteristic functions of each phase and we establish sharp estimates on the variation of its Lebesgues measure with respect to the L¹-variation of the data, in a rather general framework.
Jean-Michel Roquejoffre (Mathématiques pour l'Industrie et la Physique)
"Free Boundary Problems for the Fractional Laplacian"
We discuss here local properties - optimal regularity, nondegeneracy, smoothness - of a free boundary problem involving the fractional Laplacian, generalising the classical phase transition problem for the standard Laplacian with gradient jump. Our equations are relevant models for boundary reactions, but also to reaction-diffusion processes involving non-Gaussian diffusion.
The nonlocality of the fractional laplacian renders the problem nontrivial, and the key tool is the Caffarelli-Silvestre extension formula, which transforms the model into a codimension 2 free boundary problem.
Joint work with L. Caffarelli and Y. Sire.
Michael S. Siegel (New Jersey Institute of Technology)
"Complex Singularities in Interfacial Fluid Flow and 3D Incompressible Euler Equations"
One of the most interesting phenomena in free surface flows is singularity formation on the interface. Examples include topological singularities, such as pinch-off of a liquid thread, or the formation of a curvature singularity on an evolving vortex sheet. In the first part of this talk, we describe how singularity formation can be understood by analytically extending the variable which parameterizes the interface and analyzing the singularity motion in the complex plane. As an example, we discuss the recent construction of singular solutions for two-phase flow in porous medium or Hele-Shaw cell. In the second part, we describe a new approach for the construction of complex singular solutions to the 3D Euler equations that is motivated by the results for interfaces.
Ulisse Stefanelli (IMATI, Pavia)
"The Energy + Dissipation Functional"
The evolution of dissipative systems is governed by the interplay between energy and dissipation. Roughly speaking, evolution is driven by energy whereas dissipation sets the reference metric landscape (in other words, energy decreases along dissipation geodesics). Although energy and dissipation play such substantially different roles in the evolution, they get summed together and minimized in the standard implicit Euler time-discrete scheme.
The purpose of this talk is to report on a recent progress in the variational treatment of the dissipative evolution which basically consists in a time-continuous version of this fact. In particular, I will focus on a class of global-in-time functionals recently proposed by Mielke & Ortiz where the (weighed) sum energy+dissipation comes directly into play. The interest in this perspective is that of possibly applying the tools of the Calculus of Variations (e.g., Γ-convergence, relaxation, approximation) to evolution. I will review some flow and the rate-independent case.
This is a joint project with A. Mielke (Berlin).
Jean-Marc Vanden Broeck (University of East Anglia)
"Some New Three Dimensional Free Surface Flows"
Over the last 200 years many analytical and numerical results have been obtained for two dimensional free surface flows. A large part of this success is related to the fact that two dimensional free surface flows can be formulated in terms of analytic functions. Relatively few solutions have been calculated for three dimensional free surface flows. In this talk we present new three dimensional free surface flow solutions. These include gravity capillary solitary waves and flows generated by moving disturbances.
Juan Luis Vázquez (Universidad Autónoma de Madrid)
"Porous Medium Flow with Nonlocal Effects"
We study a model for flow in porous media including nonlocal diffusion effects. It is based on Darcy's law and the pressure is related to the density by an inverse fractional Laplacian operator. We prove existence of solutions that propagate with finite speed. The model has the very interesting property that mass preserving selfsimilar solutions can be found by solving an elliptic obstacle problem with fractional Laplacian for the pair pressure-density. We use entropy methods to show that the asymptotic behaviour is described after renormalization by these solutions which play the role of the Barenblatt profiles of the standard porous medium model.
Georg S. Weiss (University of Tokyo)
"A Nonlinear Frequency Formula and the Singular set of a Free Boundary Problem"
The degenerate singular set Σ_z of the Bernoulli free boundary problem is known to be a nullset with respect to the n-dimensional Lebesgue measure. Here we show that Σ_z can be decomposed into a Reifenberg flat part and a lower dimensional nullset. The key is a formula for a nonlinear mean frequency. We discuss generalizations to other equations.

Focus Sessions 2008

Communications 2008

fbp2012 - 08.03.2011