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# Book of abstracts

The book of abstracts is now online! It contains information about the event, a list of participants and all abstracts. Klick on the image for a download. A printout will be handed out in the beginning of the workshop, too.

# Abstracts

KEYNOTE SPEAKERS

Andreas Münch (U Oxford), Wednesday

Fingering Instabilities in Thin Film Dynamics.

Thin films with moving contact lines frequently form a capillary rim that are susceptible to an instability that corrugates the rim and the contact line in the spanwise direction and grows into finger-like structures. This instability is observed in gravity- or marangoni-driven liquids as well as in films that dewet from an hydrophobic surface. The investigation of this instability via thin film models involves a variety of interesting mathematical structures and techniques some of which will be presented and discussed in this talk, such as undercompressive shocks, the derivation and analysis of sharp interface models for the capillary rims.

Dorothee Knees (WIAS Berlin), Thursday

Weak solutions for rate-independent systems illustrated at an example for crack propagation.

Many processes in nature can be considered as rate-independent processes. Typical examples are the evolution of damage and fracture in brittle materials or elastoplasticity.

In the corresponding models it is assumed that the actual state of the material can be completely characterized by the displacement field and a collection of internal variables that characterize for instance the damage state or the plastic deformation. The model describing the evolution of these quantities with respect to external loadings consist of a (quasistatic) balance of forces resulting in a system of linear elliptic equations that is coupled with an evolutionary variational inequality describing the changes of the internal variables. This coupled system can be derived from an energy functional and a dissipation potential, which, due to rate-independence, is positively homogeneous of degree one.

Since in many applications the energy functional is not simultaneously convex with respect to the displacements and the internal variables, solutions of the evolution system might be discontinuous in time. Hence, weak formulations and suitable jump criteria are needed that lead to physically reasonable solutions.

In this lecture we discuss different possibilities to define weak solutions and illustrate the predictions with a model describing the propagation of a single crack in a brittle elastic material.

Martin Burger (Münster U), Friday

Mathematics of Collective Behaviour - Complex Patterns and Minimal Models.

The mathematical modelling of collective behaviour of organisms is receiving increasing attention in the recent years. Examples are the formation of consenus in human societies, herding of sheep and of market traders, insect swarming, bird flocking, or collective cell migration and chemotaxis. Although those processes go on on different scales and produce a variety of complex patterns, they share a surprisingly similar structure.

This talk will provide an overview of the modelling from a mathematician's viewpoint and demonstrate that (besides pre-existing patterns in the environment) the main source of pattern formation is an interaction of two effects, namely long-range attraction and short-range repulsion. In macroscopic models based on partial differential equations, this typically means a combination of a (nonlinear) diffusion term with an integral operator.

In order to understand this interplay minimal mathematical models will be derived and analyzed, which contribute strongly to the understanding of basic mechanisms of pattern formation and phase transitions with respect to system parameters as found in real data and detailed microscopic simulations.

INVITED TALKS

Tobias Ahnert (TU Berlin)

Multiphase model for dilute and non-dilute flow of suspensions.

The ensemble average process originally proposed by Drew and Passman is used to derive a multiphase model for non-Brownian suspensions. The model allows for the simulation of highly dilute up to near solid suspensions. In the limiting regimes of pure fluid and pure solid, the model becomes the Navier-Stokes equations with Newtonian stress tensor and Darcy's law, respectively. We will reduce the model for the two-dimensional shear and tube flow in order to get analytic results. Further, numeric solutions of the full 2D and 3D problem are discussed.

Marion Dziwnik (TU Berlin)

Stability analysis with non-constant base states

In this talk we consider linear stability of growing rims that appear in thin solid and liquid films as they retract from a solid substrate. The mathematical models in both cases lead to mass conserving free boundary problems for degenerate and non-degenerate thin film equations whose common feature is the presence of non-constant base states in the stability analysis. Hence the base state cannot be expressed by a simple traveling wave or self-similar form which impedes the straightfoward use of normal modes. However, large rims evolve on a slower time scale than the typical perturbations and we systematically exploit this time scale separation to derive asymptotic approximations for the base states and for solutions of the linear stability problem. Our results enable us to track the evolution of the dominant wavelength for long times and to assess the validity of the asymptotic approximations.

Matt Hennessy (University of Oxford)

Bilayer formation and topological transitions in a diffusive phasefield model

In this talk we consider the dynamics of a binary mixture that is confined between two walls of infinite horizontal extent. The evolution of the mixture is described by a purely diffusive phase-field model and we account for the energetic interactions between the walls and the constituent materials. That is, we assume both walls have a slight preference for one, but not the same, material. We show that by slowly cooling the system the onset of spinodal decomposition can be suppressed and the bias of the walls will induce the formation of a bilayer, where each layer is of nearly constant composition. The two layers are separated by a thin interfacial region where the composition changes rapidly. The long length of the bilayer "interface" suggests that this is not the lowest-energy state, and we show that locally perturbing the system can initiate a cascade of rupture events resulting in a topological transition from the bilayer state to a columnar state. The dynamics of the transition are investigated by reducing the phase-field model to a sharp-interface model which can be further simplified to a thin-film equation. This is joint work with A. Muench, B. Wagner, V. Burlakov, and A. Goriely.

Sebastian Jachalski (WIAS Berlin)

Structure formation in thin liquid bilayers

In nature we often encounter situations where two immiscible fluids move on each other, for instance the different layers in human tear films or simply droplets of fat in a chicken soup. Also for technical applications liquid bilayers play a role. Here, for example, the dewetting of a nanoscopic polymer film from a liquid substrate is an interesting topic. In this talk we consider liquid bilayers with certain geometric properties. In particular, we assume that the height of these bilayers is small compared to the characteristic length scale of evolving patterns which, for example, is true for nanoscopic polymer films. We derive systems of thin film equations for the evolution of such bilayers and discuss the dependence of stability of these systems subject to interfacial slip and intermolecular forces. Furthermore, we study stationary solutions for these thin film models and compare our results with experimental data.

Georgy Kitavtsev (MPI Leipzig)

Coarsening Rates for the Dynamics of Interacting Slipping Droplets

Reduced ODE models describing coarsening dynamics of droplets in nanometric polymer film interacting on solid substrate in the presence of large slippage at the liquid/solid interface are derived from one-dimensional lubrication equations. In the limiting case of the infinite slip length corresponding to the free suspended films a collision/absorption model then arises and is solved explicitly. The exact collision law is derived. Existence of a threshold at which the collision rates switch from algebraic to exponential ones is shown.

On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime

Around forty years ago, mixed finite element methods for the incompressible Navier-Stokes equations were introduced and are nowadays regarded as the standard discretization approach for incompressible flows in numerical mathematics. The success of mixed methods relies mainly on the relaxation of the divergence constraint. But only recently a systematic research was started, in order to understand what kind of disadvantages in practice and theory can arise by this relaxation. As a result it has turned out that mixed methods suffer from a lack of robustness, whenever large irrotational forces in the momentum equations of the Navier-Stokes equations have to balanced. In such cases, large spurious oscillations in the discrete velocities can spoil the discrete solution completely. The reason for this unpleasant behavior can be understood by investigating the Helmholtz decomposition of the discrete momentum equations. Irrotational and divergence-free forces should each build a separate force balance, but in classical mixed methods they interact in a non-physical way. Besides analysing the stability properties of classical mixed methods, a remedy is proposed which is based on the new theoretical insight, allowing the construction of new classes of cheap, divergence-free flow solvers in 2D and 3D.

Marco Morandotti (Inst. Sup. Tec., Lisbon)

Dynamics for a system of screw dislocations

We describe the dynamics for a system of screw dislocations subject to anti-plane shear. Variational techniques allow us to find minimizers for the energy functional associated with the system of dislocations in an elastic medium. Building on a model due to Cermelli and Gurtin, a weak notion of solutions (in the sense of Filippov) to ordinary differential equations is used to solve the dynamics problem. Some examples of interesting scenarios complement the presentation.

Michael Köpf (École Normale Supérieure, Paris, France)

A continuum model of wound healing in epithelial tissues

During unconstrained spreading as occurs for example in wound healing, epithelial cell monolayers can be described as a polarizable and chemo-mechanically interacting layer with weakly nonlinear elasticity. Our model reproduces the spontaneous and self-organized formation of finger-like protrusions due to the collective action of a large number of cells, that is commonly observed in experiments. Statistics of the velocity orientation obtained from numerical simulation show strong alignment in the fingers opposed to an isotropic distribution in the bulk, in good agreement with the measurements by Reffay et al. (Reffay et al., Biophysical Journal, 2011). The results faithfully reproduce faster relative advance of cells close to the leading edge of the tissue, as well as spatial velocity correlations and stress accumulation within the tissue, which proceeds in form of a "mechanical wave", travelling from the wound edge inwards (cf. Serra-Picamal et al., Nature Physics, 2012).

M. H. Koepf, L. M. Pismen: Non-equilibrium patterns in polarizable active layers, Physica D 259 (2013) 48-54

M. H. Koepf, L. M. Pismen: A continuum model of epithelial spreading, Soft Matter 9 (2013) 3727-3734

Dirk Peschka (WIAS Berlin)

Treatment of triple junctions in thin-film models

joint work with R. Huth, S. Jachalski and G. Kitavtsev

Stokes free boundary problems have a gradient flow structure which determines conditions at boundaries and in particular at triple junctions (liquid/substrate/gas). The same structure carries over to corresponding dimensionally reduced thin-film models. In order to devise a numerical algorithm we discuss the weak-form of the gradient flow and discuss different ways to treat the motion of compactly supported liquid films with nonzero contact angle, e.g. droplets. We indicate how this method can be generalised to multi-layered systems of thin-films and demonstrate the feasibility of the approach by presenting a few numerical solutions in 2D and 3D.

Filip Rindler (U Warwick)

Orientation-preserving Young measures

Young measures are widely used in the Calculus of Variations to model limits of nonlinear functions of weakly converging "generating" sequences. Often, one is interested in special classes of Young measures that are generated by constrained sequences (e.g. gradients or divergence-free vector fields). After introducing Young measures, I will outline the proof of a recent characterization result (in the spirit of the Kinderlehrer--Pedregal Theorem) for Young measures generated by gradients that have positive Jacobian almost everywhere. The argument to construct generating sequences from such Young measures satisfying the orientation-preserving constraint is based on a variant of convex integration in conjunction with an explicit lamination construction in matrix space. The resulting generating sequence is bounded in $\Lrm^p$ for $p$ less than the space dimension. Applications to relaxation of integral functionals, the theory of semiconvex hulls, and approximation of weakly orientation-preserving maps by strictly orientation-preserving ones in Sobolev spaces are given.

Hao Wu (Fudan U)

Well-posedness and stability of the full Ericksen-Leslie system for incompressible nematic liquid crystal flow

We will discuss the Ericksen-Leslie (E-L) system modeling the incompressible nematic liquid crystal flow. We prove the well-posedness and long-time behavior of the E-L system under proper assumptions on the viscous Leslie coefficients. We also discuss the connection between Parodi's relation and stability of the E-L system.

CONTRIBUTED TALKS

Vladimir Bobkov (Ufa Science Center, RAS)

On the behaviour of branch of gound state solutions to the system of elliptic equations

We consider the behaviour of solutions to the system of elliptic equations with indefinite nonlinearity depending on two spectral parameters. We introduce a curve of critical spectral points, which is the threshold of applicability of the Nehari manifolds and fibering methods. A branch of ground state is obtained and its asymptotic behavior including the blow-up phenomenon on the critical curve is studied. The differences in the behavior of similar branches of solutions for the prototype scalar equations are discussed

Karoline Disser (WIAS Berlin)

On gradient structures for Markov chains and the passage to Wasserstein gradient flows

We prove the convergence of the gradient structure for a spatially discrete reversible Markov chain introduced in [3] and [2] to the Wasserstein gradient structure of the linear Fokker-Planck equation, complementing and partially extending the results in [1].

References

[1] Nicola Gigli and Jan Maas, Gromov-Hausdorff convergence of discrete transportation metrics, SIAM J. Math. Anal. 45 (2013), no. 2, 879--899.

[2] Jan Maas, Gradient flows of the entropy for finite Markov chains, J. Funct.Anal. 261 (2011), no. 8, 2250--2292.

[3] Alexander Mielke, Geodesic convexity of the relative entropy in reversible Markov chains, Calc. Var. Partial Differential Equations 48 (2013), no. 1-2,1--31. MR 3090532

Svetlana Gurevich (Münster U)

Delayed feedback control of localized structures in reaction-diffusion systems

We are interested in stability properties of a single localized structure in a three-component reaction-diffusion system subjected to the time-delayed feedback. We shall show that variation in the product of the delay time and the feedback strength leads to complex dynamical behavior of the system, including formation of target patterns, spontaneous motion, and spontaneous breathing as well as various complex structures, arising from combination of different oscillatory instabilities. In the case of spontaneous motion, we provide a bifurcation analysis of the delayed system and derive an order parameter equation for the position of the localized structure, explicitly describing its temporal evolution in the vicinity of the bifurcation point. This equation is a subject to a nonlinear delay differential equation, which can be transformed to the normal form of the pitchfork drift bifurcation.

Patrick van Meurs (Eindhoven U Tech.)

Collective Behaviour of Walls of Dislocations

We address a scientific challenge occurring in many complex systems: predicting the evolution of the collective behaviour of particles interacting on the micro scale. More specifically, we examine dislocation networks, which have plastic deformation of metals as the emergent property. There are many dislocations in metals, and every dislocation interacts with all the others in a non-local way. The main challenge is to do upscaling (the number of dislocations goes to infinity). I will present to you how we did this in a variational framework for the static case, which is the first step towards treating dynamics.

Elias Pipping (FU Berlin)

Variational methods for rate- and state-dependent friction problems.

In geophysics, there is a need for friction laws with the property that an increase in the sliding velocity $V$ yields a decrease in the friction coefficient $\mu$, i.e. a decrease in frictional resistance. Unfortunately, a straightforward formulation, i.e. $\mu = \mu(V)$ leads to physical absurdities and mathematical issues (both on the computational and the analytical level, because of a lack of monotonicity). This model is referred to as (slip) ratedependent friction. A more complicated, experiment-based model, in which an increase in velocity first leads to an increase in velocity, and then (after a finite amount of time) an overall decrease, does not have these shortcomings. The time-delay effect is incorporated into the formulation through a state variable $\theta$, yielding $\mu = \mu(V, \theta)$; hence the name (slip) rate- and state-dependent friction.

In this presentation I will talk about how variational methods help us in solving problems of rate- and state-dependent friction and where the remaining challenges are.

Sina Reichelt (WIAS Berlin)

Two-scale homogenization in nonlinear reaction-diffusion systems with small diffusion

Alexander Mielke, Sina Reichelt*, and Marita Thomas

Many reaction-diffusion processes arising in civil engineering, biology, or chemistry take place in media with underlying microstructure, for instance concrete carbonation or the spread-out of substances in biological tissues. Such microstructures are assumed to be periodically distributed with period length $\varepsilon > 0$, which is much smaller compared to the overall size of the domain. Systems with such differences in the involved length scales are very difficult to handle numerically, because the step size of the algorithm has to be of order $\varepsilon$ in order to resolve the qualitative behavior of the system. Therefore it is the aim to derive effective equations, independent of $\varepsilon$, which are ideally simpler and qualitatively describe the properties of the original system. In the classical case, the coffecients in the effective equation are homogeneous and in this sense, the passage $\varepsilon \to 0$ is called homogenization.

In my talk I deal with a system of two coupled reaction-diffusion equations, where one species diffuses much slower than the other one and the coupling arises via nonlinear reaction terms. Using the method of two-scale convergence, I derive effective equations which are defined on a two-scale space. The two-scale space consists of the macroscopic domain and the microscopic unit cell attached to each point of the macroscopic domain.

Fabian Spill (Oxford U)

Modelling of Angiogenesis

Angiogenesis is the formation of new blood vessels from existing ones. It is a crucial process occurring in wound healing or in solid tumours. As the tumour needs to induce angiogenesis to be able to grow beyond a limited size, much medical research has been directed towards targeting angiogenesis. In this talk, I will review different approaches to the modelling of angiogenesis, such as PDE models which describe the global evolution of observable cell densities, stochastic models describing individual cells, or multiscale models.

Giles Shaw  (Cambridge U)

Continuous Extension of Integral Functionals over W^{1,1}.

In this talk, I will identify the area-strictly continuous extension of integral functionals defined on W^{1,1} to the space BV. Here, the integrand of the functional depends on the spatial domain as well as u(x) and \nabla u(x). This result is complementary to a work of Amar, De Ciccio and Fusco [ESAIM Control Optim. Calc. Var. 13 (2007), no. 2, pp. 396–412], but is valid for vector valued functions u(x) as well as just the scalar case. The main point is that, when a jump point of a BV function is encountered, the integrand must be averaged across all values between the upper and lower limits at the jump point.

Miguel Yangari (U Toulouse)

Fast propagation for fractional kpp equations with slowly decaying initial conditions.

In this talk, I present a study about the large-time behavior of solutions of one-dimensional fractional Fisher-KPP reaction-diffusion equations, when the initial condition is asymptotically frontlike and it decays at infinity more slowly than a power x^b, where b < 2α and α∈(0,1) is the order of the fractional Laplacian. We prove that the level sets of the solutions move exponentially fast as time goes to infinity. Moreover, a quantitative estimate of motion of the level sets is obtained in terms of the decay of the initial condition.

Markus Wilczek (Münster U)

Synchronization effects in Cahn-Hilliard models for Langmuir-Blodgett transfer

The pattern formation in Langmuir-Blodgett transfer experiments is theoretically studied using a generalized Cahn-Hilliard model. The influence of prestructured substrates on the patterning process is investigated in one and two dimensions. We find that the occurring synchronization effects enable a control mechanism via properties of the prestructure and facilitate the production of patterns with a broader range of features. In two dimensions, the production of a variety of complex patterns can be achieved through the competition of intrinsic properties of the pattern forming system and the external forcing introduced by a prestructure.

Martijn Zaal (Bonn U)

Variational modeling of osmotic cell swelling

A variational formulation of a class of parabolic free boundary problems is given. An example of such a problem is the swelling of a single cell due to osmosis. The variational formulation is used to show global existence of solutions.

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