(UTIA AV CR + FSv CVUT Praha):
Boundary effects and weak lower semicontinuity for signed integral functionals on BV
Abstract: We characterize lower semicontinuity of integral functionals with respect to weak convergence in BV, including integrands whose negative part has linear growth. In addition, we allow for sequences without a fixed trace at the boundary. In this case, both the integrand and the shape of the boundary play a key role. This is made precise in our newly found condition, a quasi-sublinear growth from below at points of the boundary, which compensates for possible concentration effects generated by the sequence. Some applications to relaxation of variational problems with linear growth will be outlined. It is a joint work with B. Benesová (Wurzburg) and S. Kromer (Cologne).
(KM, FJFI CVUT Praha):
Modeling of moving curves by curvature driven flow and its application in discrete dislocation dynamics
Abstract: We investigate the numerical solution of the evolution law for the mean curvature flow of open or closed non-self-intersecting curves in a plane. The model schematically reads as normal velocity = (mean) curvature + force. We treat the motion law by means of the parametric method, resulting into a system of degenerate parabolic partial differential equations for the curve parametrization. Unlike other interface-capturing methods, such as level-set method or phase-field method, the parametric approach allows us to treat the dynamics of open curves comfortably. The parametric equations are spatially discretized by means of the flowing finite volume method. To improve the quality of the numerical solution, we discuss the effect of artificial designed tangential terms redistributing the discretization points. In the second part of the lecture, we present the application of the curvature driven flow to a microscale modeling of basic mechanisms in discrete dislocation dynamics.
Prof. Victor A. Kovtunenko
(Inst. for Math. and Sci. Comp., Univ. of Graz, Austria, and Lavrent ev Inst. of Hydrodynamics, Novosibirsk, Russia):
On generalized Poisson-Nernst-Planck equations
Abstract: A strongly nonlinear system of Poisson-Nernst-Planck equations is considered. The diffusion laws are coupled with the Landau grand potential for entropy variables. The model describes electro-kinetic phenomena on multiphase medium in physical, chemical, and biological sciences. The generalized model is supplemented by the mass balance, positivity and volume constraints, quasi-Fermi statistics depending on the pressure, and inhomogeneous Robin boundary conditions representing interfacial reactions . In our research we aim at proper variational modelling, wellposedness properties and dynamic stability, as well as homogenization of the problem supported by rigorous estimate of the energy and entropy types.
Dr. Koya Sakakibara
(University of Tokyo):
Structure-preserving numerical scheme for the one-phase Hele-Shaw problems by the method of fundamental solutions combined with the uniform distribution method
Abstract: In this talk, the one-phase Hele-Shaw problems and their numerical scheme are considered. The one-phase interior Hele-Shaw problem has curve-shortening (CS), area-preserving (AP), and barycenter-fixed (BF) properties. We construct numerical scheme which satisfies the above properties in a discrete sense. As a result, computing the normal velocities by the method of fundamental solutions and the tangential velocities by the uniform distribution method, a discrete version of CS-, AP-, and BF-properties are satisfied. The one-phase exterior Hele-Shaw problem and one-phase interior Hele-Shaw problem with sink/source points can also be treated. We also show some numerical results which exemplify the effectiveness of our scheme.
(Inst. of Thermomechanics, Czech Acad. Sci.):
Constitutive model of NiTi SMA polycrystals: from experiments to simulations
Abstract: Mechanical response of polycrystals of NiTi shape memory alloys (SMA) exhibits several interesting features, e.g. strong dependence on temperature, loading mode or loading history. These effects result from interplay of deformation mechanisms of various origin with the dominant influence of martensitic phase transformation. In the talk, I will introduce a constitutive model of textured NiTi SMA which allows for a realistic description of the mechanical response under various loading conditions. Particular attention has been paid to the description of martensite reorientation, occurrence of intermediate phase and the localization effect. Simulations demonstrating capabilities of the model both at the macro- and meso- scale will be presented and compared to experimental data.
(Abt. f. angew. Math., Albert-Ludwigs-Universitat Freiburg):
Energy estimates, relaxation, and existence for strain gradient plasticity with cross hardening
Abstract: We consider a variational formulation of gradient elasto-plasticity subject to a class of single-slip side conditions. Such side conditions typically render the associated boundary-value problems non-convex. We first show that, for a large class of plastic deformations, a given single-slip condition (specification of Burgers vectors and slip planes) can be relaxed by introducing a microstructure through a two-stage process of mollification and lamination. This yields a relaxed side condition which only prescribes slip planes and allows for arbitrary slip directions. This relaxed model can be thought of as an aid to simulating macroscopic plastic behavior without the need to resolve arbitrarily fine spatial scales. We then discuss issues of existence of solutions for the relaxed model. Finally, we apply this relaxed model to a specific system, in order to be able to compare the analytical results with experiments. A rectangular shear sample is clamped at each end, and is subjected to a prescribed horizontal, modelled by an appropriate Dirichlet condition. We ask: how much energy is required to impose such a shear, and how does the energy depend on the aspect ratio of the sample? Assuming that just two slip systems are active, we show that there is a critical aspect ratio, above which the energy is strictly positive, and below which it is zero. Furthermore, in the respective regimes determined by the aspect ratio, we prove energy scaling bounds, expressed in terms of the amount of prescribed shear.
(Faculty of Mathematics, University of Vienna, Austria):
Wulff Shape Emergence in Graphene
Abstract: In this talk the problem of understanding why particles self-assemble in macroscopic clusters with overall polyhedral shape is investigated. At low temperature ground states for a general finite number n of particles of suitable phenomenological energies possibly accounting for two- and three-body atomic interactions are shown to be connected subsets of regular lattices L, such as the triangular and the hexagonal lattice. The hexagonal lattice well represents the arrangement of carbon atoms in the graphene layers.

By means of a characterization of minimal configurations via a discrete isoperimetric inequality, ground states will be seen to converge to the hexagonal Wulff shape as the number n of particles tends to infinity. Furthermore, ground states are shown to be given by hexagonal configurations with some extra particles at their boundary, and the n3/4 scaling law for the deviation of ground states from their corresponding hexagonal configurations is shown to hold. Precisely, the number of extra particles is carefully estimated to be at most KL n3/4 + o(n3/4 ), where both the rate n3/4 and the explicitly determined constant KL are proven to be sharp.

The new designed method allows to sharpen previous results [1, 5] for the triangular setting [2] and allows to provide a first analytical evidence of the zigzag-edge selectivity and the emergence of the asymptotic Wulff shape for the hexagonal setting [3] in accor- dance with what is experimentally observed in the growth of graphene flakes [4]. Results presented are in collaboration with Elisa Davoli and Ulisse Stefanelli (Vienna).

[1] Y. Au Yeung, G. Friesecke, and B. Schmidt, Minimizing atomic configurations of short range pair potentials in two dimensions: crystallization in the Wulff-shape, Calc. Var. Partial Differential Equations 44 (2012), 81-100.
[2] E. Davoli, P. Piovano, and U. Stefanelli, Sharp n3/4 law for the minimizers of the edge-isoperimetric problem on the triangular lattice, Submitted (2015).
[3] E. Davoli, P. Piovano, U. Stefanelli, Wulff shape emergence in graphene, Submitted (2015).
[4] Z. Luo, S. Kim, N. Kawamoto, A.M. Rappe, and A.T. Charlie Johnson, Growth mechanism of hexagonal-shape graphene flakes with zigzag edges, ACSNano 11 (2011), 1954-1960.
[5] B. Schmidt, Ground states of the 2D sticky disc model: fine properties and N 3/4 law for the deviation from the asymptotic Wulff-shape, J. Stat. Phys. 153 (2013), 727-738.
Dr. Diego Grandi
(University of Vienna):
Modeling shape memory alloys at finite strains: solvability and linearization
Abstract: We discuss the macroscale modeling of shape memory alloys according to a finite strain-version of the Souza-Auricchio model. Assuming the isotropy of the hyperelastic stored energy functional, a convenient formulation in terms of the Green-St-Venant transformation strain tensor can be established. For the chosen rate-independent constitutive relation, coupled to a quasi-static elastic response and with an additional regularizing interface-energy contribution, the global existence of energetic solutions to the boundary value problem (i.e. variational evolution) is proven.

A similar finite-strain approach is applied to provide a model for the magneto-elastic evolution in magnetic shape-memory materials.

The possibility of a rigorous linearization limit of the models at small strain is addressed within the framework of the evolutive variational convergence for rate independent processes.
(University of Oxford):
Finite element approximation of non-divergence form PDEs
Abstract: Non-divergence form partial differential equations with discontinuous coefficients do not generally possess a weak formulation, thus presenting an obstacle to their numerical solution by classical finite element methods. Such equations arise in many applications from areas such as probability and stochastic processes. These equations also arise as linearizations to fully nonlinear PDEs, as obtained for instance from the use of iterative solution algorithms. In such cases, it can rarely be expected that the coefficients of the operator be smooth or even continuous. For example, in applications to Hamilton-Jacobi-Bellman equations, the coefficients will usually be merely essentially bounded. In contrast to the study of divergence form equations, it is usually not possible to define a notion of weak solution when the coefficients are non-smooth. In the case of continuous but possibly non-differentiable coefficients, the Calderon-Zygmund theory of strong solutions establishes the well-posedness of the problem in sufficiently smooth domains. However, without additional hypotheses, well-posedness is generally lost in the case of discontinuous coefficients. The aim of the lecture is to survey recent developments concerning the numerical approximation of such problems by finite element methods. The lecture is based on joint work with Iain Smears (INRIA Paris).
(University of Oxford):
Numerical analysis of nonlocal Cahn-Hilliard equations
Abstract: We discuss the numerical approximation of a class of nonlinear evolution problems that arise as L2 and H-1 gradient flows for the Modica-Mortola regularization of a certain functional on BV involving the interfacial energy per unit length or unit area, the flat torus in Rd , and a nonnegative Fourier multiplier, that is continuous and symmetric, and which decays to zero at infinity. Such functionals feature in mathematical models of pattern-formation in micromagnetics and models of diblock copolymers. The resulting evolution equation is discretized by a Fourier spectral method with respect to the spatial variables and a Crank-Nicolson or an implicit midpoint scheme with respect to the temporal variable. We investigate the stability and convergence properties of the proposed numerical schemes and illustrate the theoretical results by numerical simulations. The lecture is based on joint work with Christof Melcher (RWTH Aachen), Barbora Benesova (University of Wurzburg) and Nicolas Condette (Humboldt University, Berlin).
(Inst. f. Mathematik, Technische Univ. Berlin):
Optimal control of some reaction-diffusion equations
Abstract: Results on the optimal control of the Nagumo equation and the FitzHugh-Nagumo system are surveyed. Special emphasis is laid on first- and second-order optimality conditions for sparse optimal controls. The second-order analysis is applied to explain observed numerical stability with respect to certain perturbations. The theory is illustrated by various numerical examples.
(Inst. f. Mathematik, Univ. Würzburg):
Existence of weak solutions for a model of magnetoelasticity
Abstract: In the talk, we will present a model for particles in micromagnetic fluids. Such fluids have many technological applications. They can not only be found in medical applications, but also in loud speakers and shock absorbers.

We investigate micromagnetic material in the framework of complex fluids. The system of PDEs to model the flow of the material is derived in a continuum mechanical setting. We outline the process of modeling. Moreover, we highlight the coupling between the elastic and the magnetic properties of the material.

Restricting our scope to the two dimensional setting, we then prove existence of solutions under the assumption of small initial data.

This is joint work with Carlos García-Cervera (Mathematics Department, University of California, Santa Barbara, USA), Johannes Forster, Anja Schlömerkemper (Institute for Mathematics, University of Würzburg, Germany), and Chun Liu (Department of Mathematics, Penn State University, University Park, USA).
(Katedra matematiky, FJFI CVUT):
On modelling self-organisation in real systems
Abstract: Self-organisation in nature is widely recognised and is extensively modelled. In some systems (e.g. Drosophila embryo) the spatial pattern is not self-orchestrated. On the other hand, Turing model of pattern formation is capable of breaking symmetry without pre-existing positional information. This mechanism has driven numerous experimental studies even in the context of developmental systems which suggest that Turing-like morphogen interactions and patterns can occur in such scenarios. However, a direct verification has remained elusive.

We start by introduction to the classical Turing instability. As the aim is to reveal mechanism behind the observed pattern in nature, robustness is required not only with respect to parameter sensitivity or the choice of initial or boundary conditions but also with respect to the model formulation itself. Only then are these models subjected to a detailed mathematical analysis. We illustrate the essence of these ideas on the reaction-diffusion-advection system, where we indicate that such a system should be preferred from both physical and mathematical viewpoint for self-organisation modelling. Particularly, we shall use the mixture theory within extended irreversible thermodynamics to reveal what evolution equations are relevant in real physical systems and can be considered as small perturbations of reaction-diffusion equations and mathematically analyse the possibility of the emergence of pattern in RDA systems. Note that it is required to identify plausible extensions of the Turing concept of self-organisation into more general cases. Such extensions are not unambiguous but their discussion is beneficial even for understanding the standart Turing model of spatial self-organisation.
Dr. Jan Haskovec
(King Abdullah Univ. of Sci. and Technology):
PDE-based modelling of biological network formation
Abstract: Motivated by recent papers describing rules for natural network formation in discrete settings, we propose an elliptic-parabolic system of partial differential equations. The model describes the pressure field due to a Darcy-type equation and the dynamics of the conductance network under pressure force effects with a diffusion rate representing randomness in the material structure. We prove the existence of global weak solutions and of local mild solutions and study their long term behavior. Moreover, we study the structure and stability properties of steady states that play a central role to understand the pattern capacity of the system. We show that patterns (network structures) occur in the regime of small material randomness. Moreover, we present results of systematic numerical simulations of the system that provide further insights into the properties of the network-type solutions.
Mgr. Michal Pavelka, PhD.
(VSCHT, Praha):
A hierarchy of Poisson brackets
Abstract: Reversible part of evolution equations is often governed by a Poisson bracket and energy. For example, Hamilton canonical equations are given by the canonical Poisson bracket on cotangent bundle of classical mechanics and by energy (a Hamiltonian). Similarly, Liouville equation is given by an induced Liouville Poisson bracket. Reversible parts of more macroscopic evolution equations, for example Boltzmann equation, Navier-Stokes equation, equations for polymeric fluids or turbulent flows, are also generated by Poisson brackets. The goal of this talk is to show how the more macroscopic Poisson brackets are derived from the Liouville Poisson bracket in a systematic way.
Dr. Stanislav Parez
(VSCHT, Praha):
Flow of granular materials: What can we say about landslides?
Abstract: Large landslides exhibit surprisingly long runout distances compared to a rigid body sliding from the same slope, and the mechanism of this phenomena has been studied for decades. Here we propose a scenario in which the observed long runouts are explained via a granular flow, including its spreading, but not including frictional weakening that has traditionally been suggested to cause long runouts. Kinematics of the granular flow is divided into center of mass motion and spreading due to flattening of the flowing mass. We solve the center of mass motion analytically based on a frictional law valid for granular flows, and find that the center of mass runout is similar to that of a rigid body. Based on the shape of deposits observed in experiments with collapsing granular columns and numerical simulations of landslides, we estimate the effect of spreading and derive a characteristic spreading length R_f ~ V^{1/3}. Spreading is shown to be an important, often dominating, contribution to the total runout distance. The combination of the predicted center of mass runout and the spreading length gives the runout distance in a very good match to natural landslides.
(Lecole Polytechnique Montreal):
Modeling of flows of complex fluids by modeling their internal structure
Abstract: Fluids with internal structure that evolves in time on a scale that is comparable with the scale on which the macroscopic flow evolves exhibit a complex flow behavior and are therefore called complex fluids. The internal structure can be flow-induced (e.g. the structure emerging in turbulent flows) or it can also be a mesoscopic or microscopic structure of the fluids at rest (e.g. structure of macromolecules in polymeric fluids or suspended particles in suspensions). I investigate consequences of the requirement that the time evolution of complex fluids is compatible with mechanics (i.e. it is a Hamiltonian time evolution) and compatible with thermodynamics (it obeys the second law of thermodynamics).
(Dept. of Civil Engr., IIT Madras):
Experimental Investigations on Asphalt Binders - What are the challenges?
Abstract: 90% of the highways and runways throughout the world use asphalt binders for road construction. Such materials are by-products of oil refinery. They are a complicated mixture of thousands of hydrocarbons and show diverse range of behavior as the temperature is varied. These material exhibit transitory behavior from a viscoelastic fluid to viscoelastic solid during service. In this talk, I present novel experimental techniques designed to elicit the rich behavior of these materials. These include large amplitude oscillatory shear, stress relaxation and creep and recovery. I also show some interesting observations related to manifestation of normal force and its relaxation. Such experimental data pose challenge while modeling as most of the existing models cannot describe such behavior.
On a robust DG method for the solution of compressible Euler and Navier-Stokes equations
Abstract: The lecture presents numerical method for the numerical solution of compressible flow, which is robust with respect to the Mach number and Reynolds numbers. It is based on the application of the discontinuous Galerkin method and allows the numerical simulation of compressible flow with low Mach numbers up to an incompressible limit and high speed flow, in general in time-dependent domains. The method is used for the solution of fluid-structure interaction problems.
(Faculty of Math., Univ. Vienna):
Dynamic perfect plasticity as convex minimization
Abstract: We present a novel approximation of solutions to the equations of dynamic linearized perfect plasticity, based on a global variational formulation of the problem by means of the Weighted-Inertia- Dissipation-Energy (WIDE) approach. Solutions to the system of dynamic Prandtl-Reuss perfect plasticity are identified as limit of minimizers of parameter-dependent energy functionals evaluated on trajectories (the WIDE functionals). Compactness is achieved by means of time- discretization, uniform energy estimate on minimizers of discretized WIDE-functionals, and passage to the limit in a parameter-dependent energy inequality. This is a joint work with Ulisse Stefanelli.
(Math. Inst., Charles Uni.):
Minicourse of Non-Equilibrium Thermodynamics. Part I.
Abstract: Non-equilibrium thermodynamics is the theory within which we should be able to derive evolution equations of any macroscopic or mesoscopic physical system. Although such a task is very important in modern physics and applied mathematics, it seems to be still far away from current state of the art. The theory is still under construction.

The goal of this minicourse is to review some fundamental concepts and results of non-equilibrium thermodynamics, which we can use in mathematical modeling, and to open discussion so that we find the most important open problems.

In the fist part, we will discuss the concepts of equilibrium, non-equilibrium, levels of description, geometrization of thermodynamics, reversibility and irreversibility.
(Math. Inst., Charles Uni.):
Minicourse of Non-Equilibrium Thermodynamics. Part II.
Abstract: In the second part we will discuss mainly the reversible part of evolution equations. We will start with Liouville equation in the Hamiltonian form and we will pass to lower levels of description as kinetic theory or fluid mechanics. Perhaps we will have time to start talking about the irreversible evolution.
(Univ. of California, Davis, & National Technical University of Athens):
Anisotropic Critical State Theory: Challenging a Paradigm in Granular Mechanics
Abstract: Consider the following thought experiment: load a granular specimen in triaxial compression till Critical State (CS) is reached, where shear deformation continues under fixed stress and zero volume change. At CS impose a rotation of stress Principal Axes (PA) keeping the stress principal values fixed. Will the sample continue being at CS or not?

The answer to this seemingly simple and of academic interest question can challenge the paradigm of Critical State Theory (CST) that defines failure and mechanical response of granular media in soil mechanics for more than half a century. The recently developed Anisotropic Critical State Theory (ACST) will be presented as a paradigm replacement for the classical CST. The main novel ingredient entering the new formulation is fabric, expressed in terms of a properly defined fabric tensor that evolves towards a unique norm CS value.

The presentation will be narrative providing stages of development and problems encountered and solved or still pending a solution. In the process constitutive models of soil plasticity will be presented within the framework of ACST, and the use of numerical/experimental techniques such as Discrete Element Method (DEM) and X-Ray tomography will be outlined.
(Math. Inst., Charles Uni.):
Minicourse of Non-Equilibrium Thermodynamics. Part III.
Abstract: We will derive the Poisson bracket of one-particle kinetic theory, classical hydrodynamics and a theory of mixtures. We will then start discussing entropy. It will be introduced as the Shannon entropy at the Liouville level of description. Entropy of ideal gas at the levels of kinetic theory, classical hydrodynamics and thermodynamic equilibrium will be derived by the principle of maximum entropy (MaxEnt).
(Univ. Heidelberg, Inst. of Mathematik):
Self-similar lifting and persistent touch-down point solutions in the thin-film equation
Abstract: In the talk I discuss the appearance of self-similar blow-up solutions for thin-film equations with different mobility exponents. This is related to non-uniqueness phenomena for weak solution of the same equation. The proof is based on dynamical systems arguments.
(Math. Inst., Charles Uni.):
Minicourse of Non-Equilibrium Thermodynamics. Part IV.
Abstract: After having introduced energy, entropy and Poisson brackets on different levels of description, we will discuss various forms of irreversible evolution, in particular gradient dynamics (generated by dissipation potentials). We will show connection to the method of entropy production maximization and implicit constitutive relations. All so far brought up topics will be summarized in the formulation of the GENERIC framework, particular realizations and applications (solid mechanics, plasticity, chemical reactions, electrochemistry, diffusion, nonlocal phenomena) of which will dominate the following seminars.
Mgr. Jan Stebel, PhD.
(T. U. Liberec):
Shape optimization for Stokes problem with stick-slip boundary conditions
Abstract: We consider the problem of finding an optimal shape of a domain occupied by a viscous fluid. A part of the boundary represents a solid wall to which the fluid may or need not adhere depending on the magnitude of the shear stress. Such model describes e.g. hydrophobic or microscale surfaces. The existence of an optimal shape is proved. Moreover, a regularized-penalized problem is formulated and it is proved that its solutions converge to the solution of the original shape optimization problem. Finally we present an approximation of the problem and numerical results. This is a joint work with J. Haslinger and R.A.E. M{ a}kinen.
(MU AV CR + ZCU Plzen):
My life with J.N.
Abstract: The talk will deal with a rather private description of the long lasting collaboration with my teacher, colleague and friend, professor Jindrich Necas, and will mention some facts about common events (like seminars and workshops), about some of his students and about his results.
(Universita di Roma `Tor Vergata):
A nonsmooth variant of the nonlinear diffusion equation

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