NEČAS SEMINAR ON CONTINUUM MECHANICS
organized by the Mathematical Institute of the Charles University
each Monday at 15:40
in the MFF UK building,
Sokolovská 83, lecture room K1, 2^{nd}
floor,
proceeds with the following presentations:

15:40  Prof. Dr. Christian Rohde (University of Stuttgart):  Modelling of LiquidVapour Flows: Diffuse Interface versus Sharp Interface and Mesoscopic versus Macroscopic Models  

17:20  Doc. RNDr. Jiri Langer, CSc. (UTF, MFF UK, Prague):  Je hospodin hospodarny? (in czech)  



15:40  Prof. Jens Frehse (Institut fur Angewandte Mathematik, University of Bonn):  Diagonal elliptic and parabolic systems with superquadratic Hamiltonians  Abstract: We consider elliptic or parabolic systems
(ut ) − ∆u = H (∇u) + f
where u is a vector function, periodic in the space variables. Under
additional special structure conditions for H (the ?Hamiltonian?) and
natural smoothness conditions on the data we prove the existence of
regular solutions ? without smallness conditions ? in the case that
H ( ∇u) grows like a polynomial in ∇u. Up to now, in the case of
systems of equations, only Hamiltonians growing at most quadratically
in ∇u have been treated. The work has been done in collaboration
with Alain Bensoussan (Dallas, USA). 



15:40  priv.doz. RNDr. Martin Kruzik, PhD. (UTIA, ASCR, Prague and Dep. of Physics, Civil engineering, CVUT, Prague):  Oscillations and concentrations generated by Afree mappings and weak lower semicontinuity of integral functionals  Abstract: We use a generalization of Young measures
to describe oscillations and concentrations in sequences of maps in
$L^p(\O;\R^m)$ satisfying a linear differential constraint ${\mathcal
A}u_k=0$. Applications to sequential weak lower semicontinuity of integral
functionals along ${\mathcal A}$free sequences and to weak continuity of
determinants are given, too. This is a joint work with I. Fonseca (CMU,
Pittsburgh) 

17:20  Prof. Ing. Tomas Roubicek, DrSc. (Mathematical Institute, Charles University, Prague):  Rate independent processes in viscous solids  Abstract: Abstract: Generalized standard solids with internal parameters
describing activated typically rateindependent processes such as
plasticity, damage, or phase transformations, ferromagnets or
ferroelectrics, will be described as a system of a forceequilibrium
equation and variational inequality for internal parametr variable.
Various definitions of weak solutions will be examined, especially from
the viewpoint of ability to combine rateindependent processes and other
ratedependent phenomena, as viscosity or inertia. The talk will
emphasize mathematical aspects (contrary to modelling itself, as already
exposed on Nov.12, 2007) 



15:40  (WIAS & HU Berlin):  Analysis of RateIndependent Materila Models: I. Classical rateindependent models including elastoplasticity  Abstract: Some physical processes like dry friction, elastoplasticity, damage,
hysteresis in ferromagnets and shapememory alloys can be modeled by
rateindependent material laws. We provide mathematical models for such processes and discuss general existence results based on the energetic formulation which is based on the dissipation distance and the storedenergy funtional. Several applications are given and the uestion of convergence of solutions under Gamma convergence of the unctionals is addressed. The latter theory provides convergence of umerical schemes and homogenization results. 



15:40  (WIAS & HU Berlin):  Analysis of RateIndependent Materila Models: III. Applications in material models  Abstract: Some physical processes like dry friction, elastoplasticity, damage, hysteresis in ferromagnets and shapememory alloys can be modeled by rateindependent material laws. We provide mathematical models for such processes and discuss general existence results based on the energetic formulation which is based on the dissipation distance and the storedenergy funtional. Several applications are given and the uestion of convergence of solutions under Gamma convergence of the unctionals is addressed. The latter theory provides convergence of umerical schemes and homogenization results. 

17:20  Dr. Chiara Zanini (Dep. of Mathematics and Computer Sciences, University of Udine, Italy and Institute of Physics AS CR, Prague):  Mathematical models of quasistatic crack propagation  Abstract: We consider the propagation of a single crack along a prescribed crack path and discuss two notions of evolution of brittle fractures inspired by Griffith s theory. The first one corresponds to the energetic solution for rateindependent processes introduced by Mielke, and is based on a global stability criterion. The second one, more recent, uses a different selection criterion and is based on local minimality. We compare these two notions in an explicit example. 



15:40  Prof. Dr. P. Exner (Dep of Theoretical Physics, Nuclear Physics Institute, ASCR):  Klasicke a kvantove isoperimetricke problemy resene pomoci nerovnosti pro delky tetiv  Abstract: Prednaska je motivovana isoperimetrickymi problemy, jez vznikaji v kvantove mechanice i v klasicke fyzice. Budeme se zabyvat Schrodingerovymi operatory v L2(R2) se singularni pritazlivou interakci, jejimz nosicem je uzavrena krivka.
Abstract 



15:40  (Universite Paul Sabatier, Toulouse):  Large time behavior for diffusive HamiltonJacobi equations  

17:20  (Mathematical Institute, Charles University, Prague):  Large data existence result for inhomogeneous incompressible heatconducting fluids  



15:40  (Universite Paul Sabatier, Toulouse):  Large time behavior for diffusive HamiltonJacobi equations  

16:20  Ing. Jan Zeman, PhD. (Klokneruv ustav, CVUT, Praha):  Homogenization of fibrous composites with phase debonding: Rateindependent formulation and numerical solution using FETIbased approach  Abstract: Loadinduced debonding at internal interfaces is considered to be the dominant damage phenomenon in fibrous reinforced composites, reducing the overall material properties and reduced structural integrity. In the present lecture, a rateindependent multiscale model of debonding processes is presented, following the ideas introduced recently by
Mielke, Roubicek and coworkers. It is shown that the rateindependent setting not only offers a convenient framework for studying the
qualitative properties of the solution, but also naturally leads to efficient discretization schemes. As a particular example, we examine a numerical algorithm based on the Finite Element Tearing and Interconnecting (FETI) method. The performance of the algorithm is illustrated on selected numerical examples. 



15:40  Dr.Giuseppe Tomassetti (Uni. Rome II Tor Vergata):  Coupling dynamic micromagnetics with the heat equation  Abstract: In standard micromagnetics the specific magnetization is subject to the
saturation constraint, which reflects the assumption that the temperature
is well below the Curie point. There exist, however, interesting evolution
processes, some of them of technological importance, where temperature
instantaneously rises above the Curie point in a very small portion of the
body. Modeling these processes requires a suitable nonisothermal extension of micromagnetics. In this talk, which is based on an ongoing collaboration with Paolo PodioGuidugli and Tomas Roubicek, it will illustrated how to produce such an extension using standard tools from continuum thermomechanics. Both modeling and analytical issues will be discussed. 



15:40  (School of Computing and Mathematics, Keele University, UK):  Instabilities of mixing layers  

17:20  Dr. Lucia Scardia (MPI MIS Leipzig):  Damage as Gammalimit of microfractures in linearized elasticity  Abstract: A homogenization result is given for a material having brittle
inclusions arranged in a periodic structure.
According to the relation between the softness parameter and the size
of the microstructure, three different limit models are deduced via
Gammaconvergence. In particular, damage is obtained as limit of
periodically distributed microfractures. 



15:40  (Mathematisches Institut, Universitat Leipzig, Germany):  On liquid layers in cavities  



15:40  Dr. M. Bulicek (Mathematical Institut, Charles University, Prague):  On the existence of an entropy solution to scalar hyperbolic conservation laws with discontinuous flux  Abstract: We will present an observation how the theory on existence of soution
to scalar hyperbolic conservation laws can be extended to the case
when the flux function has a finite number of monotone
jumps. We will also introduce a natural notation for entropy and
measurevalued entropy solution for such fluxes. 



15:40  Mgr. Josef Kristan (NCMM, Prague):  Modelling of elementary processes of plastic deformation of crystalline materials  



15:40  Dr. M. Lund (Institute of Organic Chemistry and Biochemistry, ASCR):  CoarseGrained Models for Molecular Systems  

16:20  Prof. Ing. Frantisek Marsik, DrSc. (Institute of Thermodynamics, ASCR):  Modeling of biochemistry of bone reconstruction  

16:40  Prof. Ing. Tomas Roubicek, DrSc. (Mathematical Institute, Charles University):  Incompressible ionized nonNewtonean fluid mixtures  

17:00  Doc. RNDr. Josef Malek, CSc. (Mathematical Institute, Charles University):  Incompressible chemicallyreacting fluids  



15:40  (Institut fuer Numerische und Angewandte Mathematik (NAM), GeorgAugustUniversitat Gottingen, Germany):  Stabilized finite element methods for thermally coupled incompressible flows  Abstract: The talk is concerned with stabilized finite element methods for the incompressible NavierStokes problem with thermal coupling. Turbulent flows are simulated based on an unsteady Reynoldsaveraged NavierStokes (URANS) model. In the discrete case, an equalorder interpolation is applied to all unknowns. Some aspects of classical residualbased stabilization techniques like streamlineupwind stabilization (SUPG), pressure stabilization (PSPG), stabilization of the divergencefree constraint (divdiv stabilization) and the treatment of boundary and interior layers will be addressed. Finally, some applications to indoorair flow simulation are given. 

17:20  (ETH Zurich, Department of Materials, Institute of Polymers, Zurich, Switzerland):  Mathematical Structure of Thermodynamics and Its Preservation in Coarse Graining  Abstract: Coarse graining is NOT a necessary evil required to solve problems which are computationally too large. Coarse graining rather is the art to simplify to the essentials and hence to provide understanding. Systematic coarse graining procedures are therefore needed, which are the topic of statistical equilibrium and nonequilibrium thermodynamics. We show that any coarse graining step should be accompanied by an increase in irreversibility and is potentially accompanied by the same kind of historical confusion that was created by the passage from reversible to irreversible equations.
The relevant thermodynamic structure and the coarse graining recipes suggested by statistical mechanics are described in detail and are illustrated by the example of hydrodynamics. We employ systematic coarse graining techniques to derive hydrodynamic equations from Grad s tenmoment equations. The coarse graining procedure is designed such that it manifestly preserves the thermodynamic structure of the equations. A number of mathematical challenges associated with structurepreserving coarse graining of evolution equations for thermodynamic systems as a generalization of Hamiltonian dynamic systems and reduction techniques are presented. Coarse graining is a key step that should always be considered before attempting to solve an equation. 



15:40  (Institute of Numerical Simulation, Hamburg University of Technology, Germany):  Finite volume evolution Galerkin method : theory and application for geophysical flow  

17:20  (Institut fuer Numerische und Angewandte Mathematik (NAM), GeorgAugustUniversitat Gottingen, Germany):  Local projection stabilization methods for singularly elliptic problems  Abstract: The concept of variational multiscale methods (VMS) is the starting point of the talk. The first goal is to describe the link of local projection stabilization (LPS) techniques to the VMS framework. Then we present recent results on the apriori analysis for LPS methods applied to the basic linear advectiondiffusionreaction problem. Moreover, a critical comparison to the standard streamlinediffusion stabilization will be given. 



15:40  (Institut fuer Numerische und Angewandte Mathematik (NAM), GeorgAugustUniversitat Gottingen, Germany):  Calibration of subgrid viscosity models for turbulent incompressible flows  Abstract: We start with some requirements for the large eddy simulation (LES) of incompressible flows. Then we consider the variational multiscale approach to LES and discuss a parameter identification problem for the corresponding subgridscale model. Finally, we present some recent results for the LES with a collocated finite volume code applied to standard benchmark problems and give some conclusions for the finite element case. 



15:40  (Dep. of Mathematical Sciences, University of Bath, UK):  From conservative lattice models for elasticity to macroscopic dissipation  Abstract: We consider travelling waves in a onedimensional chain of atoms with nearest neighbour interaction. The elastic potential is piecewise quadratic and the model is thus capable of describing phase transitions. We show that for suitable fixed subsonic waves, there is a family of ``heteroclinic travelling waves (connecting both wells of the energy). Though the microscopic picture is Hamiltonian, we derive nontrivial socalled kinetic relations on the continuum scale; they can be related to the dissipation generated by a moving phase boundary. It turns out that the microscopic asymmety determines here the macroscopic dissipation.
This is joint work with Hartmut Schwetlick (Bath). 

17:20  (University Rennes 1, France):  Simple results for 3D NSE in periodic case  Abstract: Part I  Some remarks on Sobolev spaces for periodic functions 



15:40  Dipl.Math. Marita Thomas (Humboldt Universitat zu Berlin):  Existence analysis of rateindependent damage processes  Abstract: We analyze a model describing partial damage of a solid within its energetic formulation. This approach involves the free energy functional of the body and the dissipation potential expressing the evolution of damage. We provide an existence result in the small strain as well as in the finite strain setting and give examples on energy densities covered by our theory. 

17:20  (University Rennes 1, France):  Simple results for 3D NSE in periodic case  Abstract: Part III  Deconvolution models, still in the periodic case 



15:40  (Prirodovedecka fakulta UK):  Pretvareni nasycenych porovitych teles  Abstract: 1. nasycene porosni prostredi jako dvoufazovy system,
2. rozlisovaci uroven,
3. Darcyuv zakon,
4. pretvareni,
5. Biotova teorie. 

17:20  Dr. Sebastian Franz (TU Dresden):  Superconvergence for singularly perturbed problems with parabolic layers  



14:00  (University of Stuttgart, Germany):  Stabilized semismooth Newton algorithms for variational inequalities  Abstract: Numerical simulation for partial differential equations plays an important role in many fields. Although efficient solution strategies are well known for simple settings, more complex models involving nonsmooth nonlinearities are still quite challenging. In this talk we present some applications from structural mechanics and finance where variational inequalities play an important role. Surprisingly frictional contact problems with plasticity fit into the same mathematical framework as American options. We propose fast and robust numerical approaches based on the concept of domain decomposition, adaptivity and stabilized time integration. 

15:40  (University of Bielefeld, Germany):  Existence results for contact problems with friction  Abstract: Contact problems in elasticity belong to the classical topics of applied
mathematics, but there are still many questions unsolved, in particular if
friction is taken into account. The main difficulty is the combination of
the unilateral contact condition with the Coulomb friction law that leads to a
nonmonotone and noncompact formulation. This requires special approaches to
prove the existence of solutions.
The most successful approach to analyze contact problems with Coulomb
friction was established by Jindrich Necas together with Jiri Jarusek and
Jaroslav Haslinger in a seminal paper On the solution of the variational
inequality to the Signorini problem with small friction. (Boll. Un. Mat.
Ital. B 5(17), pp. 943958). It consists in the approximation of the problem
by some convex problem, the proof of some addition regularity of the solution
by a certain translation technique and the application of a fixed point
argument. This approach was first applied to static contact problems with
Coulomb friction. It was later extended to many other types of frictional
contact problems, as e.g. quasistatic contact problems and dynamic problems
for viscoelastic and viscoplastic materials. Some results also include the
transport of heat generated by friction. In the lecture we present this
approach and give a survey on the available existence results for static,
quasistatic and dynamic frictional contact problems. 

16:50  Prof. RNDr. Igor Bock, Ph.D. (Slovak University of Technology Bratislava, Slovakia):  Dynamic Contact Problems for von Karman Plates  Abstract: We deal with systems consisting of a nonlinear hyperbolic variational inequality for a deflection and a nonlinear elliptic equation for the Airy stress function. The systems describe moderately large deflections of thin elastic and viscoelastic plates with an inner obstacle and in a dynamic action. The dynamic contact problems are not frequently solved in the framework of variational inequalities. Mainly for the elastic problems there is only limited amount of results available. The aim of the presentation is to extend these results to von K arm an plates. We neglect the rotational inertia member in the elastic case. We will consider the short memory and the long memory material in the case of a viscoelastic plate. The fist one is expressed by a pseudohyperbolic unilateral problem. The long memory material will be considered as an integrodifferential variational inequality with a singular kernel. 



15:40  (University of Warwick, UK):  Ergodic Theory for Stochastic PDEs I.  Abstract: The aim of these lectures is to present a reasonably selfcontained
theory of ergodicity for stochastic processes that is sufficiently
flexible to allow to deal with infinitedimensional problems like the
stochastic Navier Stokes equations, stochastic reactiondiffusion
equations, etc. In the first lecture, we will introduce the main
objects and problems, and remind the audience of the classical
theory of Harris chains. We will go through elementary sketches of
proofs of some of the main results of this theory. In the second
lecture, we will argue that the theory of Harris chains is not
suitable for infinitedimensional problems and we will lay down the
foundations for a modified theory that is more flexible. The remainder
of the course will be devoted to the applications of this theory to a
class of stochastic PDEs. In the third lecture, we will sketch the
proof of a general ergodicity result. The final lecture will be
devoted to showing how to leverage the bounds obtained in the third
lecture to obtain an exponential convergence result. 

17:20  (University of Warwick, UK):  Ergodic Theory for Stochastic PDEs II.  



15:40  Doc. RNDr. Lubos Pick, CSc., DSc. (Charles University, Prague):  Sobolev Spaces and their Optimality in Embeddings  Old and New  Abstract: We develop a new method that enables one to test whether a given Sobolev
embedding can or cannot be improved in the framework of the
rearrangementinvariant spaces. The method is applicable to various tasks
including Sobolev embeddings, boundary trace embeddings, logarithmic
Sobolev inequalities etc. 

17:00  Doc. RNDr. Lubos Pick, CSc., DSc. (Charles University, Prague):  The Gateway to Compactness  Abstract: We focus on finding the frontier (if only such a thing exists) between
boundedness and compactness of a Sobolev embedding. We apply the result to
obtaining a manageable condition equivalent to saying that a Sobolev
embedding involving a pair of rearrangement  invariant spaces is compact. 


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