Prof. Josef Malek
(Mathematical Institute of the Charles University, Prague):
Necas Center in 2011
Dr. Ondrej Soucek
(Charles University in Prague, Faculty of Mathematics and Physics):
Advances and open problems in physical glaciology

Glaciology is a physical discipline describing the flow and evolution of ice masses on the Earth. The evolution of grounded glaciers - ice-sheets - is primarily controlled by the climate forcing via surface temperature and precipitation/ablation rates and by the geothermal heat flux from the Earths interior. Given these inputs, the dynamics of ice-sheets on sufficiently long time and spatial scales may be viewed as a thermomechanically coupled free-surface flow of a non-Newtonian fluid subject to gravity forcing.

A large number of physical processes needs to be taken into account to provide a sufficiently realistic glacier model, some of which are still only poorly understood. Typical example is a generation and transport of meltwater in the ice matrix and its interaction with the glacier flow, that would require a mixture continuum thermodynamics rather than a single-component theory. Of the same importance are processes at the glacier base, such as the interaction of basal ice and meltwater with a sedimentary glacier bed, which strongly affects basal boundary conditions. For sufficiently low local temperatures, no-slip (frozen bed) Dirichlet boundary condition is applied, while in regions where basal temperature exceeds the local pressure-melting point, a rapid sliding occurs, governed by a Newton-type sliding law. As shown in recent numerical studies, such a type of thermally-conditioned boundary conditions may become a trigger of large-scale intrinsic instabilities of the whole ice-sheet, which is a phenomenon indicated by geological evidence. From a numerical point of view, glacier physics remains a challenging topic as even relatively crude and simplistic formulations of glacier flow problem remain rather intractable by standard numerical techniques such as finite-elements. Various approximations of the flow equations based on scaling analysis have been adopted, in order to reduce the huge computational demands.

Dr. Jan Haskovec
(Johann Radon Institute for Computational and Applied Mathematics, Linz, Austria):
Stochastic Particle Approximation to the Keller-Segel model in 2D
Abstract: We construct an approximation to the measure valued, global in time solutions to the Keller-Segel model of chemotaxis in 2D, based on systems of stochastic interacting particles. The main advantage of our approach is that it captures the solution even after the possible concentration events (blow-ups). We present a numerical method based on this approach, discuss the related technical issues and show some numerical results. The first step toward the convergence analysis of our method consists of considering a regularized particle scheme and showing that one recovers the solution of the regularized Keller-Segel system when the number of particles tends to infinity. The proof is based on the BBGKY-approach known from classical kinetic theory. The second step is to describe the limit when the regularization parameter tends to zero, which is technically much more involved and requires an application of the framework of time dependent measures with defects. Subsequently, we pass to the limit when the number of particles tends to infinity and show that the resulting object is compatible with the measure valued solutions of the Keller-Segel system, in the sense that solutions of the latter generate solutions of the former. However, from fundamental reasons it is impossible to make a rigorous statement about the equivalence of these two. Finally, we provide a detailed description of the dynamics of the particle system consisting of two particles only, and explain why the analysis of systems with three or more particles remains an open problem. This is a joint work with Christian Schmeiser (University of Vienna).
Doc. RNDr. Eduard Feireisl, DrSc.
(Institute of Mathematics AS CR):
Suitable weak solutions to the compressible Navier-Stokes system
Abstract: We introduce a class of suitable weak solutions to the Navier-Stokes system of equations governing the motion of a compressible viscous fluid. These solutions satisfy the relative entropy inequality introduced by several authors, and, in particular, they enjoy the weak-strong uniqueness property.
Dr. Ondrej Soucek
(Mathematical Institute of the Charles University):
Shallow Ice Approximation
Abstract: We present in detail the traditional physical approach towards the Shallow Ice (SI) scaling approximation of the Navier-Stokes-Fourier system for the ice-flow problem. The simplest possible (still physically relevant) setting is considered. Ice is described as a single-component continuum and described by a model of a heat-conducting non-Newtonian power-law fluid. The ice-sheet geometry is described by two smooth boundaries, the first one representing the upper free surface with a prescribed mass-production rate and surface temperature, the bottom representing the glacier base, where the heat flux is prescribed and the sliding law is specified by a Newton-type boundary condition. A-priori scaling is introduced into the stress-tensor and other field quantities and a corresponding dimensionless form of the governing equations and boundary conditions is derived. The SI approximation is obtained from the differential form of the Navier-Stokes-Fourier system as a leading-order limit of the perturbation expansion for the flatness parameter and the Froude number approaching zero.
Prof. Lorenzo Freddi
(Dipartimento di Matematica e Informatica, Universita di Udine):
From 3D non-linear elasticity to 1D elastic models for thin-walled beams
Abstract: Geometrically, a thin-walled beam is a slender structural element whose length is much larger than the diameter of the cross-section which, on its hand, is larger than the thickness of the thin wall. Beams of this kind have been used for a long time in civil and mechanical engineering and, most of all, in flight vehicle structures because of their high ratio between maximum strength and weight. Because of their slenderness thin-walled beams are quite easy to buckle and to deform and hence, in several circumstances, their study has to be conducted by means of nonlinear theories. In this talk, starting from three-dimensional nonlinear elasticity, we rigorously derive a hierarchy of one-dimensional models for a thin-walled beam, in the spirit of what has been done by Friesecke, James and Muller for plates. The different limit models are distinguished by the different scaling of the elastic energy, which, in turn, depends on the scaling of the applied loads. We can identify three main regimes. In the first the limit model is an inextensible string. In the second we obtain a Cosserat model for a thin-walled beam. Finally, in the third regime we deduce one dimensional linear/quasi-linear models for thin-walled beams.
Dr. Chiara Zanini
(Dipartimento di Matematica e Informatica, Universita di Udine):
Quasistatic delamination models for Kirchhoff-Love plates
Abstract: The talk will address a quasistatic rate-independent brittle delamination problem and also an adhesive unilateral contact problem on a cylinder with a prescribed normally-positioned surface inside. By letting the height of the cylinder go to zero, we obtain two quasistatic rate-independent crack models with prescribed path for Kirchhoff-Love plates. (This is essentially based on a joint work with L.Freddi, R.Paroni, and T.Roubicek.)
Prof. Dr. Christian Grossmann
(TU Dresden, Institute of Numerical Mathematics):
Convergence of Interior-Exterior Penalty Methods in Optimal Control
Dr. Agnieszka Swierczewska-Gwiazda
(Institute of Applied Mathematics and Mechanics, University of Warsaw, Poland):
On flows of implicitly constituted fluids characterized by a maximal monotone graph
Abstract: We study flows of incompressible fluids in which the deviatoric part of the Cauchy stress and the symmetric part of the velocity gradient are related through an implicit equation. Although we restrict ourselves to responses characterized by a maximal monotone graph, the structure is rich enough to include power-law type fluids, stress power-law fluids, Bingham and Herschel-Bulkley fluids, etc. We are interested in the development of (large-data) existence theory for internal flows subject to no-slip boundary conditions. We study first Stokes-like problems, and later we consider complete problems including the convective term. Doing so, we pay the attention to the tools involved in the analysis of the problem.
Dr. Tomasz Piasecki
(Institute of Mathematics, Polish Academy of Sciences):
Stationary compressible flow with slip boundary conditions

I am going to discuss the issue of existence of stationary solutions to the Navier Stokes Equations for compressible, barotropic flow in a cylindrical domain. More precisely, we are interested in strong solutions in a vicinity of given laminar flows. I will show existence of such solution when the perturbed flow is a constant flow or a Poiseuille profile. In the latter case we need some assumptions on the viscosity, but these turn out natural and does not provide any serious limitations.

A main problem to face in the proof is the lack of compactness in the continuity equation. To overcome this diffuculty we can introduce a kind of Lagrangian coordinates. In the new system the continuity equation simplifies enabling us to apply the Banach Fixed Point Theorem.

priv. doc. RNDr. Martin Kruzik, PhD
(UTIA, Czech Academy of Science, Prague):
Young measures supported on regular matrices
Abstract: We completely and explicitly describe Young measures generated by matrix-valued mappings ${Y_k}_{kinN} subset L^p(O;R^{n imes n})$, $OsubsetR^n$, such that ${Y_k^{-1}}_{kinN} subset L^p(O;R^{n imes n})$ is bounded, too. Moreover, the constraint $det Y_k>0$ can be easily included and is reflected in a condition on the support of the measure. These results allow us to relax minimization problems for functionals $J(Y):=int_O W(Y(x)),md x$, where $W(F)$ tends to infinity if the determinant of $F$ converges to zero. This phenomenon typically occurs in problems of nonlinear-elasticity theory for hyperelastic materials if $Y:= abla y$ for $yin W^{1,p}(O;R^n)$, for instance. We touch this particular situation with the additional condition $det abla y>0$, as well. This is a joint work with Barbora Benesova and Gabriel Patho.
Prof. Ing. Tomas Roubicek, DrSc.
(Mathematical Institute of Charles University):
Perfect plasticity at small strains and its thermodynamics
Abstract: A thermodynamically consistent model of perfect plasticity is presented and an existence of its energetic solutions is proved.
Dr. Riccarda Rossi
(University of Brescia, Italy):
Analysis of doubly nonlinear evolution equations driven by nonconvex energies
Abstract: In this talk, based on a joint collaboration with Alexander Mielke and Giuseppe Savare , we present existence results for doubly nonlinear equations featuring nonsmooth and nonconvex energies. We prove existence by passing to the limit in a time-discretization scheme, based on the Minimizing Movement approach by Ennio De Giorgi. We present some applications.
(Department of Geophysics, Faculty of Mathematics and Physics, Charles University in Prague):
Geofyzikalni modely vyvoje Zeme a dalsich teles slunecni soustavy

Geofyzikalni studium vyvoje Zeme a dalsich teles slunecni soustavy je zalozeno na numerickem reseni rovnic popisujicich prenos tepla ve vysokoviskozni kapaline. Krome tradicni Bousinesqovy aproximace se dnes jiz standardne vyuziva take rozsirena Bousinesqova aproximace (zahrnujici disipacni a adiabaticke zahrivani a teplo uvolňujici se pri fazovych prechodech) a anelasticka aproximace (uvazujici narust hustoty s hloubkou). Jelikoz se jedna o velmi pomale procesy, lze v pohybove rovnici bezpecne zanedbat setrvacny clen. Vyznamnou komplikaci pri reseni rovnic zpravidla predstavuje viskozita, ktera se muze menit v rozmezi sesti i vice radu. Viskozita je obecne chapana jako funkce tlaku, teploty a napeti (a pripadne dalsich parametru, jako je napr. velikost zrna) a je casto zavedena tak, aby simulovala i dalsi typy deformaci, jako je krehke poruseni a plasticita. Dalsim specifikem geofyzikalnich modelu jsou fazove prechody, ať uz souvisejici s tavenim materialu nebo s jeho prechodem do vysokotlake faze s odlisnymi fyzikalnimi vlastnostmi. Chovani geofyzikalnich systemu je obvykle studovano ve dvourozmerne kartezske geometrii, stale castejsi jsou ale dnes i trojrozmerne sfericke modely. Nektere soucasne modely se snazi pracovat s volnou hranici a zahrnout deje, ktere souvisi s jeji deformaci (eroze, sedimentace). Z geofyzikalnich problemu resenych na katedre geofyziky MFF budou diskutovany tri priklady:
1) model litosfericke desky zanorujici se do plaste (priklad komplikovane reologie materialu s fazovymi prechody),
2) simulace geologickeho vyvoje Ceskeho masivu v prubehu variske orogeneze (model zahrnuje nekolik nemisitelnych petrologickych fazi, krehke chovani svrchni kury a volnou erodujici hranici) a
3) generace podpovrchoveho oceanu na Saturnove mesici Enceladu v dusledku slapoveho zahrivani (trojrozmerny sfericky model, zahrnujici viskozni i viskoelastickou deformaci).

Spoluautori: Hana Cizkova, Petra Maierova a Marie Behounkova z teze katedry
Dr. Ondrej Sramek
(University of Colorado at Boulder, Department of Physics, Boulder, Colorado, USA):
Modeling of two-phase flow in geophysics: compaction, differentiation and partial melting
Abstract: Partial melting and migration of melts play an important role in the formation and evolution of the Earth other terrestrial bodies. Transport of heat, rock rheology and distribution of major, minor, as well volatile chemical species are all affected by the presence and migration of magmas. Partial melting and melt extraction are central processes for the formation of the oceanic crust and are responsible for the depletion of the bottom part of the lithosphere in incompatible elements. Migration of molten material played a major role in the dynamic evolution of the early Earth and even now plays a fundamental role in the transport of matter as well as heat in partially molten regions of deeper mantle (e.g., at the coremantle boundary). The separation of the denser metal from the lighter silicates is the most extensive differentiation process in the course of Earths evolution and the evolution of terrestrial planetary bodies in general. This process also implies the presence of distinct phases, in solid and liquid states. Gravitational energy which is released upon differentiation is a major source of heat that must be considered when assessing the thermal history of a forming planet. It is therefore essential to properly take into account the energy exchange that takes place in a multiphase medium on a large spatial scale in order to investigate early planetary evolution and to constrain the differentiation time scales. In my talk I will present a recently developed general model of two-phase flow and deformation in a two-phase medium. The model offers a self-consistent description of the mechanics and thermodynamics of a mixture of two viscous fluids, in the form of continuum mechanical equations in the limit of a slow creeping flow. The difference in pressures that exists between the two phases is generated i) by the surface tension at the interfaces between the phases which are included in the description, and ii) by the isotropic deformation (i.e., compaction or dilation) of the individual phases upon flow. In most geologic applications, one of the phases (named the liquid phase) is much less viscous than the other phase (the solid phase), which greatly simplifies the equations. I will show some results of modeling of a terrestrial planet differentiation, and of the coupling between melting and deformation.
Easter Monday - seminar canceled
Ing. Mgr. Tomas Bodnar, Ph.D.
(Department of Technical Mathematics, Czech Technical University of Prague):
Viscoelastic fluid flows at larger than small Weisenberg numbers
Abstract: The talk addresses one of the classical problems related to the mathematical modeling and numerical simulation of viscoelastic flows. A short overview of problems arrising during the simulation of viscoelastic fluid flows at moderate and high Weisenberg numbers is presented. An alternative point of view on the treatment of these issues is offered. A new simple test case is proposed to demonstrate the problem and it s possible solution. In the conclusion, a new formulation of the Johnson-Segalman model is proposed to be solved.
Dr. Giuseppe Tomassetti
(Dipt. Ingegneria Civile, Univ. di Roma II - Tor Vergata):
Energetic solution of the torsion problem in strain-gradient plasticity
Abstract: We consider elasto-plastic torsion in a thin wire in the framework of the strain-gradient plasticity theory recently proposed by Gurtin and Anand. This theory takes into account the socalled geometricallynecessary dislocations through a dependence of the free energy on the Burgers tensor G=curl E , where E is the plastic part of the linear strain. For the rate-independent case with null dissipative length scale, we construct explicitly an energetic solution of the evolution problem. We use this solution to estimate the dependence of the torque on the twist and on the material scale. Our analysis highlights some size effect, showing that thinner wires are stronger. This work is in collaboration with Maria Chiricotto and Lorenzo Giacomelli.
Dr. Jan Valdman
(University of Iceland, Reykjavik, Iceland and Max-Planck Institute Leipzig):
Computations and a posteriori error estimates in elastoplasticity
Abstract: download abstract
Dr. Martin Heida
(Institute for Applied Mathematics, University of Heidelberg, Germany):
An introduction to homogenization theory
Abstract: Homogenization theory deals with modeling of processes in media with complex micro structures, whenever this particular micro structure influences the macroscopic behavior. The homogenization theory has become a huge domain in mathematical modeling and this talk aims to give a short overview over the goals and methods in this field. We will mostly restrict to periodic micro structures and discuss formal and rigorous methods like asymptotic expansion, two-scale convergence and periodic unfolding. As an outlook we will also look at the homogenization methods for stochastic geometries.
Prof. E. Fernandez-Cara
(Dpto. EDAN, University of Sevilla, Spain):
Global Carleman inequalities and control results for systems from continuum mechanics
Abstract: This talk deals with the theoretical and numerical solution of several control problems for several PDEs from mechanics. I will present some results that rely on global Carleman inequalities and Fursikov-Imanuvilov s approach. In the linear case, according to this strategy, the (original) controllability problems can be reduced to the solution of appropriate higher-order differential problems. For similar nonlinear problems, this can be used in combination with fixed point theorems and/or iterative methods. I will also present some numerical experiments that show that this approach is very useful.
Ucastnici studentske letni staze v NCMM
Nepracuj v Tescu, ziv se vedou
Abstract: Prezentace vysledku studentske letni staze v NCMM.
Program prezentace:
15:40 Barbora Benesova: SIAM Student Chapter Prague
16:00 Dominik Mokris: Uvodni slovo ke stazim; Isogeometricka Analyza
16:20 Jan Kuratko: Vypocet hodnosti Sylvesterovy matice
16:40 Martin Rehor: Materialy ve squeezeflow geometrii
17:00 ------- Coffee Break
17:10 Miroslav Kuchta: Lapetus
17:30 Marek Netusil: Benchmarky pro anisotropni materialy
17:50 Adam Janecka: Tekutiny s viskozitou zavislou na tlaku pri povrchovem zatizeni
Prof. E. Fernandez-Cara
(Dpto. EDAN, University of Sevilla, Spain):
The control of evolution PDEs: some recent results and open problems, Part 1 of 3
Abstract: Texts for the lectures can be downloaded: Introduction, L1-OptimalControl.pdf, L2-HeatandWaves, L3_Others.pdf.

Lecture 1: Introduction. Optimal control and controllability. Basic definitions and fundamental results. Optimal control results for some nonlinear problems and related open questions. Numerical approximation, numerical results and applications.
Prof. E. Fernandez-Cara
(Dpto. EDAN, University of Sevilla, Spain):
The control of evolution PDEs: some recent results and open problems, Part 2 of 3
Abstract: Texts for the lectures can be downloaded: Introduction, L1-OptimalControl.pdf, L2-HeatandWaves, L3_Others.pdf.

Lecture 2: Controllability of parabolic equations. Unique continuation, Carleman estimates and observability. On the controllability of semilinear and nonlinear problems. Additional results and open questions: the Stokes and Navier-Stokes systems, stochastic controllability, etc.
Prof. E. Fernandez-Cara
(Dpto. EDAN, University of Sevilla, Spain):
The control of evolution PDEs: some recent results and open problems, Part 3 of 3
Abstract: Texts for the lectures can be downloaded: Introduction, L1-OptimalControl.pdf, L2-HeatandWaves, L3_Others.pdf.

Lecture 3: Controllability of linear hyperbolic equations and systems. Unique continuation, observability, the multipliers method and the geometric control condition. Semilinear hyperbolic equations. Additional results and open questions: linear elasticity, visco-elastic fluids, etc.
MUDr. Ales Hejcl, Ph.D. a Dr. med. MUDr. Amir Zolal, Ph.D.
(Neurochirurgicka klinika Univerzity J.E. Purkyne a Masarykovy nemocnice, Usti nad Labem):
Intrakranialni aneuryzma: vyvoj, hemodynamika a terapie z pohledu neurochirurga
RNDr. J. Hron, PhD. a RNDr. Martin Madlik, Ph.D.
(Matematicky Ustav UK):
Intrakranialni aneuryzma: hemodynamika a CFD z pohledu matematickeho modelovani
Prof Vlastimil Krivan
(Biology center, Ceske Budejovice):
The Lotka-Volterra predator-prey model
Abstract: In my talk I will review some crucial steps, based on mathematical reasoning, that laid foundations of today s ecology. I will start with the Lotka-Volterra predator-prey model and will review some subsequent research. I will focus on research by F. G. Gause, a Russian biologist, who made some crucial extensions of the Lotka-Volterra model. In particular, I will discuss his idea about using differential equations with discontinuities, a concept that was developed by A.F. Filippov much latter . I will show, how such models arise naturally in ecology and how they can be used to unify two major ecological fields: evolutionary and population ecology. Presentation can be downloaded from: Krivan_2011.pdf
Prof Willi Jaeger
(University of Heidelberg, Germany):
Multiscale Systems in Lifesciences - Mathematical Modelling and Simulation - Lecture 1
prof. Antonio DeSimone
(SISSA, Trieste, ITALY):
Mechanics of motility at microscopic scales: challenges and opportunities for mathematical modeling
Abstract: We will review recent progress on the mathematical modeling of crawling and swimming motility in cells, and discuss open problems and promising directions for future research.
Prof. Sergey Repin
(V.A. Steklov Institute of Mathematics at St. Petersburg, Russia):
Estimates of deviations from exact solutions of some nonlinear problems in continuum mechanics
Abstract: In the talk, we discuss estimates measuring the difference between exact solutions of boundary value problems and arbitrary functions from the corresponding (energy) space. The estimates must be computable, consistent and possess necessary continuity properties. In the context of PDE theory, deriving such type estimates present one of the general problems, which unlike, e.g., regularity theory is focused on studying neighborhoods of exact solutions. Being applied to numerical approximations these estimates imply a unified way of a posteriori error estimation. They can be also used for the analysis of modeling errors and errors caused by incomplete knowledge on the problem data. The talk contains a short introduction devoted to historical background, overview of the results obtained in the last decade (in particular for elliptic variational inequalities) and some recent results related to models with linear growth energy (as, e.g., Hencky plasticity)

S. Repin. A posteriori error estimates for PDE s, deGruyter, Berlin, 2008.
M. Fuchs and S. Repin. A Posteriori Error Estimates for the Approximations of the Stresses in the Hencky Plasticity Problem, Numer. Funct. Analysis and Optimization, 32(2011), 6, 610-640.
S. Repin and S. Sauter. Estimates of the modeling error for the Kirchhoff-Love plate model. C. R. Math. Acad. Sci. Paris 348 (2010), no. 17-18, 10391043.

Prof. Vladislav Mantic; Prof. Roman Vodicka
(University of Seville, School of Engineering; Technical University of Kosice, Civil Engineering Faculty):
A variational formulation for elastic domain decomposition problems solved by SGBEM with non-conforming discretizations
Abstract: The solution of Boundary Value Problems of linear elasticity using a Domain Decomposition approach (DDBVPs) with non-overlapping subdomains is considered. A new variational formulation based on a potential energy functional for DDBVPs expressed in terms of subdomain displacement fields is introduced. The coupling conditions between subdomains are enforced in a weak form. A novel feature of the potential energy functional is a distinct role of subdomains on both sides of the interface. The solution of a DDBVP is given by a saddle point of the potential energy functional. The present formulation of DDBVPs is solved by Symmetric Galerkin Boundary Element Method (SGBEM) considering non-conforming meshes along interfaces between subdomains if required. Finally, some numerical results are presented incuding the cases with non-conforming discretizations of curved interfaces. An excellent accuracy and convergence behaviour of our implementation of SGBEM for DDBVPs is shown providing some stability condition is fulfilled.
Prof. Henryk Petryk
(Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw, Poland):
Incremental energy minimization and microstructure formation in dissipative solids
Abstract: In elastic or pseudo-elastic solids with fully reversible microstructural changes, the energy minimization is a standard approach based on the concept of stability of equilibrium in a dynamic or thermodynamic sense. It is well known that the loss of ellipticity, rank-one convexity or quasi-convexity of a nonlinear elastic energy function can lead to formation of fine microstructures in the material. However, that approach does not take into account the dissipation of energy that invariably accompanies the transition between equilibrium states. To include the effect of rate-independent dissipation, one can minimize the total incremental energy supplied quasi-statically to the system, which incorporates both free energy and dissipation contribution to the deformation work increment. It is shown that the incremental energy minimization up to the first and second-order terms can yield an exact solution in the first-order rates, provided an appropriate symmetry restriction is imposed on the constitutive law. Loss of quasi-convexity by the rate-potential means that the mechanical work can be extracted from a deforming material element embedded in a continuum being unperturbed elsewhere, which is associated with formation of energetically preferable microstructures. Details of the approach are discussed and illustrated by examples.
Prof. Stanislaw Stupkiewicz
(Institute of Fundamental Technological Research (IPPT), Polish Academy of Sciences, Warsaw, Poland):
Interfacial energy and size effects in evolving martensitic microstructures
Abstract: Shape memory alloys (SMA) undergo phase transformation of martensitic type which is the main mechanism responsible for the interesting effects observed in these materials. The transformation is accompanied by formation and evolution of martensitic microstructures which govern the functional properties of SMA. Evolution of microstructure occurs at different scales, and micromechanical models have been developed aimed at description of this multiscale phenomenon. A promising new area of research is to include into such models the effects of interfacial energies present at different scales of martensitic microstructures. In this work, interfacial energy of two origins is accounted for, namely the atomic-scale energy of phase boundaries (taken from the materials science literature) and the elastic micro-strain energy at microstructured interfaces (e.g. at the austenitetwinned martensite interface). The latter is a bulk energy at a finer scale, however, at a higher scale it can be interpreted as the interfacial energy. This energy is predicted using micromechanical considerations. Size-dependent interfacial energy contributions introduce size effects into the multiscale modelling framework. Evolution of microstructure is then determined by applying a general evolution rule in the form of minimization of incremental energy supply. The incremental energy, being the sum of the increments in the free energy and dissipation, comprises both the bulk and the interfacial energy contributions at all levels of the microstructure. As an example, size effects are studied for the pseudoelastic CuAlNi and NiTi shape memory alloys. In particular it is shown that characteristic dimensions of the microstructure can be predicted without introducing any artificial length-scale parameters.
Dr. Vit Prusa
(Mathematical Institute, Charles University in Prague):
On a new class of models for fluids stemming from the implicit constitutive theory
Abstract: Implicit constitutive theory is a new methodological framework for developing material models. The main idea is, in the case of fluids, that one has to abandon the approach based on the fact that the Cauchy stress tensor T can be expressed as a function of the symmetric part of the velocity gradient D, and has to search for the constitutive relation in the form of an implicit tensorial relation between T and D, f(T,D)=0. We will discuss the ideas that led to the formulation of the theory, introduce some models that has been developed using the theory, and we will in brief analyze their properties.
Dr. Adrian Hirn
(IWR, Heidelberg University):
Stabilized finite elements for fluids with shear-rate- and pressure-dependent viscosity
Abstract: Non-Newtonian fluid motions are frequently modeled by a power-law ansatz that provides a nonlinear relation between the fluid s viscosity and shear rate. Such fluids play an important role in many areas of application such as engineering, blood rheology, and geology. This talk deals with the finite element (FE) approximation of the corresponding equations of motion. In order to cope with the instabilities of the Galerkin FE method resulting from violation of the inf-sup stability condition or dominating convection in case of high Reynolds numbers, we propose a stabilization method that is based on the well-known local projection stabilization method. For shear thinning fluids, we derive a priori error estimates quantifying the convergence of the method. The established error estimates provide optimal rates of convergence with respect to the supposed regularity of the solution. Finally, we consider viscosities depending on both the shear rate and pressure. We analyze the Galerkin discretization of the governing equations.
prof. E. Feireisl
(Institute of Mathematics, Academy of Sciences of the Czech Republic):
Weak solutions and weak strong uniqueness for the Navier-Stokes-Fourier system
Abstract: We introduce a concept of weak solution based on Second law of thermodynamics for the full Navier- Stokes-Fourier system describing the motion of a general viscous, compressible, and heat-conducting fluid confined to a bounded spatial domain with energetically insulating boundary. We show that the weak solutions comply with the principle of weak strong uniqueness, meaning they coincide with the strong solution emanating from the same initial data as long as the latter exists.

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