As a byproduct of a routine annual evaluation of the performance of the Mathematical Institute, we have compiled a list of papers published by members of the Division of Mathematical Modelling in 2013. If you are interested in what has been done last year, please see the list below (journals with impact factor only).

**Benešová Barbora, Kružík Martin, Roubíček Tomáš: Thermodynamically consistent mesoscopic model of the ferro/paramagnetic transition, Zeitschrift für Angewandte Mathematik und Physik 64 (1), 1-28, 2013**

A continuum evolutionary model for micromagnetics is presented that, beside the standard magnetic balance laws, includes thermomagnetic coupling. To allow conceptually efficient computer implementation, inspired by relaxation method of static minimization problems, our model is mesoscopic in the sense that possible fine spatial oscillations of the magnetization are modeled by means of Young measures. Existence of weak solutions is proved by backward Euler time discretization.

**Bulíček Miroslav, Pustějovská Petra: On existence analysis of steady flows of generalized Newtonian fluids with concentration dependent power-law index, Journal of Mathematical Analysis and Applications 402 (1), 157-166, 2013**

We study a system of partial differential equations describing a steady flow of an incompressible generalized Newtonian fluid, wherein the Cauchy stress is concentration dependent. Namely, we consider a coupled system of the generalized Navier-Stokes equations and convection-diffusion equation with non-linear diffusivity. We prove the existence of a weak solution for certain class of models by using a generalization of the monotone operator theory which fits into the framework of generalized Sobolev spaces with variable exponent. Such a framework is involved since the function spaces, where we look for the weak solution, are "dependent" of the solution itself, and thus, we a priori do not know them. This leads us to the principal a priori assumptions on the model parameters that ensure the Wilder continuity of the variable exponent. We present here a constructive proof based on the Galerkin method that allows us to obtain the result for very general class of models.

**Bulíček Miroslav, Málek Josef, Süli Endre: Existence of global weak solutions to implicitly constituted kinetic models of incompressible homogeneous dilute polymers, Communications in Partial Differential Equations 38 (5), 882-924, 2013 **

We show the existence of global weak solutions to a general class of kinetic models of homogeneous incompressible dilute polymers. The main new feature of the model is the presence of a general implicit constitutive equation relating the viscous part of the Cauchy stress and the symmetric part of the velocity gradient. We consider implicit relations that generate maximal monotone (possibly multivalued) graphs, and the corresponding rate of dissipation is characterized by the sum of a Young function and its conjugate depending on and, respectively. Such a framework is very general and includes, among others, classical power-law fluids, stress power-law fluids, fluids with activation criteria of Bingham or HerschelBulkley type, and shear-rate dependent fluids with discontinuous viscosities as special cases. The appearance of and in all the assumptions characterizing the implicit relationship is fully symmetric. The elastic properties of the flow, characterizing the response of polymer macromolecules in the viscous solvent, are modeled by the elastic part of the Cauchy stress tensor, whose divergence appears on the right-hand side of the momentum equation, and which is defined by the Kramers expression involving the probability density function, associated with the random motion of the polymer molecules in the solvent. The probability density function satisfies a FokkerPlanck equation, which is nonlinearly coupled to the momentum equation. We establish long-time and large-data existence of weak solutions to such a system, completed by an initial condition and either a no-slip or Navier’s slip boundary condition, by using properties of maximal monotone operators and Lipschitz approximations of Sobolev-space-valued Bochner functions via a weak compactness arguments based on the Div-Curl Lemma and Chacon’s Biting Lemma.

**Bulíček Miroslav, Kaplický Petr, Steinhauer Mark: On existence of a classical solution to a generalized Kelvin-Voigt model, Pacific Journal of Mathematics 262 (1), 11-33, 2013 **

We consider a two-dimensional generalized Kelvin-Voigt model describing a motion of a compressible viscoelastic body. We establish the existence of a unique classical solution to such a model in the spatially periodic setting. The proof is based on Meyers’ higher integrability estimates that guarantee the Holder continuity of the gradient of velocity and displacement.

**Bulíček Miroslav, Gwiazda Piotr, Swierczewska-Gwiazda Agnieszka: Multi-dimensional scalar conservation laws with fluxes discontinuous in the unknown and the spatial variable, Mathematical Models and Methods in Applied Sciences 23 (3), 407-439, 2013**

The paper deals with a scalar conservation law in an arbitrary dimension d with a discontinuous flux. The flux is supposed to be a discontinuous function in the spatial variable x and in an unknown function u. Under some additional hypothesis on the structure of possible discontinuities, we formulate an appropriate notion of entropy solution and establish its existence and uniqueness. The framework for proving the existence and uniqueness of entropy weak solutions is provided by the studies on entropy measure-valued solutions and may be viewed as a corollary of the uniqueness theorem for entropy measure-valued solutions.

**Feireisl Eduard: Scaling and singular limits in the equations of continuum ﬂuid mechanics, Methods and Applications of Analysis 20 (2), 115-140, 2013
**

This is a survey on some recent results concerning scaling and the related singular limits in the models of complete fluids.

**Feireisl Eduard, Novotný Antonín: Scale interactions in compressible rotating ﬂuids, Annali di Matematica Pura ed Applicata 2013 (May) **

We study a triple singular limit for the scaled barotropic Navier-Stokes system modeling the motion of a rotating, compressible, and viscous fluid, where the Mach and Rossby numbers are proportional to a small parameter ε, while the Reynolds number becomes infinite for ε tending to 0. If the fluid is confined to an infinite slab bounded above and below by two parallel planes, the limit behavior is identified as a purely horizontal motion of an incompressible inviscid fluid, the evolution of which is described by an analogue of the Euler system.

**Feireisl Eduard: Mathematical analysis of fluids in motion: from well-posedness to model reduction, Revista Matematica Complutense 26 (2), 299-340, 2013**

This paper reviews some recent results on the Navier-Stokes-Fourier system governing the evolution of a general compressible, viscous, and heat conducting fluid. We discuss several concepts of weak solutions, in particular, using the implications of the Second law of thermodynamics. We introduce the concept of relative entropy and dissipative solution and show the principle of weak-strong uniqueness. The second part of the paper is devoted to problems of model reduction and the related singular limits. Several examples of singular limits are presented: The incompressible limit, the inviscid limit, the low Rossby number limit and their combinations.

**Damanik H., Hron Jaroslav, Ouazzi A., Turek S.: Monolithic Newton-multigrid solution techniques for incompressible nonlinear flow models, International Journal for Numerical Methods in Fluids 71 (2), 208-222, 2013**

We present special Newton-multigrid techniques for stationary incompressible nonlinear flow models discretized by the high order LBB-stable Q2P1 element pair. We treat the resulting nonlinear and the corresponding linear discrete systems by a fully coupled monolithic approach to maintain high accuracy and robustness, particularly with respect to different rheological behaviors and also regarding different problem sizes and types of nonlinearity. Here, local pressure Schur complement techniques are presented as a generalization of the classical Vanka smoother. The discussed methodology is implemented for the well-known flow around cylinder benchmark configuration for generalized Newtonian as well as non-Newtonian flows including non-isothermal, shear/pressure dependent and viscoelastic effects.

**Razzaq M., Tsotskas C., Turek S., Kipouros T., Savill M., Hron Jaroslav: Multi-objective optimization of a fluid structure interaction benchmarking, CMES – Computer Modeling in Engineering and Sciences 90 (4), 303-337, 2013 **

The integration and application of a new multi-objective tabu search optimization algorithm for Fluid Structure Interaction (FSI) problems are presented. The aim is to enhance the computational design process for real world applications and to achieve higher performance of the whole system for the four considered objectives. The described system combines the optimizer with a well established FSI solver which is based on the fully implicit, monolithic formuFlation of the problem in the Arbitrary Lagrangian-Eulerian FEM approach. The proposed solver resolves the proposed fluid-structure interaction benchmark which describes the self-induced elastic deformation of a beam attached to a cylinder in laminar channel flow. The optimized flow characteristics of the aforementioned geometrical arrangement illustrate the performance of the system in two dimensions. Special emphasis is given to the analysis of the simulation package, which is of high accuracy and is the core of application. The design process identifies the best combination of flow features for optimal system behavior and the most important objectives. In addition, the presented methodology has the potential to run in parallel, which will significantly speed-up the elapsed time.

**Kulvait Vojtěch, Málek Josef, Rajagopal K. R.: Anti-plane stress state of a plate with a V-notch for a new class of elastic solids, International Journal of Fracture 179 (1-2), 59-73, 2013**

The main purpose of this study is to investigate the efficacy and usefulness of a class of recently proposed models that could be reasonable candidates for describing the response of brittle elastic materials. The class of models that are considered allows for a non-linear relationship between the linearized elastic strain and the Cauchy stress, and this allows one to describe situations wherein the stress increases while the strain yet remains small. Thus one would be in a position to model the response of brittle elastic bodies in the neighborhood of the tips of cracks and notches. In this paper we study the behavior of such models in a plate with a V-notch subject to a state of anti-plane stress. This geometrical simplification enables us to characterize the governing equation for the problem by means of the Airy stress function, though the constitutive relation is a nonlinear relation between the linearized strain and the stress. We study the problem numerically by appealing to the finite element method. We find that the numerical solutions are stable. We are able to provide some information regarding the nature of the solution near the tip of the V-notch. In particular we find stress concentration in the vicinity of the singularity.

**Mucha P. B., Pokorný Milan , Zatorska E.: Chemically reacting mixtures in terms of degenerated parabolic setting, Journal of Mathematical Physics 54 (7), 2013**

The paper analyzes basic mathematical questions for a model of chemically reacting mixtures. We derive a model of several (finite) component compressible gas taking rigorously into account the thermodynamical regime. Mathematical description of the model leads to a degenerate parabolic equation with hyperbolic deviation. The thermodynamics implies that the diffusion terms are non-symmetric, not positively defined, and cross-diffusion effects must be strongly marked. The mathematical goal is to establish the existence of weak solutions globally in time for arbitrary number of reacting species. A key point is an entropy-like estimate showing possible renormalization of the system.

**Kreml Ondřej, Nečasová Šárka, Pokorný Milan: On the steady equations for compressible radiative gas, Zeitschrift für Angewandte Mathematik und Physik 64 (3), 539-571, 2013**

We study the equations describing the steady flow of a compressible radiative gas with newtonian rheology. Under suitable assumptions on the data that include the physically relevant situations (i.e., the pressure law for monoatomic gas, the heat conductivity growing with square root of the temperature), we show the existence of a variational entropy solution to the corresponding system of partial differential equations.

Under additional restrictions, we also show the existence of a weak solution to this problem.

**Průša Vít, Rajagopal K. R.: On models for viscoelastic materials that are mechanically incompressible and thermally compressible or expansible and their Oberbeck-Boussinesq type approximations, Mathematical Models & Methods in Applied Sciences 23 (10), 1761-1794, 2013**

Viscoelastic fluid like materials that are mechanically incompressible but are compressible or expansible with respect to thermal stimuli are of interest in various applications ranging from geophysics and polymer processing to glass manufacturing. Here we develop a thermodynamical framework for the modeling of such materials. First we illustrate the basic ideas in the simpler case of a viscous fluid, and after that we use the notion of natural configuration and the concept of the maximization of the entropy production, and we develop a model for a Maxwell type viscoelastic fluid that is mechanically incompressible and thermally expansible or compressible. An important approximation in fluid mechanics that is frequently used in modeling buoyancy driven flows is the Oberbeck-Boussinesq approximation. Originally, the approximation was used for studying the flows of viscous fluids in thin layers subject to a small temperature gradient. However, the approximation has been used almost without any justification even for flows of non-Newtonian fluids induced by strong temperature gradients in thick layers. Having a full system of the governing equations for a Maxwell type viscoelastic mechanically incompressible and thermally expansible or compressible fluid, we investigate the validity of the Oberbeck-Boussinesq type approximation for flows of this type of fluids. It turns out that the Oberbeck-Boussinesq type approximation is in general not a good approximation, in particular if one considers "high Rayleigh number" flows. This indicates that the Oberbeck-Boussinesq type approximation should not be used routinely for all buoyancy driven flows, and its validity should be thoroughly examined before it is used as a mathematical model.

**Průša Vít, Rajagopal K. R.: A note on the modeling of incompressible fluids with material moduli dependent on the mean normal stress, International Journal of Non-linear Mechanics 52 (JUNE), 41-45, 2013**

In incompressible materials, both fluids and solids, a part of the stress is not prescribed by constitutive specification, that is, the part of the stress is not determined in terms of kinematical quantities, temperature, et cetera. This "indeterminate" part of the stress is variously referred to as the "constraint stress", the "reaction stress" or the "Lagrange multiplier" enforcing the constraint. In the case of an incompressible Navier-Stokes fluid, the part of the stress, that is a consequence of the constraint, also happens to coincide with the mean value of the stress which is referred to as the "mechanical pressure". However, in general non-Newtonian fluids this is not the case, and, unfortunately, in view of the widespread use of the Navier-Stokes equation, the terminology "pressure" is used interchangeably for both the part of the stress that is not constitutively specified and the mean value of the stress, leading to considerable confusion with regard to important issues concerning non-Newtonian fluids. Recognizing the distinction between the mean value of the stress and the part of the stress that is not constitutively specified becomes critical in materials whose moduli depend on the mean value of the stress. An example of the same concerns the viscosity, which depending on whether it is a function of the indeterminate part of the stress or the mean value of the stress could lead to different flow characteristics. In this short note we discuss an error that is a consequence of not recognizing the distinction between these different quantities but misidentifying them as being the same, the mechanical "pressure".

**Průša Vít, Rajagopal K. R., Saravanan U.: Fidelity of the estimation of the deformation gradient from data deduced from the motion of markers placed on a body that is subject to an inhomogeneous deformation field, Journal of Biomechanical Engineering-Transactions of the ASME 135 (8), 2013**

Practically all experimental measurements related to the response of nonlinear bodies that are made within a purely mechanical context are concerned with inhomogeneous deformations, though, in many experiments, much effort is taken to engender homogeneous deformation fields. However, in experiments that are carried out in vivo, one cannot control the nature of the deformation. The quantity of interest is the deformation gradient and/or its invariants. The deformation gradient is estimated by tracking positions of a finite number of markers placed in the body. Any experimental data-reduction procedure based on tracking a finite number of markers will, for a general inhomogeneous deformation, introduce an error in the determination of the deformation gradient, even in the idealized case, when the positions of the markers are measured with no error. In our study, we are interested in a quantitative description of the difference between the true gradient and its estimate obtained by tracking the markers, that is, in the quantitative description of the induced error due to the data reduction. We derive a rigorous upper bound on the error, and we discuss what factors influence the error bound and the actual error itself. Finally, we illustrate the results by studying a practically interesting model problem. We show that different choices of the tracked markers can lead to substantially different estimates of the deformation gradient and its invariants. It is alarming that even qualitative features of the material under consideration, such as the incompressibility of the body, can be evaluated differently with different choices of the tracked markers. We also demonstrate that the derived error estimate can be used as a tool for choosing the appropriate marker set that leads to the deformation gradient estimate with the least guaranteed error.

**Freddi Lorenzo, Roubíček Tomáš, Zanini Chiara: Quasistatic delamination of sandwich-like Kirchhoff-Love plates, Journal of Elasticity 113 (2), 219-250, 2013**

A quasistatic rate-independent adhesive delamination problem of laminated plates with a finite thickness is considered. By letting the thickness of the plates go to zero, a rate-independent delamination model for a laminated Kirchhoff-Love plate is obtained as limit of these quasistatic processes. The same dimension reduction procedure is eventually applied to processes which are sensitive to delamination modes, namely opening vs. shearing is distinguished.

**Roubíček Tomáš, Panagiotopoulos Christos, Mantič Vladislav: Quasistatic adhesive contact of visco-elastic bodies and its numerical treatment for very small viscosity, ZAMM Zeitschrift für Angewandte Mathematik und Mechanik 93, 823-840, 2013**

An adhesive unilateral contact of elastic bodies with a small viscosity in the linear Kelvin-Voigt rheology at small strains is scrutinized. The flow-rule for debonding the adhesive is considered rate-independent and unidirectional, and inertia is neglected. The asymptotics for the viscosity approaching zero towards purely elastic material involves a certain defect-like measure recording in some sense natural additional energy dissipated in the bulk due to (vanishing) viscosity, which is demonstrated on particular 2-dimensional computational simulations based on a semi-implicit time discretisation and a spacial discretisation implemented by boundary-element method.

**Roubíček Tomáš: Adhesive contact of visco-elastic bodies and defect measures arising by vanishing viscosity, SIAM Journal on Mathematical Analysis 45 (1), 101-126, 2013**

An adhesive unilateral contact of elastic bodies with a small viscosity in the linear Kelvin-Voigt rheology at small strains is scrutinized. The flow-rule for debonding the adhesive is considered rate-independent and unidirectional. The asymptotics for the viscosity or for external loading speed approaching zero is proved in some special cases, in particular when inertia is neglected or when delamination is in Mode II (pure shear). The solutions thus obtained involve certain defect-like measures recording in some sense natural additional energy dissipated in the bulk due to (vanishing) viscosity. Reflecting also the conventional engineering concept, the delamination is thus driven rather by stress than energy. An explicit example leading to a nontrivial defect measure is given.

**Panagiotopoulos Christos, Mantič Vladislav, Roubíček Tomáš: BEM solution of delamination problems using an interface damage and plasticity model, Computational Mechanics 51, 505-521, 2013**

The problem of quasistatic and rate-independent evolution of elastic-plasticbrittle delamination at small strains is considered. Delamination processes for linear elastic bodies glued by an adhesive to each other or to a rigid outer surface are studied. The energy amounts dissipated in fracture Mode I (opening) and Mode II (shear) at an interface may be different. A concept of internal parameters is used here on the delaminating interfaces, involving a couple of scalar damage variable and a plastic tangential slip with kinematic-type hardening. The so-called energetic solution concept is employed. An inelastic process at an interface is devised in such a way that the dissipated energy depends only on the rates of internal parameters and therefore the model is associative. A fully implicit time discretization is combined with a spatial discretization of elastic bodies by the BEM to solve the delamination problem. The BEM is used in the solution of the respective boundary value problems, for each subdomain separately, to compute the corresponding total potential energy. Sample problems are analysed by a collocation BEM code to illustrate the capabilities of the numerical procedure developed.

**Bartels Soeren, Roubíček Tomáš: Numerical approaches to thermally coupled perfect plasticity, Numerical Methods for Partial Differential Equations 29 (6), 1837-1863, 2013**

The partial differential equations describing viscoelastic solids in Kelvin-Voigt rheology at small strains exhibiting also stress-driven Prandtl-Reuss perfect plasticity are considered and are coupled with a heat-transfer equation through the dissipative heat produced by viscoplastic effects and through thermal expansion and corresponding adiabatic effects. Numerical discretization of the resulting thermodynamically consistent model is proposed by implicit time discretization, suitable regularization, and finite elements in space. Numerical stability is shown and computational simulations are reported to illustrate the practical performance of the method. In a quasistatic case, convergence is proved by careful successive limit passage.

**Rossi Riccarda, Roubíček Tomáš: Adhesive contact delaminating at mixed mode, its thermodynamics and analysis, Interfaces and Free Boundaries 15 (1), 1-37, 2013**

An adhesive unilateral contact problem between visco-elastic heat-conductive bodies in linear Kelvin-Voigt rheology is scrutinized. The flow-rule for debonding the adhesive is considered rate-independent, unidirectional, and non-associative due to dependence on the mixity of modes of delamination, namely of Mode I (opening) and of Mode II (shearing). Such mode-mixity dependence of delamination is a very pronounced (and experimentally confirmed) phenomenon typically considered in engineering models. An anisothermal, thermodynamically consistent model is derived, considering a heat-conductive viscoelastic material and the coupling via thermal expansion and adhesion-depending heat transition through the contact surface. We prove the existence of weak solutions by passing to the limit in a carefully designed semi-implicit time-discretization scheme.

**Roubíček Tomáš, Tomassetti Giuseppe: Phase transformations in electrically conductive ferromagnetic shape-memory alloys, their thermodynamics and analysis, Archive for Rational Mechanics and Analysis 210 (1), 1-43, 2013**

We derive a thermodynamically consistent general continuum-mechanical model describing mutually coupled martensitic and ferro/paramagnetic phase transformations in electrically-conductive magnetostrictive materials such as NiMnGa. We use small-strain and eddy-current approximations, yet large velocities and electric current injected through the boundary are allowed. Fully nonlinear coupling of magnetomechanical and thermal effects is considered. The existence of energy-preserving weak solutions is proved by showing convergence of time-discrete approximations constructed by a carefully designed semi-implicit regularized scheme.

**Roubíček Tomáš: Nonlinearly coupled thermo-visco-elasticity, Nonlinear Differential Equations and Applications 20 (3), 1243-1275, 2013**

The d-dimensional thermo-visco-elasticity system for Kelvin-Voigt-type materials at small strains with a general nonlinear coupling is considered. Thermodynamical consistency leads to a heat capacity dependent both on temperature and on the strain. Using higher-gradient theory, namely the concept of so-called second-grade non-simple materials (or of hyper-stresses), existence of a weak solution to a system arising after an enthalpy-type transformation is proved by a suitably regularized Rothe method, fine a-priori estimates for the temperature gradient performed for the coupled system, and a subsequent limit passage.

**Roubíček Tomáš, Souček Ondřej, Vodicka Roman: A model of rupturing lithospheric faults with re-occurring earthquakes, SIAM Journal on Applied Mathematics 73, 1460-1488, 2013**

An isothermal small-strain model based on the concept of generalized standard materials is devised, combining Maxwell-type rheology, damage, and perfect plasticity in the bulk. An interface analogue of the model is prescribed at the lithospheric faults, exploiting concepts of adhesive contacts with interfacial plasticity. The model covers simultaneously features such as rupturing of the fault zone accompanied with weakening/healing effects and also seismic waves emission and propagation connected with the sudden ruptures of the fault or a fluidic-like aseismic response between the ruptures. Stable numerical strategy based on semi-implicit discretisation in time is devised and its convergence is shown. Numerical simulations documenting the capacity of the model to simulate earthquakes with repeating occurrence are performed, too.