# List of papers published in 2014

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 2014. If you are interested in what has been done last year, please see the list below (journals with impact factor only).

Feireisl Eduard, Novotný Antonín, Sun Yongzong: A regularity criterion for the weak solutions to the Navier-Stokes-Fourier system, Archive for Rational Mechanics and Analysis 212 (1), pp. 219-239, 2014

We show that any weak solution to the full Navier-Stokes-Fourier system emanating from the data belonging to the Sobolev space $W^{3,2}$ remains regular as long as the velocity gradient is bounded. The proof is based on the weak-strong uniqueness property and parabolic a priori estimates for the local strong solutions.

Kalousová Klára, Souček Ondřej, Tobie Gabriel, Choblet Gael, Čadek Ondřej: Ice melting and downward transport of meltwater by two-phase flow in Europa’s ice shell, Journal of Geophysical Research-Planets 119 (3) , pp. 532-549, 2014

With its young surface, very few impact craters, and the abundance of tectonic and cryovolcanic features, Europa has likely been subjected to relatively recent endogenic activity. Morphological analyses of chaos terrains and double ridges suggest the presence of liquid water within the ice shell a few kilometers below the surface, which may result from enhanced tidal heating. A major issue concerns the thermal/gravitational stability of these water reservoirs. Here we investigate the conditions under which water can be generated and transported through Europa’s ice shell. We address particularly the downward two-phase flow by solving the equations for a two-phase mixture of water ice and liquid water in one-dimensional geometry. In the case of purely temperate ice, we show that water is transported downward very efficiently in the form of successive porosity waves. The time needed to transport the water from the subsurface region to the underlying ocean varies between approximate to 1 and 100 kyr, depending mostly on the ice permeability. We further show that water produced in the head of tidally heated hot plumes never accumulates at shallow depths and is rapidly extracted from the ice shell (within less than a few hundred kiloyears). Our calculations indicate that liquid water will be largely absent in the near subsurface, with the possible exception of cold conductive regions subjected to strong tidal friction. Recently active double ridges subjected to large tidally driven strike-slip motions are perhaps the most likely candidates for the detection of transient water lenses at shallow depths on Europa.

Bulíček Miroslav, Pustějovská Petra: Existence analysis for a model describing flow of an incompressible chemically reacting non-newtonian fluid, SIAM Journal on Mathematical Analysis 2014 (46), pp. 3223-3240, 2014

We consider a system of PDEs describing steady motions of an incompressible chemically reacting non-Newtonian fluid. The system of governing equations is composed of the convection diffusion equation for concentration and generalized Navier-Stokes equations where the generalized viscosity depends polynomially on the shear rate (the modulus of the symmetric part of the velocity gradient) and the coupling is due to the dependence of the power-law index on the concentration. This dependence of the power- law index on the solution itself causes the main difficulties in the analysis of the relevant boundary value problem. We generalize the Lipschitz approximation method and showthe existence of a weak solution provided that the minimal value of the power-law exponent is bigger than d/2.

Feireisl Eduard, Lu Yong, Novotný Antonín: Rotating compressible fluids under strong stratification, Nonlinear Analysis: Real World Applications 19, pp. 11-18, 2014

We consider the Navier-Stokes system written in the rotational frame describing the motion of a compressible viscous fluid under strong stratification. The asymptotic limit for low Mach and Rossby numbers and large Reynolds number is studied on condition that the Froude number characterizing the degree of stratification is proportional to the Mach number. We show that, at least for the well prepared data, the limit system is the same as for the problem without stratification-a variant of the incompressible planar Euler system.

Mucha Piotr Boguslaw, Pokorný Milan, Zatorska Ewelina: Approximate solutions to a model of two-component reactive flow, Discrete and Continuous Dynamical Systems – Series S 7 (5), pp. 1079-1099, 2014

We consider a model of motion of binary mixture, based on the compressible Navier-Stokes system. The mass balances of chemically reacting species are described by the reaction-diffusion equations with generalized form of multicomponent diffusion flux. Under a special relation between the two density dependent viscosity coefficients and for singular cold pressure we construct the weak solutions passing through several levels of approximation.

Roubíček Tomáš, Vodička Roman, Mantič Vlado: Energetic versus maximally-dissipative local solutions of a quasi-static rate- independent mixed-mode delamination model, Meccanica 49, pp. 2933-2963, 2014

A quasi-static rate-independent model of delamination of linearly elastic bodies at small strains, sensitive to mode of delamination, using interfacial damage and interfacial plasticity as two internal parameters, is further developed with the aim to extract representations typically employed in engineering interface-models, i.e. fracture envelope and fracture energy dependence on the mode mixity, which are suitable for the model fitting to experimental data. Moreover, two concepts of solutions are implemented: globally stable energy-conserving solutions or stress-driven maximally-dissipative local solutions, arising by the fully implicit or by a semi-implicit timestepping procedures, respectively, both yielding numerically stable and convergent time-discretizations. Spatial discretization is performed by the symmetric Galerkin boundary-element method (SGBEM). Alternating quadratic programming is implemented to cope with, respectively, global or local, energy-minimizations in the computation of the time-discretized solutions. Sample 2D numerical examples document applicability of the model as well as efficiency of the SGBEM numerical implementation and facilitate comparison of the two mentioned solution concepts.

Roubíček Tomáš, Tomassetti Giuseppe: Thermomechanics of hydrogen storage in metallic hydrides: Modeling and analysis, Discrete and Continuous Dynamical Systems – Series B 19 (7), pp. 2313-2333, 2014

A thermodynamically consistent mathematical model for hydrogen adsorption in metal hydrides is proposed. Beside hydrogen diffusion, the model accounts for phase transformation accompanied by hysteresis, swelling, temperature and heat transfer, strain, and stress. We prove existence of solutions of the ensuing system of partial differential equations by a carefully-designed, semi-implicit approximation scheme. A generalization for a drift-diffusion of multi-component ionized "gas" is outlined, too.

Roubíček Tomáš: A note about the rate-and-state-dependent friction model in a thermodynamical framework of the Biot-type equation, Geophysical Journal International 199, pp. 286-295, 2014

The conventional, phenomenological rate-and-state-dependent friction model of Dieterich- Ruina’s type is discussed and slightly modified so that, after introducing an artificial internal variable (formally in a position like effective temperature) on the fault, it is driven by a free and a dissipative energies. In contrast to the original model, it thus allows for a formulation in the framework of rational thermodynamics, including the energy balance, and for rigorous numerical analysis. This also suggests an analogous rate-and-state-dependent plastic bulk model using damage/temperature as the state variable controlling the plastic yield stress.

Panagiotopoulos C. G., Mantič V., Roubíček Tomáš: A simple and efficient BEM implementation of quasistatic linear visco-elasticity, International Journal of Solids and Structures 51 (13), pp. 2261-2271, 2014

A simple yet efficient procedure to solve quasistatic problems of special linear visco-elastic solids at small strains with equal rheological response in all tensorial components, utilizing boundary element method (BEM), is introduced. This procedure is based on the implicit discretisation in time (the so-called Rothe method) combined with a simple "algebraic" transformation of variables, leading to a numerically stable procedure (proved explicitly by discrete energy estimates), which can be easily implemented in a BEM code to solve initial-boundary value visco-elastic problems by using the Kelvin elastostatic fundamental solution only. It is worth mentioning that no inverse Laplace transform is required here. The formulation is straightforward for both 2D and 3D problems involving unilateral frictionless contact. Although the focus is to the simplest Kelvin-Voigt rheology, a generalization to Maxwell, Boltzmann, Jeffreys, and Burgers rheologies is proposed, discussed, and implemented in the BEM code too. A few 2D and 3D initial-boundary value problems, one of them with unilateral frictionless contact, are solved numerically.

Bulíček Miroslav, Frehse Jens, Steinhauer Mark: Everywhere C-alpha-estimates for a class of nonlinear elliptic systems with critical growth, Advances in Calculus of Variations 7 (2), pp. 139-204, 2014

We obtain everywhere C-alpha-regularity for vector solutions to a class of nonlinear elliptic systems whose principal part is the Euler operator to a variational integral with quadratic growth in gradient of the unknown and which satisfies a generalized splitting condition and the one-sided condition. If the leading operator is not necessarily elliptic but coercive, possible minima are everywhere Holder continuous and the same holds also for Noether solutions, i.e., extremals which are also stationary with respect to inner variations.The technique of our proof (using weighted norms and inhomogeneous hole-filling method) does not rely on L-infinity-a priori estimates for the solution.

Bulíček Miroslav, Málek Josef, Shilkin Timofei Nikolaevich: On the regularity of two-dimensional unsteady flows of heat-conducting generalized Newtonian fluids, Nonlinear Analysis: Real World Applications 19, pp. 89-104, 2014

We study regularity properties of unsteady flows of an incompressible heat- conducting fluid in a two-dimensional spatially periodic setting. Under certain structural assumptions on the Cauchy stress that include generalizations of the Ladyzhenskaya or power-law like models we establish the existence of a classical solution to such problems.

Bulíček Miroslav, Málek Josef, Rajagopal K. R., Süli Endre: On elastic solids with limiting small strain: modelling and analysis, EMS Surveys in Mathematical Sciences 2014 (1), pp. 283-332, 2014

In order to understand nonlinear responses of materials to external stimuli of different sort, be they of mechanical, thermal, electrical, magnetic, or of optical nature, it is useful to have at one’s disposal a broad spectrum of models that have the capacity to describe in mathematical terms a wide range of material behavior. It is advantageous if such a framework stems from a simple and elegant general idea. Implicit constitutive theory of materials provides such a framework: while being built upon simple ideas, it is able to capture experimental observations with the minimum number of physical quantities involved. It also provides theoretical justification in the full three-dimensional setting for various models that were previously proposed in an ad hoc manner. From the perspective of the theory of nonlinear partial differential equations, implicit constitutive theory leads to new classes of challenging mathematical problems. This study focuses on implicit constitutive models for elastic solids in general, and on its subclass consisting of elastic solids with limiting small strain. After introducing the basic concepts of implicit constitutive theory, we provide an overview of results concerning modeling within the framework of continuum mechanics. We then concentrate on the mathematical analysis of relevant boundary-value problems associated with models with limiting small strain, and we present the first analytical result concerning the existence of weak solutions in general three-dimensional domains.

Souček Ondřej, Kalousová Klára, Čadek Ondřej: Water transport in planetary ice shells by two-phase flow – a parametric study, Geophysical and Astrophysical Fluid Dynamics 108 (6), pp. 639-666, 2014

We present a two-phase model for the generation of meltwater and its propagation through the outer shells of icy satellites such as Europa, Enceladus or Titan. We exploit the analogy with the process of partial melt generation in the Earth’s interior by adopting the formalism of two-phase flow developed in the mantle-dynamics community, and by means of scaling analysis we derive a reduced system appropriate for our planetary application. The resultant system couples Darcy’s law with the deformation of the viscous ice matrix. We numerically investigate the system in a simplified one-dimensional geometry, corresponding to a laterally uniform ice layer, and analyze the role of various physical parameters. We focus on the leading-order effects, namely (i) the key importance of ice permeability, (ii) the role of complex ice rheology depending on temperature, deformation mechanisms and water content, (iii) the possible contribution of surface tension and (iv) the effects of mechanical coupling between the phases. Our analysis suggests that the global water transport through temperate ice is mainly controlled by ice permeability and can be well approximated by a model in which the complex ice rheology is parameterized in terms of a constant viscosity. While the mechanical coupling between the phases dramatically affects the flow at the local scale, the surface tension appears to be insignificant.

Souček Ondřej, Průša Vít, Málek Josef, Rajagopal K. R.: On the natural structure of thermodynamic potentials and fluxes in the theory of chemically non-reacting binary mixtures, Acta Mechanica 225 (11), pp. 3157-3186, 2014

A theory describing the behavior of chemically non-reacting binary mixtures can be based on a detailed formulation of the governing equations for the individual components of the mixture or on treating the mixture as a single homogenized continuous medium. We argue that if we accept that both approaches can be used to describe the behavior of the given mixture, then the requirement on the equivalence of these approaches places restrictions on the possible structure of the internal energy, entropy, Helmholtz potential, and also of the diffusive, energy, and entropy fluxes. (The equivalence of the approaches is understood in the sense that the quantities used in one approach can be interpreted in terms of the quantities used in the other approach and vice versa. Further, both approaches must lead to the same predictions concerning the evolution of the physical system under consideration). In the case of a general chemically non-reacting binary mixture of components at the same temperature, we show that these restrictions can indeed be obtained by purely algebraic manipulations. An important outcome of this analysis is, for example, a general form of the evolution equation for the diffusive flux. The restrictions can be further exploited in the specification of thermodynamically consistent constitutive relations for quantities such as the interaction (drag) force or the Cauchy stress tensor. As an example of the application of the current framework, we derive, among others, a generalization of Fick’s law and we recover several non-trivial results obtained by other techniques. The qualitative features of the derived generalization of Fick’s law are demonstrated by a numerical experiment.

Hron Jaroslav, Rajagopal K. R., Tůma Karel: Flow of a Burgers fluid due to time varying loads on deforming boundaries, Journal of Non-Newtonian Fluid Mechanics 210, pp. 66-77, 2014

In this paper we study three boundary-initial value problems within the context of four rate type viscoelastic constitutive models, the Maxwell model, the Oldroyd-B model, Burgers model and the generalized Burgers model. We consider challenging problems wherein the boundary is deforming with time. The flows lead to a complex system of partial differential equations that require the development of a robust numerical procedure based on the arbitrary Lagrangian-Eulerian method.

Mucha Piotr B., Pokorný Milan: The Rot-Div system in exterior domains, Journal of Mathematical Fluid Mechanics 16 (4), pp. 701-720, 2014

The goal of this paper is to reconsider the classical elliptic system rot v = f, div v = g in simply connected domains with bounded connected boundaries (bounded and exterior sets). The main result shows solvability of the problem in the maximal regularity regime in the L (p) -framework taking into account the optimal/minimal requirements on the smoothness of the boundary. A generalization for the Besov spaces is studied, too, for for . As a limit case we prove the result for , provided the boundary is merely in . The dimension three is distinguished due to the physical interpretation of the system. In other words we revised and extended the classical results of Friedrichs (Commun Pure Appl Math 8;551-590, 1955) and Solonnikov (Zap Nauch Sem LOMI 21:112-158, 1971).

Piasecki Tomasz, Pokorný Milan: Strong solutions to the Navier-Stokes-Fourier system with slip-inflow boundary conditions, ZAMM Zeitschrift für Angewandte Mathematik und Mechanik 94 (12), pp. 1035-1057, 2014

We consider a system of partial differential equations describing the steady flow of a compressible heat conducting Newtonian fluid in a three-dimensional channel with inflow and outflow part. We show the existence of a strong solution provided the data are close to a constant, but nontrivial flow with sufficiently large dissipation in the energy equation.

Jesslé Didier, Novotný Antonín, Pokorný Milan: Steady Navier-Stokes-Fourier system with slip boundary conditions, Mathematical Models and Methods in Applied Sciences 24 (4), pp. 751-781, 2014

We consider a problem modelling the steady flow of a compressible heat conducting Newtonian fluid subject to the slip boundary condition for the velocity. Assuming the pressure law of the form $p(\varrho,\vartheta) \sim \varrho^\gamma + \varrho \vartheta$, we show (under additional assumptions on the heat conductivity and the viscosity) that for any $\gamma >1$ there exists a variational entropy solution to our problem (i.e. the weak formulation of the total energy balance is replaced by the entropy inequality and the global total energy balance). Moreover, if $\gamma > \frac 54$ (together with further restrictions on the heat conductivity), the solution is in fact a weak one. The results are obtained without any restriction on the size of the data.

Roubíček Tomáš, Stefanelli Ulysse: Magnetic shape-memory alloys: thermomechanical modeling and analysis, Continuum Mechanics and Thermodynamics, pp. 783-810, 2014

A phenomenological model for the coupled thermo-electro-magneto- mechanical and phasetransformation behaviour of magnetic shape-memory alloys is advanced in small strains and eddy current approximation. The corresponding system of strongly nonlinear relations is tackled via a suitable enthalpylike transformation. A fully implicit regularized time-discretization scheme is devised and proved to be stable and convergent. In particular, the convergence proof for discrete solutions entails that a suitably weak, energyconserving solution to the continuous nonlinear system exists. Moreover, several particular models as e.g. ferro/paramagnetic transformation in ferromagnetic materials, martensitic transformation in shape memory allows, or just a simple thermistor problem are covered just as special cases.

Feireisl Eduard, Novotný Antonín: Multiple scales and singular limits for compressible rotating fluids with general initial data, Communications in Partial Differential Equations 39 (6), 1104-1127, 2014

We study the singular limit of a rotating compressible fluid described by a scaled barotropic Navier-Stokes system, where the Rossby number, the Mach number, the Reynolds number, and the Froude number are proportional to a small parameter E0. The inviscid planar Euler system is identified as the limit problem. The proof is based on the application of the method of relative entropies and careful analysis of oscillatory integrals describing the propagation of Rossby-acoustic waves.

Janečka Adam, Průša Vít: The motion of a piezoviscous fluid under a surface load, International Journal of Non-Linear Mechanics 60, pp. 23-32, 2014

Using a variant of a spectral collocation method we numerically solve the problem of the motion of a highly viscous fluid with pressure dependent viscosity under a surface load, which is a problem relevant in many applications, in particular in geophysics and polymer melts processing. We compare the results with the results obtained by the classical Navier-Stokes fluid (constant viscosity). It turns out that for a realistic parameter values the two models give substantially different predictions concerning the motion of the free surface and the velocity and the pressure fields beneath the free surface. As a byproduct of the effort to test the numerical scheme we obtain an analytical solution for the classical Navier-Stokes fluid of the surface load problem in a layer of finite depth.