Prague, Czech Republic, 17.–19. 12. 2010

In this presentation we discuss a "diffuse interface model" for the flow of two viscous Newtonian fluids in a bounded domain. Such models were introduced to describe the flow when singularities in the interface, which separates the fluids, (droplet formation/coalescence) occur. The fluids are assumed to be macroscopically immiscible, but a partial mixing in a small interfacial region is assumed in the model. Moreover, diffusion of both components is taken into account. This leads to a coupled Navier-Stokes/Cahn-Hilliard system. In the case of compressible fluids we discuss the existence of weak solutions globally in time. For incompressible fluids different models for the case of non-matched densities exist. We present two different models and compare their mathematical properties.

In this talk we present a series of qualitative results concerning the asymptotic behaviour of the solutions of a sequence of Navier-Stokes equations in domains with arbitrary roughness of vanishing amplitude. As well, we discuss the implication of these theoretical results in the drag minimization question of a fixed obstacle, with respect to the micro structure of its surface. The results were jointly obtained with E. Feireisl, Š. Nečasová and, respectively, M. Bonnivard.

We report on several basic results concerning viscous fluid flow around or past a rotating obstacle.
Consider a viscous incompressible flow around a body in $\mathbb R^3$ rotating with constant angular velocity $\omega\,||\,e_3$. Using a coordinate system attached to the body, the problem is reduced to a modified Navier-Stokes system in a fixed exterior domain.
The talk will address several basic question for the linearized case, e.g. *a priori* $L^q$-estimates, which cannot be proved by classical multiplier theory, and the structure of spectrum of the modified Stokes/Oseen operator with rotation terms; this latter problem is interesting since these operators fail to generate analytic semigroups in any $L^q$-space.
Moreover, we discuss the question of the asymptotic behavior of stationary and instationary solutions to the new system as $|x|\to\infty$. In contrast with the linear case the asymptotic behavior is shown to be different for the nonlinear system.
This results are based on joint papers with Š. Nečasová, J. Neustupa, M. Krbec, S. Kračmar, P. Penel, T. Hishida, D. Müller, G.P. Galdi, M. Kyed.

We present old and new results to solutions of the Prandtl-Reuss law and to solution of a classical model of elastic-plastic deformation with hardening. We present older results obtained with M. Bulíček and J. Málek concerning boundary differentiability of the stresses. Furtheremore, new results concerning fractional differentiability of the stress velocities are explained.

The main goal of this lecture is to review some recent results on the existence of global and exponential attractors of certain evolution systems related to a well-known diffuse interface model for (isothermal) binary fluid mixtures (aka model H). I will mainly concentrate myself on a model of immiscible incompressible fluids with shear dependent viscosity.The corresponding system consists of Navier-Stokes type equations, characterized by a nonlinear stress-strain law, which are nonlinearly coupled with a convective Cahn-Hilliard equation through the Korteweg force.

We study the Cauchy problem for scalar hyperbolic conservation laws $$u_t+{\rm div}\ F(x,u) =0, \quad u(0,x)=u_0(x)$$ with fluxes that can have countable jump discontinuities w.r.t. $x$ and $u$. We introduce a new concept of entropy weak and measure-valued solution that is consistent with the standard one for continuous fluxes. Then we answer the question what kind of properties of fluxes are needed to establish the existence and/or uniqueness of various notions of solutions. We establish existence of measure-valued entropy solution for fluxes having jump discontinuities. Under additional assumption on the Hölder continuity of flux at zero, we prove the uniqueness of entropy measure-valued solution. Finally, we establish the existence and uniqueness of weak entropy solution. We will also present relations of our solutions to solutions defined by Audusse and Perthame in the case of flux function discontinuous only w.r.t. space variable.

Joint work with Agnieszka Świerczewska-Gwiazda, Miroslav Bulíček and Josef Málek.

We provide a thermodynamic basis for compressible fluids of a Korteweg type that are characterized by the presence of the dyadic product of the density gradients $\nabla\rho\otimes\nabla\rho$ in the constitutive equation for the Cauchy stress. Following ideas by K.R. Rajagopal and A.R. Srinivasa, the derivation of the model is based on prescribing the constitutive equations for two scalars: the entropy and the entropy production. The entropy production will be given in terms of thermodynamical fluxes reflecting that these fluxes cause the changes in the thermodynamical affinities. The method will also be applied to the incompressible and the isothermal case.

This is a joint work with Eduard Feireisl (Institute of Mathematics of the Academy of Sciences of the Czech Republic) and Giulio Schimperna (University of Pavia, Italy). We present a model describing the evolution of a liquid crystal substance in the nematic phase is investigated in terms of three basic state variables: the absolute temperature, the velocity field, and the director field, representing preferred orientation of molecules in a neighborhood of any point of a reference domain. The time evolution of the velocity field is governed by the incompressible Navier-Stokes system, with a non-isotropic stress tensor depending on the gradients of the velocity and of the director field ${\bf d}$, where the transport (viscosity) coefficients vary with temperature. The dynamics of d is described by means of a parabolic equation of Ginzburg-Landau type, with a suitable penalization term to relax the constraint $|{\bf d}| = 1$. The system is supplemented by a heat equation, where the heat flux is given by a variant of Fourier’s law, depending also on the director field. The proposed model is shown compatible with First and Second laws of thermodynamics, and the existence of global-in-time weak solutions for the resulting PDE system is established, without any essential restriction on the size of the data.

In the talk we present mathematical models and numerical methods which we use for zebra-fish embryogenesis reconstruction from the large-scale 4D image sequences. Robust and efficient finite volume schemes for solving nonlinear advection-diffusion PDEs and level-set models related to filtering, object detection and segmentation of 3D images were designed to that goal and studied mathematically. They were parallelized for massively parallel computer clusters and applied to the mentioned problems in developmental biology and anticancer drug testing. The presented results were obtained in cooperation of groups at Slovak University of Technology, Bratislava, CNRS, Paris and University of Bologna.

I would like to talk about the issue of existence of traveling waves for the Boussinesq approximation. In three dimensional case we are limited by influence of the transport nonlinearity, so we are required to restrict the cross-section of channel with respect to magnitude of the Reynolds number. However thanks to an estimate controlling the the norm of commutator $[\Delta,P]$, where $P$ is the Helmholtz projector, we are able to give precise constraint of the restriction of the geometry of admissible cross-section of the channel. The talk will be based on the joint result with Marta Lewicka [On the existence of traveling waves in the 3D Boussinesq system. Comm. Math. Phys. 292 (2009), no. 2, 417–429.]

We present a strategy to describe the dynamics of crowds in heterogeneous domains. In this framework, the behavior of the crowd is considered from a two-fold perspective: both macroscopically and microscopically. On both scales we specify mass measures and their transport. Trusting recent results by Picolli, Cristiani and Tosin (arXiv:1006.0694v1 /2010) the micro and macro approaches are unified in a single model in which the distinction between the two scales enables us to capture both micro-interactions and the macro-dynamics of the crowd. Thus we benefit from the advantages of working with a continuum description, while we can also tract (i.e. zoom into) microscopic features. Apart from mass measures, the working tools include the use of porosity measures together with their transport as well as suitable application of a version of the Radon-Nikodym Theorem formulated for finite measures. We present preliminary results on the well-posedness of a particular time-discrete crowd scenario, and finally, we illustrate numerically the microscopic behavior of a crowd showing lane formation.

This is joint work with Joep Evers (TU Eindhoven, NL).)

The Cahn-Hilliard equation describes important qualitative features of the phase-separation and coarsening processes occurring in some binary materials, like, e.g., metallic alloys. Recently, a number of variants and extensions of the model have been proposed in order to cover specific situations relevant for physical applications. Often, these modified equations are characterized by the presence of more than one nonlinear term. In this talk, we will present a survey on some recent results regarding various doubly nonlinear forms of the Cahn-Hilliard equation.

For each model, we will first focus our attention on the basic question of existence of solutions in a proper weak formulation. Then, we will discuss about uniqueness, regularity and long-time behavior of weak solutions, in particular from the point of view of global attractors.

The results presented in the talk have been obtained in collaboration with various coauthors: Maurizio Grasselli (Politecnico di Milano), Alain Miranville (University of Poitiers), Irena Pawlow (Polish Academy of Sciences, Warsaw) and Riccarda Rossi (University of Brescia).

We consider the static and quasistatic Norton-Hoff modell as approximations of the Hencky respective Prandtl-Reuss law. We review known results on $H^1_{loc}$ resp. $L^{\infty}(H^1_{loc})$ regularity of the stress and show its Hölder continuity in 2D via inhomogeneous hole-filling (logarithmic Morrey condition).

We begin with a new optimal regularity result for a quasilinear elliptic problem with the p-Laplacian. We take advantage of this result to establish a suitable version of Pohozhaev's identity. Then we apply this identity to prove the nonexistence of a phase transition solution for our quasilinear elliptic phase transition model of Cahn-Hilliard-type in higher space dimension. In contrast, in one space dimension we show the existence of multidimensional submanifolds of phase transition solutions (of any dimension) in a Sobolev space.

Slides can be downloaded here.

We consider unsteady flows of incompressible fluids with a general implicit constitutive equation relating the deviatoric part of the Cauchy stress $S$ and the symmetric part of the velocity gradient $D$ in such a way that it leads to a maximal monotone (possibly multivalued) graph and the rate of dissipation is characterized by the sum of a Young function depending on $D$ and its conjugate being a function of $S$. Such a framework is very robust and includes, among others, classical power-law fluids, stress power-law fluids, fluids with activation criteria of Bingham or Herschel-Bulkley type, and shear-rate dependent fluids with discontinuous viscosities as special cases. The appearance of $S$ and $D$ in all the assumptions characterizing the implicit relationship $G(D,S) = 0$ is fully symmetric. We establish long-time and large-data existence of weak solution to such a system completed by the initial and Navier’s slip boundary conditions in both subcritical and supercritical cases. We use tools such as Orlicz functions, properties of spatially dependent maximal monotone operators and Lipschitz approximations of Bochner functions taking values in Orlicz-Sobolev spaces.

General numerical methods for solving incompressible viscous flows, which are based on LBB-stable FEM discretization techniques together with monolithic multigrid solvers are presented. We apply the fully implicit 2nd order time discretization method combined with high order Q2/P1 finite element pair for discretization in space. To treat the nonlinearities in each time step as well as for direct steady computations the resulting discrete system is solved via an approximate Newton method where the Jacobian matrices are calculate explicitly by finite differences. In each nonlinear step, the coupled linear subproblems are solved simultaneously for all quantities by means of a monolithic multigrid method with local multilevel pressure Schur complement smoothers of Vanka type. Several example applications are shown to demonstrate the robustness of this approach.

This talk represents an overview of the work on DG methods conducted by the author under the Nečas Center for Mathematical Modelling. Various theoretical results on convergence to the exact solution will be presented for nonlinear convection-diffusion problems. Especially of interest are newly derived estimates for the singularly perturbed case, which are uniform with respect to the diffusion coefficient and are valid even for the limiting nonlinear hyperbolic case. Several practical applications of the method will be presented, including fluid-structure interaction simulations concerning airfoil vibtations and flow through human vocal folds.

Numerical simulation of two-phase multicomponent flow in permeable media with species transfer between the phases often requires use of higher-order methods. Unlike first-order methods, higher-order methods may be very sensitive to problem formulation. The sensitivity to problem formulation and lack of recognition have hindered the widespread use of higher-order methods in various problems including improved oil recovery and sequestration from CO2 injection. In this work, we offer proper formulation of species balance equations and boundary conditions which overcome problems of formulations used previously that were detrimental to the efficiency of higher-order methods. We also present proper approximation of phase fluxes in the mixed finite element method. Our proposals remove major deficiencies in using higher-order methods in two-phase multicomponent flow. Numerical examples are presented to demonstrate robustness and efficiency of our approach.

We discuss stress relaxation and creep experiments of fluids that are generalizations of the classical model due to Burgers by allowing the material moduli such as the viscosities and relaxation and retardation times to depend on the stress. The physical problem, which is cast within the context of one dimension, leads to an ordinary differential equation that involves nonlinear terms like product of a function with a jump discontinuity and the derivative of a function with a jump discontinuity. As the equations are nonlinear, standard techniques that are used to study problems concerning linear viscoelastic fluids such as Laplace transforms and the theory of distributions are not applicable. We find it necessary to seek the solution in a more general setting. We discuss the mathematical and physical issues concerning the jump discontinuities and nonlinearity of the governing equation, and we show that the solution to the governing equation can be found in the sense of the generalized functions introduced by Colombeau. In the framework of Colombeau algebra we, under certain assumptions, derive jump conditions that shall be used in stress relaxation and creep experiments of fluids of the Burgers type. We conclude the talk with a discussion of the physical relevance of these assumptions.

(Joint work with K.R. Rajagopal.)

We consider the time evolution of incompressible fluids with power law-like rheology which slip at the boundary. The objective of this talk is to study the continuity and differentiability of the mapping that associates with a domain the solution of the respective system of PDEs in this domain. In the first part I will show sufficient conditions for the continuity in terms of smoothness of the boundary and discuss their optimality. The second part is devoted to the sensitivity analysis, namely the existence of material derivatives. This is joint work with Jan Sokolowski.