Piotr Gwiazda

On Unsteady Flows of Implicitly Constituted Incompressible Fluids

We consider unsteady flows of incompressible fluids with a general implicit con- stitutive 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 the Navier slip boundary conditions in both the subcrit- ical 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.

On steady flows of incompressible fluids with implicit power-law-like rheology

Abstract We consider steady flows of incompressible fluids with power-law-like rheology given by an implicit constitutive equation relating the Cauchy stress and the symmetric part of the velocity gradient in such a way that it leads to a maximal monotone (possibly multivalued) graph. Such a framework includes standard Navier–Stokes and power-law fluids, Bingham fluids, Herschel–Bulkley fluids, and shear-rate dependent fluids with discontinuous viscosities as special cases. We assume that the fluid adheres to the boundary. Using tools such as the Young measures, properties of spatially dependent maximal monotone operators and Lipschitz approximations of Sobolev functions, we are able to extend the results concerning large data existence of weak solutions to those values of the power-law index that are of importance from the point of view of engineering and physical applications.

ON NON-NEWTONIAN FLUIDS WITH A PROPERTY OF RAPID THICKENING UNDER DIFFERENT STIMULUS

The paper concerns the model of a flow of non-Newtonian fluid with nonstandard growth conditions of the Cauchy stress tensor. Contrary to standard power-law type rheology, we propose the formulation with the help of the spatially-dependent convex function. This framework includes e.g. rapidly shear thickening and magnetorheological fluids. We provide the existence of weak solutions. The nonstandard growth conditions yield the analytical formulation of the problem in generalized Orlicz spaces. Basing on the energy equality, we exploit the tools of Young measures.

STRUCTURED POPULATION EQUATIONS IN METRIC SPACES

In this paper, a framework for the analysis of measure-valued solutions of the nonlinear structured population model is presented. Existence and Lipschitz dependence of the solutions on the model parameters and initial data are shown by proving convergence of a variational approximation scheme, defined in the terms of a suitable metric space. The estimates for a corresponding linear model are used based on the duality formula for transport equations. An extension of a Wasserstein metric to the measures with integrable first moment is proposed to cope with the nonconservative character of the model. This metric is compared with a bounded Lipschitz distance, also called a flat metric, and the results are discussed in the context of applications to biological data.

Models of Discrete and Continuous Cell Differentiation in the Framework of Transport Equation

We introduce a class of structured population models describing cell differentiation that consists of discrete and continuous transitions. The model is defined in a framework of measure-valued solutions of a nonlinear transport equation with a growth term. To obtain ODE-type quasi-stationary node points we exploit the idea of non-Lipschitz zeroes in the velocity. This, in combination with the so-called measure-transmission conditions, allows us to prove the existence and uniqueness of solutions. Since the analysis has biological motivations, we provide examples of its application. (An erratum is attached.)

On the model of Bodner-Partom with nonhomogeneous boundary data

In the present paper we study existence and the uniqueness of global in time, strong, large solutions to the inelastic model of Bodner-Partom with nonhomogeneous boundary data, and with the perturbation term in the equation for the isotropic hardening function . Moreover we consider the limit case and prove the convergence result in a suitable topology to the unperturbed problem.

Elliptic problems in generalized Orlicz-Musielak spaces

We consider a strongly nonlinear monotone elliptic problem in generalized Orlicz-Musielak spaces. We assume neither a Δ2 nor ∇2-condition for an inhomogeneous and anisotropic N-function but assume it to be log-Hölder continuous with respect to x. We show the existence of weak solutions to the zero Dirichlet boundary value problem. Within the proof the L∞-truncation method is coupled with a special version of the Minty-Browder trick for non-reflexive and non-separable Banach spaces.

On steady non-Newtonian fluids with growth conditions in generalized Orlicz spaces

We are interested in the existence of weak solutions to steady non-Newtonian fluids with nonstandard growth conditions of the Cauchy stress tensor. Since the

Splitting-Particle Methods for Structured Population Models: Convergence and Applications

L^p

On flows of an incompressible fluid with a discontinuous power-law-like rheology

framework is not suitable to capture the description of strongly inhomogeneous fluids, we formulate the problem in generalized Orlicz spaces. The existence proof consists in showing that for Galerkin approximations the sequence of symmetric gradients of the flow velocity converges modularly. As an example of motivation for considering non-Newtonian fluids in generalized Orlicz spaces we recall the smart fluids.