Michael Růžička

Lebesgue and Sobolev spaces with variable exponents

1 Introduction.- 2 A framework for function spaces.- 3 Variable exponent Lebesgue spaces.- 4 The maximal operator.- 5 The generalized Muckenhoupt condition*.- 6 Classical operators.- 7 Transfer techniques.- 8 Introduction to Sobolev spaces.- 9. Density of regular functions.- 10. Capacities.- 11 Fine properties of Sobolev functions.- 12 Other spaces of differentiable functions.- 13 Dirichlet energy integral and Laplace equation.- 14 PDEs and fluid dynamics

EXISTENCE AND REGULARITY OF SOLUTIONS AND THE STABILITY OF THE REST STATE FOR FLUIDS WITH SHEAR DEPENDENT VISCOSITY

In this paper we clarify and discuss some subtle features concerning the non-Newtonian fluid models which were considered by Malek, Necas and Růžicka.19 We establish new results regarding the stability of the rest state of mechanically isolated flows of such non-Newtonian fluids for arbitrary initial disturbances. We also discuss some results concerning the existence and regularity of solutions for small data. These results are based on a new method, giving convergence almost everywhere of approximations of gradients from boundedness of fraction of the L2-norm of the second and first derivatives, developed by Necas to study existence of solutions (global in time) to a class of partial differential equations.

An example of a space Lp(x) on which the Hardy-Littlewood maximal operator is not bounded

Abstract We give an example of a function p such that the Hardy-Littlewood maximal operator is not bounded on the generalized Lebesgue space L p(x) .

ON THE NON-NEWTONIAN INCOMPRESSIBLE FLUIDS

The Navier-Stokes equations can be included as a special case into the class of non-Newtonian incompressible fluids with the nonlinear stress tensor τ=τ(e), the components of which satisfy the p-growth condition. Measure-valued solutions already exist for p>2n/(n+2). For the space periodic problem, the existence of the weak solution is then obtained for p>3n/(n+2). These solutions are regular and unique for p≥1+2n/(n+2).

Flow of shear dependent electrorheological fluids

Abstract We prove the existence of weak and strong solutions to the steady and unsteady system of partial differential equations with non-standard growth conditions describing the flow of shear dependent electrorheological fluids.

HERSCHEL–BULKLEY FLUIDS: EXISTENCE AND REGULARITY OF STEADY FLOWS

The equations for steady flows of Herschel–Bulkley fluids are considered and the existence of a weak solution is proved for the Dirichlet boundary-value problem. The rheology of such a fluid is defined by a yield stress τ* and a discontinuous constitutive relation between the Cauchy stress and the symmetric part of the velocity gradient. Such a fluid stiffens if its local stresses do not exceed τ*, and it behaves like a non-Newtonian fluid otherwise. We address here a class of nonlinear fluids which includes shear-thinning p-law fluids with 9/5 < p ≤ 2. The flow equations are formulated in the stress-velocity setting (cf. Ref. 25). Our approach is different from that of Duvaut–Lions (cf. Ref. 10) developed for classical Bingham visco-plastic materials. We do not apply the variational inequality but make use of an approximation of the Herschel–Bulkley fluid with a generalized Newtonian fluid with a continuous constitutive law.

Monotone operator theory for unsteady problems in variable exponent spaces

We introduce function spaces for the treatment of parabolic equations with variable exponents by means of the theory of monotone operators. We generalize classical results such as density of smooth functions and a formula for integration by parts to prove existence, uniqueness and L 2-continuity of weak solutions to parabolic equations involving elliptic operators A with p(τ, x)-structure, where p is a globally log-Hölder continuous variable exponent satisfying 1 < p − ≤ p + < ∞.

A Dynamic Problem of Thermoelasticity

A space periodic problem of nonlinear thermoelasticity is considered. For an elastic, linear, isotropic, homogeneous, nonviscous body in small geometry, we obtain a nonlinear system of equations. For small coefficient of the heat extension a we find a time global weak solution of the initial-value problem. The smallness of a is independent of the length of the time interval and of the datas. The space periodicity of the solution is related to the absence of reflected waves. A mixed problem for a bounded domain, even with a smooth boundary, seems to be an open problem. Our work is closely related to that by J. Necas L5J and by J. Necas, A. Novotn and V. Sverak L6J.

On the Finite Element Approximation of p-Stokes Systems

In this paper we study the finite element approximation of systems of p-Stokes type for

Large Data Existence Result for Unsteady Flows of Inhomogeneous Shear-Thickening Heat-Conducting Incompressible Fluids

p \in (1,\infty)