Grzegorz Łukaszewicz

ON PULLBACK ATTRACTORS IN

Using a recent method based on the concept of the Kuratowski measure of noncompactness of a bounded set together with some new estimates of solutions, we prove the existence of a unique minimal pullback attractor for the evolutionary process associated with a nonautonomous nonlinear reaction–diffusion system in in which the right-hand side satisfies only a certain integrability condition. In particular, we generalize a result obtained recently in [Li & Zhong, 2007] where at most an exponential growth of the right-hand side has been assumed for times going to both plus and minus infinity.

H^1_0

Abstract We consider two-dimensional nonstationary Navier–Stokes shear flow with multivalued and nonmonotone boundary conditions on a part of the boundary of the flow domain. We prove the existence of global in time solutions of the considered problem which is governed by a partial differential inclusion with a multivalued term in the form of Clarke subdifferential. Then we prove the existence of a trajectory attractor and a weak global attractor for the associated multivalued semiflow. This research is motivated by control problems for fluid flows in domains with semipermeable walls and membranes.

FOR NONAUTONOMOUS REACTION–DIFFUSION EQUATIONS

Abstract This paper is devoted to some recent results on several aspects of long time behavior of micropolar fluid flows. In particular, we consider such topics as existence and uniqueness of global in time solutions, their convergence to the stationary solution for large viscosity flows, existence of a global attractor and estimates of its Hausdorff and fractal dimensions, continuous dependence of solutions on microrotation viscosity perturbations and stability of the corresponding global attractor with respect to these perturbations, and flows in unbounded domains.

Attractors for Navier–Stokes flows with multivalued and nonmonotone subdifferential boundary conditions

The asymptotic behavior of a Stokes flow with Fourier boundary condition on one part on the boundary and Tresca free boundary friction condition on the other, when one dimension of the fluid domain tends to zero is studied. The strong convergence of the velocity is proved, a specific Reynolds equation is obtained, and the uniqueness of the limit velocity and pressure distributions is established.

Asymptotic behavior of micropolar fluid flows

Abstract We study a class of abstract nonlinear evolution equations in a separable Hilbert space for which we prove existence of strong time periodic solutions, provided the right-hand side is periodic and C1 in time, and small enough in the norm of the considered space. We prove also uniqueness and stability of the solutions. The results apply, in particular, in several models of hydrodynamics, such as magneto-micropolar and micropolar models, and classical magnetohydrodynamics and Navier–Stokes models with non-homogeneous boundary conditions. The existence part of the proof is based on a set of estimates for the family of finite-dimensional approximate solutions.

ON A LUBRICATION PROBLEM WITH FOURIER AND TRESCA BOUNDARY CONDITIONS

The asymptotic behavior of a Stokes flow with Coulomb free boundary friction condition when one dimension of the fluid domain tends to zero is studied. The strong convergence of the velocity is proved, a specific Reynolds equation is obtained, and the uniqueness of the limit velocity and pressure distributions is established.

Strong periodic solutions for a class of abstract evolution equations

We consider the bidimensional stationary Stefan problem with convection. The problem is governed by a coupled system involving a non-linear Darcys law and the energy balance equation with second member in L1. We prove existence of at least one weak solution of the problem, using the penalty method and the Schauder fixed point principle. Copyright

Asymptotic analysis of solutions of a thin film lubrication problem with Coulomb fluid–solid interface law

In this chapter we introduce some basic notions from the theory of the Navier–Stokes equations: the function spaces H, V, and V ′, the Stokes operator A with its domain D(A) in H, and the bilinear form B. We apply the Galerkin method and fixed point theorems to prove the existence of solutions of the nonlinear stationary problem, and we consider problems of uniqueness and regularity of solutions.

The stationary Stefan problem with convection governed by a non-linear Darcy's law

We consider two classes of evolution contact problems on two dimensional domains governed by first and second order evolution equations, respectively. The contact is represented by multivalued and nonmonotone boundary conditions that are expressed by means of Clarke subdifferentials of certain locally Lipschitz and semiconvex potentials. For both problems we study the existence and uniqueness of solutions as well as their asymptotic behavior in time. For the first order problem, that is governed by the Navier–Stokes equations with generalized Tresca law, we show the existence of global attractor of finite fractal dimension and existence of exponential attractor. For the second order problem, representing the frictional contact in antiplane viscoelasticity, we show that the global attractor exists, but both the global attractor and the set of stationary states are shown to have infinite fractal dimension.

Stationary Solutions of the Navier–Stokes Equations

A method is proposed to deal with some multivalued processes with weak continuity properties. An application to a nonautonomous contact problem for the Navier–Stokes flow with nonmonotone multivalued frictional boundary condition is presented.