G. Łukaszewicz

G. Łukaszewicz
University of Warsaw | UW · Institute of Mathematics

Prof. Dr hab.

About

99
Publications
7,319
Reads
How we measure 'reads'
A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more
2,715
Citations

Publications

Publications (99)
Article
Full-text available
This paper studies the nonautonomous micropolar fluid with generalized Newton constitutive law in two‐dimensional bounded domains. We first establish that the generated continuous process of the solutions operator possesses a pullback attractor. Then we verify the existence of statistical solutions by constructing the invariant Borel probability me...
Preprint
Full-text available
This paper studies the non-autonomous non-Newtonian micropolar fluids in two-dimensional bounded domains. We first establish that the generated continuous process of the solutions operator possesses a pullback attractor. Then we verify the existence of statistical solutions by constructing the invariant Borel probability measures. Further, we prove...
Article
In this article, the authors investigate the system of Schrödinger and Klein-Gordon equations with Yukawa coupling. They first prove the existence of pullback attractor and construct a family of invariant Borel probability measures. Then they establish that this family of probability measures satisfies a Liouville type theorem and is indeed a stati...
Article
Full-text available
We investigate the upper bound on the vertical heat transport in the fully 3D Rayleigh–Bénard convection problem at the infinite Prandtl number for a micropolar fluid. We obtain a bound, given by the cube root of the Rayleigh number, with a logarithmic correction. The derived bound is compared with the optimal known one for the Newtonian fluid. It...
Article
Full-text available
We consider the Rayleigh–Bénard problem for the three-dimensional Boussinesq system for the micropolar fluid. We introduce the notion of the multivalued eventual semiflow and prove the existence of the two-space global attractor AK corresponding to weak solutions, for every micropolar parameter K ⩾ 0 denoting the deviation of the considered system...
Article
In this paper, we consider a steady-state flow the thermomicropolar fluid through a thin straight channel. The flow is governed by the prescribed pressure drop between channel's ends. The heat exchange between the fluid inside the channel and the exterior medium is allowed through the upper wall, whereas the lower wall is insulated. Using the asymp...
Article
Full-text available
In this article, the authors investigate the non-autonomous magneto-micropolar fluids in a two-dimensional bounded domain. They first prove the existence of a pullback attractor for the associated process. Then, they construct a family of invariant Borel probability measures supported on the pullback attractor and prove that this family of probabil...
Chapter
In this paper we present some mutual relations between semigroup theory in the context of the theory of infinite dimensional dynamical systems and the mathematical theory of hydrodynamics. These mutual relations prove to be very fruitful, enrich both fields and help to understand behaviour of solutions of both infinite dimensional dynamical systems...
Article
Full-text available
Sometimes it is not possible to prove the uniqueness of the weak solutions for problems of mathematical physics, but it is possible to bootstrap their regularity to the regularity of strong solutions which are unique. In this paper we formulate an abstract setting for such class of problems and we provide the conditions under which the global attra...
Preprint
Full-text available
We consider the Rayleigh--B\'{e}nard problem for the three--dimensional Boussinesq system for the micropolar fluid. We introduce the notion of the multivalued eventual semiflow and prove the existence of the two-space global attractor $\mathcal{A}^K$ corresponding to weak solutions, for every micropolar parameter $K\geq 0$ denoting the deviation of...
Article
We consider the Rayleigh–Bénard problem for the two-dimensional Boussinesq system for the micropolar fluid. Our main goal is to compare the value of the critical Rayleigh number, and estimates of the Nusselt number and the fractal dimension of the global attractor with those values for the same problem for the classical Navier–Stokes system. Our es...
Article
Full-text available
In this article, we first provide a sufficient and necessary condition for the existence of a pullback-D attractor for the process defined on a Hilbert space of infinite sequences. As an application, we investigate the nonautonomous discrete Klein-Gordon-Schrödinger system of equations, prove the existence of the pullback-D attractor and then the e...
Article
Full-text available
The two-dimensional Rayleigh–Bénard problem for a thermomicropolar fluids model is considered. The existence of suitable weak solutions which may not be unique, and the existence of the unique strong solution are proved. The global attractor for the m-semiflow associated with weak solutions and the global attractor for semiflow associated with stro...
Research
The Navier-Stokes equations, which describe the movement of fluids, are an important source of topics for scientific research, technological development and innovation. Fluids are part of the systems of nature, life and society. Involved in many processes and phenomena of everyday life as an active part of the interactions within the hosting system...
Data
The Navier-Stokes equations, which describe the movement of fluids, are an important source of topics for scientific research, technological development and innovation. Fluids are part of the systems of nature, life and society. Involved in many processes and phenomena of everyday life as an active part of the interactions within the hosting system...
Chapter
In this chapter we give an overview of the equations of classical hydrodynamics. We provide their derivation, comment on the stress tensor, and thermodynamics, finally we present some elementary properties and also some exact solutions of the Navier–Stokes equations.
Chapter
Full-text available
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 pr...
Chapter
We start this chapter from necessary background on the theory of fractal dimension. Next, we formulate and study a problem which models the two-dimensional boundary driven shear flow in lubrication theory. After the derivation of the energy dissipation rate estimate and a version of Lieb–Thirring inequality we provide an estimate from above on the...
Chapter
In this chapter we consider two-dimensional nonstationary incompressible Navier–Stokes shear flows with nonmonotone and multivalued leak boundary conditions on a part of the boundary of the flow domain. Our considerations are motivated by feedback control problems for fluid flows in domains with semipermeable walls and membranes and by the theory o...
Chapter
This chapter is devoted to the study of three-dimensional nonstationary Navier–Stokes equations with the multivalued frictional boundary condition. We use the formalism of evolutionary systems to prove the existence of weak global attractor for the studied problem.
Chapter
In this chapter we consider the problem of existence and finite dimensionality of the pullback attractor for a class of two-dimensional turbulent boundary driven flows which naturally appear in lubrication theory. We generalize here the results from Chap. 9 to the non-autonomous problem.
Chapter
This chapter provides, for the convenience of the reader, an overview of the whole book, first of its structure and then of the content of the individual chapters.
Chapter
In this chapter we consider the three-dimensional stationary Navier–Stokes equations with multivalued friction law boundary conditions on a part of the domain boundary. We formulate two existence theorems for the formulated problem. The first one uses the Kakutani–Fan–Glicksberg fixed point theorem, and the second one, with the relaxed assumptions,...
Chapter
In this chapter we introduce the basic preliminary mathematical tools to study the Navier–Stokes equations, including results from linear and nonlinear functional analysis as well as the theory of function spaces. We present, in particular, some of the most frequently used in the sequel embedding theorems and differential inequalities.
Chapter
In this chapter we prove the existence of invariant measures associated with two-dimensional autonomous Navier–Stokes equations. Then we introduce the notion of a stationary statistical solution and prove that every invariant measure is also such a solution.
Chapter
In this chapter we consider two examples of contact problems. First, we study the problem of time asymptotics for a class of two-dimensional turbulent boundary driven flows subject to the Tresca friction law which naturally appears in lubrication theory. Then we analyze the problem with the generalized Tresca law, where the friction coefficient can...
Chapter
This chapter is devoted to constructions of invariant measures and statistical solutions for non-autonomous Navier–Stokes equations in bounded and certain unbounded domains in \(\mathbb{R}^{2}\).After introducing some basic notions and results concerning attractors in the context of the Navier–Stokes equations, we construct the family of probabilit...
Chapter
In this chapter we consider further non-autonomous and multivalued evolution problems, this time in the frame of the theory of pullback attractors for multivalued processes.
Chapter
In this chapter we study a typical problem from the theory of lubrication, namely, the Stokes flow in a thin three-dimensional domain \(\varOmega ^{\varepsilon }\), \(\varepsilon> 0\). We assume the Fourier boundary condition (only the friction part) at the top surface and a nonlinear Tresca interface condition at the bottom one.
Chapter
In this chapter we study the time asymptotics of solutions to the two-dimensional Navier–Stokes equations. In the first two sections we prove two properties of the equations in a bounded domain, concerning the existence of determining modes and nodes. Then we study the equations in an unbounded domain, in the framework of the theory of infinite dim...
Chapter
This chapter contains some basic facts about solutions of nonstationary Navier–Stokes equations
Book
This volume is devoted to the study of the Navier–Stokes equations, providing a comprehensive reference for a range of applications: from students to engineers and mathematicians involved in research on fluid mechanics, dynamical systems, and mathematical modeling. Equipped with only a basic knowledge of calculus, functional analysis, and partial d...
Chapter
Full-text available
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.
Chapter
Full-text available
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 p...
Article
Full-text available
Given a non-autonomous process U(.,.) on a complete separable metric space X that has a pullback attractor A(.), we construct a family of invariant Borel probability measures {mu(t)}(t is an element of R): the measures satisfy supp mu t C A(t) for all t is an element of R and the invariance property mu(t)(E) = mu(tau)(U(t,tau)E-1) for every Borel s...
Article
Full-text available
In this paper we study the global in time dynamics of a planar Bingham flow subject to a subdifferential boundary condition of Tresca's type. First, we prove the existence of a unique global in time solution of the considered problem and the existence of the global attractor. Then we show that for small driving forces the global attractor is trivia...
Article
Full-text available
A method is proposed to deal with some multivalued semiflows with weak continuity properties. An application to the reaction-diffusion problems with nonmonotone multivalued semilinear boundary condition and nonmonotone multivalued semilinear source term is presented.
Article
Full-text available
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 subdiff...
Article
Full-text available
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 $H^1_0$ in which the right-hand side...
Article
We consider a two-dimensional nonstationary Navier-Stokes shear flow with a subdifferential boundary condition on a part of the boundary of the flow domain, namely, with a boundary driving subject to the Tresca law. There exists a unique global in time solution of the considered problem which is governed by a variational inequality. Our aim is to p...
Article
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 ve...
Article
Full-text available
Inspired by a theory due to Foias and coworkers (see, for example, Foias et al. Navier–Stokes equations and turbulence, Cambridge University Press, Cambridge, 2001) and recent work of Wang (Disc Cont Dyn Sys 23:521–540, 2009), we show that the generalised Banach limit can be used to construct invariant measures for continuous dynamical systems on m...
Article
We show that the stochastic flow generated by the 2-dimensional Stochastic Navier-Stokes equations with rough noise on a Poincare-like domain has a unique random attractor. One of the technical problems associated with the rough noise is overcomed by the use of the corresponding Cameron-Martin (or reproducing kernel Hilbert) space. Our results comp...
Article
Using the method introduced in Zhong et al. (2006) [17] together with a new way of dealing with well known estimates of solutions introduced in Łukaszewicz (in press) [9] we prove the existence of a unique minimal pullback attractor for the evolutionary process associated with a nonautonomous nonlinear reaction–diffusion system in Lp, p≥2, in which...
Article
Full-text available
Using a method based on the concept of the Kuratowski measure of noncompactness of a bounded set as well as some new estimates of solutions we prove the existence of a unique minimal pullback attractor for the evolutionary process associated with nonautonomous two-dimensional micropolar fluid equations in a bounded domain, where the forces and mome...
Chapter
Recent years have seen considerable research activity at the interface of mathematics and fluid mechanics, particularly partial differential equations. The 2007 workshop at the University of Warwick was organised to consolidate, survey and further advance the subject. This volume is an outgrowth of that workshop. It consists of a number of reviews...
Article
Full-text available
In this paper we investigate some relations among the notions of pullback attractor, time-average measure and statistical solution. Using time-averages and Banach generalized limits we construct a family of probability measures {μt}tεR on the pullback attractor {A(t)}tεR of the dynamical system associated with a two-dimensional nonautonomous Navier...
Article
We consider a two-dimensional Navier-Stokes shear flow with time dependent bound- ary driving and subject to Tresca law. We establish the existence of a unique global in time solution of the considered problem and then, using a recent method based on the concept of the Kuratowski measure of noncompactness of a bounded set, we prove the existence of...
Article
We study the asymptotic behaviour of non-autonomous 2D Navier–Stokes equations in unbounded domains for which a Poincaré inequality holds. In particular, we give sufficient conditions for their pullback attractor to have finite fractal dimension. The existence of pullback attractors in this framework comes from the existence of bounded absorbing se...
Article
Full-text available
We consider a two-dimensional Navier–Stokes shear flow with time dependent boundary driving. We establish the existence of a unique global in time solution of the considered problem and then the existence of the pullback attractor for the associated evolutionary process. In the end, we estimate from above the dimension of the attractor in terms of...
Article
First, we introduce the concept of pullback asymptotically compact non-autonomous dynamical system as an extension of the similar concept in the autonomous framework. Our definition is different from that of asymptotic compactness already used in the theory of random and non-autonomous dynamical systems (as developed by Crauel, Flandoli, Kloeden, S...
Article
Full-text available
In this Note we first introduce the concept of pullback asymptotic compactness. Next, we establish a result ensuring the existence of a pullback attractor for a non-autonomous dynamical system under the general assumptions of pullback asymptotic compactness and the existence of a pullback absorbing family of sets. Finally, we prove the existence of...
Article
This research is motivated by a problem from lubrication theory. We consider a free boundary problem of a two-dimensional boundary-driven micropolar fluid flow. The existence of a unique global-in-time solution of the problem and the global attractor for the associated semigroup are known. In this paper we estimate the dimension of the global attra...
Article
We consider a free boundary problem of a two-dimensional Navier-Stokes shear flow. There exist a unique global in time solution of the considered problem as well as the global attractor for the associated semigroup. We estimate from above the dimension of the attractor in terms of given data and the geometry of the domain of the flow. This research...
Article
We consider a two-dimensional Navier–Stokes shear flow. There exists a unique global-in-time solution of the considered problem as well as the global attractor for the associated semigroup.Our aim is to estimate from above the dimension of the attractor in terms of given data and geometry of the domain of the flow. First we obtain a Kolmogorov-type...
Article
We consider a two-dimensional Navier–Stokes shear flow. There exists a unique global-in-time solution of the considered problem as well as the global attractor for the associated semigroup.Our aim is to estimate from above the dimension of the attractor in terms of given data and geometry of the domain of the flow. First we obtain a Kolmogorov-type...
Article
We investigate the flow of a magneto-micropolar fluid in an arbitrary unbounded domain on which the Poincar inequality holds. Assuming homogeneous boundary conditions and the external fields to be almost periodic in time we prove the existence of the uniform attractor by using the energy method [10] which we generalize to nonautonomous systems. We...
Article
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 particula...
Article
Full-text available
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 fr...
Article
Full-text available
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.
Article
Full-text available
We consider the Dirichlet boundary value problem for the equations of a stationary micropolar fluid in a bounded three-dimensional domain. We show the existence and uniqueness of a distributional solution with boundary values in L2.
Article
Full-text available
This paper is devoted to several aspects of long time behavior of 2D micropolar fluid flows. We prove successively: existence and uniqueness of global solutions, existence of a global attractor, convergence to the stationary solution for large viscosity flows, continuous dependence of solutions on microrotation viscosity perturbations, and stabilit...
Article
We consider a boundary value problem describing the stationary flow of a non-Newtonian fluid through the frozen ground, with a free interface between the liquid and the solid phases. We prove the existence of at least one weak solution of the problem. Copyright © 2000 John Wiley & Sons, Ltd.
Article
We consider the bidimensional stationary Stefan problem with convection. The problem is governed by a coupled system involving a nonlinear Darcy’s law and the energy balance equation with second member in L 1 . We prove existence of at least one weak solution of the problem, using the penalty method and the Schauder fixed point principle.
Article
In this chapter we study existence of solutions to initial boundary value problems for equations of micropolar fluids in the space region Q T = Ω×(0, T), where Ω is a bounded domain in R3 with smooth boundary, and T > 0 is an arbitrary number.
Article
In this chapter we consider some fundamental boundary value problems for micropolar fluids. Our main aim is to prove existence of their solutions in fairly large function spaces, determine conditions under which these solutions are unique, and study their regularity properties.
Article
The Navier-Stokes model of classical hydrodynamics has a drastic limitation—it cannot describe (by definition) fluids with microstructure, fluids that are interesting in themselves and important in applications.
Chapter
In this chapter, for the convenience of the reader, we recall some basic mathematical notions and theorems that we shall often use in the following chapters. To a number of important theorems and lemmas we provide proofs, especially when they are straightforward and simple enough or when we find them very instructive. We also clarify some notions,...
Article
We consider an initial value problem for a system of equations describing the motion and the heat convection in a viscous and incompressible fluid which occupies a smooth region Ωt⊂ℝ3 depending on time. In the equation for the distribution of temperature in the fluid we take into account not only the convective term but also the term responsible fo...
Article
We derive, for micropolar fluids, an analogue of the classical Reynolds equation of the theory of lubrication, and then discuss its particular forms depending on the assumptions imposed on both the viscosities and the data.
Article
Full-text available
The existence of a solution of the evolution inclusion u ' +∂φ(t,u)+g(t,u)-F(u)∋0on(0,T),u(0)=ξ is established. For each t in [0,T], φ(t,·) is a proper l.s.c. convex function from H to [0,∞] and F is an upper hemicontinuous set-valued mapping of L 2 (0,T;H) into its closed convex subsets. The time periodic problem u ' +∂φ(t,u)-F(u)∋0on(0,T),u(0)=u(...
Article
We consider a transmission problem for linear second order elliptic equations in a planar domain Ω, Ω=Ω + ∪Ω - ∪Γ, where the interface Γ meets the boundary of Ω in exactly two points P and Q. The boundary is not necessarily smooth at these points. We assume the homogeneous boundary condition u=(u + ,u - )=0 on ∂Ω, and u + =u - on Γ. The elliptic op...
Article
We consider an initial boundary value problem for the system of equations describing non-stationary flows of incompressible asymmetric fluids. We prove the existence of a local in time, weak solution of the problem in the case when the initial density is not separated from zero by a positive constant.
Article
Existence, uniqueness and regularity of solutions of equations describing stationary flows of viscous incompressible isotropic fluids with an asymmetric stress tensor have been considered recently.5 In this paper we extend the results of Reference 5 to include heat convection in the hydrodynamic model. We show that the boundary value problem (1.1)–...
Article
We investigate some variational inequalities associated with the equations of the stationary motion of granulated media with a constant density. These inequalities replace the usual equations of motion in the case when some additional constraints are imposed on the flow. We prove the existence of solutions of the inequalities, study their regularit...
Article
We consider an initial boundary-value problem for the system of equations describing nonstationary flows of asymmetric fluids. In an earlier article [ibid. 12, No.1, 83-97 (1988; Zbl 0668.76045)] we established the existence of a weak solutio of the problem in the interval (O,T), where T is an arbitrary positive real. In the present paper we prove...
Article
We consider a boundary-value problem describing the motion of viscous, incompressible and heat-conducting fluids in a bounded domain in ℝ3. We admit non-homogeneous boundary conditions, the appearance of exterior forces and heat sources. Our aim is to prove the existence of a solution of the problem in Sobolev spaces.
Article
In this paper we consider an initial boundary-value problem for the system of equations describing the motion of a granulated medium with constant density. We introduce the notion of a weak solution of the problem and prove the existence of such a solution.
Article
We study a variational inequality associated with a boundary-value problem for the stationary motion of viscous incompressible fluids. This inequality replaces the description of the flow given in terms of the boundary-value problem in the case when some additional constraints are imposed on the flow. We prove the existence of solutions of the ineq...
Article
We consider an initial-boundary value problem of a flow of a viscous and heat-conducting gas in a bounded domain D ⊂ R3. We assume that the boundary S of D consists of two disjoint surfaces S1 and S2 of class C2, and that the gas enters D through the surface S1 and leaves D through the surface S2. Our aim is to prove the existence (locally in time)...
Article
We consider an initial-boundary value problem for a flow of a viscous and heat conducting gas in a bounded domain D⊂ℝ 3 with the boundary S of class C 2 . Assuming that the gas leaves the domain through S, we prove the existence of a solution of the problem in a cylinder Q T0 =D×[0,T 0 ],T 0 >0. We estimate also the upper bound of such t>0 that the...
Article
We consider non-smooth solutions of two classes of initial-boundary value problems for the system of equations describing the motion of a viscous compressible fluid. Our aim is to estimate from below and from above the function of temperature (a component of these solutions) of the fluid.
Article
We study the asymptotic behaviour of dissipative non-autonomous PDEs in some unbounded domains. In particular, we give sucient conditions for a pullback attrac- tor to have finite fractal dimension. The existence of pullback attractors in this frame- work comes from the existence of bounded absorbing sets of pullback asymptotically compact processe...
Article
We consider a stationary two-phase Stefan problem with convection. The problem is governed by a coupled system involving a nonlinear Darcy law and the energy balance equation with second member in L 1 . We prove existence of at least one weak solution of the problem, using the penalty method and the Schauder point principle.

Network

Cited By