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• 1.
L. N.Gumilev Eurasian National University, Khazakstan.
L. N.Gumilev Eurasian National University, Khazakstan. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Additive weighted Lp estimates of some classes of integral operators involving generalized Oinarov Kernels2017In: Journal of Mathematical Inequalities, ISSN 1846-579X, E-ISSN 1848-9575, Vol. 11, no 3, p. 683-694Article in journal (Refereed)

Abstract. Inequalities of the formkuK f kq 6C(kr f kp +kvH f kp) , f > 0,are considered, where K is an integral operator of Volterra type and H is the Hardy operator.Under some assumptions on the kernel K we give necessary and sufficient conditions for suchan inequality to hold.1

• 2.
Department of Pure Mathematics, Eurasian National University.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Department of Mathematics, Narvik University College. Department of Pure Mathematics, Eurasian National University. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Some new results concerning a class of third-order differential equations2015In: Applicable Analysis, ISSN 0003-6811, E-ISSN 1563-504X, Vol. 94, no 2, p. 419-434Article in journal (Refereed)

We consider the following third-order differential equation with unbounded coefficients:Some new existence and uniqueness results are proved, and precise estimates of the norms of the solutions are given. The obtained results may be regarded as a unification and extension of all other results of this type

• 3.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
New insights on lubrication theory for compressible fluids2019In: International Journal of Engineering Science, ISSN 0020-7225, E-ISSN 1879-2197, Vol. 145, article id 103170Article in journal (Refereed)

The fact that the film is thin is in lubrication theory utilised to simplify the full Navier–Stokes system of equations. For incompressible and iso-viscous fluids, it turns out that the inertial terms are small enough to be neglected. However, for a compressible fluid, we show that the influence of inertia depends on the (constitutive) density-pressure relationship and may not always be neglected. We consider a class of iso-viscous fluids obeying a power-law type of compressibility, which in particular includes both incompressible fluids and ideal gases. We show by scaling and asymptotic analysis, that the degree of compressibility determines whether the terms governing inertia may or may not be neglected. For instance, for an ideal gas, the inertial terms remain regardless of the film height-to-length ratio. However, by means of a specific modified Reynolds number that we define we show that the magnitudes of the inertial terms rarely are large enough to be influential. In addition, we consider fluids obeying the well-known Dowson and Higginson density-pressure relationship and show that the inertial terms can be neglected, which allows for obtaining a Reynolds type of equation. Finally, some numerical examples are presented in order to illustrate our theoretical results.

• 4.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Homogenization of the Reynolds equation2005Report (Other academic)
• 5.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
Department of Mathematics and Statistics, University of Cape Coast. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Multiscale homogenization of a class of nonlinear equations with applications in lubrication theory and applications2011In: Journal of Function Spaces and Applications, ISSN 0972-6802, E-ISSN 1758-4965, Vol. 9, no 1, p. 17-40Article in journal (Refereed)

We prove a homogenization result for monotone operators by using the method of multiscale convergence. More precisely, we study the asymptotic behavior as epsilon -> 0 of the solutions u(epsilon) of the nonlinear equation div a(epsilon)(x, del u(epsilon)) = div b(epsilon), where both a(epsilon) and b(epsilon) oscillate rapidly on several microscopic scales and a(epsilon) satisfies certain continuity, monotonicity and boundedness conditions. This kind of problem has applications in hydrodynamic thin film lubrication where the bounding surfaces have roughness on several length scales. The homogenization result is obtained by extending the multiscale convergence method to the setting of Sobolev spaces W-0(1,p)(Omega), where 1 < p < infinity. In particular we give new proofs of some fundamental theorems concerning this convergence that were first obtained by Allaire and Briane for the case p = 2.

• 6.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Reiterated homogenization applied in hydrodynamic lubrication2008In: Proceedings of the Institution of mechanical engineers. Part J, journal of engineering tribology, ISSN 1350-6501, E-ISSN 2041-305X, Vol. 222, no 7, p. 827-841Article in journal (Refereed)

This work is devoted to studying the combined effect that arises due to surface texture and surface roughness in hydrodynamic lubrication. An effective approach in tackling this problem is by using the theory of reiterated homogenization with three scales. In the numerical analysis of such problems, a very fine mesh is needed, suggesting some type of averaging. To this end, a general class of problems is studied that, e.g. includes the incompressible Reynolds problem in both artesian and cylindrical coordinate forms. To demonstrate the effectiveness of the method several numerical results are presented that clearly show the convergence of the deterministic solutions towards the homogenized solution.Moreover, the convergence of the friction force and the load carrying capacity of the lubricant film is also addressed in this paper. In conclusion, reiterated homogenization is a feasible mathematical tool that facilitates the analysis of this type of problem.

• 7.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Reiterated homogenization of a nonlinear Reynolds-type equation2008Report (Other academic)
• 8.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
Department of Mathematics and Statistics, University of Cape Coast. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Variational bounds applied to unstationary hydrodynamic lubrication2008In: International Journal of Engineering Science, ISSN 0020-7225, E-ISSN 1879-2197, Vol. 46, no 9, p. 891-906Article in journal (Refereed)

This paper is devoted to the effects of surface roughness in hydrodynamic lubrication. The numerical analysis of such problems requires a very fine mesh to resolve the surface roughness, hence it is often necessary to do some type of averaging. Previously, homogenization (a rigorous form of averaging) has been successfully applied to Reynolds type differential equations. More recently, the idea of finding upper and lower bounds on the effective behavior, obtained by homogenization, was applied for the first time in tribology. In these pioneering works, it has been assumed that only one surface is rough. In this paper we develop these results to include the unstationary case where both surfaces may be rough. More precisely, we first use multiple-scale expansion to obtain a homogenization result for a class of variational problems including the variational formulation associated with the unstationary Reynolds equation. Thereafter, we derive lower and upper bounds corresponding to the homogenized (averaged) variational problem. The bounds reduce the numerical analysis, in that one only needs to solve two smooth problems, i.e. no local scale has to be considered. Finally, we present several examples, where it is shown that the bounds can be used to estimate the effects of surface roughness with very high accuracy.

• 9.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Homogenization of the unstationary incompressible Reynolds equation2007In: Tribology International, ISSN 0301-679X, E-ISSN 1879-2464, Vol. 40, no 9, p. 1344-1350Article in journal (Refereed)

This paper is devoted to the effects of surface roughness during hydrodynamic lubrication. In the numerical analysis a very fine mesh is needed to resolve the surface roughness, suggesting some type of averaging. A rigorous way to do this is to use the general theory of homogenization. In most works about the influence of surface roughness, it is assumed that only the stationary surface is rough. This means that the governing Reynolds type equation does not involve time. However, recently, homogenization was successfully applied to analyze a situation where both surfaces are rough and the lubricant is assumed to have constant bulk modulus. In this paper we will consider a case where both surfaces are assumed to be rough, but the lubricant is incompressible. It is also clearly demonstrated, in this case that homogenization is an efficient approach. Moreover, several numerical results are presented and compared with those corresponding to where a constant bulk modulus is assumed to govern the lubricant compressibility. In particular, the result shows a significant difference in the asymptotic behavior between the incompressible case and that with constant bulk modulus.

• 10.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
A new approach for studying cavitation in lubrication2014In: Journal of tribology, ISSN 0742-4787, E-ISSN 1528-8897, Vol. 136, no 1, article id 11706Article in journal (Refereed)

The underlying theory, in this paper, is based on clear physical arguments related to conservation of mass flow and considers both incompressible and compressible fluids. The result of the mathematical modeling is a system of equations with two unknowns, which are related to the hydrodynamic pressure and the degree of saturation of the fluid. Discretization of the system leads to a linear complementarity problem (LCP), which easily can be solved numerically with readily available standard methods and an implementation of a model problem in matlab code is made available for the reader of the paper. The model and the associated numerical solution method have significant advantages over today's most frequently used cavitation algorithms, which are based on Elrod-Adams pioneering work

• 11.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Reynolds equation flow factor estimates by means of homogenization2010In: ASIATRIB 2010: Frontiers in tribology - knowledge & friendship . proceedings of the fourth Asia International Conference on Tribology, 5-9 December 2010, Perth, Western Australia, 2010, p. 185-Conference paper (Refereed)
• 12.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Flow in thin domains with a microstructure: Lubrication and thin porous media2017In: AIP Conference Proceedings, ISSN 0094-243X, E-ISSN 1551-7616, Vol. 1798, article id 020172Article in journal (Refereed)

This paper is devoted to homogenization of different models of flow in thin domains with a microstructure. The focus is on applications connected to the effect of surface roughness in full film lubrication, but a parallel to flow in thin porous media is also discussed. Mathematical models of such flows naturally include two small parameters. One is connected to the fluid film thickness and the other to the microstructure. The corresponding asymptotic analysis is a delicate problem, since the result depends on how fast the two small parameters tend to zero relative to each other. We give a review of the current status in this area and point out some future challenges.

• 13.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Similarities and differences between the flow factor method by Patir and Cheng and homogenization2011In: Journal of tribology, ISSN 0742-4787, E-ISSN 1528-8897, Vol. 133, no 3, p. 031702-1Article in journal (Refereed)

Different averaging techniques have proved to be useful for analyzing the effects of surface roughness in hydrodynamic lubrication. This paper compares two of these averaging techniques, namely the flow factor method by Patir and Cheng (P&C) and homogenization. It has been rigorously proved by many authors that the homogenization method provides a correct solution for arbitrary roughness. In this work it is shown that the two methods coincide if and only if the roughness exhibits certain symmetries. Hence, homogenization is always the preferred method.

• 14.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Homogenization of a Reynolds equation describing compressible flow2011Report (Other academic)
• 15.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Homogenization of a Reynolds equation describing compressible flow2012In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 390, no 2, p. 456-471Article in journal (Refereed)

We homogenize a Reynolds equation with rapidly oscillating film thickness function hε, assuming a constant compressiblity factor in the pressure-density relation. The oscillations are due to roughness on the bounding surfaces of the fluid film. As shown by previous studies, homogenization is an effective approach for analyzing the effects of surface roughness in hydrodynamic lubrication. By two-scale convergence theory we obtain the limit problem (homogenized equation) and strong convergence in L2 for the unknown density ρε. By adding a small corrector term we also obtain strong convergence in the Sobolev norm.

• 16.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Homogenization of Reynolds equation2005Report (Other academic)
• 17.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
The homogenization process of the time dependent Reynolds equation describing compressible liquid flow2007In: Tribologia : Finnish Journal of Tribology, ISSN 0780-2285, Vol. 26, no 4, p. 30-44Article in journal (Refereed)

To increase the hydrodynamic performance in different machine elements during lubrication, e.g. journal bearings and thrust bearings, it is important to understand the influence of surface roughness. In this connection one encounters different approaches commonly based on some form of the Reynolds equation. They may generally be divided into deterministic- and averaging- techniques. The former regards all surface roughness information and provides a detailed understanding of the local effects that arise. The latter method is suitable when investigating how the surface roughness affects performance of the machine element as a whole. Homogenization is a rigorous mathematical concept that when applied to a certain problem may be thought of as an averaging technique also providing information about local effects. In this work the compressible time dependent Reynolds equation is homogenized. Related problems have recently been analyzed by homogenization techniques under various assumptions. In the present paper the compressibility is modeled assuming a constant lubricant bulk modulus. The formal method of multiple scale expansion is used to derive a so-called homogenized equation and a numerical solution method to solve both the deterministic problem and the homogenized problem is implemented. The numerical results clearly show that the solution of the homogenized equation is a suitable approximation to the solution of the deterministic problem. It is also demonstrated that for small values of the roughness wavelength, the homogenization technique is superior, since the solution of the deterministic problem requires an extremely fine discretization mesh. More over, the solution of the time dependent homogenized problem may in some cases be reduced to solve a stationary problem that facilitates the solution process and interpretation of results.

• 18. Almqvist, Andreas
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
The homogenization process of the time dependent Reynolds equation describing compressible liquid flow2006Report (Other academic)
• 19.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
Narvik University College, 8505 Narvik, Norway. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
New concepts of homogenization applied in rough surface hydrodynamic lubrication2007In: International Journal of Engineering Science, ISSN 0020-7225, E-ISSN 1879-2197, Vol. 45, no 1, p. 139-154Article in journal (Refereed)

This work introduces a new concept of homogenization that enables efficient analysis of the effects of surface roughness representations obtained by measurements in applications modeled by the Reynolds equation. Examples of such applications are trust- and journal-bearings. The numerical analysis of these types of applications requires an extremely dense computational mesh in order to resolve the surface roughness, suggesting some type of averaging. One such method is homogenization, which has been applied to Reynolds type equations with success recently. This approach is similar to the technique proposed by Patir and Cheng, who introduced flow factors determining the hydrodynamic action due to surface roughness. The difference is, however, that the present technique has a rigorous mathematical support. Moreover, the recipe to compute the averaged coefficients is simple without any ambiguities. Using either the technique proposed by Patir and Cheng or homogenization, the coefficients determining the averaged Reynolds equation are obtained by solving differential equations on a local scale. Unfortunately, this is detrimental when investigating the effects induced by real, measured, surface roughness, even though these local problems may be solved in parallel. The present work presents a solution by applying the technique based on bounds. This technique transforms the stationary Reynolds equation into two computationally feasible forms, one for the upper bound and one for the lower bound, where the flow factors are obtained by straightforward integration. Together with the preciseness of these bounds, the bounds approach becomes an eminent tool suitable for investigating the effect of real, measured, surface roughness on hydrodynamic performance.

• 20.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
A new concept in cavitation modelling2013In: Tribo Lyon 2013: book of abstracts : a joint event of WTC 2013, Lyon, 2013, p. 170-Conference paper (Refereed)
• 21.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Homogenization applied in rough surface hydrodynamic lubrication2007In: Svenska Mekanikdagar 2007: Program och abstracts / [ed] Niklas Davidsson; Elianne Wassvik, Luleå: Luleå tekniska universitet, 2007, p. 31-Conference paper (Other academic)
• 22.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Homogenization of the Reynolds equation2013In: Encyclopedia of Tribology, Berlin: Springer-Verlag New York Inc. , 2013, p. 1685-1690Chapter in book (Refereed)
• 23.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Modelling cavitation in (elasto)hydrodynamic lubrication2016In: Advances in tribology / [ed] Pranav H. Darji, Croatia: INTECH, 2016, p. 198-213Chapter in book (Refereed)

In this chapter we will present a derivation of a mathematical model describing how cavitation influences the pressure distribution in a thin lubricant film between two moving surfaces. The main idea in the derivation is to first describe the influence of cavitation on the mass flow and thereafter using a conservation law for the mass. This leads to a nonlinear system with two complementary variables: one is the pressure distribution and the other is related to the density, i.e. a nonlinear complementarity problem (NLCP). The proposed approach is used to derive a mass conserving cavitation model considering that density, viscosity and film thickness of the lubricant depend on the pressure. To demonstrate the applicability and evaluate the proposed model and the suggested numerical implementation, a few model problems are analysed and presented.

• 24.
Department of Mathematics, Faculty of Exact and Natural Sciences, Ivane Javakhishvili Tbilisi State University.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Sharp Hp- Lptype inequalities of weighted maximal operators of Vilenkin-Nörlund means and its applications2016In: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242X, no 1, article id 242Article in journal (Refereed)

We prove and discuss some new Hp-Lptype inequalities of weighted maximal operators of Vilenkin-Nörlund means with monotone coefficients. It is also proved that these inequalities are the best possible in a special sense. We also apply these results to prove strong summability for such Vilenkin-Nörlund means. As applications, both some well-known and new results are pointed out

• 25. Berggren, Stein
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Homogenization and computational techniques of iterated structures2001In: Proceedings of the Eight International Conference on Composites Engineering: ICCE/8 / [ed] David Hui, 2001Conference paper (Refereed)
• 26.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
A numerical study of stochastic homogenization2003In: Proceedings of the International Conference on Composites Engineering: ICCE/10 / [ed] David Hui, 2003Conference paper (Refereed)
• 27. Byström, Johan
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
A numerical study of the convergence in stochastic homogenization2004In: Journal of Analysis and Applications, ISSN 0972-5954, Vol. 2, no 3, p. 159-171Article in journal (Refereed)

This note makes the link between theoretical results on stochastic homogenization and effective computation of averaged coefficients for diffusion operators in random media. Examples of how to construct relevant random media and numerical results on the effective coefficients are given.

• 28. Byström, Johan
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Bounds and numerical results for the homogenized degenerated p-Poisson equation2004In: Applications of Mathematics, ISSN 0862-7940, E-ISSN 1572-9109, Vol. 49, no 2, p. 111-122Article in journal (Refereed)

In this paper we derive upper and lower bounds on the homogenized energy density functional corresponding to degenerated p-Poisson equations. Moreover, we give some non-trivial examples where the bounds are tight and thus can be used as good approximations of the homogenized properties. We even present some cases where the bounds coincide and also compare them with some numerical results.

• 29.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Periodic approximation of elastic properties in random media2006In: Advances in Algebra and Analysis, ISSN 0973-2306, Vol. 1, no 2, p. 103-113Article in journal (Refereed)

Most papers on stochastic homogenization either deal with theoretical aspects or with questions regarding computational issues. Since the theoretical analysis involves the solution of a problem which is stated in a abstract probability space, it is not clear how the two areas are connected. In previous works this problem has not been considered. However, recently Bourgeat and Piatnitski investigated this connection in the scalar case for second order operators of divergence form. They proved that in the limit, the method of periodic approximation gives the same effective properties as in stochastic homogenization. In this paper we prove similar results for the vector valued case, which appears in e.g. the theory of elasticity. Moreover, we provide a numerical analysis of the results.

• 30.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Reiterated homogenization of degenerate nonlinear elliptic equations2002In: Chinese Annals of Mathematics. Series B, ISSN 0252-9599, E-ISSN 1860-6261, Vol. 23, no 3, p. 325-334Article in journal (Refereed)

The authors study homogenization of some nonlinear partial differential equations of the form _ div (a (hx, h2 x, Duh)) = f , where a is periodic in the rst two arguments and monotone in the third. In particular the case where a satis es degenerated structure conditions is studied. It is proved that uh converges weakly in W0 1,1 (Ω) to the unique solution of a limit problem as h → ∞ . Moreover, explicit expressions for the limit problem are obtained.

• 31. Byström, Johan
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Reiterated homogenization of degenerate nonlinear monotone operators2004In: Frontiers in Mathematical Analysis and Numerical Methods: In Memory of Jacques-Louis Lions, Singapore: World Scientific and Engineering Academy and Society, 2004, p. 83-96Chapter in book (Other academic)
• 32.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Heat conductivity bounds by Halpin-Tsai type equations and homogenization2000In: Proceedings of the Seventh International Conference on Composites Engineering: ICCE/7 / [ed] David Hui, 2000, p. 97-98Conference paper (Refereed)
• 33.
Department of Differential Equations, Faculty of Mechanics and Mathematics, Moscow Lomonosov State University.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
On spectrum of the Laplacian in a circle perforated along the boundary: application to a Friedrichs-type inequality2011Report (Other academic)
• 34.
Moscow Lomonosov State University.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
A new weighted Friedrichs-type inequality for a perforated domain with a sharp constant2011In: Eurasian Mathematical Journal, ISSN 2077-9879, Vol. 2, no 1, p. 81-103Article in journal (Refereed)

We derive a new three-dimensional Hardy-type inequality for a cube for the class of functions from the Sobolev space $H^1$ having zero trace on small holes distributed periodically along the boundary. The proof is based on a careful analysis of the asymptotic expansion of the first eigenvalue of a related spectral problem and the best constant of the corresponding Friedrichs-type inequality.

• 35.
Department of Differential Equations, Faculty of Mechanics and Mathematics, Moscow Lomonosov State University.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
On spectrum of the Laplacian in a circle perforated along the boundary: Application to a Friedrichs-type inequality2011In: International Journal of Differential Equations, ISSN 1687-9643, E-ISSN 1687-9651, Vol. 2011, article id 619623Article in journal (Refereed)

In this paper we construct and verify the asymptotic expansion for the spectrum of a boundary-value problem in a unit circle periodically perforated along the boundary. It is assumed that the size of perforation and the distance to the boundary of the circle are of the same smallness. As an application of the obtained results the asymptotic behavior of the best constant in a Friedrichs-type inequality is investigated.

• 36. Dasht, Johan
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Numerical analysis of the convergence in homogenization of composites2002In: Proceedings of the International Conference on Composites Engineering: ICCE/9 / [ed] David Hui, 2002Conference paper (Refereed)
• 37. Engström, Jonas
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Narvik University College, 8505 Narvik, Norway.
Homogenization of random degenerated nonlinear monotone operators2006In: Glasnik Matematicki - Serija III, ISSN 0017-095X, E-ISSN 1846-7989, Vol. 41, no 1, p. 101-114Article in journal (Refereed)

This paper deals with homogenization of random nonlinear monotone operators in divergence form. We assume that the structure conditions (strict monotonicity and continuity conditions) degenerate and are given in terms of a weight function. Under proper integrability assumptions on the weight function we construct the effective operator and prove the homogenization result.

• 38.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Darcy's law for flow in a periodic thin porous medium confined between two parallel plates2016In: Transport in Porous Media, ISSN 0169-3913, E-ISSN 1573-1634, Vol. 115, no 3, p. 473-493Article in journal (Refereed)

We study stationary incompressible fluid flow in a thin periodic porous medium. The medium under consideration is a bounded perforated 3D-domain confined between two parallel plates. The distance between the plates is $$\delta$$, and the perforation consists of $$\varepsilon$$-periodically distributed solid cylinders which connect the plates in perpendicular direction. Both parameters $$\varepsilon$$, $$\delta$$ are assumed to be small in comparison with the planar dimensions of the plates. By constructing asymptotic expansions, three cases are analysed: (1) $$\varepsilon \ll \delta$$, (2) $$\delta /\varepsilon \sim \text {constant}$$ and (3) $$\varepsilon \gg \delta$$. For each case, a permeability tensor is obtained by solving local problems. In the intermediate case, the cell problems are 3D, whereas they are 2D in the other cases, which is a considerable simplification. The dimensional reduction can be used for a wide range of $$\varepsilon$$ and $$\delta$$ with maintained accuracy. This is illustrated by some numerical examples.

• 39.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Asymptotic behaviour of Stokes flow in a thin domain with amoving rough boundary2014In: Proceedings of the Royal Society. Mathematical, Physical and Engineering Sciences, ISSN 1364-5021, E-ISSN 1471-2946, Vol. 470, no 2167, article id 20130735Article in journal (Refereed)

We consider a problem that models fluid flow in a thin domain bounded by two surfaces. One of the surfaces is rough and moving, whereas the other is flat and stationary. The problem involves two small parameters ε and μ that describe film thickness and roughness wavelength, respectively. Depending on the ratio λ = ε/μ, three different flow regimes are obtained in the limit as both of them tend to zero. Time-dependent equations of Reynolds type are obtained in all three cases (Stokes roughness, Reynolds roughness and high-frequency roughness regime). The derivations of the limiting equations are based on formal expansions in the parameters ε and μ.

• 40.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
A rigorous derivation of the time-dependent Reynolds equation2013In: Asymptotic Analysis, ISSN 0921-7134, E-ISSN 1875-8576, Vol. 84, no 1-2, p. 103-121Article in journal (Refereed)

We study the asymptotic behavior of solutions of the evolution Stokes equation in a thin three-dimensional domain bounded by two moving surfaces in the limit as the distance between the surfaces approaches zero. Using only a priori estimates and compactness it is rigorously verified that the limit velocity field and pressure are governed by the time-dependent Reynolds equation.

• 41.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
On the connection between evolution Stokes equation and Reynolds equation for thin-tilm flow2012Conference paper (Refereed)
• 42.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
The transition from evolution Stokes equation in 3D-domain to the Reynolds quation2011Report (Other academic)
• 43.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Department of Mathematics and Informatics, Eduardo Mondlane University, Maputo, Mozambique. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Pressure-driven flow in thin domains2020In: Asymptotic Analysis, ISSN 0921-7134, E-ISSN 1875-8576, Vol. 116, no 1, p. 1-26Article in journal (Refereed)

We study the asymptotic behavior of pressure-driven Stokes flow in a thin domain. By letting the thickness of the domain tend to zero we derive a generalized form of the classical Reynolds–Poiseuille law, i.e. the limit velocity field is a linear function of the pressure gradient. By prescribing the external pressure as a normal stress condition, we recover a Dirichlet condition for the limit pressure. In contrast, a Dirichlet condition for the velocity yields a Neumann condition for the limit pressure.

• 44.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Homogenization of the Stokes equation with mixed boundary condition in a porous medium2017In: Cogent Mathamatics, E-ISSN 2331-1835, Vol. 4, no 1, article id 1327502Article in journal (Refereed)

We homogenize stationary incompressible Stokes flow in a periodic porous medium. The fluid is assumed to satisfy a no-slip condition on the boundary of solid inclusions and a normal stress (traction) condition on the global boundary. Under these assumptions, the homogenized equation becomes the classical Darcy law with a Dirichlet condition for the pressure.

• 45.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
A Comparison of the Roughness Regimes in Hydrodynamic Lubrication2017In: Journal of tribology, ISSN 0742-4787, E-ISSN 1528-8897, Vol. 139, no 5, article id 051702Article in journal (Refereed)

This work relates to previous studies concerning the asymptotic behavior of Stokes flow in a narrow gap between two surfaces in relative motion. It is assumed that one of the surfaces is rough, with small roughness wavelength l, so that the film thickness h becomes rapidly oscillating. Depending on the limit of the ratio h/l, denoted as k, three different lubrication regimes exist: Reynolds roughness (k-0), Stokes roughness (0<γ<1), and high-frequency roughness (γ = ∞). In each regime, the pressure field is governed by a generalized Reynolds equation, whose coefficients (so-called flow factors) depend on k. To investigate the accuracy and applicability of the limit regimes, we compute the Stokes flow factors for various roughness patterns by varying the parameter k. The results show that there are realistic surface textures for which the Reynolds roughness is not accurate and the Stokes roughness must be used instead.

• 46.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Homogenization of a compressible cavitation model2015In: European journal of applied mathematics (Print), ISSN 0956-7925, E-ISSN 1469-4425, Vol. 26, no 3, p. 383-399Article in journal (Refereed)

We develop a mathematical model in hydrodynamic lubrication that takes into account three phenomena: cavitation, surface roughness and compressibility of the fluid. Like the classical Reynolds equation, the model is mass preserving. We compute the homogenized coefficients in the case of unidirectional roughness. A one-dimensional problem is also solved explicitly

• 47. Gogatishvili, A.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
On scales of equivalent conditions characterizing weighted Stieltjes inequality2012In: Doklady Akademii Nauk, ISSN 0869-5652, Vol. 447, no 1, p. 13-14Article in journal (Refereed)
• 48.
Institute of Mathematics of the Academy of Sciences of the Czech Republic.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Department of Mathematical Analysis, Russian Peoples' Friendship University. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Some scales of equivalent conditions to characterize the Stieltjes inequality: the case q2014In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 287, no 2-3, p. 242-253Article in journal (Refereed)

We prove that the weighted Stieltjes inequality can be characterized by four different scales of conditions also for the case , . In particular, a new proof of a result of G. Sinnamon is given, which also covers the case . Moreover, for the situation at hand a new gluing theorem and also an equivalence theorem of independent interest are proved and discussed. By applying our new equivalence theorem to weighted inequalities for the Stieltjes transform we obtain the four new scales of conditions for characterization of Stieltjes inequality

• 49.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
On scales of equivalent conditions characterizing weighted Stieltjes inequality2012In: Doklady. Mathematics, ISSN 1064-5624, E-ISSN 1531-8362, Vol. 86, no 3, p. 738-739Article in journal (Refereed)
• 50. Grevholm, Barbro
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Ett mentorprojekt för gymnasieelever i Luleå2012In: Nämnaren : tidskrift för matematikundervisning, ISSN 0348-2723, Vol. 2012, no 2, p. 33-37Article in journal (Other (popular science, discussion, etc.))
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