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• 1.
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.

• 2.
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.

• 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, Mathematical Science.
Reiterated homogenization of a nonlinear Reynolds-type equation2008Report (Other academic)
• 4.
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.

• 5.
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

• 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. 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)
• 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, 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.

• 8.
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.

• 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. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Homogenization of a Reynolds equation describing compressible flow2011Report (Other academic)
• 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, 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.

• 11.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Homogenization of some problems in hydrodynamic lubrication involving rough boundaries2011Doctoral thesis, comprehensive summary (Other academic)

This thesis is devoted to the study of some homogenization problems with applications in lubrication theory. It consists of an introduction, five research papers (I–V) and a complementary appendix.Homogenization is a mathematical theory for studying differential equations with rapidly oscillating coefficients. Many important problems in physics with one or several microscopic scales give rise to this kind of equations, whence the need for methods that enable an efficient treatment of such problems. To this end several mathematical techniques have been devised. The main homogenization method used in this thesis is called multiscale convergence. It is a notion of weak convergence in  Lp spaces which is designed to take oscillations into account. In paper II we extend some previously obtained results in multiscale convergence that enable us to homogenize a nonlinear problem with a finite number of microscopic scales. The main idea in the proof is closely related to a decomposition of vector fields due to Hermann Weyl. The Weyl decomposition is further explored in paper III.Lubrication theory is devoted to the study of fluid flows in thin domains. More generally, tribology is the science of bodies in relative motion interacting through a mechanical contact. An important aspect of tribology is to explain the principles of friction, lubrication and wear. The mathematical foundations of lubrication theory are given by the Navier–Stokes equation which describes the motion of a viscous fluid. In thin domains several simplifications are possible, as shown in the introduction of this thesis. The resulting equation is named after Osborne Reynolds and is much simpler to analyze than the Navier--Stokes equation.The Reynolds equation is widely used by engineers today. For extremely thin films, it is well-known that the surface micro-topography is an important factor in hydrodynamic performance. Hence it is important to understand the influence of surface roughness with small characteristic wavelengths upon the solution of the Reynolds equation. Since the 1980s such problems have been increasingly studied by homogenization theory. The idea is to replace the original equation with a homogenized equation where the roughness effects are “averaged out”. One problem consists of finding an algorithm for computing the solution of the homogenized equation. Another problem consists of showing, on introducing the appropriate mathematical definitions, that the homogenized equation is the correct method of averaging. Papers I, II, IV and V investigate the effects of surface roughness by homogenization techniques in various situations of hydrodynamic lubrication. To compare the homogenized solution with the solution of the deterministic Reynolds equation, some numerical examples are also included.

• 12.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Homogenization theory with applications in tribology2008Licentiate thesis, comprehensive summary (Other academic)

Homogenization is a mathematical theory for studying differential equations with rapidly oscillating coefficents. Many important problems in physics with one or several microscopic length scales give rise to this kind of equations. Hence there is a need for methods that enable an efficient treatment of such problems. To this end several homogenization techniques exist, ranging from the fairly abstract ones to those that are more oriented towards applications. This thesis is concerned with two such methods, namely the "asymptotic expansion method", also known as the "method of multiple scales", and multiscale convergence. The former method, sometimes referred to as the "engineering approach to homogenization" has, due to its versatility and intutive appeal, gained wide acceptance and popularity in the applied fields. However, it is not rigorous by mathematical standards. Multiscale convergence, introduced by Nguetseng in 1989, is a notion of weak convergence in Lp spaces that is designed to take oscillations into account. Although not the most general method around, multiscale convergence has become widely used by homogenizers because of its simplicity. In spite of its success, the multiscale theory is not yet sufficiently developed to be used in connection with certain nonlinear problems with several microscopic scales. In Paper A we extend some previously obtained results in multiscale convergence that enable us to homogenize a nonlinear problem with three scales. In Appendix to Paper A we present in more detail some results that were used in the proof of some of the main theorems in Paper A. Tribology is the science of bodies in relative motion interacting through a mechanical contact. An important aspect of tribology is to explain the principles of friction, lubrication and wear. Tribological phenomena are encountered everywhere in nature and technology and have a huge economical impact on society. An important example is that of two sliding solid surfaces interacting through a thin film of viscous fluid (lubricant). Hydrodynamic lubrication occurs when the pressure generated within the lubricant, through the viscosity of the fluid, is able to sustain an externally applied load. Many common bearings, e.g. journal bearings or slider bearings, operate according to this principle. As a branch of fluid dynamics, the mathematical foundations of lubrication theory are given by the Navier-Stokes equations, describing the motion of a viscous fluid. Because of the thin film assumption several simplifications are possible, leading to various reduced equations named after Osborne Reynolds, the founding father of lubrication theory. The Reynolds equation is used by engineers to compute the pressure distribution in various situations of thin film lubrication. For extremely thin films, it has been observed that the surface micro topography is an important factor in hydrodynamic performance. Hence it is important to understand the influence of surface roughness with small characteristic wavelength upon the pressure solution. Since the 1980s such problems have been increasingly studied by homogenization theory. The idea is to replace the original equation with a homogenized equation where the roughness effects are "averaged out". One problem consists of finding an algorithm that gives the homogenized equation. Another problem, consists of showing, by introducing the appropriate mathematical defintions, that the homogenized equation really is the correct one. Papers B and C investigate the effects of surface roughness by means of multiscale expansion of the pressure in various situations of hydrodynamic lubrication. Paper B, for which Paper A constitutes a rigorous basis, considers homogenization of the stationary Reynolds equation and roughness with two characteristic wavelengths. This leads to a multiscale problem and adds to the complexity of the homogenization process. To compare the homogenized solution to the solution of the unaveraged Reynolds equation, some numerical examples are also included. Paper C is devoted to homogenization of a variational principle which is a generalization of the unstationary Reynolds equation (both surfaces are rough). The advantage of adopting the calculus of variations viewpoint is that the recently introduced "variational bounds" can be computed. Bounds can be seen as a "cheap" alternative to computing the realtively costly homogenized solution. Several numerical examples are included to illustrate the utility of bounds.

• 13.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
On Weyl decomposition of vector fields2011Report (Other academic)
• 14.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Stokes flow with kinematic and dynamic boundary conditions2019In: Quarterly of Applied Mathematics, ISSN 0033-569X, E-ISSN 1552-4485, Vol. 77, no 3, p. 525-544Article in journal (Refereed)

We review the first and second boundary value problems for the Stokes system posed in a bounded Lipschitz domain in . Particular attention is given to the mixed boundary condition: a Dirichlet condition is imposed for the velocity on one part of the boundary while a Neumann condition for the stress tensor is imposed on the remaining part. Some minor modifications to the standard theory are therefore required. The most noteworthy result is that both pressure and velocity are unique.

• 15.
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.

• 16.
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.

• 17.
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.

• 18.
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)
• 19.
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)
• 20.
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 domains2019In: Asymptotic Analysis, ISSN 0921-7134, E-ISSN 1875-8576Article 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.

• 21.
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.

• 22.
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.

• 23.
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

• 24.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Faculty of Mechanics and Mathematics, Moscow 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.
On the effects of surface roughness in thin film flow governed by the time dependent Stokes equations2014In: Doklady. Mathematics, ISSN 1064-5624, E-ISSN 1531-8362, Vol. 90, no 1, p. 445-449Article in journal (Refereed)
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