# zbMATH — the first resource for mathematics

The Stokes problem in a periodic layer. (English) Zbl 1227.35048
The authors consider the steady Stokes problem $$-\Delta v+\nabla p=f$$, $$-\nabla \cdot v=f_\nabla$$, posed in a domain $$\Omega$$ of $$\mathbb R^3$$ which is unbounded in two directions and bounded in the third one and which has a periodic structure near infinity. Outside a finite ball, the domain $$\Omega$$ has a bounded 3D pattern $$S$$ which is infinitely repeated along two directions. $$S$$ may contain some hole. The boundary conditions $$v=g$$ are imposed on $$\partial\Omega$$.
The main result of the paper proves that the solution $$(v,p)\in W_\beta^{l+1}(\Omega)^3\times W_\beta^{l,1}(\Omega)$$ of the Stokes problem has an asymptotic representation, under some hypotheses on the data of this problem. This asymptotic representation involves the solutions of Stokes problems in the reference cell $$S$$. The proof of this asymptotic representation is essentially a consequence of the properties of the solutions of the cell problems.
##### MSC:
 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 35Q30 Navier-Stokes equations 76D07 Stokes and related (Oseen, etc.) flows 76M50 Homogenization applied to problems in fluid mechanics
Full Text:
##### References:
 [1] Akimova, Asymptotics of the solution of the problem of the deformation of an arbitrary local periodic thin plate, Trudy Moskov. Mat. Obschch. 65 pp 3– (2004) · Zbl 1167.74480 [2] Borchers, On the equations rot v = g and $$\cdot$$ u = f with zero boundary conditions, Hokkaido Math. J. 19 pp 67– (1990) · Zbl 0719.35014 · doi:10.14492/hokmj/1381517172 [3] Cardone, Homogenization of the mixed boundary value problem for a formally self-adjoint system in a periodically perforated domain, Algebra i analiz. 21 pp 126– (2009) [4] Kondratiev, Boundary-value problems for elliptic equations in conical regions, Sov. Math. Dokl. 4 pp 1600– (1963) [5] Kozlov, Elliptic Boundary Value Problems in Domains with Point Singularities (1997) · Zbl 0947.35004 [6] Ladyzhenskaya, Applied Mathematical Sciences (1985) [7] Maz’ja, Weighted spaces with nonhomogeneous norms and boundary value problems in domains with conical points, in: Elliptische Differentialgleichungen (Meeting, Rostock, 1977), Wilhelm-Pieck-Univ. Rostock 161-189, English transl.: 1984, Am. Math. Soc. Transl. 123 pp 363– (1977) [8] Nazarov, Vishik-Lyusternik method for elliptic boundary-value problems in regions with conical points. 1. The problem in a cone, Sibirsk. Mat. Zh 22 pp 142– (1981) · Zbl 0479.35032 [9] Nazarov, Asymptotics of the solution to a boundary value problem in a thin cylinder with nonsmooth lateral surface, Izv. Ross. Akad. Nauk. Ser. Mat. 57 pp 202– (1993) · Zbl 0807.35031 [10] Nazarov, Asymptotic expansions at infinity of solutions to the elasticity theory problem in a layer, Trudy Moskov. Mat. Obschch. 60 pp 3– (1998) [11] Nazarov, The asymptotic properties of the solution to the Stokes problem in domains that are layer-like at infinity, J. Math. Fluid Mech. 1 pp 131– (1999) · Zbl 0940.35155 · doi:10.1007/s000210050007 [12] Nazarov, On the Fredholm property of the Stokes operator in a layer-like domain, ZAA 20 pp 155– (2001) · Zbl 0984.35127 [13] Nazarov, Elliptic Problems in Domains with Piecewise Smooth Boundaries (1994) · Zbl 0806.35001 · doi:10.1515/9783110848915 [14] Nazarov, Artificial boundary conditions for the Stokes and Navier-Stokes systems in a layer-like domain, Dokl. Ross. Akad. Nauk. 405 pp 311– (2005) · Zbl 1129.76015 [15] Nazarov, Artificial boundary conditions for Stokes and Navier-Stokes equations in domains that are layer-like at infinity, ZAA 27 pp 125– (2008) · Zbl 1144.35044 [16] Nazarov, Comportement asymptotique à l’infini d’un problème de Neumann dans une couche perforée, Comptes Rendus Mécanique 331 pp 85– (2003) · Zbl 1181.74106 · doi:10.1016/S1631-0721(02)00005-0 [17] Nazarov, Asymptotics at infinity of solutions to the Neumann problem in a sieve-type layer, Asymptotic Analysis 44 pp 259– (2005) [18] Temam, Theory and Numerical Analysis (2001)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.