The Dirichlet problem for Stokes equations outside open arcs in a half-plane and creeping flow over thin profiles

Author:
P. A. Krutitskii

Journal:
Quart. Appl. Math. **68** (2010), 537-556

MSC (2000):
Primary 35Q30, 76D07, 31A10, 45F05, 45F15

DOI:
https://doi.org/10.1090/S0033-569X-2010-01162-3

Published electronically:
May 27, 2010

MathSciNet review:
2676975

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the Dirichlet problem for Stokes equations outside curvilinear open arcs in a half-plane. We prove existence and uniqueness of a classical solution to this problem. We obtain integral representation for a solution in the form of potentials, densities in which can be found as a unique solution of the system of the Fredholm integral equations of the second kind and index zero. The creeping flow of viscous fluid over thin profiles is described by the Dirichlet problem studied in this paper.

**1.**A. N. Popov,*An application of potential theory to the solution of a linearized system of Navier-Stokes equations in the two-dimensional case*, Trudy Mat. Inst. Steklov.**116**(1971), 162–180, 237 (Russian). Boundary value problems of mathematical physics, 7. MR**0364909****2.**C. Pozrikidis,*Boundary integral and singularity methods for linearized viscous flow*, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1992. MR**1156495****3.**Werner Varnhorn,*The Stokes equations*, Mathematical Research, vol. 76, Akademie-Verlag, Berlin, 1994. MR**1282728****4.**Henry Power,*The completed double layer boundary integral equation method for two-dimensional Stokes flow*, IMA J. Appl. Math.**51**(1993), no. 2, 123–145. MR**1244192**, https://doi.org/10.1093/imamat/51.2.123**5.**Vladimirov V.S.*Equations of Mathematical Physics.*Nauka, Moscow, 1981. (In Russian; English translation of 1st edition: Marcel Dekker, N.Y., 1971.)**6.**P. A. Krutitskiĭ,*The Dirichlet problem for the Helmholtz equation in the exterior of cuts in the plane*, Zh. Vychisl. Mat. i Mat. Fiz.**34**(1994), no. 8-9, 1237–1258 (Russian, with Russian summary); English transl., Comput. Math. Math. Phys.**34**(1994), no. 8-9, 1073–1090. MR**1300397****7.**P. A. Krutitskiĭ,*On the Stokes potential for pressure*, Uspekhi Mat. Nauk**62**(2007), no. 2(374), 177–178 (Russian); English transl., Russian Math. Surveys**62**(2007), no. 2, 385–387. MR**2352372**, https://doi.org/10.1070/RM2007v062n02ABEH004403**8.**P. A. Krutitskiĭ,*On Stokes potentials for velocities*, Uspekhi Mat. Nauk**62**(2007), no. 6(378), 179–180 (Russian); English transl., Russian Math. Surveys**62**(2007), no. 6, 1212–1214. MR**2382807**, https://doi.org/10.1070/RM2007v062n06ABEH004489**9.**P. A. Krutitskii,*On properties of some integrals related to potentials for Stokes equations*, Quart. Appl. Math.**65**(2007), no. 3, 549–569. MR**2354887**, https://doi.org/10.1090/S0033-569X-07-01054-0**10.**N. I. Muskhelishvili,*Singular integral equations*, Wolters-Noordhoff Publishing, Groningen, 1972. Boundary problems of functions theory and their applications to mathematical physics; Revised translation from the Russian, edited by J. R. M. Radok; Reprinted. MR**0355494****11.**A. N. Kolmogorov and S. V. Fomin,*\cyrÈlementy teorii funktsiĭ i funktsional′nogo analiza*, 5th ed., “Nauka”, Moscow, 1981 (Russian). With a supplement “Banach algebras” by V. M. Tikhomirov. MR**630899****12.**Krein S.G. (editor).*Functional analysis*. Nauka, Moscow, 1964 (in Russian). English translation: Wolters-Noordhoff Publishing, Groningen, 1972.**13.**L. V. Kantorovich and G. P. Akilov,*Functional analysis*, 2nd ed., Pergamon Press, Oxford-Elmsford, N.Y., 1982. Translated from the Russian by Howard L. Silcock. MR**664597****14.**V. Trénoguine,*Analyse fonctionnelle*, Traduit du Russe: Mathématiques. [Translations of Russian Works: Mathematics], “Mir”, Moscow, 1985 (French). Translated from the Russian by V. Kotliar. MR**836334****15.**Pavel A. Krutitskii,*An explicit solution of the pseudo-hyperbolic initial-boundary value problem in a multiply connected region*, Math. Methods Appl. Sci.**18**(1995), no. 11, 897–925. MR**1346665**, https://doi.org/10.1002/mma.1670181105

P. Krutitskii,*Erratum: “An explicit solution of the pseudo-hyperbolic initial-boundary value problem in a multiply connected region”*, Math. Methods Appl. Sci.**19**(1996), no. 3, 253. MR**1371777**, https://doi.org/10.1002/(SICI)1099-1476(199602)19:3<253::AID-MMA1772>3.3.CO;2-J**16.**Alexander D. Bruno,*Power geometry in algebraic and differential equations*, North-Holland Mathematical Library, vol. 57, North-Holland Publishing Co., Amsterdam, 2000. Translated from the 1998 Russian original by V. P. Varin and revised by the author. MR**1773512****17.**M. Kohr,*Existence and uniqueness result for Stokes flows in a half-plane*, Arch. Mech. (Arch. Mech. Stos.)**50**(1998), no. 4, 791–803. MR**1670707****18.**Henry Power and B. Febres de Power,*Second-kind integral equation formulation for the slow motion of a particle of arbitrary shape near a plane wall in a viscous fluid*, SIAM J. Appl. Math.**53**(1993), no. 1, 60–70. MR**1202840**, https://doi.org/10.1137/0153004**19.**Hsu R., Canatos P.*The motion of a rigid body in a viscous fluid bounded by a plane wall*. J. Fluid Mech., 1989, v. 207, pp. 29-72.

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC (2000):
35Q30,
76D07,
31A10,
45F05,
45F15

Retrieve articles in all journals with MSC (2000): 35Q30, 76D07, 31A10, 45F05, 45F15

Additional Information

**P. A. Krutitskii**

Affiliation:
KIAM, Miusskaya Sq. 4, Moscow 125047, Russia

DOI:
https://doi.org/10.1090/S0033-569X-2010-01162-3

Received by editor(s):
November 24, 2008

Published electronically:
May 27, 2010

Article copyright:
© Copyright 2010
Brown University