On properties of some integrals related to potentials for Stokes equations
Author:
P. A. Krutitskii
Journal:
Quart. Appl. Math. 65 (2007), 549-569
MSC (2000):
Primary 31A10, 35Q30
DOI:
https://doi.org/10.1090/S0033-569X-07-01054-0
Published electronically:
April 19, 2007
MathSciNet review:
2354887
Full-text PDF Free Access
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Abstract: The integrals arising from potentials for two-dimensional Stokes equations are explored in the case when the potentials are defined on the smooth open arc of an arbitrary shape, while the densities in the potentials belong to weighted Hölder space and may have power singularities. The properties of smoothness of these integrals and their derivatives are studied. The singularities of the derivatives of the integrals at the ends of the arcs are examined. The integrals studied in the paper being coupled with harmonic logarithmic potential yield single layer potentials for velocities in Stokes equations. Single layer potential for pressure in Stokes equations is investigated also.
References
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- 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
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References
- Muskhelishvili, N.I. Singular integral equations. Nauka, Moscow, 1968. (In Russian; English translation: Noordhoff, Groningen, 1972.) MR 0355494 (50:7968)
- Popov, A.N. Application of potential theory to solving the linearized Navier-Stokes system of equations in a two–dimensional case. Trudy MIAN, 1971, t.116, pp.162–180. (In Russian). MR 0364909 (51:1163)
- Power, H. The completed double layer boundary integral equation method for two–dimensional Stokes flow. IMA Journal of Applied Mathematics, 1993, v.51, pp.123–145. MR 1244192 (94i:76020)
- Krutitskii, P.A. The Dirichlet problem for the Helmholtz equation in the exterior of cuts in the plane. Comput. Math. Math. Phys., 1994, v.34, No.8/9, pp.1073-1090. MR 1300397 (95f:35046)
- Vladimirov, V.S. Equations of Mathematical Physics. Nauka, Moscow, 1981. (In Russian; English translation of 1st edition: Marcel Dekker, N.Y., 1971.)
- Gakhov, F.D. Boundary value problems. Fizmatlit, Moscow, 1963. (In Russian; English translation: Pergamon Press, Oxford; Addison-Wesley, Reading, Mass., 1966.) MR 0198152 (33:6311)
- Shilov, G.E. Mathematical Analysis. Special course. Fizmatlit, Moscow, 1960. (In Russian.)
- Pozrikidis, C. Boundary integral and singularity method for linearized viscous flow. Cambridge University Press, Cambridge, 1992. MR 1156495 (93a:76027)
- Varnhorn, W. The Stokes equations. Akademie Verlag, Berlin, 1994. MR 1282728 (95e:35162)
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Additional Information
P. A. Krutitskii
Affiliation:
Department of Mathematics, Faculty of Physics, Moscow State University, Moscow 119899, Russia
Received by editor(s):
November 15, 2006
Published electronically:
April 19, 2007
Additional Notes:
The research was supported by the RFBR grants 05-01-00050, 07-01-00029
and the Bernoulli Center in Lausanne (Switzerland).
Article copyright:
© Copyright 2007
Brown University
