zbMATH — the first resource for mathematics

The boundary value problems for the scalar Oseen equation. (English) Zbl 1257.31006
Summary: The scalar Oseen equation represents a linearized form of the Navier Stokes equations, well-known in hydrodynamics. In the present paper we develop an explicit potential theory for this equation and solve the interior and exterior Oseen Dirichlet and Oseen Neumann boundary value problems via a boundary integral equation method in spaces of continuous functions on a \(C^{2}\)-boundary, extending the classical approach for the isotropic selfadjoint Laplace operator to the anisotropic non-selfadjoint scalar Oseen operator. It turns out that the solution to all boundary value problems can be presented by boundary potentials with source densities constructed as uniquely determined solutions of boundary integral equations.

31B10 Integral representations, integral operators, integral equations methods in higher dimensions
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
35C15 Integral representations of solutions to PDEs
35J25 Boundary value problems for second-order elliptic equations
45B05 Fredholm integral equations
76D07 Stokes and related (Oseen, etc.) flows
Full Text: DOI
[1] Amrouche, The scalar Oseen operator -\(\Delta\) + /x1 in \documentclassarticle\usepackageamssymb\begindocument\pagestyleempty\(\mathbb {R}^2\)\enddocument, Appl. Math. 53 (1) pp 41– (2008) · Zbl 1177.76080 · doi:10.1007/s10492-008-0012-2
[2] Amrouche, Lp-inequalities for the scalar Oseen potential, J. Math. Anal. Appl. 337 pp 753– (2008) · Zbl 1134.35021 · doi:10.1016/j.jmaa.2007.03.112
[3] Amrouche, Weighted Sobolev spaces for a scalar model Oseen equation in \documentclassarticle\usepackageamssymb\begindocument\pagestyleempty\(\mathbb {R}^3\)\enddocument, J. Math. Fluid Mech. 9 pp 181– (2007) · Zbl 1119.76015 · doi:10.1007/s00021-005-0195-1
[4] Dautray, Mathematical Analysis and Numerical Methods for Science and Technology Vol. 1 (2000) · Zbl 0944.47002 · doi:10.1007/978-3-642-58090-1
[5] Deuring, On volume potentials related to the time-dependent Oseen system, WSEAS Trans. Math. 5 (3) pp 252– (2006)
[6] P. Deuring On boundary-driven time-dependent Oseen flows, Banach Center Publ. Vol. 81 Part 1 119 132 · Zbl 1148.76016
[7] Deuring, Spatial decay of time-dependent Oseen flows, SIAM J. Math. Anal. 41 (3) pp 886– (2009) · Zbl 1189.35222 · doi:10.1137/080723831
[8] Enomoto, On the Rate of Decay of the Oseen Semigroup in Exterior Domains and its Application to Navier-Stokes Equation, J. Math. Fluid Mech. 7 pp 339– (2005) · Zbl 1094.35097 · doi:10.1007/s00021-004-0132-8
[9] Farwig, The stationary exterior 3D-problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces, Math. Z. 211 3 pp 409– (1992) · Zbl 0727.35106 · doi:10.1007/BF02571437
[10] Farwig, Weighted estimates for the Oseen equations and the Navier-Stokes equations in exterior domains, Series on Advances in Mathematics for Applied Sciences Vol. 47 pp 11– (1998) · Zbl 0934.35120
[11] Finn, Estimates at infinity for stationary solutions of the Navier-Stokes equations, Bull. Math. Soc. Sci. Math. Phys. R. P. Roumaine 3 (51) pp 387– (1959)
[12] Finn, On the exterior stationary problem for the Navier-Stokes equations, and associated perturbation problems, Arch. Ration. Mech. Anal. 19 pp 363– (1965) · Zbl 0149.44606 · doi:10.1007/BF00253485
[13] Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. I. Linearized steady problems, Springer Tracts in Natural Philosophy Vol. 38 (1994) · Zbl 0949.35004
[14] N. M. Günter
[15] W. Hackbusch
[16] Kobayashi, On the Oseen equation in the three dimensional exterior domains, Math. Ann. 310 pp 1– (1998) · Zbl 0891.35114 · doi:10.1007/s002080050134
[17] S. Kračmar D. Medková Š. Nečasová W. Varnhorn
[18] Kračmar, Estimates of Oseen kernels in weighted Lp, spaces, J. Math. Soc. Japan 53 (1) pp 59– (2001) · Zbl 0988.76021 · doi:10.2969/jmsj/05310059
[19] E. Martensen B. G. Teubner
[20] Maz’ya, Boundary Integral Equations, Analysis IV, Encyclopaedia of Mathematical Sciences Vol. 27 pp 127– (1991)
[21] C. W. Oseen
[22] Vladimirov, Equations of Mathematical Physics (1971)
[23] Walter, Einführung in die Potentialtheorie (1971)
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.