Abstract
This paper deals with the solvability of the Navier-Stokes equations on manifolds with boundary. In particular, we concentrate on the inhomogeneous slip boundary condition. Our formulation of the equations takes into account a curvature term which results from a proper derivation of the Navier-Stokes equations. This term has not been considered in prior work.
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References
Adams, R. A.:Sobolev Spaces. (Pure and Applied Mathematics; 65) New York, London: Academic Press, 1975
Aris, R.:Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Englewood Cliffs, NJ: Prentice-Hall, 1962
Aubin, T.:Nonliear Analysis on Manifolds. Monge-Ampère Equations. (Grundlehren der methematischen Wissenschaften: 252) Berlin, Heidelberg, New York: Springer-Verlag, 1982
Avez, A., Bamberger, Y.:Mouvements sphériques des fluides visqueux incompressibles. J. Mécanique17, 107–145 (1978)
Bogovskiî, M. E.:Solution for the First Boundary Value Problem for the Equation of Continuity of an Incompressible Medium. Soviet Math. Dokl.20, 1094–1098 (1979)
Cherrier, P.:Problèmes non linéaires sur les variétés riemanniennes. J. Funct. Anal.57, 154–206, (1984)
Ebin, D. G. ·Marsden J.:Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. of Math.92, 102–163 (1970)
Girault, V. ·Raviart, P.-A.:Finite Element Methods for the Navier-Stokes equations. (Springer series in computational mathematics; 5) Berlin, Heidelberg, New York: Springer-Verlag, 1986
Il'in, A. A.:The Navier-Stokes and Euler Equations on Two-Dimensional Closed Manifolds. Math. USSR-Sb.69, 559–579 (1991)
Il'In, A. A. ·Filatov, A. N.:On Unique Solvability of the Navier-Stokes Equations on the Two-Dimensional Sphere. Soviet Math. Dokl.38, 9–13 (1989)
Priebe, V. Lösung der instationären Navier-Stokes-Gleichungen auf berandeten Mannigfaltigkeiten. Diploma thesis, Rheinische Friedrich-Wilhelms-Universität Bonn, Institut für Angewandte Mathematik, 1991
Scriven, L. E.:Dynamics of a fluid interface. Equation of motion for Newtonian surface fluids. Chem. Eng. Sci.12, 98–108 (1960)
Serrin, J.:Mathematical Principles of Classical Fluid Mechanics. In: Handbuch der Physik VIII, 1; hrsg. vonS. Flügge, Berlin, Göttingen, Heidelberg: Springer-Verlag, 1959
Solonnikov, V. A. ·Ščadilov, V. E.:On a Boundary Value Problem for a Stationary System of Navier-Stokes Equations. Proc. Steklov Inst. Math.125, 186–199 (1973)
Takeshita, A.:Existence and Non-Existence of Solutions to the Stationary Navier-Stokes Equations on Compact Riemannian Manifolds. Nagoya University, Department of Mathematics, College of General Education, preprint 12 (1983)
Temam, R.:Navier-Stokes Equations. Theory and Numerical Analysis, 3rd (revised) edition. (Studies in Mathematics and its Applications; 2) Amsterdam, New York, Oxford: North-Holland, 1984
Yosida, K.:Functional Analysis, 6th edition (Grundlehren der mathematischen Wissenschaft; 123) Berlin, Heidelberg, New York: Springer-Verlag, 1980
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During the work on this version, the author received technical support through a fellowship of the DFG