Stokes flow with kinematic and dynamic boundary conditions
Author:
John Fabricius
Journal:
Quart. Appl. Math. 77 (2019), 525-544
MSC (2010):
Primary 76D03, 76D07
DOI:
https://doi.org/10.1090/qam/1534
Published electronically:
February 7, 2019
Original Version:
Posted February 7, 2019.
Corrected version:
The caption to Figure 2 was inadvertently omitted in the original posting.
MathSciNet review:
3962580
Full-text PDF
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Additional Information
Abstract: We review the first and second boundary value problems for the Stokes system posed in a bounded Lipschitz domain in $\mathbb {R}^n$. Particular attention is given to the mixed boundary condition: a Dirichlet condition is imposed for the velocity on one part of the boundary while a Neumann condition for the stress tensor is imposed on the remaining part. Some minor modifications to the standard theory are therefore required. The most noteworthy result is that both pressure and velocity are unique.
References
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- Franck Boyer and Pierre Fabrie, Mathematical tools for the study of the incompressible Navier-Stokes equations and related models, Applied Mathematical Sciences, vol. 183, Springer, New York, 2013. MR 2986590
- Philippe G. Ciarlet, On Korn’s inequality, Chin. Ann. Math. Ser. B 31 (2010), no. 5, 607–618. MR 2726058, DOI https://doi.org/10.1007/s11401-010-0606-3
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- J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969 (French). MR 0259693
- Sanja Marušić and Eduard Marušić-Paloka, Two-scale convergence for thin domains and its applications to some lower-dimensional models in fluid mechanics, Asymptot. Anal. 23 (2000), no. 1, 23–57. MR 1764338
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- J. Nečas, Sur les normes équivalentes dans $W_k^p$ et sur la coercivité des formes formellement positives, Sém. math. sup. Université Montreal, pp. 102–128, 1966.
- Jindřich Nečas, Direct methods in the theory of elliptic equations, Springer Monographs in Mathematics, Springer, Heidelberg, 2012. Translated from the 1967 French original by Gerard Tronel and Alois Kufner; Editorial coordination and preface by Šárka Nečasová and a contribution by Christian G. Simader. MR 3014461
- Ronald L. Panton, Incompressible flow, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1984. MR 802628
- Olivier Pironneau, Finite element methods for fluids, John Wiley & Sons, Ltd., Chichester; Masson, Paris, 1989. Translated from the French. MR 1030279
- James Serrin, Mathematical principles of classical fluid mechanics, Handbuch der Physik (herausgegeben von S. Flügge), Bd. 8/1, Strömungsmechanik I (Mitherausgeber C. Truesdell), Springer-Verlag, Berlin-Göttingen-Heidelberg, 1959, pp. 125–263. MR 0108116
- Luc Tartar, Topics in nonlinear analysis, Publications Mathématiques d’Orsay 78, vol. 13, Université de Paris-Sud, Département de Mathématique, Orsay, 1978. MR 532371
- Luc Tartar, An introduction to Navier-Stokes equation and oceanography, Lecture Notes of the Unione Matematica Italiana, vol. 1, Springer-Verlag, Berlin; UMI, Bologna, 2006. MR 2258988
- Roger Temam, Navier-Stokes equations, 3rd ed., Studies in Mathematics and its Applications, vol. 2, North-Holland Publishing Co., Amsterdam, 1984. Theory and numerical analysis; With an appendix by F. Thomasset. MR 769654
References
- M. E. Bogovskiĭ, Solution of the first boundary value problem for an equation of continuity of an incompressible medium, Dokl. Akad. Nauk SSSR 248 (1979), no. 5, 1037–1040 (Russian). MR 553920
- Franck Boyer and Pierre Fabrie, Mathematical tools for the study of the incompressible Navier-Stokes equations and related models, Applied Mathematical Sciences, vol. 183, Springer, New York, 2013. MR 2986590
- Philippe G. Ciarlet, On Korn’s inequality, Chin. Ann. Math. Ser. B 31 (2010), no. 5, 607–618. MR 2726058, DOI https://doi.org/10.1007/s11401-010-0606-3
- C. Conca, F. Murat, and O. Pironneau, The Stokes and Navier-Stokes equations with boundary conditions involving the pressure, Japan. J. Math. (N.S.) 20 (1994), no. 2, 279–318. MR 1308419, DOI https://doi.org/10.4099/math1924.20.279
- G. Duvaut and J.-L. Lions, Inequalities in mechanics and physics, Springer-Verlag, Berlin-New York, 1976. Translated from the French by C. W. John; Grundlehren der Mathematischen Wissenschaften, 219. MR 0521262
- J. Fabricius, E. Miroshnikova, A. Tsandzana, and P. Wall, Pressure-driven flow in thin domains, Submitted, May 2018.
- G. P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations: Steady-state problems, 2nd ed., Springer Monographs in Mathematics, Springer, New York, 2011. MR 2808162
- Roland Glowinski, Numerical methods for nonlinear variational problems, Springer Series in Computational Physics, Springer-Verlag, New York, 1984. MR 737005
- Roland Glowinski, Finite element methods for incompressible viscous flow, Handbook of numerical analysis, Vol. IX, Handb. Numer. Anal., IX, North-Holland, Amsterdam, 2003, pp. 3–1176. MR 2009826
- P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
- John G. Heywood, Rolf Rannacher, and Stefan Turek, Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations, Internat. J. Numer. Methods Fluids 22 (1996), no. 5, 325–352. MR 1380844, DOI https://doi.org/10.1002/%28SICI%291097-0363%2819960315%2922%3A5%24%5Clangle%24325%3A%3AAID-FLD307%24%5Crangle%243.0.CO%3B2-Y
- O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Second English edition, revised and enlarged. Translated from the Russian by Richard A. Silverman and John Chu. Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969. MR 0254401
- Horace Lamb, Hydrodynamics, 6th ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1993. With a foreword by R. A. Caflisch [Russel E. Caflisch]. MR 1317348
- J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969 (French). MR 0259693
- Sanja Marušić and Eduard Marušić-Paloka, Two-scale convergence for thin domains and its applications to some lower-dimensional models in fluid mechanics, Asymptot. Anal. 23 (2000), no. 1, 23–57. MR 1764338
- Jerrold E. Marsden and Thomas J. R. Hughes, Mathematical foundations of elasticity, Dover Publications, Inc., New York, 1994. Corrected reprint of the 1983 original. MR 1262126
- V. Maz’ya and J. Rossmann, $L_p$ estimates of solutions to mixed boundary value problems for the Stokes system in polyhedral domains, Math. Nachr. 280 (2007), no. 7, 751–793. MR 2321139, DOI https://doi.org/10.1002/mana.200610513
- V. Maz’ya and J. Rossmann, Mixed boundary value problems for the stationary Navier-Stokes system in polyhedral domains, Arch. Ration. Mech. Anal. 194 (2009), no. 2, 669–712. MR 2563641, DOI https://doi.org/10.1007/s00205-008-0171-z
- J. Nečas, Sur les normes équivalentes dans $W_k^p$ et sur la coercivité des formes formellement positives, Sém. math. sup. Université Montreal, pp. 102–128, 1966.
- Jindřich Nečas, Direct methods in the theory of elliptic equations, with translated from the 1967 French original by Gerard Tronel and Alois Kufner; editorial coordination and preface by Šárka Nečasová and a contribution by Christian G. Simader, Springer Monographs in Mathematics, Springer, Heidelberg, 2012. MR 3014461
- Ronald L. Panton, Incompressible flow, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1984. MR 802628
- Olivier Pironneau, Finite element methods for fluids, John Wiley & Sons, Ltd., Chichester; Masson, Paris, 1989. Translated from the French. MR 1030279
- James Serrin, Mathematical principles of classical fluid mechanics, Handbuch der Physik (herausgegeben von S. Flügge), Bd. 8/1, Strömungsmechanik I (Mitherausgeber C. Truesdell), Springer-Verlag, Berlin-Göttingen-Heidelberg, 1959, pp. 125–263. MR 0108116
- Luc Tartar, Topics in nonlinear analysis, Publications Mathématiques d’Orsay 78, vol. 13, Université de Paris-Sud, Département de Mathématique, Orsay, 1978. MR 532371
- Luc Tartar, An introduction to Navier-Stokes equation and oceanography, Lecture Notes of the Unione Matematica Italiana, vol. 1, Springer-Verlag, Berlin; UMI, Bologna, 2006. MR 2258988
- Roger Temam, Navier-Stokes equations: Theory and numerical analysis, with with an appendix by F. Thomasset, 3rd ed., Studies in Mathematics and its Applications, vol. 2, North-Holland Publishing Co., Amsterdam, 1984. MR 769654
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Additional Information
John Fabricius
Affiliation:
Department of Engineering Sciences and Mathematics, Luleå University of Technology, SE-971 87 Luleå, Sweden
MR Author ID:
843760
Email:
john.fabricius@ltu.se
Keywords:
Stokes equation,
stress condition,
traction condition,
de Rham operator,
pressure operator
Received by editor(s):
May 3, 2018
Published electronically:
February 7, 2019
Article copyright:
© Copyright 2019
Brown University
