Energetic variational approaches for incompressible fluid systems on an evolving surface
Authors:
Hajime Koba, Chun Liu and Yoshikazu Giga
Journal:
Quart. Appl. Math. 75 (2017), 359-389
MSC (2010):
Primary 37E35, 97M50, 49S05, 49Q20, 35A15
DOI:
https://doi.org/10.1090/qam/1452
Published electronically:
August 25, 2016
Erratum:
Quart. Appl. Math. 76 (2018), 147-152.
MathSciNet review:
3614501
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Additional Information
Abstract: This paper considers the equations governing incompressible fluid-flow on an evolving surface. We employ an energetic variational approach to derive the dynamical system for the motion of incompressible fluid on such an evolving surface. The focus is to understand the coupling of an incompressible fluid-flow and the evolution of a moving surface, involving both the curvature and the motion of the surface.
References
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- Dieter Bothe and Jan Prüss, On the two-phase Navier-Stokes equations with Boussinesq-Scriven surface fluid, J. Math. Fluid Mech. 12 (2010), no. 1, 133–150. MR 2602917, DOI 10.1007/s00021-008-0278-x
- M. J. Boussinesq, Sur l’existence d’une viscosité seperficielle, dans la mince couche de transition séparant un liquide d’un autre fluide contigu, Ann. Chim. Phys. 29 (1913), 349–357.
- G. Dziuk and C. M. Elliott, Finite elements on evolving surfaces, IMA J. Numer. Anal. 27 (2007), no. 2, 262–292. MR 2317005, DOI 10.1093/imanum/drl023
- David G. Ebin and Jerrold Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. (2) 92 (1970), 102–163. MR 271984, DOI 10.2307/1970699
- Yoshikazu Giga, Surface evolution equations, Monographs in Mathematics, vol. 99, Birkhäuser Verlag, Basel, 2006. A level set approach. MR 2238463
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364, DOI 10.1007/978-3-642-61798-0
- Yuri Gliklikh, Global analysis in mathematical physics, Applied Mathematical Sciences, vol. 122, Springer-Verlag, New York, 1997. Geometric and stochastic methods; Translated from the 1989 Russian original and with Appendix F by Viktor L. Ginzburg. MR 1438545, DOI 10.1007/978-1-4612-1866-1
- Hajime Koba, On derivation of compressible fluid systems on an evolving surface, preprint.
- Yoshihiko Mitsumatsu and Yasuhisa Yano, Geometry of an incompressible fluid on a Riemannian manifold, Sūrikaisekikenkyūsho K\B{o}kyūroku 1260 (2002), 33–47 (Japanese). Geometric mechanics (Japanese) (Kyoto, 2002). MR 1930362
- L. Onsager, Reciprocal Relations in Irreversible Processes. I. Physical Review 37 (1931), 405–426.
- L. Onsager, Reciprocal Relations in Irreversible Processes. II. Physical Review 38 (1931), 2265–2279.
- L. Rayleigh and J.W. Strutt, Some General Theorems Relating to Vibrations. Proceedings of the London Mathematical Society 4 (1873), 357–368.
- L.E. Scriven, Dynamics of a fluid interface Equation of motion for Newtonian surface fluids. Chem. Eng. Sci. 12 (1960), 98–108.
- Leon Simon, Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. MR 756417
- John C. Slattery, Interfacial Transport Phenomena, Springer-Verlag, New York, 1990, xvi+1159 pp., DOI 10.1007/978-1-4757-2090-7
- Michael E. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. Partial Differential Equations 17 (1992), no. 9-10, 1407–1456. MR 1187618, DOI 10.1080/03605309208820892
- Michael E. Taylor, Partial differential equations III. Nonlinear equations, 2nd ed., Applied Mathematical Sciences, vol. 117, Springer, New York, 2011. MR 2744149, DOI 10.1007/978-1-4419-7049-7
References
- Marc Arnaudon and Ana Bela Cruzeiro, Lagrangian Navier-Stokes diffusions on manifolds: variational principle and stability, Bull. Sci. Math. 136 (2012), no. 8, 857–881. MR 2995006, DOI 10.1016/j.bulsci.2012.06.007
- V. Arnol′d, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble) 16 (1966), no. fasc. 1, 319–361 (French). MR 0202082
- V. I. Arnol′d, Mathematical methods of classical mechanics, 2nd ed., translated from the Russian by K. Vogtmann and A. Weinstein, Graduate Texts in Mathematics, vol. 60, Springer-Verlag, New York, 1989. MR 997295, DOI 10.1007/978-1-4757-2063-1
- Dieter Bothe and Jan Prüss, On the two-phase Navier-Stokes equations with Boussinesq-Scriven surface fluid, J. Math. Fluid Mech. 12 (2010), no. 1, 133–150. MR 2602917, DOI 10.1007/s00021-008-0278-x
- M. J. Boussinesq, Sur l’existence d’une viscosité seperficielle, dans la mince couche de transition séparant un liquide d’un autre fluide contigu, Ann. Chim. Phys. 29 (1913), 349–357.
- G. Dziuk and C. M. Elliott, Finite elements on evolving surfaces, IMA J. Numer. Anal. 27 (2007), no. 2, 262–292. MR 2317005, DOI 10.1093/imanum/drl023
- David G. Ebin and Jerrold Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid. , Ann. of Math. (2) 92 (1970), 102–163. MR 0271984
- Yoshikazu Giga, Surface evolution equations, A level set approach, Monographs in Mathematics, vol. 99, Birkhäuser Verlag, Basel, 2006. MR 2238463
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364
- Yuri Gliklikh, Global analysis in mathematical physics, Geometric and stochastic methods; translated from the 1989 Russian original and with Appendix F by Viktor L. Ginzburg, Applied Mathematical Sciences, vol. 122, Springer-Verlag, New York, 1997. MR 1438545, DOI 10.1007/978-1-4612-1866-1
- Hajime Koba, On derivation of compressible fluid systems on an evolving surface, preprint.
- Yoshihiko Mitsumatsu and Yasuhisa Yano, Geometry of an incompressible fluid on a Riemannian manifold, Geometric mechanics (Japanese) (Kyoto, 2002), Sūrikaisekikenkyūsho Kōkyūroku 1260 (2002), 33–47 (Japanese). MR 1930362
- L. Onsager, Reciprocal Relations in Irreversible Processes. I. Physical Review 37 (1931), 405–426.
- L. Onsager, Reciprocal Relations in Irreversible Processes. II. Physical Review 38 (1931), 2265–2279.
- L. Rayleigh and J.W. Strutt, Some General Theorems Relating to Vibrations. Proceedings of the London Mathematical Society 4 (1873), 357–368.
- L.E. Scriven, Dynamics of a fluid interface Equation of motion for Newtonian surface fluids. Chem. Eng. Sci. 12 (1960), 98–108.
- Leon Simon, Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. MR 756417
- John C. Slattery, Interfacial Transport Phenomena, Springer-Verlag, New York, 1990, xvi+1159 pp.
- Michael E. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. Partial Differential Equations 17 (1992), no. 9-10, 1407–1456. MR 1187618, DOI 10.1080/03605309208820892
- Michael E. Taylor, Partial differential equations III. Nonlinear equations, 2nd ed., Applied Mathematical Sciences, vol. 117, Springer, New York, 2011. MR 2744149, DOI 10.1007/978-1-4419-7049-7
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Additional Information
Hajime Koba
Affiliation:
Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyamacho, Toyonaka, Osaka, 560-8531, Japan
MR Author ID:
1013948
Email:
iti@sigmath.es.osaka-u.ac.jp
Chun Liu
Affiliation:
Department of Mathematics, 107A McAllister Building, Pennsylvania State University, University Park, Pennsylvania 16802
MR Author ID:
362496
Email:
liuc@psu.edu
Yoshikazu Giga
Affiliation:
Department of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo, 153-8914, Japan
MR Author ID:
191842
Email:
labgiga@ms.u-tokyo.ac.jp
Received by editor(s):
April 13, 2016
Published electronically:
August 25, 2016
Additional Notes:
The work of the first author was partly supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Numbers JP25887048 and JP15K17580.
The work of the second author was partially supported by National Science Foundation grants DMS-1412005 and DMS-1216938
The work of the third author was partly supported by JSPS through the grants Kiban S number 26220702 and Kiban B number 16H03948
Article copyright:
© Copyright 2016
Brown University
