On generalized diffusion and heat systems on an evolving surface with a boundary
Author:
Hajime Koba
Journal:
Quart. Appl. Math. 78 (2020), 617-640
MSC (2010):
Primary 97M50; Secondary 49S05, 49Q20
DOI:
https://doi.org/10.1090/qam/1564
Published electronically:
December 12, 2019
MathSciNet review:
4148821
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Additional Information
Abstract: We consider a diffusion process on an evolving surface with a piecewise Lipschitz-continuous boundary from an energetic point of view. We employ an energetic variational approach with both surface divergence and transport theorems to derive the generalized diffusion and heat systems on the evolving surface. Moreover, we investigate the boundary conditions for the two systems to study their conservation and energy laws. As an application, we make a mathematical model for a diffusion process on an evolving double bubble. Specifically, this paper is devoted to deriving the representation formula for the unit outer co-normal vector to the boundary of a surface.
References
- David E. Betounes, Kinematics of submanifolds and the mean curvature normal, Arch. Rational Mech. Anal. 96 (1986), no. 1, 1–27. MR 853973, DOI https://doi.org/10.1007/BF00251411
- Philippe G. Ciarlet, An introduction to differential geometry with application to elasticity, J. Elasticity 78/79 (2005), no. 1-3, iv+215. With a foreword by Roger Fosdick. MR 2196098, DOI https://doi.org/10.1007/s10659-005-4738-8
- G. Dziuk and C. M. Elliott, Finite elements on evolving surfaces, IMA J. Numer. Anal. 27 (2007), no. 2, 262–292. MR 2317005, DOI https://doi.org/10.1093/imanum/drl023
- Morton E. Gurtin, Allan Struthers, and William O. Williams, A transport theorem for moving interfaces, Quart. Appl. Math. 47 (1989), no. 4, 773–777. MR 1031691, DOI https://doi.org/10.1090/qam/1031691
- Jürgen Jost, Riemannian geometry and geometric analysis, 6th ed., Universitext, Springer, Heidelberg, 2011. MR 2829653
- Hajime Koba, On derivation of compressible fluid systems on an evolving surface, Quart. Appl. Math. 76 (2018), no. 2, 303–359. MR 3769898, DOI https://doi.org/10.1090/qam/1491
- H. Koba, On generalized compressible fluid systems on an evolving surface with a boundary, preprint, arXiv:1810.07909
- H. Koba, C. Liu, and Y. Giga Energetic variational approaches for incompressible fluid systems on an evolving surface, Quart. Appl. Math. 75 (2017), no 2, 359–389. MR3614501. Errata to Energetic variational approaches for incompressible fluid systems on an evolving surface, Quart. Appl. Math. 76 (2018), no 1, 147–152.
- Hajime Koba and Kazuki Sato, Energetic variational approaches for non-Newtonian fluid systems, Z. Angew. Math. Phys. 69 (2018), no. 6, Paper No. 143, 28. MR 3869845, DOI https://doi.org/10.1007/s00033-018-1039-1
- L. Onsager, Reciprocal relations in irreversible processes. I, Physical Review. 37 (1931), 405–426, DOI:https://doi.org/10.1103/PhysRev.37.405
- L. Onsager, Reciprocal relations in irreversible processes. II, Physical Review 38 (1931), 2265–2279 DOI:https://doi.org/10.1103/PhysRev.38.2265
- 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
- J. W. Strutt, Some General Theorems relating to Vibrations, Proc. Lond. Math. Soc. 4 (1871/73), 357–368. MR 1575554, DOI https://doi.org/10.1112/plms/s1-4.1.357
References
- David E. Betounes, Kinematics of submanifolds and the mean curvature normal, Arch. Rational Mech. Anal. 96 (1986), no. 1, 1–27. MR 853973, DOI https://doi.org/10.1007/BF00251411
- Philippe G. Ciarlet, An introduction to differential geometry with application to elasticity, with a foreword by Roger Fosdick, J. Elasticity 78/79 (2005), no. 1-3, iv+215. MR 2196098, DOI https://doi.org/10.1007/s10659-005-4738-8
- G. Dziuk and C. M. Elliott, Finite elements on evolving surfaces, IMA J. Numer. Anal. 27 (2007), no. 2, 262–292. MR 2317005, DOI https://doi.org/10.1093/imanum/drl023
- Morton E. Gurtin, Allan Struthers, and William O. Williams, A transport theorem for moving interfaces, Quart. Appl. Math. 47 (1989), no. 4, 773–777. MR 1031691, DOI https://doi.org/10.1090/qam/1031691
- Jürgen Jost, Riemannian geometry and geometric analysis, 6th ed., Universitext, Springer, Heidelberg, 2011. MR 2829653
- Hajime Koba, On derivation of compressible fluid systems on an evolving surface, Quart. Appl. Math. 76 (2018), no. 2, 303–359. MR 3769898, DOI https://doi.org/10.1090/qam/1491
- H. Koba, On generalized compressible fluid systems on an evolving surface with a boundary, preprint, arXiv:1810.07909
- H. Koba, C. Liu, and Y. Giga Energetic variational approaches for incompressible fluid systems on an evolving surface, Quart. Appl. Math. 75 (2017), no 2, 359–389. MR3614501. Errata to Energetic variational approaches for incompressible fluid systems on an evolving surface, Quart. Appl. Math. 76 (2018), no 1, 147–152.
- Hajime Koba and Kazuki Sato, Energetic variational approaches for non-Newtonian fluid systems, Z. Angew. Math. Phys. 69 (2018), no. 6, Art. 143, 28. MR 3869845, DOI https://doi.org/10.1007/s00033-018-1039-1
- L. Onsager, Reciprocal relations in irreversible processes. I, Physical Review. 37 (1931), 405–426, DOI:https://doi.org/10.1103/PhysRev.37.405
- L. Onsager, Reciprocal relations in irreversible processes. II, Physical Review 38 (1931), 2265–2279 DOI:https://doi.org/10.1103/PhysRev.38.2265
- 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
- J. W. Strutt, Some general theorems relating to vibrations, Proc. Lond. Math. Soc. 4 (1871/73), 357–368. MR 1575554, DOI https://doi.org/10.1112/plms/s1-4.1.357
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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
Keywords:
Mathematical modeling,
energetic variational approach,
evolving surface with boundary,
boundary condition in co-normal direction,
double bubble
Received by editor(s):
October 14, 2019
Received by editor(s) in revised form:
October 31, 2019
Published electronically:
December 12, 2019
Additional Notes:
This work was partly supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number JP15K17580.
Article copyright:
© Copyright 2019
Brown University
