Advertisement

A transport theorem for nonconvecting open sets on an embedded manifold

  • Brian SeguinEmail author
Original Article
  • 3 Downloads

Abstract

Most transport theorems—that is, a formula for the rate of change of an integral in which both the integrand and domain of integration depend on time—involve domains that evolve according to a flow map. Such domains are said to be convecting. Here, a transport theorem for nonconvecting domains evolving on an embedded manifold is established. While the domain is not convecting, it is assumed that the boundary of the domain evolves according to a flow map in some generalized sense. The proof relies on considering the evolving set as a fixed set in one higher dimension and then using the divergence theorem. The domains considered can be irregular in that their boundaries need only be Lipschitz. Tools from geometric measure theory are used to deal with this irregularity.

Keywords

Irregular domains Lipschitz domains First variation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

References

  1. 1.
    Abatangelo, A., Valdinoci, E.: A notion of nonlocal curvature. Numer. Funct. Anal. Optim. 35, 793–815 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Science Publications, Oxford (2000)zbMATHGoogle Scholar
  3. 3.
    Betounes, D.E.: Kinematics of submanifolds and the mean curvature normal. Arch. Ration. Mech. Anal. 96, 1–27 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bucur, D., Buttazzo, G.: Variational Methods in Shape Optimization Problems. Birkhaüser, Basel (2005)zbMATHGoogle Scholar
  5. 5.
    Caffarelli, L., Roquejoffre, J.M., Savin, O.: Nonlocal minimal surfaces. Commun. Pure Appl. Math. 63, 1111–1144 (2010)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Cermelli, P., Fried, E., Gurtin, M.: Transport relations for surface integrals arising in the formulation of balance laws for evolving fluid interfaces. J. Fluid Mech. 544, 339–351 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Degiovanni, M., Musesti, A.: Edge-force densities and second-order powers. Ann. Mat. Pura Appl. 185, 81–103 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Estrada, R., Kanwal, R.P.: Non-classical derivation of the transport theorems for wave fronts. J. Math. Anal. Appl. 159, 290–297 (1990)CrossRefzbMATHGoogle Scholar
  9. 9.
    Falach, L., Segev, R.: Reynolds transport theorem for smooth deformations of currents on manifolds. Math. Mech. Solids 20, 770–786 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Falach, L., Segev, R.: On the role of sharp chains in the transport theorem. Contin. Mech. Thermodyn. 28, 539–559 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Federer, H.: Geometric Measure Theory. Springer, Berlin (1969)zbMATHGoogle Scholar
  12. 12.
    Fosdick, R., Tang, H.: Surface transport in continuum mechanics. Math. Mech. Solids 14, 587–598 (2009)CrossRefzbMATHGoogle Scholar
  13. 13.
    Gurtin, M.E., Fried, E., Anand, L.: The Mechanics and Thermodynamics of Continua. Cambridge University Press, Cambridge (2009)Google Scholar
  14. 14.
    Gurtin, M.E., Struthers, A., Williams, W.O.: A transport theorem for moving interfaces. Q. Appl. Math. 47, 773–777 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Harrison, J.: Operator calculus of differential chains and differential forms. J. Geom. Anal. 25, 357–420 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lidström, P.: Moving regions in euclidean space and Reynolds’ transport theorem. Math. Mech. Solids 16, 366–380 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Marzocchi, A.: Singular stresses and nonsmooth boundaries in continuum mechanics. Notes for the XXX Ravello Summer School on Mathematical Physics (2005)Google Scholar
  18. 18.
    Schuricht, F.: A new mathematical foundation for contact interactions in continuum physics. Arch. Ration. Mech. Anal. 184, 495–551 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Seguin, B.: A fractional notion of length and an associated nonlocal curvature. J. Geom. Anal. (2018).  https://doi.org/10.1007/s12220-018-00140-9 Google Scholar
  20. 20.
    Seguin, B., Fried, E.: Roughening it: evolving irregular domains and transport theorems. Math. Models Methods Appl. Sci. 24, 1729–1779 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Seguin, B., Hinz, D.F., Fried, E.: Extending the transport theorem to rough domains of integration. Appl. Mech. Rev. 66, 0508021–05080216 (2014)CrossRefGoogle Scholar
  22. 22.
    S̆ilhavý, M.: Fluxes across parts of fractional boundaries. Milan J. Math. 74, 1–45 (2006)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Walker, S.W.: The Shape of Things: A Practical Guide to Differential Geometry and the Shape Derivative. SIAM, Bangkok (2015)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsLoyola University ChicagoChicagoUSA

Personalised recommendations