Abstract
A generalized transport theorem for convecting irregular domains is presented in the setting of Federer’s geometric measure theory. A prototypical r-dimensional domain is viewed as a flat r-chain of finite mass in an open set of an n-dimensional Euclidean space. The evolution of such a generalized domain in time is assumed to follow a continuous succession of Lipschitz embedding so that the spatial gradient may be nonexistent in a subset of the domain with zero measure. The induced curve is shown to be continuous with respect to the flat norm and differential with respect to the sharp norm on currents in \({\mathbb{R}^{n}}\). A time-dependent property is naturally assigned to the evolving region via the action of an r-cochain on the current associated with the domain. Applying a representation theorem for cochains, the properties are shown to be locally represented by an r-form. Using these notions, a generalized transport theorem is presented.
Similar content being viewed by others
References
Abraham R., Marsden J.E., Ratiu T.: Manifolds, Tensor Analysis, and Applications. Springer, Berlin (1988)
Angenent S., Gurtin M.E.: Multiphase thermomechanics with interfacial structure 2. evolution of an isothermal interface. Arch. Ration. Mech. Anal. 108, 323–391 (1989)
Betounes D.E.: Kinematics of submanifolds and the mean curvature normal. Arch. Ration. Mech. Anal. 96, 1–27 (1986)
Bouchut F., Crippa G.: Uniqueness, renormalization and smooth approximations for linear transport equations. SIAM J. Math. Anal. 4, 1316–1328 (2006)
Chi K.P., Quang N.H., Van B.C.: The Lie derivative of currents on Lie groups. Lobachevskii J. Math. 33(1), 10–21 (2012)
Falach, L., Segev, R.: The configuration space and principle of virtual power for rough bodies. Math. Mech. Solids. doi:10.1177/1081286513514244 (2013)
Falach, L., Segev, R.: Reynolds transport theorem for smooth deformations of currents on manifolds. Accepted to Mathematics and Mechanics of Solids, doi:10.1177/1081286514551503 (2014)
Federer, H.: Geometric Measure Theory. Springer, (1969)
Fukui K., Nakamura T.: A topological property of Lipschitz mappings. Topol. Appl. 148, 143–152 (2005)
Giaquinta, M., Modica, G., Soucek, J.: Cartesian Currents in the Calculus of Variation I. Springer, (1998)
Golubitsky M., Guillemin V.: Stable Mappings and Their Singularities. Springer, Berlin (1973)
Gurtin M.E.: Multiphase thermomechanics with interfacial structure 1. heat conduction and the capillary balance law. Arch. Ration. Mech. Anal. 104, 195–221 (1988)
Gurtin, M.E.: Configurational Forces as Basic Concepts of Continuum Physics. Springer, (2000)
Gurtin M.E., Jabbour M.E.: Interface evolution in three dimensions with curvature-dependent energy and surface diffusion: Interface-controlled evolution, phase transition, epitaxial growth of elastic films. Arch. Ration. Mech. Anal. 163, 171–208 (2002)
Gurtin M.E., Murduch A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975)
Gurtin M.E., Struthers A.: Multiphase thermodynamics with interfacial structure 3. evolving phase boundaries in the presence of bulk deformation. Arch. Ration. Mech. Anal. 112, 97–160 (1990)
Harrison J.: Operator calculus of differential chains and differential forms. J. Geometr. Anal. 25, 357–420 (2015)
Harrison J., Pugh H.: Topological aspects of differential chains. J. Geometr. Anal. 22(3), 685–690 (2012)
Heinonen J.: Lectures on Analysis on Metric Spaces. Springer, Berlin (2000)
Hirsch M.W.: Differential Topology. Springer, Berlin (1976)
Lang S.: Fundamentals of Differential Geometry. Springer, Berlin (1999)
Struthers A., Gurtin M.E., Williams W.O.: A transport theorem for moving interfaces. Q. Appl. Math. 47, 773–777 (1989)
Michor P.W.: Manifolds of Differentiable Mappings. Shiva, New York (1980)
Noll W., Virga E.G.: Fit regions and functions of bounded variation. Arch. Ration. Mech. Anal. 102, 1–21 (1988)
Reynolds, O.: The sub-mechanics of the universe. In Papers on Mechanical and Physical Subjects, volume III. Cambridge University Press, (1903)
Segev R.: Notes on metric independent analysis of classical fields. Math. Methods Appl. Sci. 36, 497–566 (2013)
Seguin B., Fried E.: Roughening it - evolving irregular domains and transport theorem. Math. Methods Appl. Sci. 24, 1729–1779 (2014)
Seguin B., Hinz D.F., Fried E.: Extending the transport theorem to rough domains of integration. Appl. Mech. Rev. 66(5), 050802 (2014)
Truesdell, C.A., Toupin, R.: The classical field theories. In S. Flügge, editor, Handbuch der Physik, volume III/1. Springer, Berlin (1960)
Whitney H.: Geometric Integration Theory. Princeton University Press, Princeton (1957)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Victor Eremeyev, Peter Schiavone and Francesco dell'Isola.
Rights and permissions
About this article
Cite this article
Falach, L., Segev, R. On the role of sharp chains in the transport theorem. Continuum Mech. Thermodyn. 28, 539–559 (2016). https://doi.org/10.1007/s00161-015-0461-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00161-015-0461-2