A transport theorem for nonconvecting open sets on an embedded manifold
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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.
KeywordsIrregular domains Lipschitz domains First variation
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