2018/08/24 by Brian Seguin · 2 citations
Mathematics · #Analytic and geometric function theory #Boundary (topology) #Dimension (graph theory) #Divergence (linguistics) #Domain (mathematical analysis) #Flow (mathematics) #Manifold (fluid mechanics) #Mathematical Dynamics and Fractals #Nonlinear Partial Differential Equations #Open set #Set (abstract data type) #math.DG
paper · pdf · doi:10.1007/s00161-019-00777-z
published in Continuum Mechanics and Thermodynamics 32(1), 1-8 (Springer Science+Business Media) · 12 pages, 3 figures
arxiv created 2018/08/24 · openalex created_date 2018/08/31 · openalex publication_date 2019/04/16 · arxiv updated 2019/05/01 · openalex updated_date 2026/08/05
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 does evolve according to a flow map is 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 the sense that their boundaries need only be Lipschitz. Tools from geometric measure theory are used to deal with this irregularity.