分野別セミナー

　 The linear transport equation provides level-set functions to represent
moving sharp interfaces in multiphase flows as zero level-sets, where the
linear transport equation in this context is particularly called the level-set
equation. A recent development in computational fluid dynamics is to modify
the level-set equation by introducing a nonlinear term to preserve certain
geometrical features of the level-set function, where the zero level-set must
stay invariant under the modification. In this talk, based on the framework
of the initial/boundary value problem of Hamilton-Jacobi equations, we discuss
mathematical justification for a class of modified level-set equations generated
by a given smooth velocity field on a bounded domain. The first main result is
the existence of smooth solutions defined in a time-global tubular neighborhood
of the zero level-set, where the smooth solution is shown to possess the desired
geometrical feature. The second main result is the existence of time-global
viscosity solutions defined in the whole domain. In the first and second main results,
the zero level-set is shown to be identical with the original one.
The third main result is that the (continuous) viscosity solution coincides with
the local-in-space smooth solution in a time-global tubular neighborhood of the zero
level-set. This talk is based on a joint work with Dieter Bothe and Mathis Fricke
(Technische Universität Darmstadt).

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