Mathematical analysis of modified level-set equations

TL;DR

This paper provides a rigorous mathematical analysis of modified level-set equations used in multiphase flow simulations, establishing existence and uniqueness of solutions that preserve the zero level-set's geometry.

Contribution

It offers the first mathematical justification for a class of nonlinear modified level-set equations, proving existence of smooth and viscosity solutions that maintain the zero level-set invariant.

Findings

Existence of smooth solutions near the zero level-set

Global viscosity solutions in the entire domain

The zero level-set remains unchanged under the modified equations

Abstract

The linear transport equation allows to advect level-set functions to represent moving sharp interfaces in multiphase flows as zero level-sets. A recent development in computational fluid dynamics is to modify the linear transport 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. The present work establishes mathematical justification for a specific class of modified level-set equations on a bounded domain, generated by a given smooth velocity field in the framework of the initial/boundary value problem of Hamilton-Jacobi equations. The first main result is the existence of smooth solutions defined in a time-global tubular neighborhood of the zero level-set, where an infinite iteration of the method of characteristics within a fixed small time interval is demonstrated; 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, where standard Perron's method and the comparison principle are exploited. 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 viscosity solution coincides with the local-in-space smooth solution in a time-global tubular neighborhood of the zero level-set, where a new aspect of localized doubling the number of variables is utilized.