Domain Variations for Moving Boundary Problems

TL;DR

This paper introduces a new approach using flows and phantom geometry to linearize moving boundary problems, simplifying calculations and providing clearer insights into the structure of the linearization process.

Contribution

It proposes an alternative to the Hanzawa transformation by employing flows and phantom geometry, avoiding the need for a reference manifold and simplifying the linearization of boundary problems.

Findings

Simplified formulas for linear operators.

Clearer understanding of linearization structure.

Enhanced applicability to moving boundary problems.

Abstract

In the past few decades maximal regularity theory has successfully been applied to moving boundary problems. The basic idea is to reduce the system with varying domains to one in a fixed domain. This is done by a transformation (the so-called Hanzawa transformation) and yields a typically nonlocal and nonlinear coupled system of (evolution) equations. Well-posedness results can then often be established as soon as it is proved that the relevant linearization is the generator of an analytic semigroup or admits maximal regularity. To implement this program, it is necessary to somehow parametrize to space of boundaries/domains (typically the space of compact hypersurfaces $\Gamma$ in $\mathbb{R}^n$, in the Euclidean setting). This has traditionally been achieved by means of the already mentioned Hanzawa transformation. The approach, while successful, requires the introduction of a smooth manifold $\Gamma_\infty$ close to the manifold $\Gamma_0$ in which one cares to linearize. This prevents one to use coordinates in which $\Gamma_0$ lies at their "center". As a result formul{\ae} tend to contain terms that would otherwise not be present were one able to linearize in a neighborhood emanating from $\Gamma_0$ instead of from $\Gamma_\infty$. In this paper it is made use of flows (curves of diffeomorphisms) to obtain a general form of the relevant linearization in combination with an alternative coordinatization of the manifold of hypersurfaces, which circumvents the need for the introduction of a "phantom" reference manifold $\Gamma_\infty$ by, in its place, making use of a "phantom geometry" on $\Gamma_0$. The upshot is a clear insight into the structure of the linearization, simplified calculations, and simpler formul{\ae} for the resulting linear operators, which are useful in applications.