Scalar Truesdell Time Derivative and $(L^{2},H^{-1})$ -- Surface Gradient Flows

TL;DR

This paper introduces a framework for surface gradient flows that evolve surfaces and scalar quantities simultaneously, ensuring energy dissipation and scalar conservation through a proper choice of time derivative and gauge.

Contribution

It proposes a novel scalar Truesdell time derivative and analyzes coupled geometric and scalar PDEs on evolving surfaces, including special cases like surface tension flows.

Findings

Numerical demonstrations show the significance of tangential movement in surface evolution.

The framework guarantees energy dissipation and scalar conservation.

Coupled PDE system effectively models surface and scalar evolution.

Abstract

We address surface gradient flows which allow for energy dissipation by evolving the surface and a scalar quantity on it, simultaneously. A proper choice of the time derivative and the gauge of surface independence guarantees energy dissipation and ensures conservation of the scalar quantity. The resulting system of partial differential equations couples geometric evolution equations for the evolution of the surface in normal directions, equations for tangential movement and scalar-valued equations on the evolving surface. We discuss the general setting and the special case of surface tension flows and numerically demonstrate the importance of tangential movement on the evolution.