Entropy Diagnostics for Fourth Order Partial Differential Equations in Conservation Form

Phil Broadbridge

doi:10.3390/e10030365

Abstract

The entropy evolution behaviour of a partial differential equation (PDE) in conservation form, may be readily discerned from the sign of the local source term of Shannon information density. This can be easily used as a diagnostic tool to predict smoothing and non-smoothing properties, as well as positivity of solutions with conserved mass. The familiar fourth order diffusion equations arising in applications do not have increasing Shannon entropy. However, we obtain a new class of nonlinear fourth order diffusion equations that do indeed have this property. These equations also exhibit smoothing properties and they maintain positivity. The counter-intuitive behaviour of fourth order diffusion, observed to occur or not occur on an apparently ad hoc basis, can be predicted from an easily calculated entropy production rate. This is uniquely defined only after a technical definition of the irreducible source term of a reaction diffusion equation.

Highlights

Fourth order diffusion equations arise in many applications such as thin film theory [1, 2], surface diffusion on solids [3,4,5], interface dynamics [6,7,8], flow in Hele-Shaw cells, and phase field models of multiphase systems [9]
The primary purpose of this paper is to demonstrate that the entropy evolution behaviour of a partial differential equation (PDE)
Since the current-source decomposition of a reaction-diffusion equation is non-unique, we have had to develop a second key concept, that of an irreducible source term, which is uniquely defined at least for polynomial source terms and which seems to be practically useful as a diagnostic tool

Summary

Introduction

Fourth order diffusion equations arise in many applications such as thin film theory [1, 2], surface diffusion on solids [3,4,5], interface dynamics [6,7,8], flow in Hele-Shaw cells, and phase field models of multiphase systems [9]. The simplest prototype is the fourth order diffusion equation with constant coefficients, ut = −uxxxx. Fourth order diffusion equations are dissipative equations, they do not exhibit the smoothing properties usually associated with the very name “diffusion”. Unlike second order diffusion equations, they do not obey a robust maximum principle; one cannot rule out the gen-. Entropy 2008, 10 eration of new extrema in the evolving solutions of standard initial-boundary problems. With fourth order evolution we commonly observe the appearance of new structures evolving from smoother initial conditions Compared to the familiar smoothing behaviour of second order diffusion, the behaviour of some fourth order equations seems counter-intuitive. For fourth order equations, there is no robust maximum principle, entropy production rate may be useful as a simple litmus test. The primary purpose of this paper is to demonstrate that the entropy evolution behaviour of a partial differential equation (PDE)

Objectives

Conclusion