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Entropy 2008, 10(3), 365-379;

Entropy Diagnostics for Fourth Order Partial Differential Equations in Conservation Form

Australian Mathematical Sciences Institute, c/o University of Melbourne, 111 Barry St., Melbourne VIC 3010, Australia
Received: 3 May 2008 / Accepted: 21 September 2008 / Published: 25 September 2008
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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. View Full-Text
Keywords: Shannon entropy; fourth order diffusion; irreversibility Shannon entropy; fourth order diffusion; irreversibility

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Broadbridge, P. Entropy Diagnostics for Fourth Order Partial Differential Equations in Conservation Form. Entropy 2008, 10, 365-379.

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