Entropy 2008, 10(3), 365-379; doi:10.3390/e10030365
Article

Entropy Diagnostics for Fourth Order Partial Differential Equations in Conservation Form

Received: 3 May 2008; Accepted: 21 September 2008 / Published: 25 September 2008
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
Keywords: Shannon entropy; fourth order diffusion; irreversibility
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MDPI and ACS Style

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

AMA Style

Broadbridge P. Entropy Diagnostics for Fourth Order Partial Differential Equations in Conservation Form. Entropy. 2008; 10(3):365-379.

Chicago/Turabian Style

Broadbridge, Phil. 2008. "Entropy Diagnostics for Fourth Order Partial Differential Equations in Conservation Form." Entropy 10, no. 3: 365-379.

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