Entropy (Jun 2012)
Fourth Order Diffusion Equations with Increasing Entropy
Abstract
The general quasi-linear autonomous fourth order diffusion equation <em>u<sub>t</sub> = −[G(u)u<sub>xxx </sub> + h(u, u<sub>x</sub>, u<sub>xx</sub>)]<sub>x </sub></em>with positive variable diffusivity <em>G(u)</em> and lower-order flux component <em>h</em> is considered on the real line. A direct algorithm produces a general class of equations for which the Shannon entropy density obeys a reaction-diffusion equation with a positive irreducible source term. Such equations may have any positive twice-differentiable diffusivity function <em>G(u)</em>. The forms of such equations are the indicators of more general conservation equations whose entropy equation may be expressed in an alternative reaction-diffusion form whose source term, although reducible, is positive.
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