Fourth Order Diffusion Equations with Increasing Entropy

Abstract

The general quasi-linear autonomous fourth order diffusion equation <em>u_t = −[G(u)u_xxx + h(u, u_x, u_xx)]_x</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.

Keywords

• Shannon entropy
• fourth order diffusion
• irreversibility