On degenerate saturated-diffusion equations with convection
Alina E. Chertock, Alexander Kurganov, Philip Rosenau · Nonlinearity · 2004
We study a class of degenerate parabolic convection–diffusion equations, endowed with a mechanism for saturation of the diffusion flux, which corrects the unphysical gradient-flux relations at high gradients. This paper extends our previous works on the effects of diffusion with saturation on convection and the impact of saturation on porous media-type diffusion, where it has been demonstrated that a nonlinear saturating diffusion is susceptible to a self-induced formation of discontinuities. In this work we demonstrate that nonlinear convection enhances the breakdown effect. We carry both analytical and numerical studies of the model equation, u t + f ( u ) x = [φ( u ) Q ( u x , u )] x , where Q is a bounded increasing function, φ(0) = 0 and φ( u ) ∼ u n , n > 0 for u ∼ 0. Depending on a choice of n , we obtain two distinctive processes. If 0 ≤ n ≤ 1, a discontinuity forms only when the upstream–downstream disparity exceeds a critical threshold, but if n > 1, all travelling waves are found to have a sharp discontinuous front . In fact, given a compact or a semi-compact initial datum, the front will not start to move until such a discontinuity forms.