Shocks in Nonlinear Diffusion

Using two models that incorporate a nonlinear forward-backward heat equation, we demonstrate the existence of well-deened weak solutions containing shocks for diiusive problems. Occurrence of shocks is connected to multivalued inverse solutions and non-monotone potential functions. Unique viscous solutions are determined from perturbation theory by matching to a shock layer condition. Results of direct numerical simulations are also discussed. In this letter we will construct implicit weak solutions containing shocks for systems based on the nonlinear forward-backward heat equation. We will demonstrate that solutions with such jump discontinuities can generally occur in diiusive models with non-monotone potential functions. Consider the weakly parabolic nonlinear diiusion equation @u @t = @ @x D(u) @u @x ; (1) where the diiusion coeecient D(u) attains both positive and negative values for the range of u being examined (see Fig. 1). This model has many physical applications and interesting mathematical properties. Our study of (1) will include construction of a traveling wave solution for a nonlinear Kuramoto-Sivashinsky equation and a similarity solution for the Cahn-Hilliard equation. 0 2 4-2-1 0 1 2 3 u f(u) D(u) Fig. 1. Potential function f(u) and diiusion coeecient D(u). In the limit of sharp interfaces, equation (1) represents the leading order outer problem for the Cahn-Hilliard model for phase separation. For nearly immiscible uids, there is a sharp interface separating regions of almost pure uid phases. Dynamics of such phase interfaces have been extensively studied using models like the Cahn-Hilliard equation 1,8] and the lubrication equation for Hele-Shaw ow. These equations are fourth order evolution equations containing terms that account for surface tension and interfacial energy. For example, the one-dimensional Cahn-Hilliard equation 1] is @u @t = @ 2 @x 2 f(u) ? 2 @ 2 u @x 2 ;