Diffusive–dispersive travelling waves and kinetic relations V. Singular diffusion and nonlinear dispersion

Abstract

We consider scalar hyperbolic conservation laws with non-convex flux and vanishing, nonlinear and possibly singular, diffusion and dispersion terms. The diffusion has the form R(u, u_x)_x and we cover, for instance, the singular diffusion (|u_x|^pu_x)_x, where p ≥ 0 is arbitrary. We investigate the existence, uniqueness and various properties of classical and non-classical travelling waves and of the kinetic function. The latter serves to characterize non-classical shock waves, via an additional algebraic constrain called a kinetic relation. We discover that p = ⅓ is a somewhat unexpected critical value. For p ≤ ⅓, we obtain properties that are qualitatively similar to those we established earlier for regular and linear diffusion. However, for p > ⅓, the behaviour of the kinetic function is very different, as, for instance, non-classical shocks can have arbitrary small strength. The behaviour of the kinetic function near the origin is carefully investigated and depends on whether p < ½, p = ½ or p > ½. In particular, in the special case of the cubic flux-function and for the regularization (|u_x|^pu_x)_x with p = 0, ½ or 1, the kinetic function can be computed explicitly. When p = ½, the kinetic function is simply a linear function of its argument.