Scalar conservation laws with discontinuous flux function: I. The viscous profile condition

Abstract

The equation $$\frac{{\partial u}}{{\partial t}} + \frac{\partial }{{\partial x}}\left( {H(x)f(u) + \left( {1 - H(x)} \right)g(u)} \right) = 0$$ , whereH is Heaviside's step function, appears for example in continuous sedimentation of solid particles in a liquid, in two-phase flow, in traffic-flow analysis and in ion etching. The discontinuity of the flux function atx=0 causes a discontinuity of a solution, which is not uniquely determined by the initial data. The equation can be written as a triangular 2×2 non-strictly hyperbolic system. This augmentation is non-unique and a natural definition is given by means of viscous profiles. By a viscous profile we mean a stationary solution ofu t +(F δ) x =εu xx , whereF δ is a smooth approximation of the discontinuous flux, i.e.,H is smoothed. In terms of the 2×2 system, the discontinuity atx=0 is either a regular Lax, an under-or overcompressive, a marginal under- or overcompressive or a degenerate shock wave. In some cases, depending onf andg, there is a unique viscous profile (e.g. undercompressive and regular Lax waves) and in some cases there are infinitely many (e.g. overcompressive waves). The main purpose of the paper is to show the equivalence between a previously introduced uniqueness condition for the discontinuity of the solution atx=0 and the viscous profile condition.