On Scalar Conservation Laws with Point Source and Discontinuous Flux Function

Abstract

The conservation law studied is $\frac{{\partial u(x,t)}} {{\partial t}} + \frac{\partial } {{\partial t}}(F(u(x,t),x)) = s(t)\delta (x)$, where u is a concentration, s is a source, $\delta $ is the Dirac measure, and \[ F(u,x) = \left\{ \begin{gathered} f(u),\quad x > 0, \hfill \\ g(u),\quad x < 0 \hfill \\ \end{gathered} \right.\] is the flux function. The special feature of this problem is the discontinuity that appears along the t-axis and the curves of discontinuity that go into and emanate from it. Necessary conditions for the existence of a piecewise smooth solution are given. Under some regularity assumptions sufficient conditions are given enabling construction of piecewise smooth solutions by the method of characteristics. The selection of a unique solution is made by a coupling condition at $x = 0$, which is a generalization of the classical entropy condition and is justified by studying a discretized version of the problem by Godunov’s method. The motivation for studying this problem is the fact that it arises in the modelling of continuous sedimentation of solid particles in a liquid.