On scalar conservation laws with point source and discontinuous flux function

The conservation law studied is partial derivative u(x,t)/partial derivative t + partial derivative/partial derivative x (F(u(x,t),x)) = s(t)delta(x), where u is a concentration, s is a source, delta is the Dirac measure, and 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 La 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.