On Scalar Conservation Laws with Point Source and Discontinuous Flux Function

The conservation law studied is ∂ u(x,t) ∂ t + ∂ ∂ t(F(u(x,t),x)) = s(t)δ (x), where u is a concentration, s is a source, δ is the Dirac measure, and \[ F(u,x) = \{ {gathered} f(u), x > 0, \\ g(u), x < 0 \\ {gathered} .\] 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.