Well-posedness of a singular balance law
Boris Andreianov and Nicolas Seguin
Abstract: We define entropy weak solutions and establish well-posedness for the Cauchy problem for the formal equation
∂tu(t,x) + ∂x(u2(t,x)/2) = −λu(t,x)δ0(x),
which can be seen as two Burgers equations coupled in a non-conservative way through the interface located at x = 0. This problem appears as an important auxiliary step in the theoretical and numerical study of the one-dimensional particle-in-fluid model developed by Lagoutière, Seguin and Takahashi [LST08].
The interpretation of the non-conservative product “u(t,x)δ0(x)” follows the analysis of [LST08]; we can describe the associated interface coupling in terms of one-sided traces on the interface. Well-posedness is established using the tools of the theory of conservation laws with discontinuous flux ([AKR11]).
For proving existence and for practical computation of solutions, we construct a finite volume scheme, which turns out to be a well-balanced scheme and which allows a simple and efficient treatment of the interface coupling. Numerical illustrations are given.
[AKR11] Boris Andreianov, Kenneth Hvistendahl Karlsen and Nils Henrik Risebro, A Theory of L1-Dissipative Solvers for Scalar Conservation Laws with Discontinuous Flux. Arch. Ration. Mech. Anal. (2011)