Well-posedness of a singular balance law

We define entropy weak solutions and establish well-posedness for the Cauchy problem for the formal equation ∂tu(t, x) + ∂x u 2 (t, x) = −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.