Résumé
We define entropy weak solutions and establish well-posedness for the Cauchy problem for the formal equation
partial derivative(t)u(t, x) + partial derivative(x)u(2)/2(t, x) = -lambda u(t, x) delta(0)(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 Lagoutiere, Seguin and Takahashi [30].
The interpretation of the non-conservative product "u (t, x) delta(0)(x)" follows the analysis of [30]; 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 ([4]).
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.