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Preprint 2011-007

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.

[KST08]Frédéric Lagoutière, Nicolas Seguin, Takéo Takahashi, A simple 1D model of inviscid fluid-solid interaction. J. Differential Equations 245 (2008), 3503–3544
[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)
Available as PDF (387 Kbytes).
Boris Andreianov
Nicolas Seguin
Submitted by:
; 2011-03-15.