# 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

∂_{t}u(t,x) + ∂_{x}(u^{2}(t,x)/2) = −λu(t,x)δ_{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 Lagoutière, Seguin and Takahashi [LST08].

The interpretation of the non-conservative product
“u(t,x)δ_{0}(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.

*J. Differential Equations*

**245**(2008), 3503–3544

[AKR11] Boris Andreianov, Kenneth Hvistendahl Karlsen and Nils Henrik Risebro, A Theory of L

^{1}-Dissipative Solvers for Scalar Conservation Laws with Discontinuous Flux.

*Arch. Ration. Mech. Anal.*(2011)