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 HAL: hal-00576959, version 1
 Well-posedness of a singular balance law
 For the LRC Manon collaboration(s)
 (2011-03-15)
 We define entropy weak solutions and establish well-posedness for the Cauchy problem for the formal equation $$\partial_t u(t,x) + \partial_x \frac{u^2}2(t,x) = - \lambda u(t,x) \delta_0(x),$$ 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 Lagoutière, Seguin and Takahashi [LST08]. The interpretation of the non-conservative product $u(t,x) \delta_0(x)$'' 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.
 1: Laboratoire de Mathématiques (LM-Besançon) CNRS : UMR6623 – Université de Franche-Comté 2: Laboratoire Jacques-Louis Lions (LJLL) CNRS : UMR7598 – Université Pierre et Marie Curie [UPMC] - Paris VI
 Subject : Mathematics/Analysis of PDEsMathematics/Numerical Analysis
 Keyword(s): Burgers equation – Fluid-particle interaction – Non-conservative coupling – Singular source term – Well-posedness – Interface traces – Adapted entropies – Finite volume scheme – Well-balanced scheme – Convergence
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 hal-00576959, version 1 http://hal.archives-ouvertes.fr/hal-00576959 oai:hal.archives-ouvertes.fr:hal-00576959 From: Boris Andreianov <> Submitted on: Wednesday, 16 March 2011 07:57:17 Updated on: Wednesday, 16 March 2011 08:38:08