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hal-00698581, version 1

Semigroup approach to conservation laws with discontinuous flux

Boris Andreianov (, ) 1

(2012-05-16)

Abstract: The model one-dimensional conservation law with discontinuous spatially heterogeneous flux is $$u_t + \mathfrak{f}(x,u)_x=0, \quad \mathfrak {f}(x,\cdot)= f^l(x,\cdot)\char_{x<0}\!+f^r(x,\cdot)\char_{x>0}. \eqno (\text{EvPb})$$ We prove well-posedness for the Cauchy problem for (\text{EvPb}) in the framework of solutions satisfying the so-called adapted entropy inequalities. Exploiting the notion of integral solution that comes from the nonlinear semigroup theory, we propose a way to circumvent the use of strong interface traces for the evolution problem $(\text{EvPb})$ (in fact, proving existence of such traces for the case of $x$-dependent $f^{l,r}$ would be a delicate technical issue). The difficulty is shifted to the study of the associated one-dimensional stationary problem $u + \mathfrak{f}(x,u)_x=g$, where existence of strong interface traces of entropy solutions is an easy fact. We give a direct proof of this fact, avoiding the subtle arguments of kinetic formulation \cite{KwonVasseur} or of the $H$-measure approach \cite{Panov-trace}.

• 1:  Laboratoire de Mathématiques (LM-Besançon)
• CNRS : UMR6623 – Université de Franche-Comté
• Domain : Mathematics/Analysis of PDEs
• Keywords : conservation law – discontinuous flux – integral solution – boundary trace – uniqueness – Dirichlet problem
• Available versions :  v1 (2012-05-17) v2 (2012-10-16)

• hal-00698581, version 1
• oai:hal.archives-ouvertes.fr:hal-00698581
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• Submitted on: Wednesday, 16 May 2012 18:55:06
• Updated on: Thursday, 17 May 2012 19:40:59