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Semigroup approach to conservation laws with discontinuous flux


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}.
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hal-00698581 , version 1 (16-05-2012)
hal-00698581 , version 2 (16-10-2012)


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Boris Andreianov. Semigroup approach to conservation laws with discontinuous flux. G.-Q. Chen, H. Holden and K.H. Karlsen. Springer Proceedings in Mathematics & Statistics Vol.49, 49, Springer, pp. 1-22, 2013, Hyperbolic Conservation Laws and Related Analysis with Applications, 978-3-642-39006-7. ⟨10.1007/978-3-642-39007-4_1⟩. ⟨hal-00698581v2⟩
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