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Hyperbolic conservation laws of the form ut + div
f(t, x;u) = 0 with
discontinuous in (t,x) flux function f attracted much attention
in last 20 years, because of the difficulties of adaptation of the classical Kruzhkov
approach developed for the smooth case. In the discontinuous-flux case, non-uniqueness of
mathematically consistent admissibility criteria results in infinitely many different
notions of solution. A way to describe all the resulting L1-contractive
solvers within a unified approach was proposed in the work [Andreianov, Karlsen, Risebro,
2011]. We briefly recall the ideas and results developed there for the model
one-dimensional case with f(t, x; u) =
fl(u)1x<0 +
fr(u)1x>0 and highlight the main hints needed to address the
multi-dimensional situation with curved interfaces.
Then we discuss two recent developments in the subject which permit to better understand
the issue of admissibility of solutions in relation with specific modeling assumptions;
they also bring useful numerical approximation strategies. A new characterization of
limits of vanishing viscosity approximation proposed in [Andreianov and Mitrović, 2014]
permits to encode admissibility in singular but intuitively appealing entropy
inequalities. Transmission maps introduced in ([Andreianov and Cancès, 2015]) have
applications in modeling flows in strongly heterogeneous porous media and lead to a simple
algorithm for numerical approximation of the associated solutions.
Moreover, in order to embed all the aforementioned results into a natural framework, we
put forward the concept of interface coupling conditions (ICC) which role
is analogous to the role of boundary conditions for boundary-value problems. We link this
concept to known examples and techniques
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