New approaches to describing admissibility of solutions of scalar conservation laws with discontinuous flux^*,**

¹ Laboratoire de Mathématiques, CNRS UMR 6623, Université de Franche-Comté, Besançon, France
² Laboratoire J.-L. Lions CNRS UMR 7598, Université Paris 6, France

Abstract

Hyperbolic conservation laws of the form u_t + 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) = f_l(u)1_x<0 + f_r(u)1_x>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.