Optimal entropy solutions for conservation laws with discontinuous flux-functions. (English) Zbl 1093.35045

The authors study a single conservation law in one space dimension whose flux function is discontinuous at the point \(x=0\). They propose the alternative entropy framework based on a two step approach. Firstly, infinitely many classes of entropy solutions are defined depending on specific choice of an interface connection. Each of these classes generates an \(L^1\)-contractive semigroup and can be constructed by a simple Godunov type finite volume approximation scheme. The second step is to choose one of these classes. The choice has to be dictated by the physics of the problem and is based on solving an optimization problem involving the interface entropy cost functional on the set of admissible interface connections. The authors use the optimization criterion corresponding to the model of two-phase flows in a heterogeneous porous medium and prove existence and uniqueness of the optimal solution.


35L65 Hyperbolic conservation laws
35L45 Initial value problems for first-order hyperbolic systems
35B25 Singular perturbations in context of PDEs
47J30 Variational methods involving nonlinear operators
74S10 Finite volume methods applied to problems in solid mechanics
76M12 Finite volume methods applied to problems in fluid mechanics
76S05 Flows in porous media; filtration; seepage
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