Entropy formulations for a class of scalar conservations laws with space-discontinuous flux functions in a bounded domain

Abstract

In this paper, the mathematical analysis of a quasilinear parabolic-hyperbolic problem in a multidimensional bounded domain is carried out. In a region p a diffusion-advection-reaction-type equation is set, while in the complementary h ≡ \ p , only advection-reaction terms are taken into account. First, the definition of a weak solution u is provided through an entropy inequality on the whole domain Q by using the classical Kuzhkov entropy pairs and the F. Otto framework to transcribe the boundary conditions on ∂ ∩ ∂ h . Since hp contains the outward characteristics for the first-order operator set in Q h , the uniqueness proof begins by focusing on the behavior of u in the hyperbolic layer and then in the parabolic one where u fulfills a variational equality that takes into account the entered data from Q h . The existence property uses a vanishing-viscosity method.

Entropy conditions for scalar conservation laws with discontinuous flux revisited

“…Adimurthi et al [2] observed that infinitely many different, though equally mathematically consistent, notions of solution may co-exist in the discontinuous-flux problems; therefore the choice of solution notion is a part of modeling procedure (see, e.g., [6] for an exhaustive study of the vanishing capillarity limits of the onedimensional Buckley-Leverett equation, where different sets of admissibility conditions are put forward for different choices of physically relevant vanishing capillarity). In the present contribution we limit our attention to characterization of vanishing viscosity limit solutions for problems of kind (1); these approximations were studied in a huge number of works (see, e.g., [23,20,21,37,38,36,28,29,11,16,35,22]) including several works in multiple space dimensions ( [25,24,8,34,14]) and they remain relevant in several models based on discontinuous-flux conservation laws.…”

A Theory of L 1-Dissipative Solvers for Scalar Conservation Laws with Discontinuous Flux

“…Solution semigroups that are Lipschitz in L 1 but not contractive may exist and are of physical interest (see [54]), but their study will not be addressed in the present paper. A comparable theory for conservation laws with discontinuous flux is still not available, although these equations have received intense attention in last fifteen years; see [1,2,3,12,13,14,19,20,21,23,26,29,30,31,32,35,37,38,42,43,44,45,46,47,48,49,59,60,62,69,70,73,76,77] (and additional references therein) for a number of different admissibility criteria, existence and/or uniqueness results, which we partially revisit in Section 4. Recently, it was pointed out explicitly by Adimurthi, Mishra, and Veerappa Gowda in [3] that for the case f = f(x, u) with f piecewise constant in x, there may exist many different L 1 -contractive semigroups of solutions to (1.1).…”