A simple derivation of BV bounds for inhomogeneous relaxation systems

Abstract

Abstract. We consider relaxation systems of transport equations with heterogeneous source terms and with boundary conditions, which limits are scalar conservation laws. Classical bounds fail in this context and in particular BV estimates. They are the most standard and simplest way to prove compactness and convergence.We provide a novel and simple method to obtain partial BV regularity and strong compactness in this framework. The standard notion of entropy is not convenient either and we also indicate another, but closely related, notion. We give two examples motivated by renal flows which consist of 2 by 2 and 3 by 3 relaxation systems with 2-velocities but the method is more general.Key words. Hyperbolic relaxation; spatial heterogeneity; entropy condition; boundary conditions; strong compactness.Subject classifications. 35L03, 35L60, 35B40, 35Q92 IntroductionThe usual framework of hyperbolic relaxation [4,13,3] concerns the convergence of a general hyperbolic system with stiff source terms toward a conservation law, when the relaxation parameter ǫ goes to zero. More specifically, Jin and Xin [10] introduced a 2 × 2 linear hyperbolic system with stiff source term that approximates any given conservation law. The problem of interest is to prove the convergence of the microscopic quantities depending on ǫ toward the macroscopic quantities. A problem entering the framework of hyperbolic relaxation is motivated by very simplified models of kidney physiology [18,19,17] and fits the Jin and Xin framework with two major differences, boundary conditions, and spatial dependence of the source term. The type of boundary condition and the spatial dependence constitute the main novelty of the present study.The system represents two solute concentrations u ǫ (x,t) and v ǫ (x,t) and is written, for t ≥ 0 and x ∈ [0,L],

Asymptotic preserving discretization of a Jin–Xin model with implicit equilibrium manifold on a bounded domain

“…Existence and uniqueness of such entropy solution follows from the theory developed in [2]. We state the following result of convergence partially proved in [17].…”

“…The convergence of solutions of (S ε ) to solutions of (S 0 ) is provided in [17]. We complement in the present paper the analysis of [17] showing the existence of a relaxation boundary layer at x = L if the intersection between the equilibrium manifold…”

“…using BV estimates obtained in [17]. By considering test functions Φ satisfying Φ(0, t) = Φ(L, t) = 0, and letting ε go to zero in (12), we obtain that v satisfies the first item of Definition 2.…”

“…. More specifically, the well-posedness and the asymptotic analysis of the IBVP (S ε ) are given in [17]. In order to understand the IBVP when ε → 0, let us provide the assumptions on function h: there exists two positive constant β µ such that 1 < β h (v) µ, and h(0) = 0.…”

A General BV Existence Result for Conservation Laws with Spatial Heterogeneities

“…There are many models that lead to consider hyperbolic conservation laws with a flux function discontinuous in the state space, arising in fields as various as vehicular traffic flows [5] [14] [20], two-phase flows porous medias [3] (see also [19]), sedimentation [13], kidney physiology [29], cell dynamics [8], and others.…”