Singular limits for a parabolic–elliptic regularization of scalar conservation laws

Abstract

We consider scalar hyperbolic conservation laws with a nonconvex flux, in one space dimension. Then, weak solutions of the associated initial-value problems can contain undercompressive shock waves. We regularize the hyperbolic equation by a parabolic-elliptic system that produces undercompressive waves in the hyperbolic limit regime. Moreover we show that in another limit regime, called capillarity limit, we recover solutions of a diffusive-dispersive regularization, which is the standard regularization used to approximate undercompressive waves. In fact the new parabolic-elliptic system can be understood as a low-order approximation of the third-order diffusive-dispersive regularization, thus sharing some similarities with the relaxation approximations. A study of the traveling waves for the parabolic-elliptic system completes the paper.

An asymptotic-preserving method for a relaxation of the Navier–Stokes–Korteweg equations

“…where (α 1 , α 2 ) is the interval of the decreasing pressures, i.e., p (s) < 0, see . We refer to [8,9,13,19,21] for first rigorous results on the Korteweg limit. It is possible to show that (2.8) formally converges to (2.1).…”

Relaxation of the Navier–Stokes–Korteweg equations for compressible two‐phase flow with phase transition

“…; v / is the solution of the corresponding initial boundary value problem for (10). We refer to [30][31][32][33][34][35] for first rigorous results on the Korteweg limit. We underline the hypothesis by a numerical example (see Section (3) for the used discretization method).…”

“…For ϵ > 0, fixed it is expected that solutions ( ρ ϵ , α , v ϵ , α , c ϵ , α ) of an initial boundary value problem for satisfy ( ρ ϵ , α , c ϵ , α )→( ρ ϵ , ρ ϵ ) and v ϵ , α → v ϵ for the Korteweg limit where ( ρ ϵ , v ϵ ) is the solution of the corresponding initial boundary value problem for . We refer to for first rigorous results on the Korteweg limit. We underline the hypothesis by a numerical example (see Section 3) for the used discretization method).…”

A Hyperbolic–Elliptic Model Problem for Coupled Surface–Subsurface Flow

“…To prove Theorem 6 we will rely on the a priori estimates from Lemma 1 and-what concerns the limit procedure for h ε -on the Lemma of Murat (1981) in the L p -framework (Schonbek 1982), and in particular we shall refer to the arguments used in Corli and Rohde (2012) and Lu (1989). Note that the additional regularity condition on φ in Theorem 6 is needed in these papers.…”