Zero Diffusion-Dispersion Limits for Scalar Conservation Laws

Abstract

Abstract. We consider solutions of hyperbolic conservation laws regularized with vanishing diffusion and dispersion terms. Following a pioneering work by Schonbek, we establish the convergence of the regularized solutions toward discontinuous solutions of the hyperbolic conservation law. The proof relies on the method of compensated compactness in the L 2 setting. Our result improves upon Schonbek's earlier results and provides an optimal condition on the balance between the relative sizes of the diffusion and the dispersion parameters. A convergence result is also established for multi-dimensional conservation laws by relying on DiPerna's uniqueness theorem for entropy measure-valued solutions.

Hyperbolic conservation laws with vanishing nonlinear diffusion and linear dispersion in heterogeneous media

“…Similar problem was considered in [10] but under less restrictive conditions on the relative size between diffusion and dispersion parameters. More precisely, it is proved in [10] that the family of solutions of a scalar conservation law perturbed by diffusion and dispersion converges to a unique entropy admissible weak solution (see [11]) of appropriate conservation law if the diffusion parameter ε predominates the dispersion parameter δ in the sense that δ = o(ε 2 ), as ε → 0. To accomplish this, authors use the concept of measure valued solutions to conservation laws introduced by DiPerna [3].…”

“…Now, we shall analyze more closely the problem we are dealing with. If we assume that the relative size of ε and δ is weaker than in [9], then we can rely on [3] (as in [10]) to state that the family (u ε,δ ) ε,δ of solutions to (2)-(3) converges to unique entropy admissible weak solution to (1). Also, if we consider one-dimensional variant of (2)-(3) and assume that the relative size of ε and δ is the same as in [9], we can use compensated compactness, and even assume that the flux f = f (t, x, λ) from (1) is discontinuous in (t, x) ∈ R + × R to obtain the convergence result (see [6]).…”

Burgers' Equation with Vanishing Hyper-Viscosity

“…A similar approach originated with [Sch82] for the vanishing diffusion-dispersion problem where the RHS of (1.2) is replaced by εu ε xx + δ ε u ε xxx . In the particular borderline case, δ ε ∼ ε 2 , limit solutions may in fact violate Krushkov entropy condition, [KL02]. Otherwise, entropy solution limits are recovered by compensated arguments as long as diffusion dominates, δ ε ε 2 , [Sch82,KL02].…”

“…In the particular borderline case, δ ε ∼ ε 2 , limit solutions may in fact violate Krushkov entropy condition, [KL02]. Otherwise, entropy solution limits are recovered by compensated arguments as long as diffusion dominates, δ ε ε 2 , [Sch82,KL02]. We should point out that in the present context, hyper-viscosity with s > 1 yields a weaker entropy dissipation bound than in the viscosity dominated case s = 1, consult (1.6) below.…”

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

“…An extension of our results, either to general smooth fluxes with f ≥ 0 and polynomial growth [24] or to globally Lipschitz-continuous fluxes [15], is possible with minor changes.…”