Vanishing viscosity limit of a conservation law regularised by a Riesz–Feller operator

Abstract

We study a nonlocal regularisation of a scalar conservation law given by a fractional derivative of order between one and two. The nonlocal operator is of Riesz-Feller type with skewness two minus its order. This equation describes the internal structure of hydraulic jumps in a shallow water model. The main purpose of the paper is the study of the vanishing viscosity limit of the Cauchy problem for this equation. First, we study the properties of the solution of the regularised problem and then we show that solutions converge to the entropy solution of the scalar conservation law in this limit in C([0, T ]; L 1 loc (R)) for initial data in L ∞ (R), and in C([0, T ]; L 1 (R)) for initial data in L ∞ (R) ∩ BV (R). In order to prove these results we use weak entropy inequalities and the double scale technique of Kruzhkov. Such techniques also allow to show the L 1 (R) contraction of the regularised problem. For completeness, we study the behaviour in the tail of travelling wave solutions for genuinely nonlinear fluxes. These waves converge to shock waves in the vanishing viscosity limit, but decay algebraically as x − ct → ∞, rather than exponentially, the latter being a behaviour that they exhibit as x − ct → −∞, however. Finally, we generalise the results concerning the vanishing viscosity limit to Riesz-Feller operators.Observe, that this is a non-negative function, that belongs to C ∞ c ((0, ∞) × R) (the function Θ has its support in [0, T ) and (t, x) → ζ M (|x| + Lt) is regular on [0, T ) × R since, in a neighbourhood of [0, T ] × {0}, ζ M (|x| + Lt) = 1) and supp(ϕ) ⊂ (0, T ) × (−M − 1, M + 1), so we can take B = (−M − 1, M + 1). This test function satisfies ∂ t ϕ(t, x) = Lζ ′ M (|x| + Lt)Θ(t) + ζ M (|x| + Lt)Θ ′ (t), |∂ x ϕ(t, x)| = ζ ′ M (|x| + Lt) sgn(x)Θ(t) = −ζ ′ M (|x| + Lt)Θ(t),

Large time behaviour of a conservation law regularised by a Riesz-Feller operator: the sub-critical case

“…Proof. The existence and uniqueness result, the upper bound of ( 30) and ( 31) have already been proved in [15] for a regular flux function. We observe that, in order to obtain existence and regularity we can proceed as in [15, Propositions 4-5], but since the flux function is only continuous with bounded first derivative, we can only apply two steps of the argument.…”

“…The estimate (34) is a direct consequence of (30). Since now u is positive this means that |u| = u so the flux is f (u) = u q /q and belongs to [15].…”

“…where the constant C(T ) depends linearly on sup t∈(0,T ] u(t, •) ∞ , which is finite for all T (see [15]). Here we have used the second estimate with p = 1 of Lemma 2.4, the convergence f δ → f as δ → 0 and the local Lipschitz continuity of f .…”

“…which can be formally obtained using Fourier transform (see [2] for the proof). Some pertinent properties of the kernel are derived in [1] and [15]. In particular we recall that,…”

“…We also recall the viscous entropy inequality, as well as derive a comparison principle. Most of the results appear in [15] or can be adapted from [22] with the ingredients given in the previous sections, thus we shall only prove which does not appear elsewhere.…”