Hyperbolic conservation laws with nonlinear diffusion and nonlinear dispersion

Abstract

We study scalar conservation laws with nonlinear diffusion and nonlinear dispersion terms, the flux function f (u) being mth order growth at infinity. It is shown that if ε, δ = δ(ε) tend to 0, then the sequence {u ε } of the smooth solutions converges to the unique entropy solution u ∈ L ∞ (0, T * ; L q (R)) to the conservation law u t + f (u) x = 0 in L k (0, T * ; L p (R)) (k < ∞, p < q). The proof relies on the methods of compensated compactness, Young measures and entropy measure-valued solutions. Some new a priori estimates are carried out. In particular, our result includes the convergence result by Schonbek when b(λ) = λ, = 1 and LeFloch and Natalini when = 1.

Scalar Conservation Laws with Vanishing and Highly Nonlinear Diffusive-Dispersive Terms

“…In [9], we prove the same convergence property to [22] (∀u ∈ R) of the diffusion term in our scalar conservation law (1.1) imply the identity function as = 1 clearly. On the other hand, observing the domain of q for the L q (R), it is that (m <)q ∈ [4,5) in [9,22] and that (m <)q ∈ […”

“…(1.7), it is investigated by Fujino-Yamazaki [9]. In [9], we prove the same convergence property to [22] (∀u ∈ R) of the diffusion term in our scalar conservation law (1.1) imply the identity function as = 1 clearly.…”

“…In this paper, for a sequence of solutions to the equation with initial data, we give convergence results that a sequence converges to the unique entropy solution to the hyperbolic conservation law. In particular, our main theorem implies the results of and Schonbek [26], furthermore makes up for insufficiency of the results in Fujino-Yamazaki [9] and LeFloch-Natalini [22]. Applying the technique of compensated compactness, the Young measure and the entropy measure-valued solutions as main tools, we establish the convergence property of the sequence.…”

“…In this paper, we consider the scalar conservation laws with highly nonlinear diffusive-dispersive terms (1.1) without the assumptions (II) nor (II') by using the technique developed in [9,22]. Especially, we make use of the compensated compactness, the measure-valued (m.-v.) solutions of the Cauchy problem which are investigated by, for example, DiPerna [8] and Szepessy [27].…”

“…Following a detailed discussion in [9,22], by the Cauchy-Schwarz inequality and the Jensen inequality, we can easily check…”