Hyperbolic conservation laws with nonlinear diffusion and nonlinear dispersion

We study scalar conservation laws with nonlinear diffusion and nonlinear dispersion terms (any ℓ⩾1), 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 {uε} of the smooth solutions converges to the unique entropy solution u∈L∞(0,T∗;Lq(R)) to the conservation law ut+f(u)x=0 in . 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, ℓ=1 and LeFloch and Natalini when ℓ=1.