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.
We investigate the initial value problem for a scalar conservation law with highly nonlinear diffusive-dispersive terms:. 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. The final step of our proof is to show that the measure-valued mapping associated to the sequence of solutions is reduced to an entropy solution and this step is mainly based on the approach of LeFloch-Natalini [22].
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