We investigate the initial value problem for a scalar conservation law with highly nonlinear diffusive-dispersive terms: ut + f(u)x = ε(u_x_2ℓ −1)x − δ(u_x_2ℓ −1)xx (ℓ ≥ 1). 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 Kondo-LeFloch  and Schonbek , furthermore makes up for insuﬃciency of the results in Fujino-Yamazaki  and LeFloch-Natalini . 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 .
Cite this article
Naoki Fujino, Scalar Conservation Laws with Vanishing and Highly Nonlinear Diffusive-Dispersive Terms. Publ. Res. Inst. Math. Sci. 43 (2007), no. 4, pp. 1005–1022