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

We investigate the initial value problem for a scalar conservation law with highly nonlinear diffusive-dispersive terms: ut + f(u)x = e(ux2l-1)x - δ(ux2l-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 [15] 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].