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

Abstract

We investigate the initial value problem for a scalar conservation law with highly nonlinear diffusive-dispersive terms: $u_t+f(u{)}_x=\epsilon (u_x^{2\ell -1}{)}_x-\delta (u_x^{2\ell -1}{)}_{xx}$ ($\ell \ge 1$). 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].