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

Naoki Fujino
2007 Publications of the Research Institute for Mathematical Sciences  
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 Kondo-LeFloch [15] and Schonbek [26] , furthermore makes up for insufficiency of the results in Fujino-Yamazaki [9] and
more » ... ch-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] .
doi:10.2977/prims/1201012378 fatcat:bkip4bfb5farjbtawis2wbvdua