JournalsprimsVol. 43, No. 4pp. 1005–1022

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

  • Naoki Fujino

    Graduate School of Pure and Applied Sciences, Ibaraki, Tsukuba, Japan
Scalar Conservation Laws with Vanishing and Highly Nonlinear Diffusive-Dispersive Terms cover
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Abstract

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 [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].

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

DOI 10.2977/PRIMS/1201012378