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Zero diffusion-dispersion limits for scalar conservation laws. (English) Zbl 1055.35072

Summary: We consider solutions of hyperbolic conservation laws regularized with vanishing diffusion and dispersion terms. Following a pioneering work by Schonbek, we establish the convergence of the regularized solutions toward discontinuous solutions of the hyperbolic conservation law. The proof relies on the method of compensated compactness in the \(L^{2}\) setting. Our result improves upon M. E. Schonbek’s earlier results [Commun. Partial Differ. Equations 7, No. 8, 959–1000 (1982; Zbl 0496.35058)] and provides an optimal condition on the balance between the relative sizes of the diffusion and the dispersion parameters. A convergence result is also established for multidimensional conservation laws by relying on R. J. DiPerna’s uniqueness theorem for entropy measure-valued solutions [Arch. Ration. Mech. Anal. 88, No. 3, 223–270 (1985; Zbl 0616.35055)].

MSC:

35L65 Hyperbolic conservation laws
35B25 Singular perturbations in context of PDEs
35L67 Shocks and singularities for hyperbolic equations
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
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