Diffusive-Dispersive Traveling Waves and Kinetic Relations: Part I: Nonconvex Hyperbolic Conservation Laws |
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Authors: | Nabil Bedjaoui Philippe G LeFloch |
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Institution: | a Centre de Mathématiques Appliquées, UMR 7641, Ecole Polytechnique, 91128, Palaiseau Cedex, Francef1E-mail: bedjaoui@cmap.polytechnique.frf1b Centre National de la Recherche Scientifique, UMR 7641, Ecole Polytechnique, 91128, Palaiseau Cedex, Francef2E-mail: bedjaoui@cmap.polytechnique.frf2c INSSET, Université de Picardie, 48 rue Raspail, 02109, Saint-Quentin, Francef3E-mail: bedjaoui@cmap.polytechnique.frf3d Centre de Mathématiques Appliquées, UMR 7641, Ecole Polytechnique, 91128, Palaiseau Cedex, Francef4E-mail: lefloch@cmap.polytechnique.frf4e Centre National de la Recherche Scientifique, UMR 7641, Ecole Polytechnique, 91128, Palaiseau Cedex, Francef5E-mail: lefloch@cmap.polytechnique.frf5 |
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Abstract: | Motivated by the theory of phase transition dynamics, we consider one-dimensional, nonlinear hyperbolic conservation laws with nonconvex flux-function containing vanishing nonlinear diffusive-dispersive terms. Searching for traveling wave solutions, we establish general results of existence, uniqueness, monotonicity, and asymptotic behavior. In particular, we investigate the properties of the traveling waves in the limits of dominant diffusion, dominant dispersion, and asymptotically small or large shock strength. As the diffusion and dispersion parameters tend to 0, the traveling waves converge to shock wave solutions of the conservation law, which either satisfy the classical Oleinik entropy criterion or are nonclassical undercompressive shocks violating it. |
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Keywords: | hyperbolic conservation law diffusion dispersion shock wave undercompressive entropy inequality kinetic relation |
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