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Parabolic Approximations of Diffusive-Dispersive Equations

By A. Corli, C. Rohde and V. Schleper


We consider a lower-order approximation for a third-order diffusive-dispersive conservation law with nonlinear flux. It consists of a system of two second-order parabolic equations; a coupling parameter is also added. If the flux has an inflection point it is well-known, on the one hand, that the diffusive-dispersive law admits traveling-wave solutions whose end states are also connected by undercompressive shock waves of the underlying hyperbolic conservation law. On the other hand, if the diffusive-dispersive regularization vanishes, the solutions of the corresponding initial-value problem converge to a weak solution of the hyperbolic conservation law. We show that both of these properties also hold for the lower-order approximation. Furthermore, when the coupling parameter tends to infinity, we prove that solutions of initial value problems for the approximation converge to a weak solution of the diffusive-dispersive law. The proofs rely on new a-priori energy estimates for higher-order derivatives and the technique of compensated compactness

Topics: Hyperbolic conservation laws, traveling waves, singular limits, phase transitions
Year: 2014
DOI identifier: 10.1016/j.jmaa.2014.01.049
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