Dafermos regularization of a diffusive-dispersive equation with cubic flux

Stephen Schecter; Monique Richardson Taylor

doi:10.3934/dcds.2012.32.4069

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2012, Volume 32, Issue 12: 4069-4110. Doi: 10.3934/dcds.2012.32.4069

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Dafermos regularization of a diffusive-dispersive equation with cubic flux

Stephen Schecter^1, and
Monique Richardson Taylor^2,

1.
Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205
2.
Department of Mathematics, Shepherd University, Shepherdstown, WV 25443-5000

Received: June 2011

Revised: June 2012

Published: December 2012

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Abstract

We study existence and spectral stability of stationary solutions of the Dafermos regularization of a much-studied diffusive-dispersive equation with cubic flux. Our study includes stationary solutions that corresponds to Riemann solutions consisting of an undercompressive shock wave followed by a compressive shock wave. We use geometric singular perturbation theory (1) to construct the solutions, and (2) to show that asmptotically, there are no large eigenvalues, and any order-one eigenvalues must be near $-1$ or a certain number $\lambda^*$. We give numerical evidence that $\lambda^*$ is also $-1$. Finally, we use pseudoexponential dichotomies to show that in a space of exponentially decreasing functions, the essential spectrum is contained in Re$ \lambda \le -\delta <0 $.

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Mathematics Subject Classification: Primary: 35Q53; Secondary: 35C06, 35L67, 34E15.

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