A singular limit problem for conservation laws  related to the Kawahara-Korteweg-de Vries equation

Giuseppe Maria Coclite; Lorenzo di  Ruvo

doi:10.3934/nhm.2016.11.281

Article Contents

2016, Volume 11, Issue 2: 281-300. Doi: 10.3934/nhm.2016.11.281

This issue Previous Article New interaction estimates for the Baiti-Jenssen system Next Article Logarithmic estimates for continuity equations

A singular limit problem for conservation laws related to the Kawahara-Korteweg-de Vries equation

Giuseppe Maria Coclite^1, and
Lorenzo di Ruvo^2,

1.
Department of Mathematics, University of Bari, Via E. Orabona 4, I--70125 Bari
2.
Department of Science and Methods for Engineering, University of Modena and Reggio Emilia, via G. Amendola 2, 42122 Reggio Emilia

Received: March 31, 2015

Revised: September 30, 2015

Published: May 2016

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Abstract

We consider the Kawahara-Korteweg-de Vries equation, which contains nonlinear dispersive effects. We prove that as the dispersion parameter tends to zero, the solutions of the dispersive equation converge to discontinuous weak solutions of the Burgers equation. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the $L^p$ setting.

Keywords:

Mathematics Subject Classification: Primary: 35G25, 35L65; Secondary: 35L05.

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