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

Giuseppe Maria Coclite; Lorenzo di  Ruvo; Giuseppe Maria Coclite; Lorenzo di  Ruvo

doi:10.3934/nhm.2016.11.281

Networks and Heterogeneous Media

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

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

Giuseppe Maria Coclite ^{1
,},
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: 01 April 2015 Revised: 01 October 2015
Primary: 35G25, 35L65; Secondary: 35L05.

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.
- Singular limit,
- compensated compactness,
- Kawahara-Korteweg-de Vries equation,
- entropy condition
Citation: Giuseppe Maria Coclite, Lorenzo di Ruvo. A singular limit problem for conservation laws related to the Kawahara-Korteweg-de Vries equation[J]. Networks and Heterogeneous Media, 2016, 11(2): 281-300. doi: 10.3934/nhm.2016.11.281

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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.

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