A note on the convergence of the solution of the high order Camassa-Holm equation to the entropy ones of a scalar conservation law

Giuseppe Maria Coclite; Lorenzo Di Ruvo

doi:10.3934/dcds.2017052

Article Contents

2017, Volume 37, Issue 3: 1247-1282. Doi: 10.3934/dcds.2017052

This issue Previous Article Functional envelopes relative to the point-open topology on a subset Next Article A remark on the partial regularity of a suitable weak solution to the Navier-Stokes Cauchy problem

A note on the convergence of the solution of the high order Camassa-Holm equation to the entropy ones of a scalar conservation law

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

1.
Dipartimento di Matematica, Università di Bari, via E. Orabona 4,70125 Bari, Italy
2.
Dipartimento di Scienze e Metodi dell'Ingegneria, Università di Modena e Reggio Emilia, via G. Amendola 2,42122 Reggio Emilia, Italy

* Corresponding author: G. M. Coclite
* Corresponding author: G. M. Coclite

Received: April 2016

Revised: May 2016

Published: May 2017

The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilitàe le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

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Abstract

We consider the high order Camassa-Holm equation, which is a non linear dispersive equation of the fifth order. We prove that as the diffusion and dispersion parameters tends to zero, the solutions converge to the entropy ones of a scalar conservation law. 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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Mathematics Subject Classification: Primary:35G25, 35L65;Secondary:35L05.

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