Weakly singular shock profiles for a non-dispersive regularization of shallow-water equations

Yue Pu; Robert L. Pego; Denys Dutykh; Didier Clamond

doi:10.4310/CMS.2018.v16.n5.a09

Weakly singular shock profiles for a non-dispersive regularization of shallow-water equations

Yue Pu, Robert L. Pego, Denys Dutykh, Didier Clamond

Mathematics

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12 Scopus citations

Abstract

We study a regularization of the classical Saint-Venant (shallow-water) equations, recently introduced by D. Clamond and D. Dutykh [Commun. Nonl. Sci. Numer. Simulat., 55:237-247, 2018]. This regularization is non-dispersive and formally conserves mass, momentum and energy. We show that, for every classical shock wave, the system admits a corresponding non-oscillatory traveling wave solution which is continuous and piecewise smooth, having a weak singularity at a single point where energy is dissipated as it is for the classical shock. The system also admits cusped solitary waves of both elevation and depression.

Original language	British English
Pages (from-to)	1361-1378
Number of pages	18
Journal	Communications in Mathematical Sciences
Volume	16
Issue number	5
DOIs	https://doi.org/10.4310/CMS.2018.v16.n5.a09
State	Published - 2018

Keywords

Cuspons
Energy loss
Green-Naghdi equations
Long waves
Peakons
Serre equations
Shallow water
Weak solutions

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10.4310/CMS.2018.v16.n5.a09

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title = "Weakly singular shock profiles for a non-dispersive regularization of shallow-water equations",

abstract = "We study a regularization of the classical Saint-Venant (shallow-water) equations, recently introduced by D. Clamond and D. Dutykh [Commun. Nonl. Sci. Numer. Simulat., 55:237-247, 2018]. This regularization is non-dispersive and formally conserves mass, momentum and energy. We show that, for every classical shock wave, the system admits a corresponding non-oscillatory traveling wave solution which is continuous and piecewise smooth, having a weak singularity at a single point where energy is dissipated as it is for the classical shock. The system also admits cusped solitary waves of both elevation and depression.",

keywords = "Cuspons, Energy loss, Green-Naghdi equations, Long waves, Peakons, Serre equations, Shallow water, Weak solutions",

author = "Yue Pu and Pego, \{Robert L.\} and Denys Dutykh and Didier Clamond",

note = "Funding Information: This work was initiated during the 2017 ICERM semester program on Singularities and Waves in Incompressible Fluids. The work of RLP and YP is partially supported by the National Science Foundation under NSF grant DMS 1515400, the Simons Foundation under grant 395796 and the Center for Nonlinear Analysis (through NSF grant OISE-0967140). Publisher Copyright: {\textcopyright} 2018 International Press.",

year = "2018",

doi = "10.4310/CMS.2018.v16.n5.a09",

language = "British English",

volume = "16",

pages = "1361--1378",

journal = "Communications in Mathematical Sciences",

issn = "1539-6746",

publisher = "International Press, Inc.",

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TY - JOUR

T1 - Weakly singular shock profiles for a non-dispersive regularization of shallow-water equations

AU - Pu, Yue

AU - Pego, Robert L.

AU - Dutykh, Denys

AU - Clamond, Didier

N1 - Funding Information: This work was initiated during the 2017 ICERM semester program on Singularities and Waves in Incompressible Fluids. The work of RLP and YP is partially supported by the National Science Foundation under NSF grant DMS 1515400, the Simons Foundation under grant 395796 and the Center for Nonlinear Analysis (through NSF grant OISE-0967140). Publisher Copyright: © 2018 International Press.

PY - 2018

Y1 - 2018

N2 - We study a regularization of the classical Saint-Venant (shallow-water) equations, recently introduced by D. Clamond and D. Dutykh [Commun. Nonl. Sci. Numer. Simulat., 55:237-247, 2018]. This regularization is non-dispersive and formally conserves mass, momentum and energy. We show that, for every classical shock wave, the system admits a corresponding non-oscillatory traveling wave solution which is continuous and piecewise smooth, having a weak singularity at a single point where energy is dissipated as it is for the classical shock. The system also admits cusped solitary waves of both elevation and depression.

AB - We study a regularization of the classical Saint-Venant (shallow-water) equations, recently introduced by D. Clamond and D. Dutykh [Commun. Nonl. Sci. Numer. Simulat., 55:237-247, 2018]. This regularization is non-dispersive and formally conserves mass, momentum and energy. We show that, for every classical shock wave, the system admits a corresponding non-oscillatory traveling wave solution which is continuous and piecewise smooth, having a weak singularity at a single point where energy is dissipated as it is for the classical shock. The system also admits cusped solitary waves of both elevation and depression.

KW - Cuspons

KW - Energy loss

KW - Green-Naghdi equations

KW - Long waves

KW - Peakons

KW - Serre equations

KW - Shallow water

KW - Weak solutions

UR - https://www.scopus.com/pages/publications/85059268233

U2 - 10.4310/CMS.2018.v16.n5.a09

DO - 10.4310/CMS.2018.v16.n5.a09

M3 - Article

AN - SCOPUS:85059268233

SN - 1539-6746

VL - 16

SP - 1361

EP - 1378

JO - Communications in Mathematical Sciences

JF - Communications in Mathematical Sciences

IS - 5

ER -

Weakly singular shock profiles for a non-dispersive regularization of shallow-water equations

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Keywords

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