Please confirm you are human
(

__Sign Up__for free to never see this)
← Back to Search

# Global Conservative Solutions Of The Camassa–Holm Equation—A Lagrangian Point Of View

H. Holden, X. Raynaud

Published 2007 · Mathematics

We show that the Camassa–Holm equation ut − uxxt + 3uux − 2uxuxx − uuxxx = 0 possesses a global continuous semigroup of weak conservative solutions for initial data u|t=0 in H1. The result is obtained by introducing a coordinate transformation into Lagrangian coordinates. To characterize conservative solutions it is necessary to include the energy density given by the positive Radon measure μ with . The total energy is preserved by the solution.

This paper references

10.1090/S0025-5718-07-01919-9

Convergent difference schemes for the Hunter-Saxton equation

H. Holden (2007)

10.1007/BF02392586

Wave breaking for nonlinear nonlocal shallow water equations

A. Constantin (1998)

10.1103/PHYSREVLETT.71.1661

An integrable shallow water equation with peaked solitons.

Camassa (1993)

10.5802/AIF.2375

Periodic conservative solutions of the Camassa–Holm equation

H. Holden (2008)

10.5802/AIF.1757

Existence of permanent and breaking waves for a shallow water equation: a geometric approach

A. Constantin (2000)

Camassa–Holm, Korteweg–de Vries and related models for water

R S. (2002)

10.1007/978-3-540-75712-2_35

Global Weak Solutions for a Shallow Water Equation

G. Coclite (2008)

10.3934/DCDS.2006.14.505

A convergent numerical scheme for the Camassa--Holm equation based on multipeakons

H. Holden (2005)

10.1016/0167-2789(81)90004-X

Symplectic structures, their B?acklund transformation and hereditary symmetries

B. Fuchssteiner (1981)

10.1016/S0165-2125(98)00014-6

Exact travelling-wave solutions of an integrable equation arising in hyperelastic rods

H. Dai (1998)

Symplectic structures, their Bäcklund transformations

Raton (1981)

10.2991/jnmp.2001.8.s.5

Peakon-Antipeakon Interaction

R. Beals (2001)

10.1088/0305-4470/38/4/007

Conservation laws of the Camassa–Holm equation

J. Lenells (2005)

Global multipeakon solutions of the Camassa-Holm equation

H Holden (2007)

10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5

On the weak solutions to a shallow water equation

Zhouping Xin (2000)

10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L

Stability of peakons

A. Constantin (2000)

Functional Analysis. Reprint of the sixth

K Yosida (1980)

10.1098/rspa.2000.0520

Solitary shock waves and other travelling waves in a general compressible hyperelastic rod

H. Dai (2000)

10.1088/0305-4470/35/32/201

TOPICAL REVIEW: On the geometric approach to the motion of inertial mechanical systems

A. Constantin (2002)

Well-posedness for a parabolic-elliptic system. Discrete Cont

M Coclite (2005)

10.4310/MAA.2005.V12.N2.A7

An Optimal Transportation Metric for Solutions of the Camassa-Holm Equation

A. Bressan (2005)

Global weak solutions to a generalized hyperelasticrod wave equation

G. M. Coclite (2005)

Conservative solution of the Camassa Holm Equation on the real line

M. Fonte (2005)

10.1201/9780203747940

Measure theory and fine properties of functions

L. Evans (1992)

and K

G. M. Coclite (2005)

10.1142/S0219891607001045

GLOBAL CONSERVATIVE MULTIPEAKON SOLUTIONS OF THE CAMASSA–HOLM EQUATION

H. Holden (2007)

10.1016/S0065-2156(08)70254-0

A New Integrable Shallow Water Equation

R. Camassa (1994)

Convergent difference schemes

K. H. Karlsen (2007)

10.1007/BF01170373

Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod

H. Dai (1998)

On the peakonantipeakon interaction

E. Wahlén. (2002)

10.1146/annurev.fl.24.010192.001045

Topological methods in hydrodynamics

V. Arnold (1998)

10.1137/040611975

Convergence of a Finite Difference Scheme for the Camassa-Holm Equation

H. Holden (2006)

10.1017/S0022112001007224

Korteweg-de Vries and related models for water waves

A. H. Johnson (2002)

Analyse Fonctionnelle

H. Brezis (1983)

Well - posedness for a parabolicelliptic system

G. M. Coclite (2005)

10.1016/J.JDE.2006.09.007

Global conservative solutions of the generalized hyperelastic-rod wave equation ✩

H. Holden (2007)

Wave Motion 28(4):367–381

hyperelastic rods (1998)

Camassa–Holm

R. S. Johnson (2002)

10.1137/040616711

Global Weak Solutions to a Generalized Hyperelastic-rod Wave Equation

G. Coclite (2005)

Global existence and blow-up for a shallow water equation

A. Constantin (1998)

10.1007/S00205-006-0010-Z

Global Conservative Solutions of the Camassa–Holm Equation

A. Bressan (2007)

10.1098/rspa.2000.0701

On the scattering problem for the Camassa-Holm equation

A. Constantin (2001)

10.1081/PDE-120016129

ON THE UNIQUENESS AND LARGE TIME BEHAVIOR OF THE WEAK SOLUTIONS TO A SHALLOW WATER EQUATION

Z. Xin (2002)

Analyse fonctionnelle : théorie et applications

H. Brezis (1983)

10.2307/j.ctt130hk3w.9

Real Analysis

Kellen Petersen August (2009)

Raynaud) Department of Mathematical Sciences

(1053)

10.3934/DCDS.2005.13.659

WELLPOSEDNESS FOR A PARABOLIC-ELLIPTIC SYSTEM

G. Coclite (2005)

Mooney–Rivlin rod

Dai (2000)

Functions of Bounded Variation and Free Discontinuity Problems

L. Ambrosio (2000)

This paper is referenced by

10.1155/2014/754976

Nonlinear functional analysis of boundary value problems 2013

Y. Wu (2014)

10.1016/j.jde.2019.08.042

Blow-up phenomena, ill-posedness and peakon solutions for the periodic Euler-Poincaré equations

Wei Luo (2019)

10.1155/2013/107450

On the Multipeakon Dissipative Behavior of the Modified Coupled Camassa-Holm Model for Shallow Water System

Zhixi Shen (2013)

10.1007/s10231-020-00980-9

Well-posedness and continuity properties of the new shallow-water model with cubic nonlinearity

Yong-sheng Mi (2020)

10.3934/DCDS.2017052

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

G. Coclite (2015)

The Cauchy Problem for the Camassa-Holm Equation with Quartic Nonlinearity in Besov Spaces

Hammad Khalil (2016)

10.3934/DCDS.2015.35.25

Uniqueness of Conservative Solutions to the Camassa-Holm Equation via Characteristics

A. Bressan (2014)

10.3934/DCDSS.2016084

On the Cauchy problem of the modified Hunter-Saxton equation

Yongsheng Mi (2016)

10.1142/S0219891618500182

Existence and Lipschitz stability for α-dissipative solutions of the two-component Hunter–Saxton system

K. Grunert (2016)

10.1016/J.NA.2009.09.008

Global weak solutions for the Dullin–Gottwald–Holm equation

S. Zhang (2010)

10.1155/2014/613851

Global Conservative Solutions of a Generalized Two-Component Camassa-Holm System

F. Zhang (2014)

10.1080/00036811.2012.667083

Initial boundary value problem for a coupled Camassa–Holm system with peakons

Qiaoyi Hu (2013)

10.3934/DCDS.2013.33.1713

Global conservative and dissipative solutions of the generalized Camassa-Holm equation

Shouming Zhou (2012)

10.1016/J.JDE.2013.06.008

On the solutions of a model equation for shallow water waves of moderate amplitude

Yongsheng Mi (2013)

10.1142/S0219891617500242

On measures of accretion and dissipation for solutions of the Camassa-Holm equation

Grzegorz Jamróz (2016)

10.1017/FMS.2014.29

A continuous interpolation between conservative and dissipative solutions for the two-component Camassa-Holm system

K. Grunert (2014)

10.1063/1.3675900

Non-uniform dependence for a modified Camassa-Holm system

G. Lv (2012)

10.1007/S00028-019-00533-5

Global existence and propagation speed for a generalized Camassa–Holm model with both dissipation and dispersion

Qiaoyi Hu (2015)

On the isospectral problem of the Camassa-Holm equation

Jonathan Eckhardt (2011)

10.1007/s00205-018-1234-4

Lipschitz Metric for the Novikov Equation

H. Cai (2016)

10.5802/AIF.2375

Periodic conservative solutions of the Camassa–Holm equation

H. Holden (2008)

10.1016/J.JDE.2015.03.020

A note on the Camassa–Holm equation

G. Coclite (2015)

10.3934/DCDS.2017065

On an $N$-Component Camassa-Holm equation with peakons

Y. Mi (2016)

10.4236/JAMP.2017.56108

Optimal Distributed Control Problem for the b -Equation

Chunyu Shen (2017)

10.1016/j.jmaa.2020.123933

The global Gevrey regularity of the rotation two-component Camassa-Holm system

Y. Guo (2020)

10.4236/JAMP.2017.56106

Modeling for Collapsing Cavitation Bubble near Rough Solid Wall by Mulit-Relaxation-Time Pseudopotential Lattice Boltzmann Model

Minglei Shan (2017)

10.1093/IMRN/RNP100

Symmetric Waves Are Traveling Waves

Mats Ehrnstrom (2009)

10.1007/s00605-020-01430-7

On the Cauchy problem of a new integrable two-component Novikov equation

Yong-sheng Mi (2020)

10.3934/DCDS.2012.32.4209

Global conservative solutions of the Camassa-Holm equation for initial data nonvanishing asymptotics

K. Grunert (2011)

10.1080/00036811.2015.1073265

On weak solutions to a shallow water wave model of moderate amplitude

Y. Guo (2016)

10.1155/2014/808214

On the Study of Global Solutions for a Nonlinear Equation

Haibo Yan (2014)

10.1137/16M1063009

Generic Regularity of Conservative Solutions to Camassa-Holm Type Equations

Mingjie Li (2017)

See more