← Back to Search

# On Measures Of Accretion And Dissipation For Solutions Of The Camassa-Holm Equation

Grzegorz Jamróz

Published 2016 · Mathematics, Physics

We investigate the measures of dissipation and accretion related to the weak solutions of the Camassa-Holm equation. Demonstrating certain properties of nonunique characteristics, we prove a new representation formula for these measures and conclude about their structural features, such us singularity with respect to the Lebesgue measure. We apply these results to gain new insights into the structure of weak solutions, proving in particular that measures of accretion vanish for dissipative solutions of the Camassa-Holm equation.

This paper references

10.1007/3-540-26367-5_1

A

A. Spring (2005)

10.3934/DCDS.2015.35.25

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

A. Bressan (2014)

10.1016/b978-0-12-384931-1.00016-7

P

J. Lackie (2013)

10.1016/J.JDE.2010.07.006

Lipschitz metric for the periodic Camassa–Holm equation☆

K. Grunert (2010)

10.1090/S0894-0347-1991-1086966-0

Well-posedness of the initial value problem for the Korteweg-de Vries equation

C. Kenig (1991)

10.1515/9783111576855-015

J

Seguin Hen (1824)

10.1515/9783050077338-026

Y

E. M. S. J. xviii (1824)

10.1142/S0219891611002366

GENERALIZED CHARACTERISTICS AND THE HUNTER–SAXTON EQUATION

C. Dafermos (2011)

10.1142/S0219530507000857

GLOBAL DISSIPATIVE SOLUTIONS OF THE CAMASSA–HOLM EQUATION

A. Bressan (2007)

Constantin Global Dissipative Solutions of the Camassa - Holm Equation Anal

G. Chen Bressan (2007)

10.1515/9783111576855-009

D

Saskia Bonjour (1824)

10.1007/BF02392586

Wave breaking for nonlinear nonlocal shallow water equations

A. Constantin (1998)

10.1016/J.JDE.2004.09.007

Traveling wave solutions of the Camassa-Holm equation

J. Lenells (2005)

Global Existence

A. Constantin (1998)

10.1017/FMS.2014.29

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

K. Grunert (2014)

10.1098/rsta.2007.2009

Classification of all travelling-wave solutions for some nonlinear dispersive equations

J. Lenells (2007)

Constantin Global Conservative Solutions of the Camassa - Holm Equation Arch

A. Bressan (2007)

Teschl On the substitution rule for Lebesgue - Stieltjes integral Expo

B. Fuchssteiner Fokas (2012)

10.3934/DCDS.2013.33.2809

Lipschitz metric for the Camassa-Holm equation on the line

K. Grunert (2010)

10.1016/J.JDE.2016.10.036

Transfer of energy in Camassa-Holm and related models by use of nonunique characteristics

Grzegorz Jamróz (2016)

10.1080/03605300601088674

Global Conservative Solutions of the Camassa–Holm Equation—A Lagrangian Point of View

H. Holden (2007)

10.1016/J.JDE.2011.08.006

Maximal dissipation in equations of evolution

C. Dafermos (2012)

10.1007/s00205-008-0128-2

The Hydrodynamical Relevance of the Camassa–Holm and Degasperis–Procesi Equations

A. Constantin (2009)

10.1515/9783111419787-003

H

Yu-Qin Cao (1824)

Zhang On the Weak Solutions to a Shallow Water Equation Comm

P. Xin (1974)

10.1016/S1874-5717(06)80010-1

Chapter 7 On the Global Weak Solutions to a Variational Wave Equation

P. Zhang (2005)

Escher Global Existence and Blow - up for a Shallow Water Equation Ann

J. Constantin (1998)

10.1016/0167-2789(81)90004-X

Symplectic structures, their B?acklund transformation and hereditary symmetries

B. Fuchssteiner (1981)

10.1016/j.exmath.2012.09.002

On the Substitution Rule for Lebesgue-Stieltjes Integrals

Neil Falkner (2012)

10.1515/9783111413426-013

L

Il Liceo (1824)

10.3934/DCDS.2009.24.1047

Dissipative solutions for the Camassa-Holm equation

H. Holden (2009)

McKean A shallow water equation on the circle Comm

H. P. Constantin (1999)

Dafermos The entropy rate admissibility criterion for solutions of hyperbolic conservation laws

G. Falkner

10.1007/s00220-014-1958-4

Local-in-Space Criteria for Blowup in Shallow Water and Dispersive Rod Equations

L. Brandolese (2014)

Brandolese Local - in - Space Criteria for Blowup in Shallow Water and Dispersive Rod Equations Commun

A. Bressan (2014)

10.1007/S00205-006-0010-Z

Global Conservative Solutions of the Camassa–Holm Equation

A. Bressan (2007)

10.1016/0022-0396(73)90043-0

The entropy rate admissibility criterion for solutions of hyperbolic conservation laws

C. Dafermos (1973)

10.1515/9783111548050-028

Q

Chlorpromazine Thorazine (1824)

X

H. Holden (2009)

10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D

A shallow water equation on the circle

A. Constantin (1999)

Raynaud Lipschitz metric for the periodic Camassa - Holm equation

H. Holden Grunert (2013)

10.1515/9783111576855-012

G

G.V.T.V. Weerasooriya (1824)

Raynaud Lipschitz metric for the Camassa - Holm equation on the line Discrete Contin

H. Holden Grunert (1981)

X

K. Grunert (2013)

10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5

Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation

A. Constantin (1998)

On uniqueness of dissipative solutions of the Camassa-Holm equation

Grzegorz Jamróz (2016)

Lannes The hydrodynamical relevance of the Camassa - Holm and Degasperis - Procesi equations Arch

D. Constantin (2009)

Symplectic structures

A. S. Fokas (1981)

Fonte An optimal transportation metric for solutions of the Camassa - Holm equation

D. Camassa (2005)

Generalized Characteristics and the Structure of Solutions of Hyperbolic Conservation Laws

C. Dafermos (1976)

Partial Differential Equations Graduate Studies in Mathematics, Volume 19

L. C. Evans (1998)

10.3934/DCDS.2015.35.4149

Continuous Riemann solvers for traffic flow at a junction

A. Bressan (2015)

10.1016/J.AIM.2015.09.040

Maximal dissipation in Hunter-Saxton equation for bounded energy initial data

T. Cie'slak (2014)

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

On the weak solutions to a shallow water equation

Z. Xin (2000)

10.1515/9783111548050-024

M

M. Sankar (1824)

10.1103/PHYSREVLETT.71.1661

An integrable shallow water equation with peaked solitons.

Camassa (1993)

This paper is referenced by