scholarly journals Lipschitz metric for the modified two-component Camassa–Holm system

2018 ◽  
Vol 16 (02) ◽  
pp. 159-182 ◽  
Author(s):  
Chunxia Guan ◽  
Kai Yan ◽  
Xuemei Wei
Keyword(s):  
Energy Density ◽  
Real Line ◽  
Weak Solutions ◽  
Lipschitz Continuity ◽  
Distance Functions ◽  
Lagrangian Coordinates ◽  
Global Weak Solutions ◽  
Positive Radon Measure ◽  
Two Component ◽  
Lipschitz Metric

This paper is devoted to the existence and Lipschitz continuity of global conservative weak solutions in time for the modified two-component Camassa–Holm system on the real line. We obtain the global weak solutions via a coordinate transformation into the Lagrangian coordinates. The key ingredients in our analysis are the energy density given by the positive Radon measure and the proposed new distance functions as well.

scholarly journals Energy of wave and solutions of C-Holm equation under Lagrangian point of view

2018 ◽  
Vol 6 (11) ◽  
Author(s):  
Shkelqim Hajrulla ◽  
L Bezati ◽  
F Hoxha
Keyword(s):  
Energy Density ◽  
Initial Data ◽  
Coordinate Transformation ◽  
Radon Measure ◽  
Continuous Semigroup ◽  
Point Of View ◽  
Lagrangian Point ◽  
Lagrangian Coordinates ◽  
Positive Radon Measure ◽  
Holm Equation

     Abstract: We deal with the Camassa-Holm equation   possesses a global continuous semigroup of weak conservative solutions for initial data. 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.

scholarly journals Invariance of Gibbs measures under the flows of Hamiltonian equations on the real line

2019 ◽  
Vol 22 (02) ◽  
pp. 1950012
Author(s):  
Federico Cacciafesta ◽  
Anne-Sophie de Suzzoni
Keyword(s):  
Initial Data ◽  
Real Line ◽  
Weak Solutions ◽  
Gibbs Measures ◽  
Direct Consequence ◽  
Dispersive Equations ◽  
Global Weak Solutions ◽  
Hamiltonian Equations ◽  
The Real ◽  
Nonlinear Dispersive Equations

We prove that the Gibbs measures [Formula: see text] for a class of Hamiltonian equations written as [Formula: see text] on the real line are invariant under the flow of [Formula: see text] in the sense that there exist random variables [Formula: see text] whose laws are [Formula: see text] (thus independent from [Formula: see text]) and such that [Formula: see text] is a solution to [Formula: see text]. Besides, for all [Formula: see text], [Formula: see text] is almost surely not in [Formula: see text] which provides as a direct consequence the existence of global weak solutions for initial data not in [Formula: see text]. The proof uses Prokhorov’s theorem, Skorohod’s theorem, as in the strategy in [N. Burq, L. Thomann and N. Tzvetkov, Remarks on the Gibbs measures for nonlinear dispersive equations, preprint (2014); arXiv:1412.7499v1 [math.AP]] and Feynman–Kac’s integrals.

On the existence of global weak solutions to an integrable two-component Camassa–Holm shallow-water system

2013 ◽  
Vol 56 (3) ◽  
pp. 755-775 ◽  
Author(s):  
Chunxia Guan ◽  
Zhaoyang Yin
Keyword(s):  
Shallow Water ◽  
Weak Solutions ◽  
Burgers Equation ◽  
Water System ◽  
Global Weak Solution ◽  
Rarefaction Waves ◽  
Global Weak Solutions ◽  
Two Component ◽  
The Cauchy Problem ◽  
Shallow Water System

AbstractIn this paper, we investigate the existence of global weak solutions to an integrable two-component Camassa–Holm shallow-water system, provided the initial datau0(x)andρ0(x)have end statesu± andρ±, respectively. By perturbing the Cauchy problem of the system around rarefaction waves of the well-known Burgers equation, we obtain a global weak solution for the system under the assumptionsu− ≤ u+andρ− ≤ ρ+.

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