scholarly journals On measures of accretion and dissipation for solutions of the Camassa–Holm equation

2017 ◽  
Vol 14 (04) ◽  
pp. 721-754
Author(s):  
Grzegorz Jamróz
Keyword(s):  
Lebesgue Measure ◽  
Weak Solutions ◽  
Representation Formula ◽  
Structural Features ◽  
Dissipative Solutions ◽  
Holm Equation

We investigate the measures of dissipation and accretion related to the weak solutions of the Camassa–Holm equation. Demonstrating certain novel properties of nonunique characteristics, we prove a new representation formula for these measures and conclude about their structural features, such as the fact that they are singular 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.

On the weak solutions for the rotation-two-component Camassa–Holm equation

2020 ◽  
Vol 61 (6) ◽  
pp. 061514
Author(s):  
Li Yang ◽  
Chunlai Mu ◽  
Shouming Zhou ◽  
Xinyu Tu
Keyword(s):  
Weak Solutions ◽  
Holm Equation ◽  
Two Component

Global weak solutions for a generalized Camassa-Holm equation

2018 ◽  
Vol 291 (16) ◽  
pp. 2457-2475
Author(s):  
Xi Tu ◽  
Zhaoyang Yin
Keyword(s):  
Weak Solutions ◽  
Global Weak Solutions ◽  
Holm Equation

scholarly journals TheH1(R)Space Global Weak Solutions to the Weakly Dissipative Camassa-Holm Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Zhaowei Sheng ◽  
Shaoyong Lai ◽  
Yuan Ma ◽  
Xuanjun Luo
Keyword(s):  
Cauchy Problem ◽  
Weak Solutions ◽  
Space Variable ◽  
Global Weak Solutions ◽  
Initial Value ◽  
Dissipative Term ◽  
First Order ◽  
Holm Equation ◽  
The Cauchy Problem ◽  
Derivatives Of

The existence of global weak solutions to the Cauchy problem for a generalized Camassa-Holm equation with a dissipative term is investigated in the spaceC([0,∞)×R)∩L∞([0,∞);H1(R))provided that its initial valueu0(x)belongs to the spaceH1(R). A one-sided super bound estimate and a space-time higher-norm estimate on the first-order derivatives of the solution with respect to the space variable are derived.

scholarly journals GLOBAL DISSIPATIVE SOLUTIONS OF THE CAMASSA–HOLM EQUATION

2007 ◽  
Vol 05 (01) ◽  
pp. 1-27 ◽  
Author(s):  
ALBERTO BRESSAN ◽  
ADRIAN CONSTANTIN
Keyword(s):  
Initial Data ◽  
Wave Breaking ◽  
Integral Transformation ◽  
Local Source ◽  
Dissipative Solutions ◽  
Holm Equation ◽  
Directional Variation ◽  
The Cauchy Problem ◽  
Non Local ◽  
New Variables

This paper is devoted to the continuation of solutions to the Camassa–Holm equation after wave breaking. By introducing a new set of independent and dependent variables, the evolution problem is rewritten as a semilinear hyperbolic system in an L∞space, containing a non-local source term which is discontinuous but has bounded directional variation. For a given initial condition, the Cauchy problem has a unique solution obtained as fixed point of a contractive integral transformation. Returning to the original variables, we obtain a semigroup of global dissipative solutions, defined for every initial data [Formula: see text], and continuously depending on the initial data. The new variables resolve all singularities due to possible wave breaking and ensure that energy loss occurs only through wave breaking.

scholarly journals Patched peakon weak solutions of the modified Camassa–Holm equation

2019 ◽  
Vol 390 ◽  
pp. 15-35 ◽  
Author(s):  
Yu Gao ◽  
Lei Li ◽  
Jian-Guo Liu
Keyword(s):  
Weak Solutions ◽  
Holm Equation

Global weak solutions for the Dullin–Gottwald–Holm equation

2010 ◽  
Vol 72 (3-4) ◽  
pp. 1690-1700 ◽  
Author(s):  
Shuanghu Zhang ◽  
Zhaoyang Yin
Keyword(s):  
Weak Solutions ◽  
Global Weak Solutions ◽  
Holm Equation
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