scholarly journals Lipschitz metric for the periodic second-order Camassa–Holm equation

2017 ◽  
Vol 451 (2) ◽  
pp. 990-1025
Author(s):  
Danping Ding ◽  
Shuanghua Zhang
2020 ◽  
Vol 8 ◽  
Author(s):  
JOSÉ A. CARRILLO ◽  
KATRIN GRUNERT ◽  
HELGE HOLDEN

We analyze stability of conservative solutions of the Cauchy problem on the line for the Camassa–Holm (CH) equation. Generically, the solutions of the CH equation develop singularities with steep gradients while preserving continuity of the solution itself. In order to obtain uniqueness, one is required to augment the equation itself by a measure that represents the associated energy, and the breakdown of the solution is associated with a complicated interplay where the measure becomes singular. The main result in this paper is the construction of a Lipschitz metric that compares two solutions of the CH equation with the respective initial data. The Lipschitz metric is based on the use of the Wasserstein metric.


2011 ◽  
Vol 250 (3) ◽  
pp. 1460-1492 ◽  
Author(s):  
Katrin Grunert ◽  
Helge Holden ◽  
Xavier Raynaud

2013 ◽  
Vol 33 (7) ◽  
pp. 2809-2827 ◽  
Author(s):  
Katrin Grunert ◽  
◽  
Helge Holden ◽  
Xavier Raynaud ◽  

Author(s):  
W. L. Bell

Disappearance voltages for second order reflections can be determined experimentally in a variety of ways. The more subjective methods, such as Kikuchi line disappearance and bend contour imaging, involve comparing a series of diffraction patterns or micrographs taken at intervals throughout the disappearance range and selecting that voltage which gives the strongest disappearance effect. The estimated accuracies of these methods are both to within 10 kV, or about 2-4%, of the true disappearance voltage, which is quite sufficient for using these voltages in further calculations. However, it is the necessity of determining this information by comparisons of exposed plates rather than while operating the microscope that detracts from the immediate usefulness of these methods if there is reason to perform experiments at an unknown disappearance voltage.The convergent beam technique for determining the disappearance voltage has been found to be a highly objective method when it is applicable, i.e. when reasonable crystal perfection exists and an area of uniform thickness can be found. The criterion for determining this voltage is that the central maximum disappear from the rocking curve for the second order spot.


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