A semi-discrete scheme derived from variational principles for global conservative solutions of a Camassa–Holm system
Peer reviewed, Journal article
Published version
![Thumbnail](/ntnu-xmlui/bitstream/handle/11250/2740531/Galtung_2021_Nonlinearity_34_2220.pdf.jpg?sequence=6&isAllowed=y)
Åpne
Permanent lenke
https://hdl.handle.net/11250/2740531Utgivelsesdato
2021Metadata
Vis full innførselSamlinger
- Institutt for matematiske fag [2438]
- Publikasjoner fra CRIStin - NTNU [38015]
Sammendrag
We define a kinetic and a potential energy such that the principle of stationary action from Lagrangian mechanics yields a Camassa–Holm system (2CH) as the governing equations. After discretizing these energies, we use the same variational principle to derive a semi-discrete system of equations as an approximation of the 2CH system. The discretization is only available in Lagrangian coordinates and requires the inversion of a discrete Sturm–Liouville operator with time-varying coefficients. We show the existence of fundamental solutions for this operator at initial time with appropriate decay. By propagating the fundamental solutions in time, we define an equivalent semi-discrete system for which we prove that there exists unique global solutions. Finally, we show how the solutions of the semi-discrete system can be used to construct a sequence of functions converging to the conservative solution of the 2CH system.