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Convergence of a Particle Method and Global Weak Solutions of a Family of Evolutionary PDEs

Alina Chertock North Carolina State University Jian-Guo Liu Duke University Terrance Pendleton North Carolina State University

Analysis of PDEs mathscidoc:1702.03051

SIAM Journal on Numerical Analysis, 50, (1), 1-21, 2012.2
The purpose of this paper is to provide global existence and uniqueness results for a family of fluid transport equations by establishing convergence results for the particle method applied to these equations. The considered family of PDEs is a collection of strongly nonlinear equations which yield traveling wave solutions and can be used to model a variety of flows in fluid dynamics. We apply a particle method to the studied evolutionary equations and provide a new self-contained method for proving its convergence. The latter is accomplished by using the concept of space-time bounded variation and the associated compactness properties. From this result, we prove the existence of a unique global weak solution in some special cases and obtain stronger regularity properties of the solution than previously established.
Camassa–Holm equation, Degasperis–Procesi equation, Euler–Poincar´eequation,global weak solution, particle method, space-time BV estimates, peako
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@inproceedings{alina2012convergence,
  title={Convergence of a Particle Method and Global Weak Solutions of a Family of Evolutionary PDEs},
  author={Alina Chertock, Jian-Guo Liu, and Terrance Pendleton},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170208230631582049334},
  booktitle={SIAM Journal on Numerical Analysis},
  volume={50},
  number={1},
  pages={1-21},
  year={2012},
}
Alina Chertock, Jian-Guo Liu, and Terrance Pendleton. Convergence of a Particle Method and Global Weak Solutions of a Family of Evolutionary PDEs. 2012. Vol. 50. In SIAM Journal on Numerical Analysis. pp.1-21. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170208230631582049334.
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