Journal
SIAM JOURNAL ON NUMERICAL ANALYSIS
Volume 50, Issue 1, Pages 1-21Publisher
SIAM PUBLICATIONS
DOI: 10.1137/110831386
Keywords
Camassa-Holm equation; Degasperis-Procesi equation; Euler-Poincare equation; global weak solution; particle method; space-time BV estimates; peakon solutions; conservation laws; completely integrable systems
Categories
Funding
- NSF [DMS-0712898, DMS 10-11738]
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1115682, 1011738] Funding Source: National Science Foundation
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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.
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