Blow-up, Zero alpha Limit and the Liouville Type Theorem for the Euler-Poincar, Equations
- Authors
- Chae, Dongho; Liu, Jian-Guo
- Issue Date
- Sep-2012
- Publisher
- SPRINGER
- Citation
- COMMUNICATIONS IN MATHEMATICAL PHYSICS, v.314, no.3, pp 671 - 687
- Pages
- 17
- Journal Title
- COMMUNICATIONS IN MATHEMATICAL PHYSICS
- Volume
- 314
- Number
- 3
- Start Page
- 671
- End Page
- 687
- URI
- https://scholarworks.bwise.kr/cau/handle/2019.sw.cau/15162
- DOI
- 10.1007/s00220-012-1534-8
- ISSN
- 0010-3616
1432-0916
- Abstract
- In this paper we study the Euler-Poincar, equations in . We prove local existence of weak solutions in , and local existence of unique classical solutions in , k > N/2 + 3, as well as a blow-up criterion. For the zero dispersion equation (alpha = 0) we prove a finite time blow-up of the classical solution. We also prove that as the dispersion parameter vanishes, the weak solution converges to a solution of the zero dispersion equation with sharp rate as alpha -> 0, provided that the limiting solution belongs to with k > N/2 + 3. For the stationary weak solutions of the Euler-Poincar, equations we prove a Liouville type theorem. Namely, for alpha > 0 any weak solution is u=0; for alpha= 0 any weak solution is u=0.
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