Generalized surface quasi-geostrophic equations with singular velocities
- Authors
- Chae, Dongho; Constantin, Peter; Cordoba, Diego; Gancedo, Francisco; Wu, Jiahong
- Issue Date
- Aug-2012
- Publisher
- WILEY-BLACKWELL
- Citation
- COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, v.65, no.8, pp 1037 - 1066
- Pages
- 30
- Journal Title
- COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS
- Volume
- 65
- Number
- 8
- Start Page
- 1037
- End Page
- 1066
- URI
- https://scholarworks.bwise.kr/cau/handle/2019.sw.cau/20170
- DOI
- 10.1002/cpa.21390
- ISSN
- 0010-3640
1097-0312
- Abstract
- This paper establishes several existence and uniqueness results for two families of active scalar equations with velocity fields determined by the scalars through very singular integrals. The first family is a generalized surface quasigeostrophic (SQG) equation with the velocity field u related to the scalar theta by u=del(perpendicular to)Lambda(beta-2)theta, where 1<beta <= 2 and Lambda=(-Delta)(1/2) is the Zygmund operator. The borderline case beta = 1 corresponds to the SQG equation and the situation is more singular for beta > 1. We obtain the local existence and uniqueness of classical solutions, the global existence of weak solutions, and the local existence of patch-type solutions. The second family is a dissipative active scalar equation with u=del(perpendicular to)(log(I-Delta))(mu)theta for mu > 0, which is at least logarithmically more singular than the velocity in the first family. We prove that this family with any fractional dissipation possesses a unique local smooth solution for any given smooth data. This result for the second family constitutes a first step towards resolving the global regularity issue recently proposed by K. Ohkitani. (c) 2012 Wiley Periodicals, Inc.
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