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|Abstract:||This paper establishes several existence and uniqueness results for two fam- ilies of active scalar equations with velocity fields determined by the scalars through very singular integrals. The first family is a generalized surface quasi- geostrophic (SQG) equation with the velocity field u related to the scalar by u D r ? ƒ ˇ 2 , where 1 < ˇ 2 and ƒ D ./ 1=2 is the Zygmund operator. The borderline case ˇ D 1 corresponds to the SQG equation and the situation is more singular for ˇ > 1. We obtain the local existence and uniqueness of classical solutions, the global existence of weak solutions, and the local exis- tence of patch-type solutions. The second family is a dissipative active scalar equation with u D r ? .log.I // for > 0, which is at least logarithmi- cally 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 to- wards resolving the global regularity issue recently proposed by K. Ohkitani.|
|Electronic Publication Date:||23-Feb-2012|
|Citation:||Chae, Dongho, Constantin, Peter, Cordoba, Diego, Gancedo, Francisco, Wu, Jiahong. (2012). Generalized surface quasi-geostrophic equations with singular velocities. COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, 65 (1037 - 1066. doi:10.1002/cpa.21390|
|Pages:||1037 - 1066|
|Type of Material:||Journal Article|
|Journal/Proceeding Title:||COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS|
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