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Generalized surface quasi-geostrophic equations with singular velocities

Opened Access Generalized surface quasi-geostrophic equations with singular velocities

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Autor: Chae, Dongho
Constantin, Peter
Córdoba Gazolaz, Diego
Gancedo García, Francisco
Wu, Jiahong
Departamento: Universidad de Sevilla. Departamento de Análisis Matemático
Fecha: 2012
Publicado en: Communications on Pure and Applied Mathematics, 65 (8), 1037-1066.
Tipo de documento: Artículo
Resumen: 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 quasi-geostrophic (SQG) equation with the velocity field u related to the scalar θ by u = ∇⊥Λ β−2 θ, where 1 < β ≤ 2 and Λ = (−∆)1/2 is the Zygmund operator. The borderline case β = 1 corresponds to the SQG equation and the situation is more singular for β > 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 = ∇⊥(log(I − ∆))µθ for µ > 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 const...
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Cita: Chae, D., Constantin, P., Córdoba Gazolaz, D., Gancedo García, F. y Wu, J. (2012). Generalized surface quasi-geostrophic equations with singular velocities. Communications on Pure and Applied Mathematics, 65 (8), 1037-1066.
Tamaño: 307.9Kb
Formato: PDF

URI: http://hdl.handle.net/11441/45197

DOI: 10.1002/cpa.21390

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