Generalized surface quasi‐geostrophic equations with singular velocities |
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Authors: | Dongho Chae Peter Constantin Diego Córdoba Francisco Gancedo Jiahong Wu |
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Institution: | 1. Department of Mathematics, Chung‐Ang University, Dongjak‐gu Heukseok‐ro 84, Seoul 156‐756, Korea;2. Department of Mathematics, Program in Applied and Computational Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544‐1000;3. Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones, Científicas C. Nicolás Cabrera, 13‐15, Campus Cantoblanco UAM, 28049, Spain;4. Departamento de Análisis Matemático, Universidad de Sevilla, 41012 Sevilla, Spain;5. Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK 74078 |
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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 θ by $u=\nabla^\perp\Lambda^{\beta-2}\theta$ , where $1<\beta\le 2$ and $\Lambda=(-\Delta)^{1/2}$ 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 $u=\nabla^\perp (\log(I-\Delta))^\mu\theta\ {\rm for}\ \mu>0$ , 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. © 2012 Wiley Periodicals, Inc. |
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