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Generalized surface quasi-geostrophic equations with singular velocities. (English) Zbl 1244.35108
Summary: 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 $u=\nabla^{\bot}\Lambda^{\beta -2}\theta$, where $1<\beta \leq 2$ 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 $u=\nabla^{\bot}(\log (I-\delta ))^{\mu}\theta$ 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 {\it K. Ohkitani} [Dissipative and ideal surface quasi-geostrophic equations. Lecture presented at ICMS, Edinburgh (2010)].

35Q35PDEs in connection with fluid mechanics
35D30Weak solutions of PDE
35A02Uniqueness problems for PDE: global uniqueness, local uniqueness, non-uniqueness
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