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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 $u={\nabla }^{\perp }{{\Lambda }}^{\beta -2}\theta$, where $1<\beta \le 2$ and ${\Lambda }={\left(-{\Delta }\right)}^{1/2}$ 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 $u={\nabla }^{\perp }{\left(log\left(I-\delta \right)\right)}^{\mu }\theta$ 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 [Dissipative and ideal surface quasi-geostrophic equations. Lecture presented at ICMS, Edinburgh (2010)].
##### MSC:
 35Q35 PDEs in connection with fluid mechanics 35D30 Weak solutions of PDE 35A02 Uniqueness problems for PDE: global uniqueness, local uniqueness, non-uniqueness