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Title: Well-Posedness for Ohkitani Model and Long-Time Existence for Surface Quasi-geostrophic Equations

Abstract: We consider the Cauchy problem for the logarithmically singular surfacequasi-geostrophic (SQG) equation, introduced by Ohkitani, partial derivative(t)theta-del(perpendicular to)log(10 +(-Delta)(1/2))thetadel theta=0, and establish local existence and uniqueness of smooth solutions in the scale of Sobolev spaces with exponent decreasing with time. Such a decrease of the Sobolev exponent is necessary, as we have shown in the companion paper (Chae et al. in Illposed nessvia degenerate dispersion for generalized surface quasi-geostrophic equations with sin-gular velocities,arXiv:2308.02120) that the problem is strongly ill-posed in any fixed Sobolev spaces. The time dependence of the Sobolev exponent can be removed whenthere is a dissipation term strictly stronger than log. These results improve well posed-ness statements by Chae et al. (Comm Pure Appl Math 65(8):1037-1066, 2012). Thiswell-posedness result can be applied to describe the long-time dynamics of the delta-SQGequations, defined by partial derivative(t)theta-del(perpendicular to)log(10 +(-Delta)(1/2))thetadel theta=0, for all sufficiently small delta>0 depending on the size of the initial data. For the same range of delta, we establish global well-posedness of smooth solutions to the dissipative SQG equations.

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