### Abstract

Any classical solution of the two-dimensional incompressible Euler equation is global in time. However, it remains an outstanding open problem whether classical solutions of the surface quasi-geostrophic (SQG) equation preserve their regularity for all time. This paper studies solutions of a family of active scalar equations in which each component u_{j} of the velocity field u is determined by the scalar θ through u_{j} = RΛ^{-1} P(Λ)θ, where R is a Riesz transform and Λ = (-Δ)^{1/2}. The two-dimensional Euler vorticity equation corresponds to the special case P(Λ) = I while the SQG equation corresponds to the case P(Λ) = Λ. We develop tools to bound {double pipe}▽u{double pipe}L^{∞} for a general class of operators P and establish the global regularity for the Loglog-Euler equation for which P(Λ) = (log(I + log(I - Δ)))^{γ} with 0 ≦ γ ≦ 1. In addition, a regularity criterion for the model corresponding to P(Λ) = Λ^{β} with 0 ≦ β ≦ 1 is also obtained.

Original language | English (US) |
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Pages (from-to) | 35-62 |

Number of pages | 28 |

Journal | Archive for Rational Mechanics and Analysis |

Volume | 202 |

Issue number | 1 |

DOIs | |

State | Published - Oct 1 2011 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Analysis
- Mathematics (miscellaneous)
- Mechanical Engineering