### Abstract

We consider the initial value problem for the fractionally dissipative quasi-geostrophic equation (Equation presented) on T^{2} = [0,1]^{2} with γ ∈ (0,1). The coefficient of the dissipative term Λ^{γ} = (-Δ)^{γ/2} is normalized to 1. We show that, given a smooth initial datum with ∥θ_{0}∥_{L}2^{γ/2} ∥θ0∥_{H}^{2}γ/2 ≤ R, where R is arbitrarily large, there exists γ_{1} = γ_{1}(R) ∈ (0,1) such that, for γ ≥ γ_{1}, the solution of the supercritical SQG equation with dissipation λ^{γ} does not blow up in finite time. The main ingredient in the proof is a new concise proof of eventual regularity for the supercritical SQG equation, which relies solely on nonlinear lower bounds for the fractional Laplacian and the maximum principle.

Original language | English (US) |
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Pages (from-to) | 535-552 |

Number of pages | 18 |

Journal | Indiana University Mathematics Journal |

Volume | 65 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 2016 |

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### Keywords

- Eventual regularity
- Global regularity
- Lower bounds for fractional laplacian
- Supercritical SQG

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Indiana University Mathematics Journal*,

*65*(2), 535-552. https://doi.org/10.1512/iumj.2016.65.5807