### 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) |
---|---|

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 |

### Fingerprint

### 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

**On the global regularity for the supercritical SQG equation.** / Zelati, Michele Coti; Vicol, Vlad.

Research output: Contribution to journal › Article

*Indiana University Mathematics Journal*, vol. 65, no. 2, pp. 535-552. https://doi.org/10.1512/iumj.2016.65.5807

}

TY - JOUR

T1 - On the global regularity for the supercritical SQG equation

AU - Zelati, Michele Coti

AU - Vicol, Vlad

PY - 2016/1/1

Y1 - 2016/1/1

N2 - We consider the initial value problem for the fractionally dissipative quasi-geostrophic equation (Equation presented) on T2 = [0,1]2 with γ ∈ (0,1). The coefficient of the dissipative term Λγ = (-Δ)γ/2 is normalized to 1. We show that, given a smooth initial datum with ∥θ0∥L2γ/2 ∥θ0∥H 2γ/2 ≤ R, where R is arbitrarily large, there exists γ1 = γ1(R) ∈ (0,1) such that, for γ ≥ γ1, the solution of the supercritical SQG equation with dissipation λγ 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.

AB - We consider the initial value problem for the fractionally dissipative quasi-geostrophic equation (Equation presented) on T2 = [0,1]2 with γ ∈ (0,1). The coefficient of the dissipative term Λγ = (-Δ)γ/2 is normalized to 1. We show that, given a smooth initial datum with ∥θ0∥L2γ/2 ∥θ0∥H 2γ/2 ≤ R, where R is arbitrarily large, there exists γ1 = γ1(R) ∈ (0,1) such that, for γ ≥ γ1, the solution of the supercritical SQG equation with dissipation λγ 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.

KW - Eventual regularity

KW - Global regularity

KW - Lower bounds for fractional laplacian

KW - Supercritical SQG

UR - http://www.scopus.com/inward/record.url?scp=84965036799&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84965036799&partnerID=8YFLogxK

U2 - 10.1512/iumj.2016.65.5807

DO - 10.1512/iumj.2016.65.5807

M3 - Article

VL - 65

SP - 535

EP - 552

JO - Indiana University Mathematics Journal

JF - Indiana University Mathematics Journal

SN - 0022-2518

IS - 2

ER -