TY - JOUR

T1 - On the global regularity for the supercritical SQG equation

AU - Zelati, Michele Coti

AU - Vicol, Vlad

N1 - Funding Information:
The work of V. V. was in part supported by the NSF grant DMS-1348193 and by an Alfred P. Sloan fellowship.
Publisher Copyright:
© Indiana University Mathematics Journal.

PY - 2016

Y1 - 2016

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∥H2γ/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∥H2γ/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

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U2 - 10.1512/iumj.2016.65.5807

DO - 10.1512/iumj.2016.65.5807

M3 - Article

AN - SCOPUS:84965036799

SN - 0022-2518

VL - 65

SP - 535

EP - 552

JO - Indiana University Mathematics Journal

JF - Indiana University Mathematics Journal

IS - 2

ER -