TY - JOUR
T1 - On the global regularity for the supercritical SQG equation
AU - Zelati, Michele Coti
AU - Vicol, Vlad
N1 - 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 -