In [18], Kazuo obtains a unique global solution to equation (1.2) in dimension three provided the initial **data** is in the Sobolev space H m,2 (R 3 ), with m > max{5/2, 1 + 2γ 1 }, provided γ 1 and γ 2 satisfy the inequality 2γ 1 + γ 2 ≥ 5 and that
(1.3) ∞ 1 ds sg 2 1 (s)g 2 (s) = ∞.
The goal of this paper is to obtain a much wider array of existence results, specifically existence results for initial **data** with low regularity and for initial **data** outside the L 2 setting. We will also, when applicable, use the energy bound from [18] to extend these local solutions to global solutions. Our plan is to follow the general contraction-mapping based procedure outlined by Kato and Ponce in [7] for the Navier-Stokes equation, with two key modification.