where u : I × R n → R n for some time strip I = [0, T ), ν > 0 is a constant due to the viscosity of the fluid, p : I × R n → R n denotes the fluid pressure, and u 0 : R n → R n . The requisite differential operators are defined by △ = n i=1
∂ 2 ∂ 2 x i and ∇ = ∂ ∂x i
, ..., ∂ ∂x n . In dimension n = 2, local and global existence of solutions to the Navier-Stokes equation are well known (see [9]; for a more modern reference, see Chapter 17 of [15]). For dimension n ≥ 3, the problem is significantly more complicated. There is a robust collection of local existence results, including [5], in which Kato proves the existence of local solutions to the Navier-Stokes equation with initial **data** in L n (R n ); [7], where Kato and Ponce solve the equation with initial **data** in the Sobolev space H n/p−1,p (R n ); and [8], where Koch and Tataru establish local existence with initial **data** in the space BMO −1 (R n ) (for a more complete accounting of local existence theory for the Navier-Stokes equation, see [10]). In all of these local results, if the initial datum is assumed to be sufficiently small, then the local solution can be extended to a global solution. However, the issue of global existence of solutions to the Navier-Stokes equation in dimension n ≥ 3 for arbitrary initial **data** is one of the most challenging open problems remaining in analysis.