  As one of the seven open problems in the addendum to their 1989 book
"Computability in Analysis and Physics", Pour-El and Richards proposed ``...
the recursion theoretic study of particular nonlinear problems of classical
importance. Examples are the Navier-Stokes equation, the KdV equation, and the
complex of problems associated with Feigenbaum's constant.'' In this paper, we
approach the question of whether the Navier-Stokes Equation admits recursive
solutions in the sense of Weihrauch's Type-2 Theory of Effectivity. A natural
encoding (``representation'') is constructed for the space of divergence-free
vector fields on 2-dimensional open square $\Omega = (-1, 1)^2$. This
representation is shown to render first the mild solution to the Stokes
Dirichlet problem and then a strong local solution to the nonlinear
inhomogeneous incompressible Navier-Stokes initial value problem uniformly
computable. Based on classical approaches, the proofs make use of many subtle
and intricate estimates which are developed in the paper for establishing the
computability results.
