  Based on Leray's formulation of the Navier-Stokes equations and the
conditions of the exact linear representation of the nonlinear problem found in
this paper, a compact explicit expression for the exact operator solution of
the Navier-Stokes equations is given. It is shown that the introduced linear
operator for Leray's equations is the generator of one-parameter contraction
semigroup. This semigroup yields the existence of a unique and smooth classical
solution of the associated Cauchy problem of Navier-Stokes equations in space
$\mathbb{R}^3$ under smooth initial conditions.
