  We develop a new approach for regularity estimates, especially vorticity
estimates, of solutions of the three-dimensional Navier-Stokes equations with
periodic initial data, by exploiting carefully formulated linearized vorticity
equations. An appealing feature of the linearized vorticity equations is the
inheritance of the divergence-free property of solutions, so that it can
intrinsically be employed to construct and estimate solutions of the
Navier-Stokes equations. New regularity estimates of strong solutions of the
three-dimensional Navier-Stokes equations are obtained by deriving new explicit
a priori estimates for the heat kernel (i.e., the fundamental solution) of the
corresponding heterogeneous drift-diffusion operator. These new a priori
estimates are derived by using various functional integral representations of
the heat kernel in terms of the associated diffusion processes and their
conditional laws, including a Bismut-type formula for the gradient of the heat
kernel. Then the a priori estimates of solutions of the linearized vorticity
equations are established by employing a Feynman-Kac-type formula. The
existence of strong solutions and their regularity estimates up to a time
proportional to the reciprocal of the square of the maximum initial vorticity
are established. All the estimates established in this paper contain known
constants that can be explicitly computed.
