New Approach for Vorticity Estimates of Solutions of the Navier-Stokes Equations

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.
Comment: 27 pages