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

TL;DR

This paper introduces a novel approach to estimate vorticity in solutions of the 3D Navier-Stokes equations, utilizing linearized vorticity equations and explicit heat kernel estimates to improve regularity analysis.

Contribution

It develops a new method leveraging linearized vorticity equations and explicit heat kernel bounds for better regularity estimates of Navier-Stokes solutions.

Findings

Established existence of strong solutions with regularity up to a time inversely proportional to initial vorticity squared.

Derived explicit a priori estimates for the heat kernel of the associated diffusion operator.

Provided computable constants in the regularity estimates.

Abstract

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.