On Green's function of the vorticity formulation for the 3D Navier-Stokes equations

TL;DR

This paper introduces a new vorticity formulation for the 3D Navier-Stokes equations, derives Green's function bounds using resolvent methods, and extends results to the Stokes problem with mixed boundary conditions.

Contribution

It presents a novel vorticity formulation and Green's function estimates for the 3D Navier-Stokes and Stokes problems, extending previous results to more general boundary conditions.

Findings

Established Green's function bounds for the 3D Navier-Stokes equations.

Derived similar results for the Stokes problem with mixed boundary conditions.

Provided a resolvent-based approach for analyzing the vorticity formulation.

Abstract

We give a novel vorticity formulation for the 3D Navier-Stokes equations with Dirichlet boundary conditions. Via a resolvent argument, we obtain Green's function and establish an upper bound, which is the 3D analog of [24]. Moreover, we prove similar results for the corresponding Stokes problem with more general mixed boundary conditions.