Weighted energy estimates for the incompressible Navier-Stokes equations and applications to axisymmetric solutions without swirl

TL;DR

This paper develops weighted energy estimates for the 3D incompressible Navier-Stokes equations, enabling the existence of global regular solutions for axisymmetric initial data without swirl in weighted L2 spaces.

Contribution

It introduces a new class of weights to extend the Leray approach, proving global regularity for axisymmetric solutions without swirl with initial data in weighted L2 spaces.

Findings

Existence of global regular solutions under weighted initial data conditions.

Extension of Leray's method using weighted energy estimates.

Applicable to axisymmetric flows without swirl in weighted spaces.

Abstract

We consider a family of weights which permit to generalize the Leray procedure to obtain weak suitable solutions of the 3D incom-pressible Navier-Stokes equations with initial data in weighted L 2 spaces. Our principal result concerns the existence of regular global solutions when the initial velocity is an axisymmetric vector field without swirl such that both the initial velocity and its vorticity belong to L 2 ((1 + r 2) -- $\gamma$ 2 dx), with r = x 2 1 + x 2 2 and $\gamma$ $\in$ (0, 2).