Global wellposedness for a certain class of large initial data for the 3D Navier-Stokes Equations

TL;DR

This paper proves global well-posedness for a special class of large initial data in the 3D Navier-Stokes equations on the torus, using orthogonality conditions in the initial data components.

Contribution

It introduces a new class of initial data with orthogonality properties that guarantees global solutions for the 3D Navier-Stokes equations, even with large initial norms.

Findings

Global well-posedness established for the special initial data class.

Existence of large initial data satisfying the orthogonality conditions.

Solutions persist for all time under the specified conditions.

Abstract

In this article, we consider a special class of initial data to the 3D Navier-Stokes equations on the torus, in which there is a certain degree of orthogonality in the components of the initial data. We showed that, under such conditions, the Navier-Stokes equations are globally wellposed. We also showed that there exists large initial data, in the sense of the critical norm $B^{-1}_{\infty,\infty}$ that satisfies the conditions that we considered.