Remarks on the global large solution to the three-dimensional incompressible Navier-Stokes equations

TL;DR

This paper establishes a new smallness condition on initial data for the 3D incompressible Navier-Stokes equations, ensuring global solutions, and constructs solutions with arbitrarily large norms in a specific Besov space.

Contribution

It introduces a novel smallness hypothesis involving Besov space norms that guarantees global solutions to the Navier-Stokes equations, expanding understanding of initial data conditions.

Findings

Existence of a new smallness condition for initial data.

Proof of global unique solutions under this condition.

Construction of solutions with large Besov norms.

Abstract

In this paper, we derive a new smallness hypothesis of initial data for the three-dimensional incompressible Navier-Stokes equations. That is, we prove that there exist two positive constants $c_0,C_0$ such that if \begin{equation*} \|u_0^1+u^2_0,u^3_0\|_{\dot{B}_{p,1}^{-1+\frac{3}{p}}} \|u^1_0,u^2_0\|_{\dot{B}_{p,1}^{-1+\frac{3}{p}}} \exp\{C_0 (\|u_0\|^{2}_{\dot{B}_{\infty,2}^{-1}}+\|u_0\|_{\dot{B}_{\infty,1}^{-1}})\} \leq c_0, \end{equation*} then \eqref{NS} has a unique global solution. As an application we construct two family of smooth solutions to the Navier-Stokes equations whose $\B^{-1}_{\infty,\infty}(\mathbb{R}^3)$ norm can be arbitrarily large.