Global smooth solutions to Navier-Stokes equations with large initial data in critical space

TL;DR

This work addresses global regularity for the 3D Navier–Stokes equations with large initial data in critical spaces. The authors decompose the solution into a linear heat-flow part $u_L=e^{t\Delta}u_0$ and a nonlinear remainder, and impose a nonlinear smallness condition on $U_0=-\nabla\cdot(u_L\otimes u_L)$ measured by a composite norm that combines $\dot{H}^{-1}$ and $L^2_tL^2_x$ controls. A bootstrap framework and robust nonlinear-term estimates bound the interaction $Nu=-\nabla\cdot(u\otimes u)$, together with an energy inequality for the corrected velocity $v$, to establish global existence and uniqueness. The result extends global well-posedness to arbitrarily large initial data in the Besov space $\dot{B}^{-1}_{\infty,\infty}$ under a nonlinear smallness premise, advancing critical-space theory for Navier–Stokes.

Abstract

In this paper, we investigate the existence of a unique global smooth solution to the three-dimensional incompressible Navier-Stokes equations and provide a concise proof. We establish a new global well-posedness result that allows the initial data to be arbitrarily large within the critical space $\dot{B}^{-1}_{\infty,\infty}$, while still satisfying the nonlinear smallness condition.

Theorems & Definitions (11)

• Theorem 1.1
• Remark 1.2
• Lemma 2.1
• proof
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3
• proof