Alternative Theorem of Navier-Stokes Equations in $\mathbb{R}^3$

TL;DR

This paper establishes an alternative regularity criterion for the Navier-Stokes equations in three dimensions, identifying conditions under which solutions are global or blow up, with explicit bounds and computable criteria.

Contribution

It introduces a new regularity criterion based on a scaling-invariant norm class, providing explicit bounds for maximal existence time and a verifiable alternative theorem.

Findings

Existence of a unique solution for initial data in certain function classes.

An alternative theorem characterizing finite or infinite maximal existence time.

Explicit expressions for bounds on blow-up time and decay rates.

Abstract

We consider Cauchy problem of the incompressible Navier-Stokes equations with initial data $u_0\in L^1(\mathbb{R}^3)\cap L^{\infty}(\mathbb{R}^3)$. There exist a maximum time interval $[0,T_{max})$ and a unique solution $u\in C\big([0,T_{max}); L^2(\mathbb{R}^3) \cap L^p(\mathbb{R}^3)\big)$ ($\forall p>3$). We find one of function class $S_{regular}$ defined by scaling invariant norm pair such that $T_{max}=\infty$ provided $u_0\in S_{regular}$. Especially, $\|u_0\|_{L^p}$ is arbitrarily large for any $u_0\in S_{regular}$ and $p>3$. On the other hand, the alternative theorem is proved. It is that either $T_{max}= \infty$ or $T_{max}\in(T_l,T_r]$. Especially, $T_r<T_{max}<\infty$ is disappearing. Here the explicit expressions of $T_l$ and $T_r$ are given. This alternative theorem is one kind of regular criterion which can be verified by computer. If $T_{max}=\infty$, the solution $u$ is regular for any $(t,x)\in(0,\infty) \times \mathbb{R}^3$. As $t\rightarrow\infty$, the solution is decay. On the other hand, lower bound of blow up rate of $u$ is obtained again provided $T_{max}\in (T_l,T_r]$.