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

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 $T_{max}\in (T_l,T_r]$.