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 $u\in C([0,T_{max});L^2({\mathbb{R}}^3)\cap L^p({\mathbb{R}}^3))$ ($\forall p>3$). 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, $\Vert u_0{\Vert }_{L^p}$ 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 $t\to \infty$, the solution is decay. On the other hand, lower bound of blow up rate of $u$ is obtained again provided $T_{max}\in (T_l,T_r]$.

There are errors which can not be corrected in the proof of main theorem

References in corpus (1)

• Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations II