论文标题
$ \ mathbb {r}^3 $中Navier-Stokes方程的替代定理
Alternative Theorem of Navier-Stokes Equations in $\mathbb{R}^3$
论文作者
论文摘要
我们考虑了不可压缩的Navier-Stokes方程的Cauchy问题，其中具有初始数据$ u_0 \ in L^1（\ Mathbb {r}^3）\ Cap l^{\ cap l^{\ infty}（\ Mathbb {r}^3）$。存在最大的时间间隔$ [0，t_ {max}）$和一个唯一的解决方案$ u \ c \ big（[[0，t_ {max}）; l^2（\ mathbb {r}^3）\ cap l^p（\ mathbb {r}^3）\ big）$（$ \ forall p> 3 $）。我们找到一个函数类$ s_ {常规} $通过缩放不变标准对定义的$ s_ {常规} $，以便$ t_ {max} = \ infty $提供了$ u_0 \ in S_ {常规} $。尤其是，对于任何$ u_0 \，$ \ | U_0 \ | _ {l^p} $对于任何$ u_0 \ in s_ {prorival} $和$ p> 3 $都是任意的。另一方面，证明了替代定理。是$ t_ {max} = \ infty $或$ t_ {max} \ in（t_l，t_r] $。尤其是，$ t_r <t_t_ {max} <\ infty $正在消失。在这里，$ t_l $和$ t_r $的明显表达方式。 $ t_ {max} = \ infty $，解决方案$ u $对于任何$（t，x）\ in（0，\ infty）\ times \ times \ times \ mathbb {r}^3 $ as $ t \ rightArlow \ rightarrow \ infty \ infty $。 （T_L，T_R] $。
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 $T_{max}\in (T_l,T_r]$.
