## Finite blow up of NS solution

$\displaystyle \begin{cases}u_t+u\cdot\nabla u-\nu\Delta u+\nabla p=0\\div u=0\\u(x,0)=u_0\end{cases}\text{ on }\mathbb{R}^3$

Thm: If NS 3-D has an initial data leading to an infinite time blow up then it also has an initial data leading to finite blow up.

Proof: Suppose ${u\in C([0,\infty);W^{1,2})}$ is a weak solution of NS 3-D. Assume ${u}$ blows up at infinity, then ${\exists\,t_n\rightarrow \infty}$, ${||u(t_n)||_{H^1}\rightarrow \infty}$. NS equation implies

$\displaystyle \frac{1}{2}\frac{d}{dt}|u|_{L^2}^2+|u|_{\dot{H}^1}^2\leq 0$

where ${\dot{H}^1}$ is the homogeneous ${H^1}$ space, ${|u|_{\dot{H}^1}^2=\int |\nabla u|^2dx}$. Denote ${E=\int_{\mathbb{R}^3}|u_0|^2dx}$, then

$\displaystyle |u|_{L^2}^2\leq E$

$\displaystyle \int_t^{t+1}|u|_{\dot{H}^1}^2\leq 2E\quad \forall\, t>0$

Then ${\exists\, \tau_n\in (t_n-1,t_n)}$ such that ${|u(\tau_n)|_{\dot{H}^1}\leq \sqrt{2E}}$. Consider ${\tilde{u}(t)=u(t+\tau_n)}$, note that ${|\tilde{u}_n(0)|_{H^1}\leq |u(\tau_n)|_{H^1}\leq C}$. Then there exists subsequence converging weakly in ${H^1}$, suppose ${\tilde{u}_n(0)\rightarrow \tilde{u}_0}$ in ${L^2}$ and converge weakly in ${H^1}$.

Claim: Start NS with initial data ${\tilde{u}_0}$, the solution must blow up in finite time. Suppose not. Let ${v}$ be the solution with initial data ${\tilde{u}_0}$ and it is nice in ${[0,1]}$. Let ${w=v-\tilde{u}_n}$, then

$\displaystyle \begin{cases}w_t-\nu\Delta w+v\cdot\nabla w+w\cdot\nabla v+w\cdot\nabla w+\nabla p=0 \\ div w=0\\w(x,0)\text{ small in } L^2\end{cases}\\$

This equation implies

$\displaystyle \frac{1}{2}\frac{d}{dt}|w|_{L^2}^2+|w|_{\dot{H}^1}^2\leq C|w|^2_{L^2}$

Since ${|w|_{L^2}}$ can arbitrarily small, we can find ${t_0\in (0,1)}$ such that ${|w(0)|_{{H}^1}^2\ll 1}$. Multiplying the equation of ${w}$ by ${\Delta w}$, one can get

$\displaystyle \frac{d}{dt}|w|_{H^1}^2\leq C(|u|^3_{H^1}+|u|^2_{H^1})$

This implies ${|w(t)|_{H^1}\ll 1}$ for ${t\in [t_0,1]}$. This contradicts ${|u(t_n)|_{H^1}\rightarrow \infty}$. $\Box$

Remark: Follows from my note of lectures given by Peter Constantin. As he said, this proof is done by Foias and Temam.