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.

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