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arXiv 1503.04323math.AP
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三维斯托克斯方程在$\dot H^{3/2}$、$\dot H^{5/2}$和$\dot B^{5/2}_{2,1}$中爆破解的下界

Lower bounds on blowing-up solutions of the 3D Navier--Stokes equations in $\dot H^{3/2}$, $\dot H^{5/2}$, and $\dot B^{5/2}_{2,1}$

David S. McCormick, Eric J. Olson, James C. Robinson, Jose L. Rodrigo, Alejandro Vidal-Lopez, Yi Zhou

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摘要：

如果$u$是一个光滑的解决方案的Navier-斯托克斯方程的${\mathbb R}^3$与第一blowup时间$T$，我们证明下界$u$在Sobolev空间$\dot H^{3/2}$，$\dot H^{5/2}$，和Besov空间$\dot B^{5/2}_{2,1}$，最佳的爆破率：我们证明了强下界$\|u(t)\|_{\dot H^{3/2}}\ge c(T-t)^{-1/2}$和$\|u(t)\|_{\dot B^{5/2}_{2,1}}\ge c(T-t)^{-1}$，但在$\dot H^{5/2}$中，我们只得到较弱的结果$\limsup_{t\to T^-}(T-t)\|u(t)\|_{\dot H^{5/2}}\ge c$。证明涉及新的不等式的非线性项在Sobolev和Besov空间，这两个都是使用$u$的并矢分解。

英文摘要：

If $u$ is a smooth solution of the Navier--Stokes equations on ${\mathbb R}^3$ with first blowup time $T$, we prove lower bounds for $u$ in the Sobolev spaces $\dot H^{3/2}$, $\dot H^{5/2}$, and the Besov space $\dot B^{5/2}_{2,1}$, with optimal rates of blowup: we prove the strong lower bounds $\|u(t)\|_{\dot H^{3/2}}\ge c(T-t)^{-1/2}$ and $\|u(t)\|_{\dot B^{5/2}_{2,1}}\ge c(T-t)^{-1}$, but in $\dot H^{5/2}$ we only obtain the weaker result $\limsup_{t\to T^-}(T-t)\|u(t)\|_{\dot H^{5/2}}\ge c$. The proofs involve new inequalities for the nonlinear term in Sobolev and Besov spaces, both of which are obtained using a dyadic decomposition of $u$.

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