Dynamical Behavior for the Solutions of the Navier-Stokes Equation

Abstract

We study the Cauchy problem for the incompressible Navier-Stokes equations (NS) in three and higher spatial dimensions: \begin{align} u_t -Δu+u\cdot \nabla u +\nabla p=0, \ \ {\rm div} u=0, \ \ u(0,x)= u_0(x). \label{NSa} \end{align} Leray and Giga obtained that for the weak and mild solutions $u$ of NS in $L^p({\mathbb{R}}^d)$ which blow up at finite time $T>0$, respectively, one has that for $d<p\le \infty$, $$\Vert u(t){\Vert }_p\gtrsim (T-t{)}^{-(1-d/p)/2},0<t<T.$$ We will obtain the blowup profile and the concentration phenomena in $L^p({\mathbb{R}}^d)$ with $d\le p\le \infty$ for the blowup mild solution. On the other hand, if the Fourier support has the form $\mathrm{supp}\widehat{u_0}\subset \{\xi \in {\mathbb{R}}^n:{\xi }_1\ge L\}$ and $\Vert u_0{\Vert }_{\infty }\ll L$ for some $L>0$, then \eqref{NSa} has a unique global solution $u\in C({\mathbb{R}}_{+},L^{\infty })$. Finally, if the blowup rate is of type I: $$\Vert u(t){\Vert }_p\sim (T-t{)}^{-(1-d/p)/2},for0<t<T<\infty ,d<p<\infty$$ in 3 dimensional case, then we can obtain a minimal blowup solution $\Phi$ for which $$ \inf \{\limsup_{t \to T}(T-t)^{(1-3/p)/2}\|u(t)\|_{L^p_x}: \ u\in C([0,T); L^p) \mbox{\ solves \eqref{NSa}}\} $$ is attainable at some $\Phi \in L^{\infty }(0,T;{\dot{B}}_{p/2,\infty }^{-1+6/p})$.

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