Dynamical behavior for the solutions of the Navier-Stokes equation

We study several quantitative properties of solutions to the incompressible Navier-Stokes equation in three and higher dimensions:$ {align} u_t -Δ u+u· ∇ u +∇ p = 0, \ \ { div} u = 0, \ \ u(0, x) = u_0(x). \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ( 1 ){NSa} {align}$More precisely, for the blow up mild solutions with initial data in $L∞(R^d)$ and $Hd/2 -1(R^d)$, we obtain a concentration phenomenon and blowup profile decomposition respectively. On the other hand, if the Fourier support has the form ${ supp} \ {u_0} ⊂ \{ξ∈ R^n: ξ_1≥ L \}$ and $ \|u_0\|∞ L$ for some $L >0$, then (1) has a unique global solution $u∈ C(R_+, L^∞)$. In 3D, we show the compactness of the set consisting of minimal-$L^p$ singularity-generating initial data with $3<p< ∞$, furthermore, if the mild solution with data in $L^p({{R}³})$ blows up in a Type-Ⅰ manner, we prove the existence of a blowup solution which is uniformly bounded in critical Besov spaces $ B-1+6/pp/2, ∞({{R}³})$.