Dynamical behavior for the solutions of the Navier-Stokes equation

Abstract

We study several quantitative properties of solutions to the incompressible Navier-Stokes equation in three and higher dimensions:$ \begin{align} u_t -Δ u+u· \nabla u +\nabla p = 0, \ \ {\rm div} u = 0, \ \ u(0, x) = u_0(x). \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( 1 \right)\label{NSa} \end{align}$More precisely, for the blow up mild solutions with initial data in $L^{∞}(\mathbb{R}^d)$ and $H^{d/2 -1}(\mathbb{R}^d)$, we obtain a concentration phenomenon and blowup profile decomposition respectively. On the other hand, if the Fourier support has the form ${\rm supp} \ \widehat{u_0} \subset \{ξ∈ \mathbb{R}^n: ξ_1≥ L \}$ and $ \|u_0\|_{∞} \ll L$ for some $L >0$, then (1) has a unique global solution $u∈ C(\mathbb{R}_+, L^∞)$. 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 $L^p({{\mathbb{R}}^{3}})$ blows up in a Type-Ⅰ manner, we prove the existence of a blowup solution which is uniformly bounded in critical Besov spaces $\dot B^{-1+6/p}_{p/2, ∞}({{\mathbb{R}}^{3}})$.