Dynamical behavior for the solutions of the Navier-Stokes equation

Kuijie Li; Tohru Ozawa; Baoxiang Wang

doi:10.3934/cpaa.2018073

Article Contents

2018, Volume 17, Issue 4: 1511-1560. Doi: 10.3934/cpaa.2018073

This issue Previous Article Blow up for initial boundary value problem of critical semilinear wave equation in two space dimensions Next Article On special regularity properties of solutions of the Zakharov-Kuznetsov equation

Dynamical behavior for the solutions of the Navier-Stokes equation

1.
LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China
2.
Department of Applied Physics, Waseda University, Tokyo 169-8555, Japan

^* Corresponding author: Baoxiang Wang
^* Corresponding author: Baoxiang Wang

Received: July 31, 2016

Revised: March 31, 2017

Published: April 2018

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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 $\dot B^{-1+6/p}_{p/2, ∞}({{\mathbb{R}}^{3}})$.

Keywords:

Navier-Stokes equation,
concentration phenomena,
blowup profile,
Type-Ⅰ blowup solution,
$L^p$-minimal singularity-generating data.

Mathematics Subject Classification: Primary: 35Q30; Secondary: 76D03.

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