A profile decomposition approach to the $L^\infty _t(L^{3}_x)$ Navier–Stokes regularity criterion

Abstract

In this paper we continue to develop an alternative viewpoint on recent studies of Navier–Stokes regularity in critical spaces, a program which was started in the recent work by Kenig and Koch (Ann Inst H Poincaré Anal Non Linéaire 28(2):159–187, 2011). Specifically, we prove that strong solutions which remain bounded in the space ${L^3(\mathbb R ^3)}$ do not become singular in finite time, a known result established by Escauriaza et al. (Uspekhi Mat Nauk 58(2(350)):3–44, 2003) in the context of suitable weak solutions. Here, we use the method of “critical elements” which was recently developed by Kenig and Merle to treat critical dispersive equations. Our main tool is a “profile decomposition” for the Navier–Stokes equations in critical Besov spaces which we develop here. As a byproduct of this tool, assuming a singularity-producing initial datum for Navier–Stokes exists in a critical Lebesgue or Besov space, we show there is one with minimal norm, generalizing a result of Rusin and Sverak (J Funct Anal 260(3):879–891, 2011).

Appendix: A perturbation result

Let us state (without proof) a perturbation result for the $d$-dimensional Navier–Stokes system.

Proposition 6.1

Let $s_p=-1+\frac{d}{p}$, $r\in [1,\frac{2p}{p+1}]$ and define $s:=s_p+\frac{2}{r}$. Assume finally that $p<2d+3$. There are constants $\varepsilon _0$ and $C$ such that the following holds. Let $w_0 \in \dot{B}^{s_p}_{p,p}$, $f \in F:={\mathcal L }^r([0,T];\dot{B}^{s-2}_{p,p})+{\mathcal L }^{\frac{2p}{p+1}}([0,T];\dot{B}^{s_p-1+\frac{1}{p}}_{p,q})$ and $v \in D:=\mathcal L ^p([0,T];\dot{B}^{s_p+\frac{2}{p}}_{p,p})$ be given, with

$$\begin{aligned} \Vert w_0\Vert _{\dot{B}^{s_p}_{p,p}} +\Vert {f}\Vert _{F} \le \varepsilon _0 \exp \left( - C \Vert v\Vert _{D} \right)\!. \end{aligned}$$

Suppose moreover that $\text{ div} \, v=0$, and let $w$ be a solution of

$$\begin{aligned} \partial _t w-\Delta w+w\cdot \nabla w +v \cdot \nabla w +w \cdot \nabla v =-\nabla \pi + f \end{aligned}$$

with $\text{ div} \, w=0.$ Then $w$ belongs to $E_{p,p}(T)$ and the following estimate holds:

$$\begin{aligned} \Vert w\Vert _{E_{p,p}(T)}\lesssim (\Vert w_0\Vert _{\dot{B}^{s_p}_{p,q}} + \Vert {f}\Vert _F) \exp C \Vert {v}\Vert _D. \end{aligned}$$

The proof of that proposition follows the estimates of [9] (see in particular Propositions 4.1 and Theorem 3.1 of [9]). The two main differences are

• the absence of an exterior force in [9], but that force is added with no difficulty to the estimates;

• the rather weak estimate on the drift term $v$, which accounts for the restricted numerology on time exponents in the definition of $E_{p,p}$. The reader should note that closing estimates on $w$ in our setting amounts to doing again the same estimates that were done in the proof of Lemma 3.7.