Blow-up of dynamically restricted critical norms near a potential Navier–Stokes singularity

Abstract

In this paper we develop new methods to obtain regularity criteria for the three-dimensional Navier–Stokes equations in terms of dynamically restricted endpoint critical norms: the critical Lebesgue norm in general or the critical weak Lebesgue norm in the axisymmetric case. This type of results is inspired in particular by a work of Neustupa (Arch Ration Mech Anal 214(2):525–544, 2014), which handles certain non endpoint critical norms. Our work enables to have a better understanding of the nonlocal effect of the pressure on the regularity of the solutions.

Appendix A. Known results about Lorentz spaces

Given a measurable subset $\Omega \subseteq \mathbb {R}^{d}$, let us define the Lorentz spaces. For a measurable function $f:\Omega \rightarrow \mathbb {R}$ define:

$$\begin{aligned} d_{f,\Omega }(\alpha ):=\mu (\{x\in \Omega : |f(x)|>\alpha \}), \end{aligned}$$

(A.1)

where $\mu$ denotes the Lebesgue measure. The Lorentz space $L^{p,q}(\Omega )$, with $p\in [1,\infty )$, $q\in [1,\infty ]$, is the set of all measurable functions g on $\Omega$ such that the quasinorm $\Vert g\Vert _{L^{p,q}(\Omega )}$ is finite. Here:

$$\begin{aligned}{} & {} \Vert g\Vert _{L^{p,q}(\Omega )}:= \Bigg (p\int \limits _{0}^{\infty }\alpha ^{q}d_{g,\Omega }(\alpha )^{\frac{q}{p}}\frac{d\alpha }{\alpha }\Bigg )^{\frac{1}{q}}, \end{aligned}$$

(A.2)

$$\begin{aligned}{} & {} \Vert g\Vert _{L^{p,\infty }(\Omega )}:= \sup _{\alpha >0}\alpha d_{g,\Omega }(\alpha )^{\frac{1}{p}}. \end{aligned}$$

(A.3)

It is known there exists a norm, which is equivalent to the quasinorm defined above, for which $L^{p,q}(\Omega )$ is a Banach space. For $p\in [1,\infty )$ and $1\le q_{1}< q_{2}\le \infty$, we have the following continuous embeddings

$$\begin{aligned} L^{p,q_1}(\Omega ) \hookrightarrow L^{p,q_2}(\Omega ) \end{aligned}$$

(A.4)

and the inclusion is known to be strict.

Our main tool in this paper is the following Hölder’s inequality for Lorentz spaces. The statement below and proof can be found in Hunt’s paper [17, Theorem 4.5, p. 271]; see also [26, Theorems 3.4$-$3.5, page 141].

Proposition A.1

Suppose that $1\le p,q,r\le \infty$ and $1\le s_1,s_2\le \infty$. Furthermore, suppose that p, q, r, $s_1$ and $s_2$ satisfy the following relations:

$$\begin{aligned} \frac{1}{p}+\frac{1}{q}=\frac{1}{r} \end{aligned}$$

and

$$\begin{aligned} \frac{1}{s_1}+\frac{1}{s_2}=\frac{1}{s}. \end{aligned}$$

Then the assumption that $f\in L^{p,s_1}(\Omega )$ and $g\in L^{q,s_2}(\Omega )$ imply that $fg \in L^{r,s}(\Omega )$, with the estimate

$$\begin{aligned} \Vert fg\Vert _{L^{r,s}(\Omega )}\le C(p,q,s_1,s_2)\Vert f\Vert _{L^{p,s_1}(\Omega )}\Vert g\Vert _{L^{q,s_2}(\Omega )}. \end{aligned}$$

(A.5)