On the Serrin-Type Condition on One Velocity Component for the Navier–Stokes Equations

Abstract

In this paper we consider the regularity problem of the Navier–Stokes equations in ${\mathbb {R}}^{3}$. We show that the Serrin-type condition imposed on one component of the velocity $u_3\in L^p(0,T; L^q({\mathbb {R}}^{3} ))$ with $\frac{2}{p}+ \frac{3}{q} <1$, $3<q \le +\infty$ implies the regularity of the weak Leray solution $u: {\mathbb {R}}^{3} \times (0,T) \rightarrow {\mathbb {R}}^{3}$, with the initial data belonging to $L^2({\mathbb {R}}^3) \cap L^3({\mathbb {R}}^{3})$. The result is an immediate consequence of a new local regularity criterion in terms of one velocity component for suitable weak solutions.

A Appendix

The aim of this appendix is to provide an estimate which will be used various times for the estimation of integrals involving the pressure during the proof of our main result. We start our discussion by defining a singular integral operator, which is necessary for the decomposition of the pressure. Throughout this appendix, let $0< R \le 1$ be fixed. Let $Q_0(R) = U_0(R) \times (-1,0)$, where $U_0(R) = B'(R)\times (-1,1)$. Given $f_{ ij}\in L^p(Q_0(R)), 1<p< +\infty , i,j=1,2,3,$ we define

$$\begin{aligned} {\mathscr {J}}(f)(x,t) = P.V. \int \nolimits _{ {\mathbb {R}}^{3} } K(x-y) :f(y,t) \chi _{ U_0(R)}(y)\mathrm{d}y,\quad (x,t)\in {\mathbb {R}}^{3}\times (-1,0), \end{aligned}$$

with the Calderón-Zygmund kernel $K_{ ij}= \partial _{ i}\partial _j N, i,j=1, 2,3,$ where

$$\begin{aligned} N(x) = \frac{1}{4\pi |x|},\quad x\in {\mathbb {R}}^{3} \setminus \{0\}. \end{aligned}$$

Clearly, by virtue of Calderón-Zygmund’s inequality, ${\mathscr {J}}: L^p(Q_0(R)) \rightarrow L^p({\mathbb {R}}^{3} \times (-1,0))$ defines a bounded linear operator. In particular,

$$\begin{aligned} \Vert {\mathscr {J}}(f)\Vert _{ L^p({\mathbb {R}}^{3} )} \le c \Vert f\Vert _{ L^p(U_0(R))}. \end{aligned}$$

(A.1)

Furthermore, setting $\pi _0= {\mathscr {J}}(f)$, it holds that

$$\begin{aligned} -\Delta \pi _0 = \nabla \cdot \nabla \cdot f\quad \text {in}\quad Q_0(R), \end{aligned}$$

(A.2)

in the sense of distributions. As in Section 2 we use the following notation:

$$\begin{aligned} r_j= 2^{ -j},\quad U_j(R) = B'(1)\times (-r_j, r_j),\quad Q_j(R) = U_j(R) \times (-r_j^2, 0)\quad j\in {\mathbb {N}}_0. \end{aligned}$$

Lemma A.1

Let $n\in {\mathbb {N}}$. Let $\Psi \in C^\infty ({\mathbb {R}}^3\times (-\infty , 0))$ such that for constants $\alpha>0, c>0$ and $C>0$ it holds that

$$\begin{aligned} {\left\{ \begin{array}{ll} c r_j^{ -\alpha } \le \Psi (x, t) \le Cr_j^{ -\alpha } \quad \forall (x, t)\in A_j =Q_j(R) \setminus Q_{ j+1}(R),\quad \forall j=0, \ldots , n-1 \\ c r_n^{ -\alpha } \le \Psi (x, t) \le Cr_j^{ -\alpha } \quad \forall (x, t)\in Q_n(R). \end{array}\right. } \end{aligned}$$

(A.3)

Let $1< p,q<+\infty , 1 < l \le q'$. Let $v\in L^{ p}(-1, 0; L^q(U_0(R)))$, and $f\in L^{ p'}(-1, 0; L^{ q'}(U_0(R)))$, $1< m, l < +\infty$. Then, setting $\pi _0 = {\mathscr {J}}(f)$ it holds that

$$\begin{aligned}&\int \nolimits _{-1}^{ t}\int \nolimits _{U_0(R)} \pi _0 v \Psi \eta \mathrm{d}x\mathrm{d}s \nonumber \\&\quad \le c \sup |\eta | \sum _{k=0}^{n} r_k^{ -\alpha } \Vert f\Vert _{ L^{ p'}(-r_k^2, 0; L^{ q'}(U_k(R)))} \Vert v\Vert _{ L^{p}(-r_k^2, 0; L^{ q}(U_k(R)))} \nonumber \\&\qquad + c\sup |\eta |\sum _{j=0}^{n} \sum _{k=j}^{n} r_k^{ \frac{1}{q'}-\alpha } r_{ j}^{ \frac{2}{q'}- \frac{3}{l}} \Vert f\Vert _{L^{ p'}(-r_k^2, 0; L^{ l}(U_j(R)))} \Vert v\Vert _{ L^{p}(-r_k^2, 0; L^{ q}(U_k(R)))}, \end{aligned}$$

(A.4)

where $\eta \in C^\infty _c(U_0(R)\times (-1, 0])$ stands for a cut off function. The constant in (A.4) depends only on p, q and l.

Proof

Let $f\in L^p(Q_0(R))$. Set $\pi _0 = {\mathscr {J}}(f)$. For $j\in {\mathbb {N}}_0$ let $\chi _j\in C^\infty _c(U_j(R) \times (-r_j^2, 0] )$ with $\chi _j=1$ on $Q_{ j+1}(R)$ such that $0 \le \chi _j \le 1$, and $|\partial _3\chi _j| \le c r_j^{ -1}$, and $|\nabla ' \chi _j| \le c R^{ -1}$. We set

$$\begin{aligned} \phi _j={\left\{ \begin{array}{ll} 1- \chi _0\quad &{}\text {if}\quad j=0, \\ \chi _{ j}- \chi _{ j+1} &{}\text {if}\quad j=1, \ldots , n-1, \\ \chi _n\quad &{}\text {if}\quad j=n. \end{array}\right. } \end{aligned}$$

We have $\displaystyle \sum _{j=0}^n\phi _j= 1- \chi _0 + \chi _0- \chi _1 + \cdots +\chi _{ n-1}- \chi _n + \chi _n=1$. Accordingly, $f = \displaystyle \sum _{j=0}^{n} f \phi _j$, and therefore it holds that

$$\begin{aligned} \pi _0 = {\mathscr {J}}(f) = \sum _{j=0}^{n} {\mathscr {J}}(\phi _j f) = \sum _{j=0}^{n} \pi _{ 0,j}. \end{aligned}$$

This yields

$$\begin{aligned}&\int \nolimits _{-0}^{ t}\int \nolimits _{U_0(R)} \pi _0 v\Psi \eta \mathrm{d}x\mathrm{d}s = \sum _{k=0}^{n}\int \nolimits _{-1}^{ t}\int \nolimits _{U_0(R)} \pi _0 v \Psi \phi _k\eta \mathrm{d}x\mathrm{d}s \\&\quad = \sum _{j=0}^{n}\sum _{k=0}^{n} \int \nolimits _{-1}^{ t}\int \nolimits _{U_0(R)} \pi _{ 0,j} v \Psi \phi _k \eta \mathrm{d}x\mathrm{d}s = \sum _{k=0}^{n}\sum _{j=k}^{n}\int \nolimits _{-1}^{ t}\int \nolimits _{U_0(R)} \pi _{ 0,j} v\Psi \phi _k \eta \mathrm{d}x\mathrm{d}s \\&\quad \qquad + \sum _{j=0}^{n} \sum _{k=j+1}^{n}\int \nolimits _{-1}^{ t}\int \nolimits _{U_0(R)} \pi _{ 0,j} v \Psi \phi _k \eta \mathrm{d}x\mathrm{d}s = I+II. \end{aligned}$$

First, we calculate

$$\begin{aligned} I = \sum _{k=0}^{n}\int \nolimits _{-1}^{ t}\int \nolimits _{U_0(R)} \Pi _{ 0,k} v\Psi \phi _k\eta \mathrm{d}x\mathrm{d}s, \end{aligned}$$

where

$$\begin{aligned} \Pi _{ 0,k}= {\left\{ \begin{array}{ll} \pi _0\quad &{} \text {if}\quad k=0, \\ {\mathscr {J}} (\phi _k f)\quad &{}\text {if}\quad k=1, \ldots , n. \end{array}\right. } \end{aligned}$$

Observing (A.3), applying Hölder’s inequality along with (A.1), we get

$$\begin{aligned} I&\le c \sup |\eta | \sum _{k=0}^{n} r_k^{ -\alpha } \Vert \Pi _{ 0,k}\Vert _{ L^{ p'}(-r_k^2, 0; L^{ q'}(U_k(R)))} \Vert v\Vert _{ L^{ p}(-r_k^2, 0; L^{ q}(U_k(R)))} \\&\le c \sup |\eta |\sum _{k=0}^{n} r_k^{ -\alpha } \Vert f\Vert _{ L^{ p'}(-r_k^2, 0; L^{ q'}(U_k(R)))} \Vert v\Vert _{ L^{p}(-r_k^2, 0; L^{q}(U_k(R)))}. \end{aligned}$$

For the second integral we find that

$$\begin{aligned} II&= \sum _{j=n-2}^{n} \sum _{k=j}^{n} \int \nolimits _{-1}^{ t}\int \nolimits _{U_0(R)} \Pi _{ 0,j} v\Psi \phi _{ k} \eta \mathrm{d}x\mathrm{d}s \\&\qquad +\sum _{j=0}^{n-3} \sum _{k=j}^{j+3} \int \nolimits _{-1}^{ t}\int \nolimits _{U_0(R)} \Pi _{ 0,j} v\Psi \phi _{k} \eta \mathrm{d}x\mathrm{d}s \\&\qquad + \sum _{j=0}^{n-3} \sum _{k=j+4}^{n}\int \nolimits _{-1}^{ t}\int \nolimits _{U_0(R)} \Pi _{ 0,j} v\Psi \phi _k \eta \mathrm{d}x\mathrm{d}s = II_1+ II_2+ II_3. \end{aligned}$$

Arguing as above, observing (A.3) and applying Hölder’s inequality and (A.1), we see that

$$\begin{aligned} II_1+II_2 \le c \sup |\eta | \sum _{k=1}^{n} r_k^{ -\alpha } \Vert f\Vert _{ L^{ p'}(-r_k^2, 0; L^{ q'}(U_k(R)))} \Vert v\Vert _{ L^{ p}(-r_k^2, 0; L^{ q}(U_k(R)))}. \end{aligned}$$

It remains to estimate $II_3$. We calculate

$$\begin{aligned} II_3 = \sum _{j=0}^{n-2} \sum _{k=j+4}^{n}\int \nolimits _{-1}^{ t}\int \nolimits _{U_0(R)} \Pi _{ 0,j} v \Psi \phi _k \mathrm{d}x\mathrm{d}s = \sum _{j=0}^{n-2} \sum _{k=j+4}^{n} J_{ jk} . \end{aligned}$$

(A.5)

Let $j+4 \le k \le n$ be fixed. Applying Hölder’s inequality, together with (A.3), we find that

$$\begin{aligned} J_{ jk}&\le c\sup |\eta | r_k^{ -\alpha } \Vert \Pi _{ 0, j}\Vert _{ L^{ p'}(-r_k^2, 0; L^{ q'}(U_k(R)))} \Vert v\Vert _{ L^{p}(-r_k^2, 0; L^{ q}(U_k(R)))}. \end{aligned}$$

From the definition of $\Pi _{ 0, j}$ it follows that $\Delta \pi _{ 0, j} = \nabla \cdot \nabla \cdot (f\phi _j)$ in the sense of distributions. Since ${\text {supp}}(\phi _j) \subset Q_{ j}(R) \setminus Q_{ j+2}(R)$ the function $\pi _{ 0, j}$ is harmonic in ${\mathbb {R}}^{2}\times (-r_{ j+2}, r_{ j+2})\times (-r_{ j+2}^2, 0)$. Applying Lemma A.2 below for $h=\pi _{ 0, j}, r=r_{ j+2}$ and $\rho = r_k$, we get, for almost all $s\in (-r_k^2, 0)$,

$$\begin{aligned} \Vert \Pi _{ 0, j}(s)\Vert _{ L^{ q'}(U_k(R))}&=\Vert \pi _{ 0, j}(s)\Vert _{ L^{ q'}(B'(R)\times (-r_k, r_k))} \\&\le c\sup |\eta | r_k^{ \frac{1}{q'}} r_{ j+2}^{ \frac{2}{q'}- \frac{3}{l}} \Vert \Pi _{ 0, j}(s)\Vert _{ L^{ l}({\mathbb {R}}^{3} )}. \end{aligned}$$

Taking the $L^{ p'}$ norm with respect to s, and employing (A.1), we find that

$$\begin{aligned} \Vert \Pi _{ 0, j}\Vert _{L^{ p'}(-r_k^2,0; L^{ q'}(U_k(R)))} \le c\sup |\eta | r_k^{ \frac{1}{q'}} r_{ j+2}^{ \frac{2}{q'}- \frac{3}{l}} \Vert f\Vert _{L^{ p'}(-r_k^2, 0; L^{ l}(U_j(R)))}. \end{aligned}$$

Accordingly,

$$\begin{aligned} J_{ jk}&\le c \sup |\eta | r_k^{ \frac{1}{q'}-\alpha } r_{ j+2}^{ \frac{2}{q'}- \frac{3}{l}} \Vert f\Vert _{L^{ p'}(-r_k^2, 0; L^{ l}(U_j(R)))} \Vert v\Vert _{ L^{p}(-r_k^2, 0; L^{ q}(U_k(R)))}. \end{aligned}$$

Inserting this inequality into (A.5), we arrive at

$$\begin{aligned} II_3&= c\sup |\eta |\sum _{j=1}^{n-2} \sum _{k=j+3}^{n} r_k^{ \frac{1}{q'}-\alpha } r_{ j+2}^{ \frac{2}{q'}- \frac{3}{l}} \Vert f\Vert _{L^{ p'}(-r_k^2, 0; L^{ l}(U_j(R)))} \Vert v\Vert _{ L^{p}(-r_k^2, 0; L^{ q}(U_k(R)))} \\&\le c\sup |\eta |\sum _{j=0}^{n} \sum _{k=j}^{n} r_k^{ \frac{1}{q'}-\alpha } r_{ j}^{ \frac{2}{q'}- \frac{3}{l}} \Vert f\Vert _{L^{ p'}(-r_k^2, 0; L^{ l}(U_j(R)))} \Vert v\Vert _{ L^{p}(-r_k^2, 0; L^{ q}(U_k(R)))}. \end{aligned}$$

Combining the above estimates, we get the claim. $\quad \square$

Lemma A.2

Let $0< r \le R <+\infty$. Let $h: B'(2R)\times (-r, r) \rightarrow {\mathbb {R}}$ be harmonic. Then for all $0< \rho \le \frac{r}{4}$ and $1 \le l \le p \le +\infty$ we get

$$\begin{aligned} \Vert h\Vert _{ L^p(B'(R)\times (-\rho , \rho ))}^p \le c\rho r^{2 -3\frac{p}{l}} \Vert h\Vert _{ L^l(B'(2R)\times (-r, r))}^p, \end{aligned}$$

(A.6)

where c stands for a positive constant depending only on p and l.

Proof

Let $k\in {\mathbb {N}}, k \ge 2$. Set $\rho _k = 2^{ -k} r$. Since $B'(2R)\times (- r, r)$ is a non isotropic cylinder, in order to apply the mean value property of harmonic functions we use a covering argument. We may choose a finite family of points $\{x'_{\nu }\}$ in $B'(R)$ such that $\{B'(x_{\nu }', r/4)\}$ is a covering of $\overline{B'(R)}$, and it holds that

$$\begin{aligned} \sum _{\nu } \chi _{ B'(x'_{ \nu }, r)} \le N,\quad |x_{\nu }- x_{ \mu }| \ge \frac{r}{4}\quad \forall \nu \ne \mu , \end{aligned}$$

(A.7)

where N stands for an absolute number. Setting $x_{\nu }= (x'_{ \nu }, 0)$, we see that $B'(x'_{\nu }, r/4 )\times (- r/4, r/4) \subset B(x_{\nu }, r/2)$. With this notation we have that

$$\begin{aligned} \Vert h\Vert _{ L^p(B'(R)\times (-\rho _k, \rho _k))}^p&\le \sum _{\nu }\Vert h\Vert _{ L^p(B'(x'_{ \nu }, r/4)\times (- \rho _k, \rho _k))}^p\nonumber \\&\le c r^2 \rho _k\sum _{\nu }\Vert h\Vert _{ L^\infty (B'(x'_{ \nu }, r/4)\times (- r/4, r/4))}^p \nonumber \\&\le c r^2 \rho _k\sum _{\nu }\Vert h\Vert _{ L^\infty (B(x_{ \nu }, r/2))}^p. \end{aligned}$$

(A.8)

Since h is harmonic, using the mean value property, we find that

$$\begin{aligned} \Vert h\Vert ^p_{ L^\infty (B(x_{\nu }, r/2 ))}&\le c r^{ -\frac{3p}{l}} \Vert h\Vert ^p_{ L^l(B(x_{\nu }, r ))} \\&\le c r^{ -\frac{3p}{l}} \Vert h\Vert ^{ p-l}_{ L^l( B'(2R)\times (-r,r))} \Vert h\Vert ^l_{ L^l(B'(x'_{\nu }, r )\times (-r,r))} \\&= c r^{ -\frac{3p}{l}} \Vert h\Vert ^{ p-l}_{ L^l( B'(2R)\times (-r,r))} \int \nolimits _{-r}^r \int \nolimits _{B'(2R)} |h|^l \chi _{ B'(x'_{\nu }, r )} \mathrm{d}x'\mathrm{d}x_3. \end{aligned}$$

Taking the sum over $\nu$ and using (A.7), we obtain that

$$\begin{aligned} \sum _{\nu }\Vert h\Vert _{ L^\infty (B(x_{ \nu }, r/2))}^p \le cN r^{ -\frac{3p}{l}} \Vert h\Vert ^{ p}_{ L^l( B'(2R)\times (-r,r))}. \end{aligned}$$

(A.9)

Combining (A.8) and (A.9), we get

$$\begin{aligned} \Vert h\Vert _{ L^p(B'(R)\times (-\rho _k , \rho _k))}^p \le c\rho _k r^{2 -3\frac{p}{l}} \Vert h\Vert _{ L^l(B'(2R)\times (-r, r))}^p. \end{aligned}$$

(A.10)

Let $0< \rho \le \frac{r}{4}$. Then there exists a unique integer $k\ge 2$ such that $\rho _{ k+1} < \rho \le \rho _k$, Thus, (A.10) implies

$$\begin{aligned} \Vert h\Vert _{ L^p(B'(R)\times (-\rho , \rho ))}^p&\le c\rho _k r^{2 -3\frac{p}{l}} \Vert h\Vert _{ L^l(B'(2R)\times (-r, r))}^p \\&\le 2c\rho r^{2 -3\frac{p}{l}} \Vert h\Vert _{ L^l(B'(2R)\times (-r, r))}^p, \end{aligned}$$

whence we get (A.6). $\quad \square$

The following iteration lemma has been used in the sequel of the proofs above, and its proof can be found in [16, V. Lemma 3.1]:

Lemma A.3

Let f(t) be a nonnegative bounded function defined in $[r_0, r_1], 0 \le r_0< r_1 <+\infty$. Suppose that for $r_0 \le t < s \le r_1$ we have

$$\begin{aligned} f(t)\le [A(s-t)^{-\alpha } + B] + \theta f(s), \end{aligned}$$

(A.11)

where $A,B, \alpha , \theta$ are nonnegative constants with $0 \le \theta <1$. Then, for all $r_0 \le \rho <R \le r_1$, we have

$$\begin{aligned} f(\rho )\le c[A(R-\rho )^{-\alpha } + B], \end{aligned}$$

(A.12)

where c is a constant depending on $\alpha$ and $\theta$.