Localized non blow-up criterion of the Beale-Kato-Majda type for the 3D Euler equations

Abstract

We prove a localized non blow-up theorem of the Beale–Kato–Majda type for the solution of the 3D incompressible Euler equations.

Appendices

Gronwall type lemma

Lemma A.1

Let $-\infty<a<b<+\infty$. Let $f\in L^1(a,b)$ and $k\in {\mathbb {N}}$. Then it holds

$$\begin{aligned} \int \limits _{a}^{b} \int \limits _{a}^{t_1} \cdots \int \limits _{a}^{t_{ k-1}} \prod _{ j=1}^{ k} f(t_j) d t_k d t_{ k-1} \ldots d t_1 = \frac{1}{k!} \left( \int \limits _{a}^{b} f(t) dt\right) ^k. \end{aligned}$$

(A.1)

Proof

We prove the assertion by induction. For $k=1$ (A.1) is obvious. Assume the assertion holds for $k-1$. Using the hypothesis of induction, we find

$$\begin{aligned} \int \limits _{a}^{b} \int \limits _{a}^{t_1} \cdots \int \limits _{a}^{t_{ k-1}} \prod _{ j=1}^{ k} f(t_j) d t_k d t_{ k-1} \ldots d t_1&= \int \limits _{a}^{b} f(t_1)\left\{ \int \limits _{a}^{t_1} \cdots \int \limits _{a}^{t_{ k-1}} \prod _{ j=2}^{ k} f(t_j) d t_k d t_{ k-1} \ldots dt_2 \right\} d t_1 \nonumber \\&= \frac{1}{(k-1)!}\int \limits _{a}^{b} f(s) \left( \int \limits _{a}^{s} f(t) dt\right) ^{ k-1} ds. \end{aligned}$$

(A.2)

Applying integration by parts, we calculate

$$\begin{aligned} \int \limits _{a}^{b} f(s) \left( \int \limits _{a}^{s} f(t) dt\right) ^{ k-1} ds&=\int \limits _{a}^{b} \frac{d}{ds} \int \limits _{a}^s f(\tau ) d\tau \left( \int \limits _{a}^{s} f(t) dt\right) ^{ k-1} ds\\&= \left( \int \limits _{a}^{b} f(t) dt\right) ^k - (k-1) \int \limits _{a}^{b} f(s) \int \limits _{a}^s f(\tau ) d\tau \left( \int \limits _{a}^{s} f(t) dt\right) ^{ k-2} ds\\&= \left( \int \limits _{a}^{b} f(t) dt\right) ^k - (k-1) \int \limits _{a}^{b} f(s) \left( \int \limits _{a}^{s} f(t) dt\right) ^{ k-1} ds. \end{aligned}$$

This yields

$$\begin{aligned} \int \limits _{a}^{b} f(s) \bigg (\int \limits _{a}^{s} f(t) dt\bigg )^{ k-1} ds = \frac{1}{k} \left( \int \limits _{a}^{b} f(t) dt\right) ^k. \end{aligned}$$

(A.3)

Replacing the integral on the right-hand side of (A.2) by the right-hand side of (A.3), we obtain (A.1) for k. Hence by induction (A.1) holds for all $k\in {\mathbb {N}}$. $\square$

Lemma A.2

(Iteration lemma) Let $\beta _m: [t_0, t_1] \rightarrow {\mathbb {R}}$, $m\in {\mathbb {N}}\cup \{ 0\}$ be a sequences of bounded functions. Suppose there exists $0<K=K(t) <+\infty$ for each $t\in [t_0, t_1)$ such that

$$\begin{aligned} |\beta _m(t)| < K(t)^m \qquad \forall t\in [t_0, t_1), \forall m\in {\mathbb {N}}. \end{aligned}$$

(A.4)

Furthermore let $a\in L^1(t_0, t_1)$ with $a(t)\ge 0$ for almost every $t\in [t_0, t_1]$. We assume that the following recursive integral inequality holds true for a constant $C>0$

$$\begin{aligned} \beta _{ m}(t) \le Cm + \int \limits _{t_0}^{t} a(s) \beta _{ m+1}(s) ds, \quad m\in {\mathbb {N}}\cup \{ 0\}. \end{aligned}$$

(A.5)

Then the following inequality holds true for all $t\in [t_0,t_1]$

$$\begin{aligned} \beta _0(t) \le C \int \limits _{t_0}^{t} a(s) ds \, e^{\int \limits _{t_0}^{t} a(s ) ds }. \end{aligned}$$

(A.6)

Proof

Iterating (A.5) m-times, and applying Lemma A.1, we see that for each $t\in [t_0, t_1)$ it follows

$$\begin{aligned} \beta _0(t)&\le C\int \limits _{t_0}^{t} a(s_1) ds_1 + 2C\int \limits _{t_0}^{t} \int \limits ^{s_1}_{t_0} a(s_1) a(s_2) ds_2 ds_1 \nonumber \\&\quad + \cdots + Cm\int \limits _{t_0}^{t} \int \limits ^{s_1}_{t_0} \ldots \int \limits ^{s_{ m-1}}_{t_0} a(s_1) a(s_2) \ldots a(s_m)ds_{ m} \ldots ds_2 ds_1 \nonumber \\&\quad + \int \limits _{t_0}^{t} \int \limits ^{s_1}_{t_0} \ldots \int \limits ^{s_{ m}}_{t_0} a(s_1) a(s_2) \ldots a(s_{ m+1}) \beta _{ m+1}(s_{ m+1}) ds_{ m+1} \ldots ds_2 ds_1 \nonumber \\&= \sum _{k=1}^{m}\frac{C}{(k-1) !} \bigg (\int \limits _{t_0}^{t} a(s) ds \bigg )^{ k}+ J_m(t), \end{aligned}$$

(A.7)

where

$$\begin{aligned} |J_m (t)|&\le \sup _{0<s< t} |\beta _{m+1}(s)| \int \limits _{t_0}^{t} \int \limits ^{s_1}_{t_0} \ldots \int \limits ^{s_{ m}}_{t_0} a(s_1) a(s_2) \ldots a(s_{ m+1}) ds_{ m+1} \ldots ds_2 ds_1\\&\le \frac{ 1}{(m+1)!} \bigg (\sup _{0<s< t} K(s) \int \limits _{t_0}^{t} a(s ) ds\bigg )^{ m+1} \rightarrow 0\quad \hbox { as}\ \quad m\rightarrow +\infty \end{aligned}$$

for each $t\in [t_0, t_1)$. Therefore,

$$\begin{aligned} \beta _0(t)\le \sum _{k=1} ^\infty \frac{C}{(k-1) !} \bigg (\int \limits _{t_0}^{t} a(s) ds \bigg )^{ k}= C \int \limits _{t_0}^{t} a(s) ds\, e^{\int \limits _{t_0}^{t} a(s ) ds}. \end{aligned}$$

$\square$

Gagliardo–Nirenberg’s inequality with cut-off

Lemma B.1

Let $\psi \in C^{\infty }_{\mathrm{c}}({\mathbb {R}}^{n})$ with $0 \le \psi \le 1$, and $k \ge 1$. For all $u\in W^{1,\, q}({\mathbb {R}}^{n})\cap L^2({\mathbb {R}}^{n})$, $2< q < +\infty$, and for $m \ge k\frac{n(q-2)+ 2q}{2q}$ it holds

$$\begin{aligned} \Vert u \psi ^{ m-k}\Vert _{ q} \le c \Vert u \psi ^{ m - k\frac{n(q-2)+ 2q}{2q}}\Vert _{ 2}^{ \frac{2q}{n(q-2)+ 2q}}\Vert \nabla u \psi ^m\Vert _{ q}^{ \frac{n(q-2)}{n(q-2)+ 2q}} + c\Vert \nabla \psi \Vert _{ \infty }^{\frac{n(q-2)}{2q}}\Vert u \psi ^{ m - k\frac{n(q-2)+ 2q}{2q}}\Vert _{ 2} . \end{aligned}$$

(B.1)

Proof

Let $\psi \in C^{\infty }_{\mathrm{c}}( {\mathbb {R}}^{n})$ with $0 \le \psi \le 1$, and $k \ge 1$. Let $m \ge k\frac{n(q-2)+ 2q}{2q}$. By means of Hölder’s inequality we estimate

$$\begin{aligned} \Vert u \psi ^{ m-k} \Vert _{ q}&\le \Vert u \psi ^{m - k- k\frac{n(q-2)}{2q} }\Vert _{ 2}^{ \frac{2}{n(q-2)+2}} \Vert u \psi ^{m- k + \frac{k}{q}} \Vert _{ \frac{nq}{n-1}}^{ \frac{n(q-2)}{n(q-2)+2}}\\&\le c\Vert u \psi ^{m - k- k\frac{n(q-2)}{2q} }\Vert _{ 2}^{ \frac{2}{n(q-2)+2}} \Vert w\Vert _{ \frac{n}{n-1}}^{ \frac{n(q-2)}{qn(q-2)+2q}}, \end{aligned}$$

where $w = | u|^q \psi ^{qm- kq + k}$. By Sobolev’s inequality along with Hölder’s inequality we infer

$$\begin{aligned} \Vert w\Vert _{ \frac{n}{n-1}}&\le \Vert \nabla w\Vert _{ 1}\\&\le c\int \limits _{ {\mathbb {R}}^{n}} | u|^{ q-1}| \nabla u| \psi ^{qm- kq + k} dx + c \Vert \nabla \psi \Vert _{ \infty }\Vert u \psi ^{ m-k}\Vert ^q_{ q}\\&\le c\Vert u \psi ^{ m-k}\Vert _{ q}^{ q-1} \Vert \nabla u \psi ^m\Vert _{ q} + c \Vert \nabla \psi \Vert _{ \infty }\Vert u \psi ^{ m-k}\Vert ^q_{ q}. \end{aligned}$$

Combining the last two estimates and applying Young’s inequality, we obtain the assertion (B.1). $\square$

Lemma B.2

Let $u \in W^{1,\, q}(B(\rho ))$, $n< q < +\infty$ such that $\nabla \cdot u =0$ almost everywhere in $B(\rho )$. Then for every $\psi \in C^{\infty }_c(B(\rho ))$ with $0 \le \psi \le 1$ and $m \ge \frac{q}{2}$ it holds

$$\begin{aligned} \Vert u \psi ^{ m}\Vert _{ \infty } \le c\Vert u \psi ^{ m- \frac{q}{2}}\Vert _{ 2}^{ \frac{2(q-n)}{2q-2n+nq } } \Vert \nabla u \psi ^{ m}\Vert ^ { \frac{nq}{2q-2n+nq}}_{ q} + c\Vert \nabla \psi \Vert _{ \infty }^{ \frac{n}{2}} \Vert u \psi ^{ m-\frac{q}{2}}\Vert _{ 2}. \end{aligned}$$

(B.2)

Proof

By virtue of Gagliardo-Nirenberg’s inequality we estimate

$$\begin{aligned} \Vert u \psi ^{ m}\Vert _{ \infty }&\le c \Vert u \psi ^{ m}\Vert _{ q}^{ 1- \frac{n}{q}} \Vert \nabla ( u \psi ^{ m})\Vert _{ q}^{ \frac{n}{q}} \\&\le c\Vert u \psi ^{ m-1} \Vert _{ q}^{ 1- \frac{n}{q}} \Vert \nabla u \psi ^{ m}\Vert ^{ \frac{n}{q}}_{ q} + c\Vert \nabla \psi \Vert _{ \infty }^{ \frac{n}{q}} \Vert u \psi ^{ m-1}\Vert _{ q}. \end{aligned}$$

From $L^q$-interpolation we find

$$\begin{aligned} \Vert u \psi ^{ m-1}\Vert _{ q} \le \Vert u \psi ^{ m- \frac{q}{2}} \Vert _{ 2}^{ \frac{2}{q }} \Vert u \psi ^{ m}\Vert _{ \infty }^{ 1- \frac{2}{q}}. \end{aligned}$$

Combining the two estimates above, and applying Young’s inequality, we obtain the assertion of the lemma. $\square$

Although the following result of embedding of BMO(B(r)) into $L^p(B(r))$ might be well known, for readers convenience we give a short proof.

Lemma B.3

Let $u\in BMO(B(r))$. Then $u\in \cap _{ 1 \le q< \infty } L^q(B(r))$, and it holds

$$\begin{aligned} \Vert u\Vert _{ L^q(B(r))} \le c r^{ \frac{n}{q}} | u|_{BMO(B(r))}+ c r ^{ \frac{n}{q}-n}\Vert u\Vert _{ L^1(B(r))} = c r^{ \frac{n}{q}} \Vert u\Vert _{ BMO(B(r))}. \end{aligned}$$

(B.3)

Proof

Let $u\in BMO(B(1))$. By $U\in BMO$ we denote the extension defined in Section 2. By John–Nirenberg’s inequality [9] it holds

$$\begin{aligned} m(\{x\in Q(2) \,|\, | U - U_{ Q(2)} | > \lambda \}) \le c_1 e^{ - \frac{c_2 \lambda }{| U| _{ BMO(Q(2))}}}\quad \forall \,\lambda \in (0, +\infty ) \end{aligned}$$

with constants $c_1, c_2$, depending on n and q only, where $m(\cdot )$ denotes the Lebesgue measure in ${\mathbb {R}}^n$, and Q(r) denotes the cube with the center at origin and the side length 2r. Multiplying both sides by $q\lambda ^{ q-1}$, integrating the result over $(0, +\infty )$, using a suitable change of coordinates, and employing (2.4), we arrive at

$$\begin{aligned} \Vert U- U _{ Q(2)} \Vert _{ L^q(Q(2))}^q&= q \int \limits _{0}^{\infty } \lambda ^{ q-1}m(\{x\in Q(2) \,|\, | U- U _{ Q(2)} | > \lambda \} )\\&\le q | U| _{ BMO(Q(2))}^q \int \limits _{0}^{\infty } \lambda ^{ q-1}c_1 e^{ - c_2 \lambda } \\&\le c \Vert u\Vert ^q_{ BMO(B(1))}. \end{aligned}$$

Accordingly,

$$\begin{aligned} \Vert u\Vert _{ L^q(B(1))} \le \Vert U- U_{ Q(2)} \Vert _{ L^q(Q(2))}+ c| U_{ Q(2)}| \le c \Vert u\Vert _{ BMO(B(1))}. \end{aligned}$$

Hence, (B.3) follows from the above estimate by using a standard scaling argument. $\square$