Energy Concentrations and Type I Blow-Up for the 3D Euler Equations

Abstract

We exclude Type I blow-up, which occurs in the form of atomic concentrations of the $L^2$ norm for the solution of the 3D incompressible Euler equations. As a corollary we prove nonexistence of discretely self-similar blow-up in the energy conserving scale.

Some Auxiliary Lemmas

Here we prove fundamental properties of a harmonic function used in the proof of the main theorem.

Lemma A.1

Let $p\in L^2(\Omega )$ be a harmonic function on $\Omega$. Then for every $\phi \in C^{\infty }_c(\Omega )$ and for all $m\in \mathbb {N}$ it holds

$$\begin{aligned} \int \limits _{\Omega } | \nabla ^m p|^2 \phi dx = \frac{1}{2^m} \int \limits _{\Omega } p^2 \Delta ^m \phi dx. \end{aligned}$$

(A.1)

Proof

We show (A.1) by an inductive argument. First, (A.1) with $m=1$ is clear by the integration by parts. Assume (A.1) holds for $m\in \mathbb {N}$. Then we have $| \nabla ^{ m+1} p|^2 = \sum _{i=1}^{n} | \nabla ^{ m} \partial _i p|^2$. Using the assumption that (A.1) holds for $\partial _i p$ in place of p, and integrating it by parts, we infer

$$\begin{aligned} \int \limits _{\Omega } | \nabla ^{ m+1} p|^2 \phi dx&= \sum _{i=1}^{n}\int \limits _{\Omega } | \nabla ^{ m} \partial _ip|^2 \phi dx\\&=\frac{1}{2^m} \int \limits _{\Omega } | \nabla p|^2 \Delta ^m \phi dx =\frac{1}{2^{ m+1}} \int \limits _{\Omega } p^2 \Delta ^{ m+1} \phi dx. \end{aligned}$$

$\quad \square$

Lemma A.2

Let $U = \mathbb {R}^{n} \setminus \overline{B(r)}, 0< r<+\infty$. Let $\zeta \in C^{\infty }(U)$ denote a cut off function such that $0 \le \zeta \le 1$ with supp $\zeta \subset U$, and $| \nabla ^k\zeta | \le c r^{ -k}$, $k=1, \ldots , n+1$. Then for every $u\in L^{ \frac{2n}{n-1}}(U)$ which is harmonic in U it holds

$$\begin{aligned} \Vert \nabla u \zeta \Vert _{ \infty } \le c r^{ -\frac{n+1}{2}} \Vert u \Vert _{ L^{ \frac{2n}{n-1}}(U)}. \end{aligned}$$

(A.2)

Proof

First, let $x\in \mathbb {R}^{n} \setminus B(2r)$. Applying the mean value property of harmonic functions and Jensen’s inequality, we get

$$\begin{aligned} | \nabla u(x) \zeta (x)| \le | \nabla u(x)| \le c r^{ -n-1} \int \limits _{B(x, r)} | u| dx \le c r^{ -\frac{n+1}{2}} \Vert u\Vert _{ L^{ \frac{2n}{n-1}}(U)}. \end{aligned}$$

Secondly, let $x\in U\cap B(2r)$. By $\eta \in C^{\infty }_{ c}(B(4r))$ we denote a cut off function such that $0 \le \eta \le 1$ in $B(4r), \eta \equiv 1$ on B(2r) and $| \nabla ^k \eta | \le c r^{ -k}$, $k=1, \ldots , n+1$. Using Sobolev’s embedding theorem, and applying Lemma A.1 with $\phi = | \nabla ^j(\zeta \eta )|^2$, $j=1, \ldots , n$, we estimate

$$\begin{aligned} | \nabla u(x) \zeta (x)|&\le \Vert \nabla u \zeta \eta \Vert _{ L^\infty (B(4r))} \\&\le c \sum _{k=0}^{n} r^{ - \frac{n}{2} +k}\Vert \nabla ^k (\nabla u \zeta \eta )\Vert _{ L^{2}(B(4r))} \\&\le c\sum _{k=0}^{n} \sum _{j=0}^{k} r^{ - \frac{n}{2} +k}\Vert \nabla ^{ k-j+1} u \nabla ^{ j}(\zeta \eta )\Vert _{ L^{2}(B(4r))} \\&\le cr^{ - \frac{n+2}{2}} \Vert u\Vert _{ L^{2}(U\cap B(4r))} \le cr^{ - \frac{n+1}{2}} \Vert u\Vert _{ L^{ \frac{2n}{n-1}}(U)}. \end{aligned}$$

The assertion now follows from the above two estimates. $\quad \square$

Lemma A.3

Let $\{ p_k\}$ be a sequence of harmonic functions in $L^2(\Omega )$, which converges weakly to some limit p in $L^2(\Omega )$ as $k \rightarrow +\infty$. Then p is harmonic and for every compact set $K \subset \Omega$ and every multi index $\alpha = (\alpha _1, \ldots , \alpha _n)$ it holds

$$\begin{aligned} D^{ \alpha } p_k \rightarrow D^{ \alpha } p \quad { uniformly~ on~}K\quad { as}\quad k \rightarrow +\infty . \end{aligned}$$

(A.3)

Proof

By virtue of Weyl’s lemma it is clear that p is harmonic in $\Omega$. Let $x\in \Omega$, and let $B(x, r) \subset \Omega$ be a ball. By the weak convergence and the mean value property of harmonic functions we obtain

This shows that $p_k \rightarrow p$ pointwise as $k \rightarrow +\infty$. In particular, $p_k \rightarrow p$ in $L^2(\Omega ')$ as $k \rightarrow +\infty$ for every $\Omega ' \Subset \Omega$. Applying the identity (A.1) for a suitable cut off function $\phi \ge 0$ it follows that $p_k \rightarrow p$ in $H^m(\Omega ')$ as $k \rightarrow +\infty$ for every $\Omega ' \Subset \Omega$ and for all $m\in \mathbb {N}$. The uniform convergences (A.3) is now an immediate consequence of Sobolev’s embedding theorem. $\quad \square$

Lemma A.4

For $0< R< +\infty$ define $Q(R)= B(R) \times I(R), I(R)= (- R^{ \frac{n+2}{2}}, 0)$. Let $f\in L^p(Q(R); \mathbb {R}^{n^2})$, $1< p< +\infty$. Let $u\in L^p(Q(R))$, solving the equation

$$\begin{aligned} -\Delta u = \sum _{i,j=1}^{n} \partial _i \partial _j f_{ ij}\quad \text { in}\quad Q(R) \end{aligned}$$

(A.4)

in the sense of distributions. Assume for some $\lambda \in (0,n)$ it holds

$$\begin{aligned} \sup _{ 0< \rho< R} \rho ^{ - \lambda } \Vert f\Vert ^p_{ L^p(Q(\rho ))} < +\infty . \end{aligned}$$

(A.5)

Then there exists a constant $c>0$ depending only on n, p and $\lambda$ such that

$$\begin{aligned} \sup _{ 0< \rho< R} \rho ^{ - \lambda } \Vert u\Vert ^p_{ L^p(Q(\rho ))} \le c \Big (R^{ -\lambda }\Vert u\Vert ^p_{ L^p(Q(R))}+ \sup _{ 0< \rho < R} \rho ^{ - \lambda } \Vert f\Vert ^p_{ L^p(Q(\rho ))}\Big ). \end{aligned}$$

(A.6)

Proof

By a routine scaling argument we may assume that $R=1$. We extend f(t) by zero outside B(1), and denote this extension again by f. Clearly, the family of annalus $U_j = B(2^{ j+1}) \setminus \overline{B(2^{ j-1})} , j\in \mathbb {Z}, j \le 1$, cover $\overline{B(2)}$. By $\{ \psi _j\}$ we denote a corresponding partition of unity of smooth radial symmetric functions, such that $\sum _{j=-\infty }^{1} \psi _j =1$ on B(2) together with $| \nabla \psi _j | \le c 2^{ -j}$ and $| \nabla ^2 \psi _j | \le c 2^{ -2j}$ for all $j\in \mathbb {Z}, j \le 1$. Let $m\in \mathbb {Z}$, with $m \le 0$ be arbitrarily chosen, but fixed. We write $u= u_1+ u_2+ u_3$, where

$$\begin{aligned} u_1(x, t)&= \sum _{j=-\infty }^{m} P.V. \int \limits _{ \mathbb {R}^{n}} f(x-y, t): \nabla ^2 N(y) \psi _j(y)dy, \\ u_2 (x, t)&= \sum _{j=m+1}^{1} P.V. \int \limits _{ \mathbb {R}^{n}} f(x-y, t) : \nabla ^2 N(y) \psi _j(y)dy, \\ u_{ 3}(x,t )&= u(x, t)- u_1(x, t)- u_2(x,t),\quad (x, t)\in Q(1), \end{aligned}$$

where N stands for the Newton potential in $\mathbb {R}^{n}$.

Our aim will be to estimate the $L^p$ norm of $u_1, u_2$ and $u_3$ over $Q(2^m)$ separately.

First, by triangle inequality we see that for $x\in B(2^{ m})$ and $| x-y| \ge 2^{ m+2}$ we get $| y| \ge 2^{ m+1}$. Thus, by Calderón-Zygmund inequality we find for almost every $t\in (-1, 0)$

$$\begin{aligned} \Vert u_1(t)\Vert _{ L^p(B(2^m))}^p \le c \Vert f (t) \Vert ^p_{ L^p(B(2^{ m+2}))}. \end{aligned}$$

Integration of both sides over $I(2^m)$ with respect to time along with (A.5) yields

$$\begin{aligned} \Vert u_1\Vert _{ L^p(Q(2^m))}^p \le c \Vert f \Vert ^p_{ L^p(Q(2^{ m+2}))} \le c 2 ^{ m \lambda } \sup _{ 0< \rho < R} \rho ^{ - \lambda } \Vert f\Vert ^p_{ L^p(Q(\rho ))}. \end{aligned}$$

Next, fix $x\in B(2^{ m})$. It is readily seen that for all $j \ge m+1$ it holds $B(x, 2^{j+1 }) \subset B(2^{ j+1}+ 2^m) \subset B(2^{ j+2})$. Noting that $| k(y)| \le c | y|^{ -n}$ it follows $| k|\psi _j \le c 2^{ -jn}$. Accordingly, by the aid of Jensen’s inequality, and observing (A.5), we estimate

$$\begin{aligned} | u_2(x,t)|&\le c \sum _{j=m+2}^{1}\int \limits _{U_j} f(x-y, t) 2^{ -jn} dx \le \sum _{j=m+2}^{1} 2^{ j \frac{n}{p}}\bigg (\int \limits _{B(2^{ j+2})} | f(y, t)|^p dy \bigg )^{ \frac{1}{p}} \end{aligned}$$

Taking the ${\text {ess sup}}$ over $x\in B(2^m)$, and taking the $\Vert \cdot \Vert _{ L^p(I(2^m))}$ of both sides with respect to t, using Minkowski’s inequality, and observing (A.5), we are led to

$$\begin{aligned} \bigg ( \int \limits _{I(2^m)}\Vert u_2(t)\Vert ^p_{ L^\infty (B(2^m))} dt \bigg )^{ \frac{1}{p}}&\le \bigg (\sum _{j=m+2}^{1} 2^{ j n}\int \limits _{Q(2^{ j+2})} | f(y, t)|^p dy \bigg )^{ \frac{1}{p}} \\&\le c\Big (\sum _{j=m+2}^{1} 2^{ - j(n-\lambda ) } \sup _{ 0< \rho< R} \rho ^{ - \lambda } \Vert f\Vert ^p_{ L^p(Q(\rho ))}\Big )^{ \frac{1}{p}} \\&\le c 2 ^{ -m \frac{n-\lambda }{p}}\Big (\sup _{ 0< \rho < R} \rho ^{ - \lambda } \Vert f\Vert ^p_{ L^p(Q(\rho ))}\Big )^{ \frac{1}{p}}. \end{aligned}$$

Consequently,

$$\begin{aligned} \Vert u_2\Vert ^p_{ L^p(Q(2^m))} \le c 2^{ mn } \int \limits _{I(2^m)}\Vert u_2(t)\Vert ^p_{ L^\infty (B(2^m))} dt \le c 2 ^{ m \lambda } \sup _{ 0< \rho < R} \rho ^{ - \lambda } \Vert f\Vert ^p_{ L^p(Q(\rho ))}. \end{aligned}$$

In only remains to estimate $u_3$. By the definition of $u_1$ and $u_2$, recalling that $f(t) \equiv 0$ on $\mathbb {R}^{n} \setminus B(1)$, we see that for almost all $t\in (-1,0)$ and for all $x\in B( 1)$ it holds

$$\begin{aligned} u_1(x, t) + u_2(x,t)&= \sum _{j=-\infty }^{1} P.V. \int \limits _{B(2)} f(x-y, t): \nabla ^2 N(y) \psi _j(y)dy \nonumber \\&= \int \limits _{ \mathbb {R}^{n}} f(x-y,t): \nabla ^2 N(y) dy. \end{aligned}$$

(A.7)

In particular, $u_1 + u_2$ solves (A.4) in the sense of distributions. By Weyl’s lemma we deduce that $u_3(t) = u(t)- u_1(t)-u_2(t)$ is harmonic. Thus,

$$\begin{aligned} \Vert u_3\Vert _{ L^p(Q(2^m))}^p&\le c 2^{ m n} \Vert u_3\Vert _{ L^p(I(2^m); L^\infty ( B(2^m))}^p \\&\le c 2^{ m \lambda } \Big (\Vert u\Vert _{ L^p(Q(2^m))}^p + \Vert f\Vert _{ L^p(Q(1))}^p\Big ). \end{aligned}$$

Combining the estimates of $u_1, u_2$ and $u_3$ we get for all $m\in \mathbb {Z}$, $m \le 0$,

$$\begin{aligned} 2^{ - m \lambda }\Vert u\Vert ^p_{ L^p(Q(2^m ))} \le c \Big (\Vert u\Vert ^p_{ L^p(Q(1))}+ \sup _{ 0< \rho < R} \rho ^{ - \lambda } \Vert f\Vert ^p_{ L^p(Q(\rho ))}\Big ). \end{aligned}$$

Taking the supremum over all $m\in \mathbb {Z}$, $m \le 0$ on the left-hand side, we obtain the assertion (A.6). $\quad \square$