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\def\rightheadline{EJDE--1999/48\hfil Smallness of the singular set \hfil\folio}
\def\leftheadline{\folio\hfil Zoran Gruji\'c
\hfil EJDE--1999/48}
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\vbox {\eightrm\noindent\baselineskip 9pt %
Electronic Journal of Differential Equations,
Vol. {\eightbf 1999}(1999) No.~48, pp. 1--8.\hfill\break
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\hfil\break ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)}
\footnote{}{\vbox{\hsize=10cm\eightrm\noindent\baselineskip 9pt %
1991 {\eighti Subject Classification:} 35Q30, 76D03. \hfil\break
{\eighti Key words and phrases:} Navier-Stokes equations, singular
set, turbulence. \hfil\break
\copyright 1999 Southwest Texas State
University and
University of North Texas.\hfil\break
Submitted
August 20, 1999. Published December 3, 1999.} }
\bigskip\bigskip
\centerline{ON THE SMALLNESS OF THE (POSSIBLE) SINGULAR SET IN SPACE}
\centerline {FOR 3D NAVIER-STOKES EQUATIONS}
\medskip
\centerline{Zoran Gruji\'c}
\bigskip\bigskip
{\eightrm\baselineskip=10pt \narrower
\centerline{\eightbf Abstract}
We utilize $L^\infty$ estimates on the complexified solutions of 3D
Navier-Stokes equations via a plurisubharmonic measure type maximum principle
to give a short proof of the fact that the Hausdorff dimension of the
(possible) singular set in space is less or equal 1 assuming chaotic,
Cantor set-like structure of the blow-up profile.
\bigskip}
\bigbreak
\centerline{\bf 1. Introduction} \medskip\nobreak
The problem of global regularity for 3D Navier-Stokes equation (NSE) is one
of the most challenging problems in the mathematical theory of
fluid dynamics. Since the fundamental work of Leray [L] in 1930's,
we know the existence of global weak solutions; however, the
existence of strong (regular) solutions is known only locally in
time. Some partial regularity results appeared already in the
Leray's work - later on, various partial regularity results were
obtained in [FT1], [Sch1], [Sch2], [CKN] and [St]. The best
result up to date is in [CKN] - it implies that for every $T>0$,
one-dimensional Hausdorff measure of the singular set in $\Omega
\times (0,T)$ is $0$. If we are looking at a snapshot, i.e. at
the singular set in space for some fixed singular time $T_s$, we
do not have a better estimate. The best we can say is again that
one-dimensional Hausdorff measure of the singular set $S_{T_s}$ in
$\Omega \times \{T_s\}$ is $0$. The proof of the [CKN] result is
based on a local theory - local blow-up estimates on families of
shrinking space-time cylinders. The main tools in the proof are
localized energy inequality, local interpolation and localized
estimates on the pressure. A simplified proof using essentially
the same tools appeared recently in [Li].
In this paper, we present a completely different, short proof of the fact
that $d_H (S_{T_s}) \le 1$ ($d_H$ denotes the Hausdorff dimension
of a set) assuming chaotic, Cantor set-like structure. Instead of
developing local theory, we utilize $L^\infty$ estimates on the
complexified solutions via a plurisubharmonic measure type maximum
principle. In fact, we prove a more general result, namely that
$u \in L^\infty (0,T_s; L^\alpha_w(\Omega)),$ coupled with the
Cantor set-like geometry implies that $d_H (S_{T_s}) \le 3 -
\alpha,$ for all $2 \le \alpha < 3$. We would like to point out
that chaotic structure of the blow-up profile is a physically
interesting case in the sense that some theories explain 3D
turbulence via the existence of a chaotic singular set [L], [M].
The paper is organized as follows. In Chapter 2, we recall some analyticity
properties of 3D NSE, as well as some basic estimates. Chapter 3 contains a
plurisubharmonic measure-type maximum principle, and Chapter 4 the main result.
\bigbreak
\centerline{\bf 2. Some known properties of solutions} \medskip\nobreak
We consider normalized (unit viscosity, period = $2 \pi$) NSE in
$\Omega = [0,2 \pi]^3$ with periodic boundary conditions and potential force.
$$ \displaylines{
{{\partial u} \over {\partial t}} - \triangle u + (u \cdot
\nabla)u + \nabla \pi = 0 \,, \cr
\hfill \nabla \cdot u = 0 \hfill\llap{(2.1)} \cr}
$$
for $(x,t) \in \Omega \times (0, \infty)$, supplemented with
the initial condition
$$ u(x,0) = u_0(x), \ x \in \Omega , $$
where the ${\Bbb R}^3$-valued function $u$ is the velocity and the
${\Bbb R}$-valued function {$\pi$ is the pressure. Also, we
require that
$$ \int_{\Omega} u(x,t) \ dx = 0, \ t \ge 0, \eqno(2.2)$$
and $\nabla \cdot u_0 = 0$.
A self-contained presentation of various aspects of the mathematical
theory of the NSE can be found in [CF].
The basic energy estimate
for the NSE is obtained (formally) multiplying the equations by
$u$ in $L^2(\Omega)$. Integrating by parts, utilizing $\nabla
\cdot u = 0$ and applying Poincar{\'e} inequality one arrives at
$$ \|u(t)\|_{L^2}^2 \le \|u_0\|_{L^2}^2 e^{-ct} \,, \eqno(2.3)$$
for all $t>0$, i.e. $L^2$ norm decays exponentially in time. For a
weak solution, exponential decay is valid only for a large enough
$t$; however, $$ \|u(t)\|_{L^2} \le \|u_0\|_{L^2} \eqno(2.3)'$$ is
valid {\it a.e.} in $t$.
Let now $M, t > 0$ and denote by $\Omega_t(M)$ a super-level set
$$ \{ x \in \Omega : \ |u(x,t)| \ge M \} . $$
Then, $(2.3)'$
implies the following weak-$L^2$ estimate on any weak solution of
(2.1). $$ \lambda^3 (\Omega_t(M)) \le { {\|u_0\|_{L^2}^2} \over
{M^2} } \,, \eqno(2.4)$$ for all $M > 0$, {\it a.e.} in $t$, where
$\lambda^3$ denotes Lebesgue measure on ${\Bbb R}^3$. }
The following theorem is
an $L^\infty$ description of local in time analytic smoothing of 3D NSE.
$H^1$- version was previously obtained in [FT2] via Gevrey-class
technique.
\proclaim{Theorem 2.1 [GK]}. Let $T=1/\big( {c {\|u_0\|^2_{L^\infty}}
(1+ \log _+{\|u_0\|_{L^\infty}})^2}\big)$.
Then, the solution $u$ of (2.1), (2.2) on (0,T) satisfies the following
property: for every
$t \in (0,T), \ u$ is a restriction of an analytic function
$u(x,y,t) + i v(x,y,t)$ in the region
$$ {\cal R}_t = \{ x+iy \in {\Bbb C}^3 : |y| \le c^{-1} t^{1/2} \}. $$
Moreover, there exists an absolute constant $K$ such that
$$ \|u(\cdot,y,t)\|_{L^\infty} + \|v(\cdot,y,t)\|_{L^\infty}
\le K {\|u_0\|_{L^\infty}}\,, $$
for $t \in (0,T)$ and $(x,y) \in {\cal R}_t$.
\bigbreak \centerline{\bf 3. A plurisubharmonic measure type
maximum principle in ${\Bbb C}^n$} \medskip\nobreak
The following ${\Bbb C}^n$ version of ``2-constants" theorem for product
domains follows from a general theorem that is expressed in terms of
plurisubharmonic measures [Ga] and the fact that for product domains
plurisubharmonic measure can be expressed in terms of one-dimensional harmonic
measures [GaKa].
\proclaim{Theorem 3.1}. Let $D_1, D_2, \dots, D_n$ be open sets in $\Bbb C$,
and let $E_1, E_2, \dots , E_n$ be such that
$E_i \subset \partial D_i$, for $i=1,2, \dots ,n$. Denote by
$\omega_{E_i}$ harmonic measures of $E_i$ with respect to $D_i$,
and assume that $f$ is a bounded analytic function on $D = D_1
\times D_2 \times \dots \times D_n$ satisfying the following
property:
$$\|f\|_{L^\infty (E_i)} \le M_2,$$
for $i=1,2, \dots ,n$, and
$$ \|f\|_{L^\infty (\complement E_i)} \le M_1,$$
for $i=1,2,\dots ,n$ (the inequalities are in the sense of $\limsup$
through the interior of $D_i$). Then
$$ |f(z)| \le M_2^{\inf (\omega_{E_1}(z),
\omega_{E_2}(z), \dots , \omega_{E_n}(z))} M_1^{1 - \inf
(\omega_{E_1}(z), \omega_{E_2}(z),
\dots , \omega_{E_n}(z))}, $$
for all $z \in D$.
\bigbreak
\centerline{\bf 4. Uniform Cantor sets and the main result} \medskip\nobreak
We start with a standard construction of a uniform one-dimensional
Cantor set (c.f. [F]).
Let $m \ge 2$ be an integer and $s > 1$. Start with an interval $[0,L]$,
and then construct a uniform Cantor set $C_{m,s}$ inductively in the following
way. In every step, each basic interval $I$ is replaced by $m$ equally spaced
subintervals of lengths ${1 \over {sm}} I$, the ends of $I$ coinciding with the
ends of the extreme subintervals. The limit set is $C_{m,s}$.
Using standard techniques, one can prove the following result.
\proclaim{Proposition 4.1}. Let $C_{m,s}$ be an one-dimensional uniform
Cantor set with parameters $m \in {\Bbb N}, \ m \ge 2$ and
$s > 1$. Then
$$ d_{H} C_{m,s} = {\log m \over {\log s + \log m}}. $$
\noindent{\bf Remark 4.2} \ For every fixed $m$, the range of Hausdorff dimension
of $C_{m,s}$ is $(0,1)$, i.e. it covers all fractal dimensions.
\medskip
We will construct our three-dimensional Cantor sets as products of uniform
one-dimensional Cantor sets.
The estimate in Theorem 3.1 implies that the worst case scenario is when all
component sets have the same harmonic measure, and so we will assume that all
three one-dimensional Cantor sets in the product are of the same type. Hence,
our singular set in space $S_{m,s}$ will have the form
$$S_{m,s} = C_{m,s} \times C_{m,s} \times C_{m,s}. \eqno (4.1)$$
Although Hausdorff dimension of a product is not always equal to the sum of
Hausdorff dimensions of the components, it is true if the Hausdorff and the
upper box dimensions of the component sets coincide. One can easily check that
for one-dimensional uniform Cantor sets, and thus Proposition 4.1 implies the
following.
\proclaim{Proposition 4.3}. Let $S_{m,s}$ be the set defined in (4.1). Then
$$ d_{H} S_{m,s} = {3 \log m \over {\log s + \log m}}. $$
\noindent{\bf Remark 4.4} \ Two-parameter family of the sets $S_{m,s}$ is a
reasonable model for a chaotic singular set - the Hausdorff dimension has
the range $(0,3)$.
A weak-type estimate (e.g. (2.4)) imposes a decay rate on the super-level
sets $\Omega_t(M)$. On the other hand, we assumed a chaotic structure of the
singular set which imposes a dispersion of the sets $\Omega_t(M)$. The idea is
to combine these two properties via Theorem 3.1 - shortly, harmonic measure
should eliminate a blow-up scenario in which the singular set is too chaotic,
i.e. if $d_H S_{m,s}$ is too big.
Before we precisely formulate our assumption on the geometry of the blow-up
profile, we recall the following local in time existence and uniqueness result.
\proclaim{Theorem 4.5 [L]}. Let $\nabla u_0 \in L^2(\Omega)$. Then,
there exists $T^*(\|\nabla u_0\|_{L^2}) > 0$ such that (1.1), (1.2) has a
unique regular solution on $(0,T^*)$.
Start with the initial data $u_0, \|\nabla u_0\|_{L^2} < \infty$, and let
$T_s \ge T^*$ be the first (possible) singular time. Assume that $S_{T_s} =
S_{m,s}$, and consider the following build-up of the flow compatible with the
standard construction of $S_{m,s}$.
{\it Let $0 < \epsilon < T_s$, and let $[T_s - \epsilon, T_s) =
\cup_{k=1}^{\infty} I_k$, where $\{I_k\}_{k=1}^{\infty}$ is a disjoint
decomposition of $[T_s - \epsilon, T_s)$ in the intervals $I_k = [a_k,b_k)$,
$(b_k-a_k) \rightarrow 0, \ k \rightarrow \infty$.
Denote by $S^k_{m,s}$ the kth generation in the standard construction of
$S_{m,s}$, and assume that for $\tau \in I_k$,
$$\Omega_{\tau}(M) \subset S^k_{m,s}, \eqno(A1)$$
and
$${1 \over c} ({1 \over s})^k \le {\lambda}^1 ( {\Pi}^i (\Omega_{\tau}(M)) ),
\eqno(A2)$$
$i=1,2,3$, all $M$ satisfying
$${1 \over K^4}\|u(\tau)\|_{L^\infty} \le M \le {1 \over K^3}
\|u(\tau)\|_{L^\infty}. $$
Above, $c \ge 1$ is a suitable absolute constant, $K > 1$ is the absolute
constant from Theorem 2.1, ${\lambda}^1$ denotes $1D$ Lebesgue measure, and
${\Pi}^i$ the projection on the ith coordinate.
}
\medskip
\noindent{\bf Remark 4.6} \ $ 2 \pi (1/s)^k$ is the linear measure of
the kth generation in the standard construction of $S_{m,s}$.
\medskip
We are now ready to state the main result.
\proclaim{Theorem 4.7}. Let $\nabla u_0 \in L^2(\Omega)$, and let $T_s$ be the
first (possible) singular time. Assume that the blow-up profile is given by
(A1)-(A2), and $u \in L^\infty (0,T_s; L^\alpha_w(\Omega))$, for some $2 \le
\alpha < 3$. Then
$$ d_H S_{T_s} \le 3 - \alpha.$$
\noindent{Remark 4.8} \ $u \in L^\infty (0,T_s; L^2_w(\Omega))$ is not an
assumption - it follows from (2.4).
\medskip\noindent
{\it Proof:} \ We argue by contradiction. Assume that
$ d_H S_{T_s} = 3 - \alpha + \eta$, for some $0 < \eta < \alpha$. Then, a simple
calculation utilizing Proposition 4.3 implies the following relation between
the parameters $s,m$,
$$s={(sm)}^{(\alpha - \eta) / 3}. \eqno(4.2)$$
Translating the estimate from Theorem 2.1 in time, we obtain that for any
$t \in (0,T_s)$, the solution $u$ is a restriction of an analytic
function $u=u+iv$ with the uniform radius of analyticity at time
$$ \tau (t) = t + {c \over {{\|u(t)\|^2_{L^\infty}}
(\log \|u(t)\|_{L^\infty})^2}} \eqno(4.3)$$
at least
$$ \rho (\tau (t)) = {c \over {{\|u(t)\|_{L^\infty}}
(\log \|u(t)\|_{L^\infty})}} . \eqno(4.4)$$
Also,
$$\|u(\tau (t))\|_{L^\infty ( \{z=x+iy \in {\Bbb C}^3: \ |y| \le \rho
(\tau (t)) \} )} \le K \|u(t)\|_{L^\infty}. \eqno(4.5)$$
Let $t \in [T_s - \epsilon,T_s)$ be an ``escaping time",
i.e. $\|u(t')\|_{L^\infty} > \|u(t)\|_{L^\infty}$, for all $t < t' < T_s$,
and define
$$ M(t) = {1 \over {K^3}} \|u(t)\|_{L^\infty}.$$
Consider now $\tau = \tau (t)$, where $\tau (t)$ is given by (4.3). Then,
$\tau \in I_k$ for some $k \in {\Bbb N}$, and one can easily check (using (4.5)
and the fact that $t$ is an escaping time) that $M(t)$ satisfies both
inequalities required in our geometric assumption.
Hence, $(A1)-(A2)$ coupled with $u \in L^\infty (0,T_s; L^\alpha_w(\Omega))$
(we can choose an escaping time $t$ such that $L^\alpha_w$ bound holds at
$\tau$) imply
$$({1 \over s})^k \le c {1 \over {M(t)^{\alpha / 3}}}.\eqno(4.6)
$$
Inserting (4.2), we arrive at
$$({1 \over c} M(t))^{\alpha \over {\alpha - \eta}} \le (sm)^k. \eqno(4.7)
$$
Since ${\alpha \over {\alpha - \eta}} > 1$, for every $c^* > 0$, there exists
$M^* (c^*; \alpha, \eta) > 0$ such that
$$c^* M(t) \log M(t) \le (sm)^k, \eqno(4.8)$$
for all $M(t) \ge M^*$. $c^*$ will be chosen later in the proof. Notice that we can
always choose an escaping time $t$ such that $M(t)$ is large enough.
To be able to successfully apply Theorem 3.1, harmonic measure
$$\omega_{\Pi^1 ((\Omega_{\tau (t)}(M(t)))_{per})}(w),
\eqno(4.9)$$ with respect to $$D^1_{\tau (t)} = \{z_1=x_1+iy_1 \in
{\Bbb C} : \ 0 \le y_1 \le \rho(\tau (t)) \}$$ (we could work with
any coordinate projection), computed at
$$w \in \{z_1=x_1+iy_1 \in {\Bbb C}: \ y_1 = {\rho(\tau (t)) \over 2} \}$$
should stay
uniformly bounded away from ${1 \over 2}$.
Since
$$ \rho (\tau (t)) = {c \over {{\|u(t)\|_{L^\infty}} (\log
\|u(t)\|_{L^\infty})}} = {c \over {K^3 M(t) \ \log (K^3 M(t))}}
\ge {c \over {M(t) \ \log (M(t))}}, \eqno(4.10)$$ for $M(t)$ large
enough, and since the length of an interval in $S^k_{m,s}$ is
$$ 2\pi ({ 1 \over {sm}})^k , $$
an elementary harmonic measure
computation (taking into account $(A1)$) implies that it is enough
to require that
$${M(t) \log M(t) \over (sm)^k} \le {1 \over c^*}, \eqno (4.11)$$
for a sufficiently large
$c^*(m,s)$ (this is true for all $k \ge k^*(s)$, and we can always
assume that $k$ is large enough). More precisely, there exists
$c^*(m,s)$ such that (4.11) implies
$$\omega_{\Pi^1 ((\Omega_{\tau
(t)}(M(t)))_{per})} (w) \le {1 \over 4} , \eqno(4.12)$$
for all $w\in \{z_1=x_1+iy_1 \in {\Bbb C}: \ y_1 = {\rho(\tau (t)) \over 2}
\}$.
Harmonic measure condition (4.11) is equivalent to (4.8), and hence
satisfied for an appropriate escaping time $t$.
Consider now positive part of the domain of analyticity of $u(\tau (t))$,
$$D_{\tau (t)} = \{z=x+iy \in {\Bbb C}^3 : \ 0 \le y_i \le
\rho(\tau (t)), \ i=1,2,3 \}.$$
Then, the estimate (4.5) implies
$$\|u(\tau (t))\|_{L^{\infty}(\partial D_{\tau (t)})} \le K^4
M(t). \eqno(4.13)$$ Also, $$|u(\tau (t), z)| \le M(t),
\eqno(4.14)$$
for $z \in G_{\tau (t)} = \partial D_{\tau (t)} -
\{ \{z=x+iy \in {\Bbb C}^3 : \ y_i= \rho (\tau (t)), \ i=1,2,3 \}
\cup (\Omega_{\tau (t)} (M(t)))_{per} \} $.
Since the harmonic measure of the projection of
$B_{\tau (t)} = \partial D_{\tau (t)} - G_{\tau (t)}$ is by (4.12) less or
equal to ${3 \over 4}$ uniformly in a set containing
$P_{\tau (t)} = \{z=x+iy \in {\Bbb C}^3 : \ y_i =
{\rho(\tau (t)) \over 2}, \ i=1,2,3 \}$, Theorem 3.1 yields
$$ \|u(\tau (t)\|_{L^\infty (P_{\tau (t)})}
\le { (K^4 M(t)) }^{3/4} { M(t) }^{1/4} = \|u(t)\|_{L^\infty}. \eqno(4.15)$$
By symmetry ($u(\overline z) = \overline {u(z)}$), the same estimate holds on
negative part of domain $D_{\tau (t)}$ as well, and thus the
maximum principle gives
$$ \|u(\tau (t))\|_{L^\infty} \le
\|u(t)\|_{L^\infty}, \eqno(4.16)$$
contradicting $t$ being an escaping time. \hfil $\square$\medskip
\noindent{\bf Acknowledgments}\
The author thanks Professors L. Caffarelli and C. Foias for their interest
and stimulating discussions.
\bigbreak
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\bigskip
\noindent Zoran Gruji\'c \hfill\break
Department of Mathematics \hfill\break
University of Texas \hfill\break
Austin, TX 78712, USA \hfill\break
e-mail: grujic@math.utexas.edu
\bye