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\numsec=1\numfor=1%\pgn=1
\centerline{\bf SOME RIGOROUS RESULTS ABOUT 3D NAVIER-STOKES.}
\vskip1.truecm
\centerline{Giovanni Gallavotti}\vskip0.5truecm
\centerline{Dipartimento di Fisica}
\centerline{Universit\`a ``La Sapienza'' }
\centerline{P.le A. Moro 2, 00185 Roma, Italia}
{\bf Plan}\footnote{$^*$}{
{\it Text of two lectures delivered at the Les Houches NATO-ASI meeting
``Turbulence in spatially ordered systems'', january, 1992}.}
There are very few rigorous results on 3D Navier-Stokes equations.
Here I select and review a few among the most remarkable:
\item{1.} Leray's local theorem
\item{2.} Scheffer's local theorem
\item{3.} Caffarelli, Kohn, Nirenberg regularity theorem.
I provide an interpretation of the scaling ideas in 1, 2, 3 with
particular attention to the similarity of 3) with the theory of
relevant variables in the renormalization group methods. The
connection of 3) with Kolmogorov- Obuchov theory (quoted as (KO) below)
is attempted.
Finally I present a comment inspired by Chorin's ideas on the
vorticity filaments dynamics (3D). To simplify a subject, quite
intricate in itself, only a fluid in a periodic container $\O$
is considered.
\vskip0.5truecm
{\it Key words:\rm\ Fluid dynamics, Renormalization group, Kolmogorov Obuchov
theory, vorticity filaments, Hausdorff dimension, Non linear PDE's.}
\vskip0.5truecm
{\bf \S1. Leray's formulations of the theory.}
\vskip0.5truecm
The first disturbing and striking feature of 3DNS is that only a {\it
non constructive} existence proof of a solution is available for
general smooth data and forces. This amounts to saying that, unless
one invokes statements about something being ``physically obvious'',
the numerical experiments concerning turbulence phenomena are in fact
experiments on the program generating them, rather than on the
idealized NS fluid (this fact does not have, necessarily, negative
implications, but it has to be kept in mind).
The NS equations are, with usual or obvious notations:
%
$$\eqalign{
&\dot{\V u}+(\W u\cdot\W\dpr)\V u=
-\V\dpr p+\n\D{\V u}+\V f\cr &\V\dpr\cdot\V u=0\
,\quad\ig_\O\V u=\V 0\ ,\quad\O=[0,L]^3_{\hbox{\sette periodic}}\cr
&\V u|_{t=0}=\V u_0\cr}\Eq(1.1)$$
%
The force $\V f$ will be supposed time independent and the force as
well as the initial datum $\V u_0$ will be supposed very smooth; $\V f$,
$\V u\in C^\io(\O)$, $\ig\V f=\V0$, $\ig\V u_0=\V0$, $\V\dpr\cdot\V f=0$.
There is {\it nothing to gain nor to loose\/} in weakening such
regularity assumptions (e.g. most of what follows could be adapted to
$u_0\in H_2(\O)$, \ie to very wild $\V u_0$'s, and one would
just have to climb on mirrors to show that nothing really changes.
To the initial datum and to $\V f$ we associate a few characteristic
parameters:
%
$$E_0=\ig_\O|\V u_0(\V x)|^2\ ,\quad
D_1=\ig_\O|\W\dpr\,\V u|^2,\quad F_j=\ig_\O|\V\dpr^j\W f|^2\Eq(1.2)$$
%
and:
%
\vglue-3.pt
$$\vbox{\halign{#&\ \strut#\hfil&#&\ #\hfil&#&\ #\hfil\ &#&\ #\hfil\ &#\cr
&\omit&&\omit&&\omit&&\omit&\cr
&$T$&& =&&$ L^2 \n^{-1}$&&(viscous time scale)&\kern0.8truecm\cr
&$V$&&=&&$ \n L^{-1}$&&(viscous velocity scale)&\cr
&$V_1 $&&=&&$ (D_1 L^{-1})^{1/2}$&&(initial velocity variation
scale)\qquad\qquad\qquad\qquad\qquad&\hfil$\eq(1.3)$\cr
&$W_1$ &&=&&$ (L F_1)^{1/4}$&&(forcing velocity scale)&\cr
&$R$&&=&&$(V^2_1 + W^2_1)^{1/2}/ V$&&(Reynold's number)&\cr
&\omit&&\omit&&\omit&&\omit&\cr}}$$
\vglue-4.pt
Note that, if for instance $\V f=\V 0$, the number $R=V_1/V=\n^{-1}
\Big(L \ig|\W\dpr\V u|^2\Big)^{1/2}$ is the ``usual''
$(\d u)L\n^{-1}$ where $\d u=V_1$ is a ``velocity variation'' on scale $L$,
\ie a velocity formed with an average of $\W\dpr\,\V u$ over the whole
$\O$.
Other possibilities could be considered to define the Reynold's number
(for instance one could set: $ R = L^{-2}\ig_\O |\V\dpr\W u| d\V\x$)
but, as we shall see, the above \equ(1.3) is a good one as it allows us
to make mathematically rigorous many statements formulated by using the
concept of Reynold's number and usually believed to be heuristically
true.
The (KO) length scale, frequency and number of degrees of freedom
can be likewise given a precise definition in terms of the quantities
in \equ(1.3):
%
$$L_k={L\over R^{3/4}}\ ,\quad
\n_k={V_1+W_1\over L}R^{3/4}\ ,\quad N_k={4\pi\over 3}R^{9/4}\Eq(1.4)$$
%
(see [LL], \S31,\S32, for a concise exposition of (KO)).
The mathematical problems of \equ(1.1) are better illustrated
if one uses the following auxiliary ``regularized'' equations:
%
$$\dot{\V u}+\langle\W u\rangle_l\cdot\W\dpr\,\V u=\n\D\V u-\V\dpr p+\V
f + \hbox{\sette supplementary and initial conditions as in
\equ(1.1)}\Eq(1.5)$$
%
where the field $\V u$ is transported by the field:
%
$$\langle \W u\rangle_l(\V x)=l^{-3}\ig_{R^3}\l(l^{-1}(\V x-\V y))
\,\V u(\V y))d\V y\Eq(1.6)$$
%
\ie by an ``average velocity'' on scale $l$, rather than by $\V u$
itself. Here $\l\in C^\io(R^3)$, $\l\ge 0$, $\ig_{R^3}\l=1$, and
$\l$ vanishes outside a finite sphere; the field $\V u$ is extended
periodically outside $\O$.
We call \equ(1.5) the ``Leray's regularization'' of the Navier-Stokes
equations.
\vskip0.5truecm
\noindent{\it Remark:}
If $\V u(\V x)=L^{-3/2}\sum\limits_{k\not=0}\V\gamma_{\V k}e^{i\V {kx}}$;
$\ \V\gamma_{\V k}={\overline{\V \gamma}_{-\V k}}$, $\,\V k\cdot\V\gamma_{\V k}=0$,
the \equ(1.5) becomes:
%
$$\dot{\V\gamma}_{\V k}=-\n\V k^2\V\gamma_{\V
k}+\V f_{\V k}-iL^{-3/2}\sum_{\V k_1+\V k_2=\V k}\hat\l(l\V
k_1)\,\W \g_{\V k_1}\cdot\W k_2{\bf\Pi}_{\V k_2}\,
\V\g_{\V k_2}\Eq(1.7)$$
%
where $({\bf \P}_{\V k})_{ij}=(\d_{ij}-k_ik_j\V k^{-2})$ is the
projection operator on the plane orthogonal to $\V k$, and the pressure
is given in terms of $\V u$ by:
%
$$p=-\sum_{i,j=1}^3\D^{-1}\dpr_i\dpr_j\,(u_iu_j)=
-L^{-3}\sum_{i,j}{k_i k_j\over \kk^2}e^{i\V k\cdot\V
x}\sum_{\kk_1+\kk_2=\kk}\g_{\kk_1i}\g_{\kk_2j}\Eq(1.8)$$
%
The \equ(1.7) is ``different'' from the ``sharp momentum cut off'' NS
equation with cut off at $l^{-1}$: the latter corresponds to setting
$\hat\l\equiv 1$ if $|\V k_1|\le l^{-1}$ and $\hat\l\equiv 0$ if $|\V
k|>l^{-1}$ and this means that $\l$ is oscillating and slowly decaying
so that in particular the key inequality:
%
$$|\langle\W u\rangle_l(\V x)|\le\max_{\V y}|\W u(\V y)|\,
\qquad{\rm for\ all\ } l\Eq(1.9)$$
%
does not necessarily hold.
The main local result of Leray is the following:
\vskip0.5truecm
\noindent THEOREM 1:
\item{1)}{\sl Equation \equ(1.5), \ie the
regularized NS equation on scale $l>0$, with
initial data $\V u_0\in C^\io(\O)$, and stationary force $\V f\in
C^\io(\O)$ admits a $C^\io$ solution, denoted $\V u^l$, for all times
$t$ and verifies, for all $T_0>0$, the following "energy" and
"vorticity" bounds:
%
$$\ig_\O|\V u^l(\V x,t)|^2d\V x\le\ov E_0\ ,\quad
\n\ig_{[0,T_0]\times\O} d\th\,d\V\x\,|\V\dpr\W u|^2
\le 2\ov E_0+T_0\sqrt{\ov E_0F_0}\equiv \n T_0 S(T_0)\Eq(1.10)$$
%
with $\ov E_0=\max\left(E_0,\left({L\over 2\pi}\right)^2{F_0^{1/2}\over
\n} \right)$, $F_0=\ig_\O|\V f|^2$, and $S(T_0)$ can be bounded in terms
of the initial Reynolds number $R_0$ (\eg $S(T_0)\le{T V^3\over
2\pi^2}R_0\Big(R_0^2+\big({L\over T_0 V}\big)^2\Big)$, se \equ(1.3)).
\item{2)} There exist dimensionless constants $\LL$, $\LL_{pq}$ such
that:
%
$$|\dpr^p_t\V\dpr^q\W u^l|\le\LL_{pq}VR \left({t\over
T}\right)^{-{1\over4}-p-2q} ,\quad\hbox{for all } 0<{t\over T}\le\LL
R^{-4}\Eq(1.11)$$
%
where $V,T$ are the characteristic scales in \equ(1.3).
\item{3)} The limit $\V u^0$ of $u^l$,
and of all its derivatives, as $l\to 0$,
exists uniformly for $0\le t\le T\LL R^{-4}$ and defines a $C^\io$
solution of 3DNS.
\item{4)} If $\V u_0$ is only assumed to be such that $\|\V
u_0\|_2<\io$, $\|\W\dpr\V u_0\|_2<\io$ the regularized equation admits a
solution verifying:
%
$$\align{
\hbox{\rm a)}&\kern0.5truecm\hbox{bounds \equ(1.11) for}\ t > 0&
\kern0.7truecm l\ge0&\cr
\hbox{\rm b)}&\kern0.5truecm \ig^{T_0}_0||\V u^l(t)||^2_2d t** 0$. If one wanted more uniform
bounds one just could not obtain them in terms of $R$ alone. In fact
Leray proved a large number of corollaries of the technique of proof
that he used: and among them a bound uniform as $t\to\io$ of $\V u^l$, in
terms of $R$ {\it as well as} of the maximum of $\V u_0$ and $\V f$, see
the next remark.
5) If one added the assumptions that $V_1$ and $\max_{\V x}|\V u(\V
x_0)|$ are small enough then one could prove $l$-uniform bounds on $\V
u^l$ and its derivatives for {\it all\/} times. This yields a global
existence, uniqueness and regularity theorem for \equ(1.1). The drawback
is that the conditions are so stringent (\ie the dimensionless
constants appearing in the bounds so big) that, whenever they hold, one
would consider the fluid to be in a ``very laminar'' regime, hence
uninteresting.
6) The proof of the above theorem is very simple. Such a theorem
cannot, however, be proved with a sharp cut off in momentum, see comment to
\equ(1.7), because the validity of the inequality \equ(1.9) plays a key
role.
7) The Leray's cut off equations as well as the sharp cut off ones have
solutions still verifying the energy conservation; this leads quite
easily to \equ(1.10). But {\it neither\/} verifies, in the $0$
viscosity limit, the vorticity conservation laws formally valid for
\equ(1.1). We refer here to the vorticity, transported by a closed
contour $\gamma (t)$ moving with the velocity field,
$\Gamma(\gamma(t))=\oint_{\gamma(t)}\V u\cdot d\V l$ which is a
constant in the $0$ viscosity limit: ``Thomson's circulation
theorem''.
This is an extremely important point as no regularization appears to be
known allowing a proof of the above theorem 1 and preserving the
above circulation theorem.
8) The second \ap bound in \equ(1.10) can be written as a property of the
``instantaneous Reynolds number'', $R(t)=(V_1(t)^2+W^2_1)^{1/2}/V$. One
finds:
%
$$\eqalignno{
&\ig^{T_0}_0{d\th\over T_0}R^2(\th)\equiv\ig^{T_0}_0
{d\th\over T_0}\, (L^{-1}\ig_\O|\W\dpr \V u|^2d^3\V\x+W^2_1)V^{-2}\le\cr
&\le\left(W^2_1+\left({2\ov E_0\over T_0}+\sqrt{\ov
E_0F_0}\right){1\over L\n}\right){1\over V^2}
\equiv B^2<\io&\eq(1.13)\cr
&B^2\le {1\over 2\p^2}R_0\left(R_0^2+\big({L\over VT_0}\big)^2\right)
\buildrel\hbox{\sette def}\over=R^2_0B^2_0\cr}$$
%
directly from the \equ(1.10), bounding everything in term of $R$ for
simplicity.
9) The above theorem is not as well known as its soft, ``abstract
nonsense'', consequence which we proceed to discuss, after setting up
a formal definition.
The following definition is so natural that some might be led to cede
to the temptation of eliminating the qualifier ``weak'':%
\vskip0.5truecm
\noindent DEFINITION: {\sl A ``weak solution'' of NS with initial
condition $\V u_0\in L_2(\O)$ is a solution of
\equ(1.7) with $\hat\l=1$ and with finite energy. In other words it is a
sequence of functions $t\to\V\g_{\V k}(t)$, differentiable almost
everywhere in $t$, and such that;
%
$$\eqalignno{
\V\g_{\V 0}&=0\ ,\ \V\g_{\V k}(t)\equiv\ov{\V\g_{-\V
k}(t)}\ ,\quad \sum_{\V k\not=\V 0}|\V\g_{\V k}(t)|^2\le\ov E_0\
,\quad\V k\cdot\V\g_{\V k}\equiv 0\cr
\dot{\V\g}_k&=-\n\V k^2\V\g_{\V k}-iL^{-3/2}\sum_{\V k_1+\V k_2=\V k}
\W\g_{\V k_1}\cdot\W k_2\,{\bf \Pi}_{\V k}
\W\g_{\V k_2}+\V f_{\V k}&\eq(1.14)\cr
\V\g_{\V k}(0)&\equiv\V\g_{0\V k}=\hbox{ Fourier transform of }\V
u_0\cr}$$
%
for some $\ov E_0 > 0$.}
\vskip3.pt
\noindent {\it Remarks:}
1) We note that, if $\ov E_0<\io$, the r.h.s. of the second of
\equ(1.14) makes sense and is bounded, (by using the Schwartz
inequality) by:
%
$$\n\V k^2\sqrt{\ov E_0}+|\V k|\,\ov E_0+\sqrt{F_0}\Eq(1.15)$$
%
as one can see because $\W k_2\equiv\W k-\W k_1$ can be replaced by
$\W k$ in $\W\g_{\V k_1}\cdot\W k_2$ (by the incompressibility relation $\V
k\cdot\V\g_{\V k}=0)$ so that \equ(1.15) follows by bounding the transport
term via the Schwartz inequality. Hence it always makes sense to write
the NS equations, whenever an energy bound is assumed, in the weak
form.
2) The remark 1) is not sufficient (hence the name "weak") to write the
equations verified by a weak solution in the form \equ(1.1) because the
finiteness of the kinetic energy (or even of the dissipation) does not
even imply the (pointwise) existence of the derivatives appearing in
the equations.
3) In the case $\V f=\V0$ one expects that the fluid will slow down and
stop, because of the dissipation. In fact \equ(1.14) imply, easily:
%
$$||\V u(t)||^2_2\le e^{-\n(2\p L^{-1})^2t}||\V u_0||^2_2\Eq(1.16)$$
%
Such a "physically obvious" result is not known to hold if $L=\io$, \ie
in the case of a fluid filling the whole space: it is therefore a sign
of how little we know about the solutions of the NS equations.
4) It is then easy to see that the regularized equation in weak form
and with $\V u_0\in C^\io(\O)$, or more generally with finite intial
Reynolds number (\ie $\ig_\O|\W\dpr\,\V u|^2<\io$) admits a solution
verifying theorem 1, part 1) or 3) (not 2) which is not as soft: see
[L] or [G] for a proof). This implies that $\g^l_{\V k}(t)$ exist and
verify \equ(1.7) and, hence, \equ(1.15) (which also follows from
\equ(1.7) replacing $\hat\l$ by 1). But this means that $\g^l_{\V
k}(t)$ form a family of equicontinous equibounded functions (in the
parameter $l$). Hence trivially there is a convergent subsequence
converging to some $\V\g^0_{\V k}(t)$.
The vorticity bound can be used to check that, indeed, such a limit
verifies \equ(1.11) and to deduce the following:
%
\vskip0.5truecm
\noindent THEOREM 2 (Leray):
\vskip-0.5truecm
\\{\sl Under the same assumptions of
theorem 1 the 3DNS admits a weak solution, limit of regularized Leray
solutions and global in time.}
%
\vskip0.5truecm
\noindent{\it Remarks:}
1) The ``abstract nonsense'' proof of theorem 2 is quite easy (see [L]
or [G]). It is so soft that it does not even distinguish the Leray
regularization from the sharp cut off regularization and, hence, weak
solutions could also be constructed as limits of the sharp cut off
regularization solutions.
2) Thus we have an ``existence theorem'': but totally useless and
illusory. In fact {\it no\/} prescription has been provided to
identify the convergent subsequence. This is a deep problem of
principle as one {\it just cannot\/} write a computer program apt to
tabulate the subsequence nor to compute $\V\g^0_{\V k}(t)$ for fixed $t
> 0$ and with a prefixed precision $\e$ (for $t\gg\LL T R^{-4}$, (unless
the initial data and external force are so small and regular to verify the
laminarity conditions for global existence, see remark 5) to theorem 1,
of course).
3) Another bad aspect of weak solutions is that the energy balance is no
longer automatic. By multiplying both sides of \equ(1.1) by
$\f\V u$, where $\f\ge0$ is an arbitrary $C^\io$ function of $\V
x$, $t$ for $\V x\in\O$, $t\in(0,s)$, one finds:
%
$$\eqalign{
&{1\over2}\ig_\O d\V\x|\V u(\V\x,t)|^2\f(\V x,s)+\n\ig_{t\le
s}\ig_\O\f(x,t)|\V\dpr\W u|^2d\V x d t=\cr
&=\ig_{t\le s}\ig_\O\Big[{1\over2}(\f_t+\D\f)|\V u|^2+
|\V u|^2\langle\W u\rangle_l\cdot\W\dpr\f
+p^l\V u\cdot\V\dpr\f+\V f\cdot\V u\f\Big]\,dt\,d\V\x\cr}\Eq(1.17)$$
%
at least formally, if $p^l$ is the pressure of the approximate solution
given by the integral: $p^l=-\sum_{ij}\D^{-1}\dpr_i\dpr_j (u^l_i u^l_j$).
But in general the interchanges of integrals and limits will not be
allowed if the solution is just a weak solution: with the consequence
that a weak solution will not necessarily obey the non regularized
version of \equ(1.17) (with $\W u$ instead of $\langle\W u\rangle_l$).
It might, however, verify it with $\le$ replacing $=$: the latter form
of \equ(1.17) is called the ``energy inequality''. Thus we say:
%
\vskip0.5truecm
\noindent DEFINITION: {\sl A Leray solution of \equ(1.1) is a weak
solution of \equ(1.1) verifying also the
energy inequality defined (for all $\f\in C^\io(\O\times(0,s])$, $\f(\V
x,0)=0$, $\f\ge0$, and for all $s>0$) by:
%
$$\eqalign{
&{1\over2}\ig d\V\x\,|\V u(\V\x,s)|^2\f(\V x,s)+\n\ig_{t\le
s}\f(x,t)|\V\dpr\W u|^2d\V xd t\le\cr
&\le\ig_{t\le s}\ig_\O\Big[{1\over2}(\f_t+\D\f)|\V u|^2+
|\V u|^2(\W u\cdot\W\dpr\f+p^0\,\W u\cdot\W\dpr\f+\V
f\cdot\V u\,\f)\Big]\,d\V x\, dt\cr}\Eq(1.18)$$
%
if $p^0$ is the pressure of the weak solution: $p^0=-\sum_{ij}
\D^{-1}\dpr_i\dpr_j (u_i u_j)$.}
%
\vskip0.5truecm
\noindent {\it Remarks:}
1) If $\f$ is chosen to be the characteristic function of a sphere $|\V
x- \V x_0|$ sign, from time to time), (Scheffer).
\penalty-10000
The most well known theorem of Leray, on the above weak solutions,
is:
\penalty10000
\vskip0.5truecm
\penalty10000
\noindent THEOREM 3:
\vskip-0.3truecm
\\{\sl One can find a weak solution of the
\equ(1.1), \ie a solution to \equ(1.14), such that:
\item{\rm1)} It is a limit, as $l\to0$, of a subsequence of solutions of
the Leray's regularized NS equation, \equ(1.7).
\item{\rm2)} $\W\dpr\,\V u(t)\in L_2(\O\times[0,T_0])$ (for all $T_0$)
and verifies the second of \equ(1.10) as well as \equ(1.18);
\item{\rm 3)} For almost all $t$'s the velocity $\V u (t, \V x)$ is
$C^\io$ (in $t$ and $\V x$) to the right of $t$ for a short enough
($t$-dependent) time.}
\vskip0.5truecm
\noindent{\it Remarks:}
1) In fact all the weak limits of sequences of solutions of the
Leray's regularized equations are "Leray' solutions", \ie they verify
the above properties 1,2 of theorem 3.
2) Apart from some logical exercises necessary to deal with
``subsequences'' (\ie with non constructed objects, which therefore in
some sense do not really exist), the part of the theorem not dealing
with the energy inequality is a trivial consequence of the vorticity
bound in \equ(1.10). Recalling that the Reynolds number
$R$ is $(V_1^2 + W_1^2)^{1/2}/V$, see \equ(1.3), can be written in the
form \equ(1.13):
%
$$\ig^{T_0}_0R(t)^2d t<\io\quad\hbox{for all }T_0>0\Eq(1.19)$$
%
where $R(t) = (V_1 (t)^2 + W_1^2)^{1/2}/V$.
Hence, heuristically, \equ(1.10) implies that for almost all times the
Reynold's number is finite: $R(t) < \io$ and the (constructive part of)
Leray's theorem (points 1,2), implies that $\V u(t)$ is $C^\io$ for
$t\in(t_0,t_0+\LL TR(t_0)^{-4})$ for any $t_0$ for which $R (t_0) <
\io$; \ie for almost all $t$'s. See below for a more precise analysis.
3) The reason why the Leray's approximations play a special role is
that theorems 1,2 above hold only for Leray's regularized solutions:
therefore the weak solution in the above theorem has to be constructed
from Leray's method (not a serious limitation as, in any case, we are
talking of objects that, in principle, cannot be constructed). This
might be confusing as even the weak solutions obtained as limits of the
equations regularized with the sharp cut off verify \equ(1.14): but in
general they might not have the other properties.
4) The difficulty in making rigorous the heuristic argument in remark 1
is that, strictly speaking, from the above argument we do not know that
the weak solution after a time $t_0$ when $R(t_0) <\io$ coincides with
the smooth solution with the same $\V u (t_0)$ constructed by Leray's
local theorem. An abstract argument is necessary, however, to insure
this property: see remark 1) after the proof of theorem 3' below.
Such a delicate point arises simply because we have essentially lost
control on the construction of solutions and we are proceeding on
abstract grounds (in the axiom of choice realm) where every subtlelty
becomes possible and important.
5) The proof of \equ(1.18) follows from \equ(1.17) by passage to the
limit on a convergent subsequence $\V u^{l_j}$. The l.h.s. of \equ(1.17)
is (obviously, being a norm) upper semicontinous if $\V u^{l_j}\to\V
u^0$ weakly (\ie componentwise in the Fourier transform components). And
the r.h.s. is in general continous: hence the inequality \equ(1.18).
The latter continuity is a non trivial but standard consequence of two
important inequalities that will play a major role below, namely the
Sobolev and the Calderon- Zygmund inequalities, see \equ(2.10),
\equ(2.11) below. The proof of the continuity property is quite
interesting as it shows a few general regularity properties of the weak
solutions of the NS equations and it is reported in appendix B,
for completeness.
Most reasonable people will, {\it however}, refuse to talk about
properties of solutions for which no constructive algorithm is
provided. For such people one can just avoid introducing the theorems
2, 3 above and formulate them as uniform properties of the
approximating solutions.
For instance theorem 3 can be rephrased in the apparently less elegant
but far clearer form:%
\vskip0.5truecm
\noindent THEOREM 3':
\vskip-0.3truecm
\\{\sl For all $l>0$, $T_0>0$ the solution $\V u^l$
of Leray's regularized equation with initial Reynold's number $R_0$ and
the initial quantity $B_0^2$ (see [1.11]), verifies:
%
$$|\dpr^p_t\V\dpr^q\V u^l(\V x,t)|\le\LL_{pq}{R_0Vn\over
((t-t_0)/T)^{1/4+p+q/2}},\quad 0\le t-t_0\le{\LL T\over
n^4}R_0^{-4}\Eq(1.20)$$
%
for a set $\D_n$ of times $t_0\le T_0$ with complement in $[0,T_0]$
with measure:
%
$$\ig_{\D _n}{d\t\over T_0}\le{B_0\over n^2}\Eq(1.21)$$}
%
\noindent {\it Remarks:}
1) This is much better; it shows that the set of times where the solution
does not exceed by a factor $n$ the bounds it verifies in the happy initial
interval (where it can be really constructed by theorem 1, part 2)) has a
large measure as $n\to\io$
2) The above form has the advantage of making it clearer what we still
do not understand: no information is given about the location of
$\D_n$, ({\it it is an open problem}).
3) It also shows clearly that the Leray's solutions play a privileged role.
They are such that the boundedness of the Reynolds number implies a bound
on $\V u$ and its derivatives. A property which is not proved to hold
for weak solutions constructed starting with the sharp cut off equations.
4) The proof of theorem 3' is trivial and this shows that the aura of
mistery surrounding theorem 3 is entirely due to the logical problems
which usually accompany any use the axiom of choice (to construct
evanescent converging subsequences). Here is the proof:
%
\vskip0.5truecm
\noindent{\it Proof (of theorem 3'):} From theorem 1, \equ(1.10),
\equ(1.13) we see that:
%
$$\ig^{T_0}_0R_l^2(\t){d\t\over T_0}\equiv\ig^{T_0}_0{d\t\over
T_0}\left(\ig_\O|\W\dpr\V u^l|^2{d^3\V x\over
L}+W^2_1\right){1\over V^2}\le R_l^2B^2_0\Eq(1.22)$$
%
hence the set $\D_n$ of the points $\t$ where $R_l(\t) < nR_0$ has
complement with measure $\le B_0/n^2$. If $t_0\in\D_n$ then, by
theorems 1, 2, the \equ(1.20) holds! Had we used sharp cut
off regularizations the above argument would not have worked (because
2) of theorem 1 might not hold).
\vskip0.5truecm
{\it Remarks:}
1) The above proof is essentially all there is to say to get theorem 3
as well. In fact to prove theorem 3 it will be sufficient, if $R_l(t)$
is the Reynold's number of $\V u^l(t)$, to interpret $R(t)^2$ as
$R(t)^2=\liminf_{l\to0} R_l(t)^2$ and note that $R(t)^2\ge \ig_\O
\,|\W\dpr\,\V u(t)|^2$ (by Fatou's lemma). One has, however, to worry
about the following problem: how can one be sure that, if
$R(\t)^2<\io$, the weak solution limit of the approximating $\V u^l$ as
$l\to0$ is in fact converging to the strong (\ie $C^\io$- smooth)
solution existing in the interval $(\t,\t+\LL T R(\t)^{-4})$? for
completeness we discuss this (minor) point in appendix A.
We conclude the introduction with an important theorem, corollary of
the proof of Leray's theorem:
\vskip0.5truecm
\noindent THEOREM 4' (Serrin):
\vskip-0.3truecm
\\{\sl Let $t\to\V u^l(\V x,t)$ be a solution of the Leray's
regularized equations \equ(1.5), \equ(1.7). Then if in the
$r$-vicinity of $(\V x,t_0)$ it is $|u^l(\V x,t)\le C$ (\ie this holds
for $|\V x-\V x_0|0$, let $\d>0$
and consider the coverings $C_\d$ of $A$ with sets $F$ with positive
diameter $\le\d$. A ``best covering'' is one which minimizes
$\sum_{F\in C_\d}$ (diameter $F$)$^\a$ and one sets:
%
$$\m_\a(A)=\lim_{\d\to0}\,\inf_{C_\d}\sum_{F\in C_\d}
\hbox{(diameter $F$)}^\a\Eq(2.2)$$
%
It can be shown that the limit exists and for any $\a$ the function
$A\to\m_\a(A)$ is countably additive on the smallest family $\BB$
of sets containing the closed sets and invariant with respect to the
operations of complement and countable union, see [DS], III.9.47.
It is easy to check that given $A\in\BB$ there is $\a_c\ge0$ such that:
%
$$\eqalign{
&\m_\a(A)=+\io\quad\hbox{if}\quad\a<\a_c\cr
&\m_\a(A)=0\quad\kern0.5truecm\hbox{if}\quad\a>\a_c\cr}\Eq(2.3)$$
%
and $\a_c$ is called the ``Hausdorff dimension'', while, $\m_{\a_c}(A)$ is
the "Hausdorff measure".
The dimension of a smooth $k$- dimensional surface is $\a_c=k$. The
subsets of the interval $[0,1]$ consisting of the points which, when
expanded in base $q = 2,3\ldots$, do {\it not\/} contain the digit $1$
(such sets are the "Cantor sets" $C_q$) have Hausdorff dimension
$\a_c=\log_q(q-1)$ and $\a_c$-Hausdorff measure $1$, as well as length
zero (\ie zero Lebesgue measure).
There are sets $C$ in $[0, 1]$ with Hausdorff dimension 1 but zero length
as well as zero Hausdorff measure: \eg $C={\bigcup}_{q\ge3}C_q$. And,
therefore, the Hausdorff dimension and measure can provide interesting
properties of zero measure sets.
A simple, but conceptually important for the methods set up by the proof,
theorem is the following:
\vskip0.5truecm
\noindent THEOREM 5, (Scheffer):
\vskip-0.3truecm
\\{\sl The Hausdorff dimension of $S_0$, the
time singularity set of a Leray's solution of Navier Stokes equation, is
$\a_c\le{1\over2}$ and $\m_{1/2}(S)=0$.}
\vskip0.5truecm
\noindent {\it Remarks:}
1) The proof is given below as it illustrates a general method used to
convert \ap estimates into Hausdorff dimension estimates. The very
same method will be used, later, to study the space time singularities.
2) Clearly we are very far from $\a_c=0$ (if true).
3) Of course one could formulate theorem 5 without talking of the non
algorithmically constructible Leray's solutions. It would become a theorem
on Leray's approximations. We regard theorem 5 just as a short way to
formulate such results; the results on the approximations can be easily set
up from the following proof, but we omit the discussion (bowing to social
pressure, requiring easy to remember, possibly shocking, statements).
4) To simplify the situation we take, below, $\V f=\V0$: nothing
is really different when $\V f\not=\V0$.
\vskip0.5truecm
\noindent{\it Proof:} Let $t\in[0,T_0]$ ($T_0$ being an arbitrary
length of time), and suppose that for some $\s>0$:
%
$$\left({\s\over T}\right)^{-\a}\ig^t_{t-\s}d\th
R(\th)^20$ and
such that \equ(2.4) with $A=\LL^{1/2}T$ is violated, {\it i.e.}:
%
$$\limsup_{\s\to0}\left({\s\over T}\right)^{-1/2}\ig^t_{t-\s}d\th
R(\th)^2\ge T\LL^{1/2}\Eq(2.5)$$
%
otherwise $t$ would be a ``regularity time''.
We fix $\d_0>0$ and consider the covering $\FF$ of $S_0$ with {\it all}
the intervals $F$ violating \equ(2.4). By \equ(2.5) {\it every point
of $S_0$ is covered by such intervals with arbitrarily small diameter}.
In the latter situation a simple ``covering theorem'' (by Vitali)
provides the existence of a countable family of {\it pairwise disjoint}
elements of $\FF$, denoted $F_1$, $F_2,\ldots$, such that the family $4F_1$,
$4F_2, \ldots$ of the intervals, obtained from $F_1,F_2,\ldots$ by
dilating each by a factor $4$ about its center, is still a covering of
$S_0$.
We have, therefore, a covering $\{4F_j\}_{j=1}$ of $S_0$ with sets whose
diameter is $\le 4\d_0$. Then \equ(2.5) implies:
%
$$\eqalignno{
&\sum_j\left({4\over T}\hbox{ diameter }F_j\right)^{1/2}=
2\sum_j\left({\hbox{diam} F_j\over T}\right)^{1/2}\le&\eq(2.6)\cr
&\quad\le {2\over T\LL^{1/2}}\sum_j\ig_{F_j}R(\th)^2d\th\le
{2\over\LL^{1/2}T}\ig_{S_0^{4\d_0}\cap[0,T_0+4\d_0]}R(\th)^2d\th
<\io\cr}$$
%
where $S_0^{4\d_0}$ denotes the set of points within a distance
$4\d_0$ from $S_0$, and because we have the estimate \equ(1.13)
for any $T_0$. This shows that $\a_i(S_0)\le{1\over2}$.
We could even replace $S_0^{4\d_0}$ with an open set $G_{\d_0}$
containing $S_0$ and {\it with measure} within, say, $4\d_0$ from that
of $S_0$ (which is zero): in fact we could consider an open set with
measure $4\d_0$ containing $S_0$ and use as covering the sets $4F_j\cap
G_{\d_0}$. And we see that the last integral in \equ(2.6) could be
replaced by the integral to the set $G_{\d_0}\cap[0,T_0]$ which,
letting $\d_0\to0$, converges to the integral over $S_0\cap[0,T_0]$ of
$R(\th)^2$, which is zero because $S_0$ has zero measure. Hence
$\m_{1/2}(S_0)=0$.
The analysis of the set $S$ of the space time singularities is much
more interesting. The main result of Scheffer is a new local
regularity theorem, which we formulate in terms of a family of
``observables'' or ``operators'' describing the local properties of $\V
u$ relevant for our singularity theory.
The attention of the reader should be focused on the analogy of what
follows with the ``renormalization group'' methods for the analysis of
a singularity. In fact the reader familiar with the methods can think
of the words ``operators'' and ``observables'' to have the ``same''
meaning (see [G2]).
Our operators corresponding to a ``box $Q_r(\V x,t)$ of scale $r$''
around $(\V x,t)$, or ``describing an eddie of scale $r$'' with:
%
$$Q_r(\V x,t)=\left\{(\V\x,\th)|\ |\V\x-\V x|0$:
%
$$\ig^{T_0}_0\ig_\O d\th\,d\V\x(|\V u|^{10/3}+|p|^{5/3})\le C_L
LV^2E^{2/3}_0T_0\ig^{T_0}_0{d\th\over T_0}R(\th)^2<\io\Eq(2.12)$$
%
with the notations \equ(1.3), giving the best known \ap bounds on the
$L_q$-norms of $\V u$ and $p$. It also implies (by H\"older
inequality) the finiteness of $J_r,K_r,G_r$.
In fact (by H\"older inequality, see appendix C), there is a constant
$C_0>0$ such that:
%
$$S_r\ge (C_0J_r),\ (C_0K_r),\ (C_0G_r)\Eq(2.13)$$
%
and $S_r$ is obviously bounded because of \equ(2.12). Also $A_r,S_r$
are, obviously finite because of the \ap bounds on energy and
dissipation \equ(1.10), \equ(1.13).
The basic theorem of Scheffer is the following:
\vskip0.5truecm
\noindent THEOREM 6 (Scheffer):
\vskip-0.3truecm
\\{\sl There exists a dimensionless constant $\e_s$ such that
$(G_r+J_r+K_r)<\e_s$ implies that $\V u$ is regular in the box
$Q_{r/2}(\V x,t)$; more precisely $\V u$ is uniformly bounded there; by
$C_s\e_s^{1/3}r^{-1}$ for some dimensionless constant $C_s$.}
%
\vskip0.5truecm
\noindent{\it Remarks:}
1) This theorem is a consequence of $\V\dpr\cdot\V u=0$ and \equ(1.18)
with $p$ given by \equ(2.9). No other properties are used. Hence one can
say that it is a consequence of the energy conservation alone. In
particular no use is made of Thomson's theorem, \ie of the property that
in the $\n=0$ limit the vorticity of a closed loop transported by the
fluid is conserved.
2) Let $Q_r^*(\V x^*,t^*)$ be the box of scale $r$ such that $(\V
x,t)$ is the center of $Q_{r/2}(\V x^*,t^*)$: this is the box with $\V
x=\V x^*$, $t^*=t+r^2/8\n$. Denote with a $*$ superscript the \equ(2.8)
evaluated in $Q_r^*(\V x^*,t^*)$. Clearly if $(\V x,t)\in S$ it must
be $J_r+G_r+K_r\ge\e_s$ for all $r>0$: otherwise the above theorem
would imply regularity in $Q_{r/2}^*(\V x^*,t^*)\ni(\V x,t)$. Hence
$S\ge\e_sC_0$, or:
%
$$\ig_{Q^*_r}(|\V u|^{10/3}+|\r|^{5/3})\ge\e_sC_0\n^{4/3}r^{5/3}
\Eq(2.14)$$
%
We can therefore repeat the proof of theorem 5 by replacing the
intervals $(t-\s,t]$ with the boxes $Q^*_r$. A covering theorem of the
type of Vitali's theorem can be proved also for such sets. This means
that there is a omothety constant $\l>0$, such that out of a covering
{\it with boxes $F=Q^*_r$ of $S$ with boxes of arbitrarily small diameter
covering each single point of} $S$ one can extract a countable family of
pairwise disjoint sets $F_1,F_2,\ldots$ such that diam $F_j\le\d_0$, for
any prefixed $\d_0$, and such that $\{\l F_j\}\ldots$, is a covering of
$S$. One can check that the omothety (about the center) factor $\l$
can be taken $\l = 6$. Therefore we deduce, proceeding as in the proof
of theorem 5, that $\a_c(S)\le{5\over3}$ and $\m_{5/3}(S)=0$:
%
\vskip0.5truecm
\noindent COROLLARY 7 (Scheffer):
\vskip-0.3truecm
\\{\sl The singularity set $S$ has
$\a_c(S)\le{5\over3}$, $\m_{5/3}(S)=0$.}
%
\vskip0.5truecm
3) This is remarkable as the previous estimate, theorem 5, would have
allowed the possibility that, at a given time, {\it all\/} the points
$\V x\in\O$ became singular (\ie a Hausdorff dimension for $S$ equal to
$3$) or even worse (up to the formation of a singularity set of
dimension $3.5$, taking theorem 5 into account).
4) As we shall see this is not ``optimal'' as far as Hausdorff
dimension is concerned.
5) We consider a regularized Leray's solution at scale $l$, and the set
of points $(\V x,t)$ for which one cannot find a small enough vicinity
$Q_r^*(\V x,t)$ with $r>\h_s$ with $\h_s$ a suitable function
of $l/L$ infinitesimal as $l\to0$, and such that inside $Q_r^*(\V x,t)$
the velocity $\V u$ {\it cannot\/} be bounded by $C_s\e_s^{1/3}r^{-1}$.
Then one can prove that the set $S_l$ of such points has Hausdorff
dimension $\a_c(S_l)\le{5\over3}$, $\m_{5/3}(S_l)=0$. This is a
constructive form of corollary 7. The computation of $\h_s$ requires
some extra work (without new ideas), with respect to the following
description of the proof.
The description of the proof of theorem 6 is as follows (see [CKN]).
Given a Leray's solution and a point $(\V x,t)$ we see that if $(\V
x,t)\not\in S$, \ie if $\V u$ is $C^\io$ near $(\V x,t)$, then
$A_r,\,S_r,\,G_r,\,J_r,\,K_r\tende{r\to 0}0$ at
speed $r^2,\,r^4,\,r^3,\,r^3,\,r^{5/2}$, respectively.
Hence we try to show that $A_r\to0$ at least fast as $r^2$: this will
imply that $\V u(\V x,t)$ is bounded (at least, to be precise,
if $(\V x,t)$ is a Lebesgue point\footnote{$^*$}{Recall that a Lebesgue
point of $f\in L$, $(R^3)$ is a point $\V x$ such that
$|B_r^{-1}\ig_{B_r}f(\x)d\V\x\to f(\V x)$ as the radius $r$ of the sphere
$B_r$ around $\V x$ tends to $0$: this happens necessarily
almost everywhere "Lebesgue theorem".} for $\V u$).
Furthermore the argument will be ``uniform'' for all points $(\V x',t'
)\in Q_{r/2}(\V x,t)$, hence $\V u(\V x,t)$ will be almost everywhere
bounded in $Q_{r/2}(\V x,t)$, (\ie in all Lebesque points of
$Q_{r/2}(\V x,t)$), hence $C^\io$ in the interior of $Q_{r/2}(\V x,t)$
by Serrin's theorem.
Scheffer's idea is that we can infer that $A_r\to0$ as $r^2$ simply by
showing that $A_r$ verifies a recursive relation linking its value on a
certain scale to the values on the ``previous scales''.
We fix a sequence $r_n=L2^{-n}$ of length scales, and we denote
$G_n=G_{r_n}$, $A_n=A_{rn}$, etc.
At this point one can try several paths. We can, for instance, try to
find a relation linking $A_n$, $\d_n$ or, to simplify, $T_n=A_n+\d_n$
to $T_{n-1}$ and to $T_{n_0}$, if $r_{n_0}=$ scale where the assumption
$G_{n_0}+J_{n_0}+K_{n_0}<\e_s$ is verified.
This amounts to say, from the renormalization group viewpoint, that we
regard $T_n=A_n+\d_n$ as the ``operator'' describing the phenomenon
(of absence or presence of a singularity in $\V x,t$) and we look for a
``beta functional'' $\BB_n$:
%
$$T_n=\BB_n(T_{n-1},\ldots,T_{n_0};\ G_{n_0},\ J_{n_0},\ K_{n_0}),\qquad
n\le n_0 \Eq(2.15)$$
%
defined on small enough scales (\ie for $n\le n_0$ for some $n_0$ large
enough). The property of theorem 6 should be expressed by the fact
that \equ(2.15) implies $T_n\le\ov\e r^2_n$ for all $n\le n_0$ if
$G_{n_0}$, $J_{n_0}$, $K_{n_0}$ are small enough, and for a suitable
$\bar\e>0$.
The tools to build, or estimate, the beta functional $\BB_n$ (assumed to
exist) are very limited.
A first tool is to rely on some simple relations between quantities on
various scales which are just of ``kinematical nature'', \ie
independent on the fact that $\V u$ verifies NS equations. They are
just expressing (S) or (CZ) or H\"older inequalities (denoted (H) below).
For instance (see lemma 3.1 in [CKN]):
%
$$G_{n+1}\le CG_n\ ,\quad A_{n+1}\le CA_n$$
$$G_n\le C(A_n^{3/2}+A_n^{3/4}\d_n^{3/4})\le
C(A_n+\d_n)^{3/2}\equiv CT_n^{3/2}\Eq(2.16)$$
%
are either trivial or consequences of (S) inequality on the integration
over $\V\x\in B_n$ followed by (H) inequality for the $\th\in\D_n$
integration. Some details on the techniques to obtain such inequalities
from the (S) and (CZ) inequalities are illustrated in appendix C).
The \equ(2.16) permits us to eliminate, from any relation linking the
$T_j$'s, the quantities $G_n$ by bounding them in terms of $T_n$.
A second tool is to derive
relations between $T_n$ and $T_j$, $G_j$ by using
the ``dynamical relation'' \equ(1.18) expressing the energy
conservation: it is however clear that not only $G_j$ but $J_j$ and perhaps
$K_j$ as well will appear in any relation linking $T_n$ to $T_{n-1},\ldots$.
Hence the attempt to find a kinematical relation between $J_n,K_n$ and
$J_j,K_j$, $j0$ denotes a suitable positive constant (not
necessarily the same everywhere), (see lemmata 5.3, 5.4, 5.2 in [CKN]).
To such kinematical inequalities one can add one more "dynamical"
inequality:
%
$$A_n\le
C\,(2^pG^{2/3}_{n-p}+2^pA_{n-p}\d_{n-p}+2^pJ_{n-p})\Eq(3.8)$$
%
which follows from the energy inequality \equ(1.18) combined with (H),
(S), (CZ), (P) inequalities (see lemmata 5.5, 5.1 in [CKN]
recalling that we took the density of external force $\V f=\V0$
for simplicity). The \equ(3.7), \equ(3.8) yield
\equ(3.5).
%
The \equ(3.4) and \equ(3.5) are quite remarkable: they induce us to
consider the ``operators'':
%
$$\a_n=A_n\ ,\quad \k_n=K^{8/5}_n\Eq(3.9)$$
%
and we see that the operators $J,K$ can be eliminated
from \equ(3.7), \equ(3.8) and it follows:
%
$$\eqalign{
\a_n&\le
2^{-p}C\a_{n-p}+2^{3p}C(\a^{1/2}_{n-p}\d^{1/2}+\a_{n-p}\d)+
C2^{p/5}\a^{1/2}_{n-2p}\k^{1/2}_{n-2p}\cr
\k_n&\le
2^{-8p/15}C\k_{n-p}+2^{3p}C\d^{5/8}\a_{n-p}\cr}\Eq(3.10)$$
%
where $\d$ denotes a common bound or $\d_{n-p}$,
$\d_{n-2p}$.
By using the inequality $(ab)^{1/2}\le{1\over\sqrt2}(\m a+\m^{-1}b)$,
valid for any $\m>0$, we can ``linearize'' the two terms
$(\a_{n-p}\d)^{1/2}$ and $(\a_{n-2p}\k_{n-2p})^{1/2}$. We take $\m$ such
that $2^{-1/2}\m 2^{3p}C= 2^{-p}C$ for the first and $2^{-1/2}\m
2^{p/5}C=2^{-p}C$ for the second. We find:
%
$$\eqalign{
\a_n&\le(22^{-p}C+2^{3p}C\d)\a_{n-p}+2^{-p}C\a_{n-2p}+C
2^{7/5p}\k_{n-2p}+2{}^{4p}C\d\cr
\k_n&\le
2^{-8p/15}C\k_{n-p}+2^{3P}C\d^{5/8}\a_{n-p}\cr}\Eq(3.11)$$
%
for all $p>0$, and $\d$ denotes a common bound or $\d_{n-p}$,
$\d_{n-2p}$.
Given $\e < 0$ we can fix first $p$ large (so that $2^{-p}C$ is as small
as needed) and then $\d$ small, say $p =
p_0$, $\d=\d_0$, so that the r.h.s. of \equ(3.11) can be bounded by:
%
$$\eqalign{
\a_n&\le(2\,2^{-p}C+2^{3p}C\d)\a_{n-p}+2^{-p}C\a_{n-2p}+C2^{7/5}
\k_{n-2p}+2^{4p}C\d\cr \k_n&\le
2^{-8p/15}C\k_{n-p}+2^{3p}C\d^{5/8}\a_{n-p}\cr}\Eq(3.12)$$
%
with the coefficients of $\a_\cdot,\k_\cdot$ in the r.h.s. all smaller
than $\e$ except the one equal to $\tilde C= C 2^{7/5}$ which, of
course, will be (very) large:
%
$$\eqalign{
\a_n\le& \e\a_{n-p}+\e\a_{n-2p}+\tilde C\k_{n-2p}+\bar C\d\cr
\k_n\le& \e\k_{n-p}+\e\a_{n-p}\cr}\Eq(3.13)$$
%
with $\bar C=2^{4p} C$ and $\d$ denotes a common bound or $\d_{n-p}$,
$\d_{n-2p}$.
Thus, introducing the vectors:
%
$$\eqalign{ \V a'&=\V
a_n=(\a_n,\k_n,\a_{n-p},\k_{n-p})=(\a',\k',\h',\beta')\cr \V a&=\V
a_{n-p}=(\a_{n-p},\k_{n-p},\a_{n-2p},\k_{n-2p})=(\a,\k,\h,\beta)\cr
\V \r&=(1,0,0,0)\cr}\Eq(3.14)$$
%
we can rewrite \equ(3.13) as:
%
$$\V a_n\le M\V a_{n+p}+\ov C\d\V\r\Eq(3.15)$$
%
where the matrix $M$ can be read from \equ(3.13) and it has all its
matrix elements non negative and bounded by $\e$ except one of them,
which is off diagonal. Such a $4\times4$ matrix has norm less than $1$
if $\e$ is small enough and therefore \equ(3.15) implies:
%
$$\limsup_{n\to\io}\V a_n\le\ov C\d{M\over 1-M}\V\r\Eq(3.16)$$
%
componentwise.
Hence we see that if $\d_n$ is small enough for all $n\ge n_0$, the
\equ(3.5) generates a trajectory which ``flows' within $O(\d)$ of the
origin. In fact one should remark that as soon as
$\d_n<\e_{\hbox{ckn}}$, \ie for $n\ge n_0$, the recursion relation
\equ(3.5) (or, better, its simplified form \equ(3.16)) starts
contracting. At the scale $r_{n_0}$ the
$G_{n_0}+J_{n_0}+K_{n_0}+A_{n_0}$ can be bounded in terms of the \ap
bounds: the bounds will be huge but explicit.
Therefore, since the contraction rate is bounded by that of $M$, within
a large but explicitly controlled number of steps the values of such
observables shrink to become $<\e_s$. In other words knowing
$r_{n_0}$, the scale at which the Reynolds number becomes
$<\e_{\hbox{ckn}}$ ({\it which is a quantitative way of saying that it
becomes ``of $O(1)$''}) we can find, in term of the \ap bounds, on which
lower scale the velocity field becomes bounded by an \ap controlled bound
(via theorem 6).
%
Hence if $\d<\e_{\hbox{ckn}}$ we see that $\a_n$, $\k_n$ and,
by \equ(3.7), also $G_n+J_n+K_n$ become smaller than $\e_s$, the
constant in Scheffer's theorem (theorem 6). Clearly this means that
$\V u$ is $C^\io$ in $Q_{\bar r/2}^*(\V x,r)$ if $\bar r$ is the value
of $r_n$ for which $G_n+J_n+K_n<\e_s$ for $r_n=\ov r$.
This is a very small information, but at least it makes clear the
mathematical meaning and the validity of one of the assumption of (KO)
theory. We also see that, from (KO) theory, we expect that the local
Reynold's number $\d_{n}$ should decrease according to a well defined
law, at least if on the scale $L$ it is very large. The law is, (see
[LL], p. 153):
%
$$\d_n\approx R^2\left({r_n\over L}\right)^{8/3}=R^22^{-n8/3}
\Eq(3.17)$$
%
which would suggest that one should look for a relation of the $\d_n$ on
various scales of the form:
%
$$\d_n\le 2^{-p8/3}\d_{n-p}+?\ .\Eq(3.18)$$
%
It is however probable that any relation of this type can only come from
taking appropriately into account the Thomson's law (valid in the limit
of $\n=0$) and, possibly, by putting some restriction on the initial
data so that they look like ``typical'' turbulent data. %
(Un)fortunately the problem is completely open.
\vskip1.truecm
{\bf\S4 Comments on filament dynamics in the Euler regime.}
\vskip0.5truecm
\numsec=4\numfor=1%\pgn=1
Some remarkable numerical experiments have been performed by Chorin
who studies the filaments evolution under Euler and Navier Stokes
equations.
He proposes a method of integration in which the filaments are considered
very irregular to start with: this allows to bypass effectively the
\ap obstacle due to the well known divergence of the velocity of smooth
filaments of zero width.
Another way to avoid the problem is to give a finite width to the
filaments or to form sheets of dimension $2$. The disadvantage is first
of all of numerical nature as one has to evolve two or three
dimensional objects with the obvious storage and memory problems.
Here we present a mathematical analysis of the velocity at a given time
for a filament of vorticity concentrated on a {\it one dimensional
curve $\g$}. By this we mean that $\g$ is a one parameter continuous
closed curve everywhere tangent to the vorticity field $\V\o=\hbox{\rm
rot}\,\V u$.
The main problem is to understand if one can find a way around the fact
that if $\g$ is smooth then the Biot--Savart formula gives, for the
velocity $\V u(\V x)$ in a point $\V x\in\g$:
%
$$\V u(\V x)={\G\over 4\pi}\oint_\g{d\V\r\wedge(\V x-\V\r)\over |\V
x-\V\r|^3}\Eq(4.1)$$
%
where $d\V\r$ is the line element on $\g$, and $\G$ is the vortex
"intensity".
The \equ(4.1) gives $\V u$ a logarithmically infinite value at every
point $\V x$ of $\g$ where the curvature $C(\V x)$ does not vanish:
this means that if $\g$ is replaced by a tube of width $\d$ then $\V u$
is proportional to $C(\V x)\log C(\V x)\d$: this velocity is directed
orthogonally to the osculating plane (determined by the tangent vector to
$\g$ and the principal normal vector, pointing towards the center of
curvature). %
The idea, which seems to be a possible interpretation of the continuous
version of Chorin's lattice fluids, is that a filament can exist and
move at finite speed only if it is so ``rugged'' that it has {\it no}
well defined curvature. This is also suggested by the fact that a
really thin filament will be deformed in dramatically different ways
and directions, because of the fact that the velocity is proportional
to the curvature; and the larger the thinner is the filament.
Hence even if initially smooth it will be immediately deformed and look
``like a brownian sample path'': unless the filament is a circle: a
case in which it would zoom to infinity (disappearing together with our
problem, so we could forget about it). %
{\it The conjecture is, thus, that \equ(4.1) gives finite velocity to
the points of the filaments when the filaments are so rugged to be a
sample of a (suitable class of) stochastic processes.}
%
To put the conjecture in the form of a theorem we must give a meaning
to \equ(4.1). We therefore begin by considering a plane $z=0$ and, on it,
a curve $\g$ which is roughly a circle centered at the origin and with
radius $R$.
The filament $\g$ parametric equations in cylindrical
coordinates will be, for $\a\in[0,2\pi]$:
%
$$r(\a)=R(1+\e(\a)),\quad z=R\th(\a)\Eq(4.2)$$
%
with $\e(\a)$, $\th(\a)$, for the time being, arbitrary but periodic.
Then we can calculate the three components of the line element $d\V{\r}$:
%
$$d\V{\r}=(-R(1+\e)s+R\e'c,\ R(1+\e)c+R\e's,\quad R\th')$$
$$\V\r-\V\r_0=(R(1+\e)c-R(1-\e_0),\ R(1+\e)s,\
R(\th-\th_0))\Eq(4.3)$$
%
where $c\=\cos\a,\,s\=\sin\a$, $\e_0 \equiv\e(0)$, $\th_0 \equiv\th(0)$,
$\V\r_0=\V\r(0)$, $\V\r=\V\r(\a)$ and $\e'$,
$\th'$ are the derivatives with respect to $\a$ of $\e$, $\th$.
Setting $\m=\th-\th_0$, $\h=\e-\e_0$ we get:
%
$$|\V\r-\V\r_0|=\Big(2(1-c)(1+\e+\e_0+\e\e_0)+\h^2+\m^2\Big)R^2
\Eq(4.4)$$
%
and the three components of $d\V\r\wedge(\V\r-\V\r_0)$ are:
%
$${d\V\r\over d\a}\wedge(\V\r-\V\r_0)=
\cases{ R^2[(1+\e)c\m+\e's\m-(1+\e)\th' s]\cr
R^2[(1+\e)(c-1)\th'+\th'\h+(1+\e)\m s-\m\e'c]\cr
R^2[-(1+\e)\h+(1+\e_0)(1-c)+(1+\e)\e'cs+(1+\e_0)\e's]\cr}\Eq(4.5)$$
%
To simplify the discussion we set $(1+\e)\sim 1$ as we imagine $\e,\th$
to be very small (but we could carry them around without this
assumption).
The three components of the velocity at $\V\r_0$ are then given by:
%
$$\eqalign{
\V u=&{\G\over 4\pi R}\ig^{2\pi}_0{d\a\over
(2(1-c)+\h^2+\m^2)^{3/2}}\cdot\cr
&\cdot\Big(\m c+\e'\m s-\th's,\,(c-1)\th'+\th'\h+\m s-\m\e'c,\,
-\h +(1-c)+2\e'cs\Big)\cr}\Eq(4.6)$$
%
The quantities $\e'$, $\th'$ present interpretation problems because
we want, eventually, to suppose that $\a\to\e(\a)$, $\a\to\th(\a)$ are
``brownian paths'', \ie samples of a gaussian process generated on the
circle by the operator $-D\dpr^2_\a$ acting on the functions with zero
average. This is locally the same as the unconstrained brownian
motion, and one can think of it as an independent increments process
(the corrections due to the periodicity of the boundary conditions can
be easily introduced).
The formal expression $\e'(\a)=(\e(\a+\d)-\e(\a))$ $\d^{-1},\d\to0$,
shows that, by the independence\footnote{$^1$}{neglecting the periodic
boundary conditions corrections} of the increments of $\e,\th$
and by the smoothness of $\e,\th$ (as they will typically be H\"older
continuous of order $\sim1/2$), the terms with the derivatives give zero
contribution to the average of \equ(4.6) over the samples. %
Hence the proposal is to define:
%
$$\V u={\G\over 4\pi
R}\ig^{2\pi}_0{d\a\over(2(1-c)+\h^2+\m^2)^{3/2}}\quad (\m
c,\m s,-\h+(1-c))\Eq(4.7)$$
%
setting to zero the terms with the derivatives, which average to zero.
One can then check easily that $\langle\V u\rangle$, $\langle\V
u^2\rangle$ are finite. For instance the only non zero component of $\V u$
is in the $z$ direction and has value:
%
$$v={\G\over 8\pi
R}\ig d\a d\h d\m{1-\cos\a\over(1-\cos\a+\h^2+\m^2)^{3/2}}
f_\a(\h)g_\a(\m)\Eq(4.8)$$
%
where $f_\a(\h)$, $g_\a(\m)$ are the gaussian\footnote{$^2$}{neglecting
the periodic boundary conditions corrections} distribution functions of
$\h=\e(\a)-\e(0)$ and $\m=\th(\a)-\th(0)$. The integral \equ(4.8) is
finite as the gaussians $f_\a$, $g_\a$ have dispersion $|\a|\,D$, for
some $D>0$, assuming $\e$ and $\th$ to be the same process, and one
easily finds:
%
$$V={\G\over RD}C\Eq(4.9)$$
%
where $C$ is a finite computable constant. %
To formalize the above heuristic analysis we consider a filament:
%
$$\g:\qquad \a\to\V\r(\a)+\V\x(\a)\Eq(4.10)$$
%
where $\V\r\in C^\io$ and $\V\x$ is a sample of a stochastic
process, \eg a brownian motion on the circle.
The evolution equation for the
filaments then is:
%
$$\eqalign{
{d\over dt}(\V\r_0+\V\x_0)&={}^{''}\ {\G\over
4\pi}\oint{( d(\V\r+\V\x)\wedge(\V\r+\V\x-\V\r_0-\V\x_0)
\over|\V\r-\V\r_0+\V\x-\V\x_0|^3}\ {}^{''}\buildrel\hbox{def}\over\=\cr
&\buildrel\hbox{def}\over\={\G\over
4\p}\oint{ d\V\r\wedge(\V\r+\V\x-\r_0-\x_0)\over|\V\r-\V
\r_0+\V\x-\V\x_0^3}=\V u_0(\V\r_0),\qquad \V\r_0\in\g\cr}\Eq(4.11)$$
%
where the $ d\V\x$ terms are {\it eliminated}, by definition. This is
in some sense the analogue of the elimination of the self interaction
terms in the vortex dynamics in $d=2$; and it is justified empirically
by the fact that everything in the integrand in \equ(4.11) is smooth
except $d\V\x$, which however has zero average. Note, however, that in
two dimensions the physical and mathematical meaning of the elimination
of the self interaction terms is {\it perfectly} understood, see [MP]. %
The theorem is:
\vskip0.5truecm
\noindent THEOREM 10:
\vskip-0.3truecm
\\{\sl The function $\V u_0(\V\r)$ in the r.h.s. of \equ(4.11)
is a well defined random variable if the $\V\x$ are samples of brownian
motion on the circle. Such is also $|\V u_0|^2$.}
%
\vskip0.5truecm
Continuing in a heuristic fashion we see that \equ(4.11) generates two
equations, one for the average shape $\V\r$ and one for its
fluctuations $\V\x$:
%
$$\eqalign{
{d\V\r_0\over dt}&=\media{{\G\over
4\pi}\oint{d\V\r\wedge(\V\r+\V\x-\r_0+\x_0)\over
|\V\r+\V\x-\V\r_0-\V\x_0^3}}\cr
{d\V\x_0\over dt}&={r\over
4\pi}\oint{d\V\r\wedge(\V\r+\V\x-\r_0+\x_0)\over
|\V\r+\V\x-\V\r_0-\V\x_0^3}-{d\V\r_0\over dt}\cr}\Eq(4.12)$$
%
where $\V\x\=\V\x(\V\r),\,\V\x_0\=\V\x(\V\r_0),\ \V\r,\V\r_0\in \g$ and
the average $\media\cdot$ is over the distribution of the fluctuation
field at time $t=0$. $\V\x$.
It would be very nice if one could say that the evolution \equ(4.12)
does not change the statistics of the fluctuations (\ie that the
statistics of the flutuation field is time independent), at least for
some initial fluctuation distribution. This would decouple the two
equations and one would have a well defined time evolution of the (very
smooth) average filament: the velocity scale will be set by the
dispersion $D$ of the distribution of $\V\x$.
This invariance of the fluctuations distributions might be
approximately true, or even exact in some limit situations (\eg
$C\to0,D\to0$ with $D^a/C=$ constant for some suitable $a$). And one
can imagine to check the following analogue of a well known $2D$ vortex
dynamics results.
Let us suppose that $\V\o$ be a vorticity field with flux line {\it all
closed}.
We approssimate $\V\omega$ by a finite family of smooth filaments
(which is ``weakly'' possible by our closed flux lines assumption)
$\g_j$, $j=1,2\ldots$ separated by a distance of order $e>0$. Then we
attribute a fluctuations field $\V\x^l_j$ to each filament with (say)
brownian distribution with dispersion $D^l_j$.
Is it possible to fix $D^l_j$ so that as $l\to0$ the evolution (via the
first of \equ(4.12)) of the local average velocity field approaches a
limit $\V u(\xx,t)$, with the corresponding vorticity field approaching
a limit $\V\o(\xx,t)$, and $\V u(\xx,t)$ is a solution of Euler's
equations? Of course the problem can be modified by allowing $D^l_j$ to
depend on $t$ or by avoiding the assumption of invariance of the
fluctuations and by using the two \equ(4.12) as a coupled system of
equations.
But it is clear that it is perhaps too early to draw any conclusions:
we think that the above analysis can suggest several numerical
experiments which could perhaps help to a better understanding of the
important works of Chorin.
%
\vskip1.truecm
%
{\bf Appendix A: An abstract argument.}
%
\vskip0.5truecm \def\TT{{\cal T}}
The following is a version of the abstract argument necessary to prove
theorem 3. Below $R(t)^2\=\liminf_{l\to0}\n^{-2}L\ig_\O|\V\dpr\W
u^l(t)|^2$.
Let $\TT_n$ be the set of times in the prefixed interval $[0,T_0]$
where $R(t)^40$ :
%
$$\eqalign{
\lim_{l\to0}&\ig_0^{T_0}dt\ig_\O |\V u^l-\V u^0|^2\,d\V
x\=\ig_0^{T_0} dt \sum_{0<|\V k|}|\V\g_\kk^l(t)-\V\g^0_\kk(t)|^2\le\cr
\le\lim_{l\to0}&\sum_{0<|\kk|**