0.$ We shall derive this result below in
the context of forced, steady-states. However, it
remains possible that enstrophy conservation may fail in the class
$\omega\in L^2$ with spectrum decaying less steeply.
The previous results on weak solutions do not cover all the cases of
physical interest. For example, a common
situation considered for numerical simulation is decaying turbulence
with initial data chosen from a Gaussian random
ensemble of initial vorticity fields $\omega_0$ with the Kraichnan
$k^{-3}$ energy spectrum: see
Farge et al. \cite{28,29}, Benzi and Vergassola \cite{30}. In that
case, the initial vorticity fields
have {\em infinite enstrophy} with finite probability, and
$\omega_0\in L^2$ fails. In this case, as well as others,
it is more natural to consider $\omega\in B^{s,\infty}_p,$ the
{\em Besov spaces}, with suitable $s$ and $p.$
These spaces will be very important for our later analyses, but here
we just remark that they correspond to
classes of functions with H\"{o}lder index $s$ in the space
$L^p$-mean sense. It is a consequence of Theorem 4 in
\cite{36} for the case $p=2$ that the individual realizations of a
homogeneous ensemble of random vorticity fields
with enstrophy spectrum $\Omega(k)=O\left(k^{-(1+2s)}\right)$ will
belong to $B^{s-,\infty}_2$ with probability
one. Therefore, with the Kraichnan spectrum, for which $s=0,$
it will be true that $\omega_0\in B^{0-,\infty}_2.$ This
space is ``marginally worse'' than $L^2$ and existence within
this class is not given by any of the previous
theorems.
Recently, Shnirelman \cite{42a} has established solvability
of the Euler equations for $\omega\in B^{s,\infty}_2$
with $s>2.$ Actually, solutions with such large values of $s$
are classical and Shnirelman's results are only
a slight improvement of the well-known results of Ebin and
Marsden \cite{15g} and others on solvability with
$\omega\in H^{s},\,\,\,s>2.$ The result on enstrophy conservation
of Sulem and Frisch \cite{34} is best
stated in terms of the Besov spaces, and implies that $\omega
\in B^{s,\infty}_2,\,\,\,s>{{1}\over{3}}$
is a sufficient condition for enstrophy conservation to hold.
Clearly, the Besov index in Shnirelman's construction
is too large to be of interest for 2D enstrophy cascades, but
we conjecture that Euler solutions in fact exist globally
in $B^{0,\infty}_2.$ This conjecture has a very natural
physical interpretation that a bound of the enstrophy spectrum
by Kraichnan's spectrum, $\Omega(k)=O\left(k^{-1}\right),$
will be dynamically preserved in time. It would be very natural
to look for enstrophy dissipation at finite time within this class.
There is an issue raised in the works \cite{29,30} which shall
be important in our discussion later of the steady-states.
Both of these works reported that ``negative exponents'' occur
due to ``cusps'' in vortex cores, with $h\approx -1/2$
the most typical value and $h_{\mn}\approx -1.$ Since there
seemed to be no internal mechanism in 2D for generation of
such singularities from regular initial data, it was conjectured
in \cite{29,30} that the ``cusps'' observed
in those works were present initially. Actually, a homogeneous
{\em Gaussian} random field with enstrophy spectrum
$\sim k^{-(1+2s)},$ from which initial data were selected in
\cite{29,30} for $s=0,$ have for {\em all} $s\in \bR$
realizations $\omega\in B^{s-,\infty}_{\infty}$ a.s. and not
merely $\omega\in B^{s-,\infty}_{2}$ as given by Theorem 4
of \cite{36}. This is a simple generalization to nonpositive
$s\in \bR$ of the well-known Wiener-L\'{e}vy theorem, which
states that the minimum H\"{o}lder singularity for realizations
of such a Gaussian ensemble is $h_{\mn}=s-$ a.s.
(G. Eyink, unpublished). \footnote{Because of this fact, our
classification of initial data into ``Type A'' and
``Type B'' for vorticity fields chosen at random from Gaussian
ensembles coincides with that used by She et al.
(for velocity fields) in the study of decaying Burgers
turbulence \cite{42aa}.} Therefore, the initial data of \cite{29,30}
are distributional, but with no singularities as severe as $h=-1/2$
and the ``cusps'' observed, if real and not
numerical artefacts, must have been produced in the course of the
dynamical evolution. Such a phenomenon would not
be dissimilar to what happens for $h>0.$ As discussed above, the
exponent in that case can decay, but at most to
$[\![h]\!],$ the greatest integer less than $h$ \cite{15a,1,15b}.
Similarly, one might suppose that if the initial data
$\omega\in B^{0,\infty}_\infty(\bT^2),$ then a solution exists
globally in $B^{-1,\infty}_\infty(\bT^2).$
Again, the exponent may deteriorate in magnitude by 1. This
behavior would be consistent with our conjecture
that a solution exists globally in $B^{0,\infty}_2(\bT^2),$
because $B^{0,\infty}_2(\bT^2)\subset
B^{-1,\infty}_\infty(\bT^2)$ in 2D as a consequence of Besov
space embedding theorems (see Appendix III).
\noindent {\em Remark on nonlocality}
A few remarks are in order regarding the issue of ``locality''
of the enstrophy cascade. It is
possible to make an analysis of the contributions to the
enstrophy flux from wavevector
triads in distinct octave bands, analogous to that made
for energy transfer in \cite{16,12}.
Except for a single class of contributions, all triads
yield an enstrophy flux $Z_{\ell_K}$ for
$\ell_K=2^{-K}$ which is actually $O(\ell_K^{3h}).$ This
is the magnitude of the flux
contributed by the ``local triads'' with all wavevectors
from octave bands near the $K$th.
However, the nonlocal class of terms like $\omega_N
(\bv_M\bdot\grad)\omega_L$ with $N,L\approx K$
and $M\ll K$ contributes at the order
\be Z^{{\rm nloc.}}_{\ell_K}\sim 2^{K(1-2h)}2^{-M(1+h)}
{{2^M}\over{2^K}}=2^{-hM}2^{-2hK}. \lb{24} \ee
Note that $\bv_M\sim 2^{-(1+h)M}$ since the velocity field
is in $C^{1+h}$ when $\omega\in C^h$
and that the factor $2^M/2^K$ comes from cancellations due
to detailed conservation, as described
in \cite{16,12}. This class of contributions is dominated
by the triads with $M\approx 0,$ i.e. by the
largest-scale modes, and gives the leading term in the flux
$\sim \ell_K^{2h}.$ Hence, the enstrophy cascade
is {\em infrared-dominated} and nonlocal when $0 1.$ In fact, if $\bv\in L^p,$ then
\be \|\Delta_\bl^2\bv\|_p=O(\ell^{1+s})\longleftrightarrow
\bv\in B^{1+s,\infty}_p
\longleftrightarrow
\grad\bv\in B^{s,\infty}_p. \lb{III16} \ee
The first equivalence follows from Theorem 2.5.12 in \cite{35}
on characterization of Besov spaces by differences.
The second equivalence is a well-known result for all $s\in \bR.$
For $0~~1.$ The middle equivalence is another consequence
of Theorem 2.5.12 in \cite{35} on characterization
by differences, which states that both are equivalent to
$\bv\in B^{1+s,\infty}_p$ when $0<1+s<1.$
To obtain the last $\longrightarrow$ implication, we recall
$\bv\in B^{1+s,\infty}_p
\longleftrightarrow \grad\bv\in B^{s,\infty}_p$ and
$\omega$ is, of course, just a component of $\grad\bv.$
We should emphasize that the Theorem 2.5.12 in \cite{35}
on characterization by differences does not
apply when $s\leq d\left({{1}\over{{\rm min}\{1,p\}}}-1\right)$
and, in particular, when $s\leq 0$ for
$p\geq 1.$ Therefore, $\|\Delta_\bl\omega\|_p=O(\ell^{s})$ is only a sufficient condition for
$\omega\in B^{s,\infty}_p$ and not a necessary one when $s\leq 0.$
We next show that the condition
\be \langle |\Delta_\bl\omega|^p\rangle=O(\ell^{\zeta_p})
\lb{III29} \ee
with $\zeta_p<0$ implies that $\bv\in B^{1+s,\infty}_p$ a.s.
for each $s<\sigma_p$ and, therefore,
\be \|\Delta_\bl\bv\|_p\leq \ell^{1+s}\cdot
\|\bv\|_{B^{s,\infty}_p}\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\mus-{\rm a.s.} \lb{III30} \ee
Theorem 4 in \cite{36} does not apply when $\sigma_p<0.$
Instead we shall show directly that $\bv\in B^{1+s,\infty}_p$
by means of the following characterization. Define the
{\em truncated-ball means}
\be ({\cal V}_\ell^M f)(\br)\equiv
\overline{(\Delta_{\ell\cdot \bh}^Mf)(\br)}, \lb{III31} \ee
where the line indicates a space-average over the
``truncated-ball'' $\{\bh:1<|\bh|<2\}.$ Then it is
known that
\be \|f\|^\#_{B^{s,\infty}_p}=\sup_{N\geq 0}\left(2^{sN}
\|{\cal V}_{2^{-N}}^M f\|_p\right) \lb{III32} \ee
defines a seminorm for the Besov space $B^{s,\infty}_p$
with any integer $M>s>0.$ See Sections 2.5.11-12
of \cite{35}. To employ this result, we first average
the condition Eq.(\ref{III29}) over the truncated-ball:
\be \overline{\langle |\Delta_{\ell\cdot\bh}\omega|^p\rangle}
=O(\ell^{\zeta_p}). \lb{III33} \ee
This implies for each fixed $\ell,$ or for any fixed countable set
of $\ell$'s, that
\be \Delta_{\ell\cdot\bh}\omega\in L_p \,\,\,\,\,\,\,\,\,\,\,\,\,\,
\mus-{\rm a.e.}\,\,\,
\omega,\lambda-{\rm a.e.}\,\,\,\bh, \lb{III34} \ee
where $\lambda$ is Lebesgue measure on the truncated-ball. We can
then invoke Eq.(\ref{III14}) to obtain
\be \|\Delta_{\ell\cdot\bh}^2\bv\|_p\leq C\cdot\ell\cdot
\|\Delta_{\ell\cdot\bh}\omega\|_p\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\mus-{\rm a.e.}
\,\,\,\omega,\lambda-{\rm a.e.}\,\,\,\bh. \lb{III35} \ee
This may be averaged jointly over the truncated-ball and
the ensemble to obtain, with Eq.(\ref{III33}),
\be \overline{\langle\|\Delta_{\ell\cdot\bh}^2\bv\|_p\rangle}
=O(\ell^{1+\sigma_p}). \lb{III36} \ee
However, it follows from Jensen's inequality that
\be \|{\cal V}_\ell^2\bv\|_p\leq
\overline{\|\Delta_{\ell\cdot\bh}^2\bv\|_p} \lb{III37} \ee
and thus
\be \langle \|{\cal V}_\ell^2\bv\|_p\rangle \leq
\overline{\langle\|\Delta_{\ell\cdot\bh}^2\bv\|_p\rangle}
= O(\ell^{1+\sigma_p})
\lb{III38} \ee
for each $\ell$ in a countable set ${\cal L}.$ Let us choose
the latter set to be ${\cal L}=\{\ell_N=2^{-N}:N\geq 0\},$
or, in other words,
\be \langle\|{\cal V}_{2^{-N}}^2\bv\|_p\rangle
= O\left(2^{-N(1+\sigma_p)}\right) \lb{III39} \ee
for all $N\geq 0.$ Then the same Borel-Cantelli
argument used in the proof of Theorem 4 of \cite{36} implies
that
\be \|{\cal V}_{2^{-N}}^2\bv\|_p= O\left(2^{-N(1+s)}\right)
\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\mus-{\rm a.s.}
\lb{III40} \ee
for each $s<\sigma_p.$ We conclude, by the characterization
theorem, that $\bv\in B^{1+s,\infty}_p$ a.s.
and that Eq.(\ref{III30}) holds.
The last result we needed in our arguments is the inequality
\be \|\bv\|_{B^{1+s,\infty}_p}\leq C\|\omega\|_{B^{s,\infty}_p}
\lb{III41} \ee
for some constant $C>0,$ which was used in establishing
sufficiency of the moment conditions.
For Eq.(\ref{III41}) we may refer to the literature. Since
Eq.(\ref{III9}) relates $\grad\bv$ to $\omega$
by a classical singular-integral operator $T,$ we may appeal
to Theorem A of Lemarie \cite{91}:
\be \|T\omega\|_{B^{s,q}_p}\leq C' \|\omega\|_{B^{s,q}_p}
\lb{III42} \ee
for $1>s>0$ and $p,q\in [1,+\infty].$ This is a result of
Calder\'{o}n-Zygmund type in the Besov spaces.
See also \cite{92}. In our case,
\be \|\grad\bv\|_{B^{s,\infty}_p}\leq C''
\|\omega\|_{B^{s,\infty}_p}. \lb{III43} \ee
>From this we conclude at once Eq.(\ref{III41}).
\noindent {\bf Acknowledgements:} This paper had its beginnings
in some conversations with Z.-S. She,
who made many useful suggestions and criticisms. Several other
UA colleagues---D. Levermore, S. Malham,
S. Nazarenko, J. Watkins, and X. Xin---provided either
references or helpful remarks. U. Frisch and
A. Shnirelman very kindly sent me their unpublished works.
I am grateful to all of the above.
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