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\section{Introduction}

In a recent paper (\cite{cw1}) we discussed the $L^2$ limit of solutions $u^{(NS)}$
of the Navier-Stokes equations in the case of vortex patch initial
data. We proved that, if the initial vorticity is a vortex patch with
smooth boundary, then the difference  $u^{(NS)} - u^{(E)}$ between the 
the Navier-Stokes and Euler velocities corresponding to this initial
datum is in $L^2$ and converges to zero at a rate proportional to 
$\sqrt{\nu}$. This is a slower rate of convergence than the rate
(O$(\nu ))$ of
the inviscid limit for smooth solutions (\cite{mcg}, \cite{k}, \cite{bm}, \cite{c1}). The fact
that
there is a drop in the rate of convergence when one passes from the
smooth to the non-smooth regime is not an artifact: there are
elementary examples providing lower bounds.

In the present work we investigate the $L^p$ inviscid limit for vorticities.
We are motivated in our study by the statistical
equilibrium theory of vortices (\cite{miller}, \cite{robert}). The initial
vorticities are taken in the phase space ${\mbox{\bf Y}}$ of bounded functions that vanish outside
a compact set. We are mostly interested in long time, uniform bounds,
i.e., bounds that are valid for many turnover times and that have an
explicit rate of vanishing. The smoothing effect that is 
present in the Navier-Stokes equations is
absent in the Euler equations. Because of this, 
internal transition layers prevent
{\it uniform } $L^p$ bounds for the difference between vorticities of
solutions with the same non-smooth initial data. Therefore, it seems that a pathwise uniform
Eulerian inviscid limit in this phase space
is not possible. The term pathwise refers here to the comparison of
individual solutions, paths that start from the same initial
data. We find that in order to obtain uniform bounds we need to
consider non-pathwise bounds: the most convenient close 
companion to a solution of the Navier-Stokes equation might be a
mollified Euler solution. To be more precise, if $S^{NS}(t)b$
represents the solution (vorticity)
of the Navier-Stokes equation with initial vorticity $b\in{\mbox{\bf
    Y}}$, if $S^E(t)b$ represents the solution
of the Euler equation and if we denote by $f_\delta = f*\phi_{\delta}$
the convolution with a mollifier $\phi_{\delta}$, then
a pathwise estimate concerns the difference $S^{NS}(t)b - S^E(t)b$ and
non pathwise estimates concern differences $S^{NS}(t)b -
S^E(t)b_{\delta}$ and $S^{NS}(t)b -\left ( S^E(t)b\right )_{\delta}$.
We find that the latter is better suited for long time
estimates. While $S^E(t)b_{\delta}$ solves the Euler equations,
$\left ( S^E(t)b\right )_{\delta}$ solves suitably modified Euler equations.

We prove uniform $L^p$ bounds that
vanish as $\nu^{\frac{s}{2p}}$ for the difference between  Navier-Stokes
and modified Euler solutions corresponding to  initial
data in Besov spaces 
$b\in{\mbox{\bf Y}}\cap B^{s,\infty}_2(R^2)$. 
We find that the optimal mollification is over
a distance of order $\delta\sim\sqrt{\nu}$, a fact that is consistent
with the estimate for the smallest length scales in two
dimensional turbulence. In order to obtain a short time result it
is enough to mollify the initial datum for the Euler evolution. However, in
order to obtain a long time result we have to mollify the solution. Thus,
the long time approximation follows slightly modified Eulerian dynamics.
The assumptions we require for the long time results are satisfied by
vortex patches with smooth boundaries. 

The main
difficulty is due to the fact that one needs to estimate 
gradients of the Eulerian vorticity. We use the method of (\cite{cw1})
to show that velocity differences are small and we
obtain estimates for the gradients
of the Eulerian vorticity; the smallness of velocity
differences counterbalances the large vorticity gradients.
The {\it non pathwise uniform} results can be used to obtain {\it non-uniform
pathwise} 
results (that is, pathwise results without rates of convergence).
In particular we prove the strong pathwise convergence in $L^p$, $1<p<\infty$. 

\vspace{1cm}

\section{Previous Results}

The Navier-Stokes equations and the Euler equations in $R^2
$ are
$$
\frac{\partial u}{\partial t} +u\cdot \nabla u
=-\nabla p+\nu \Delta u
$$
$$\nabla\cdot u=0$$
where $\nu>0$ in the case of the Navier-Stokes equations, $\nu = 0$ in
that of 
the Euler equation. The corresponding vorticity 
$$
\omega = \nabla^{\perp}\cdot u
$$
satisfies 
$$
\frac{\partial \omega}{\partial t} +u\cdot\nabla \omega= \nu \Delta\omega
$$
and $u$ can be recovered from $\omega $ via
$$
u= \frac{1}{2\pi}\left (\nabla^{\perp}\log (|\cdot |)\right )* \omega .
$$
The notation $\nabla^{\perp}$ refers to the gradient rotated by 90 degrees.

We consider the evolution in the vorticity space ${\mbox{\bf Y}}$
$$
{\mbox{\bf Y}} = L^1(R^2)\cap L^{\infty}_{c}(R^2)
$$
of bounded functions with compact support; the norm is the sum of the
$L^1$ and $L^{\infty}$ norms.
The solutions 
$$
S^{NS}(t)a_0 = \omega^{NS}(x,t) 
$$
and
$$
S^{E}(t)a_0 = \omega^{E}(x,t) 
$$
of the Navier-Stokes and, respectively Euler equation,
corresponding to initial datum $\omega (x,0) =a_0\in {\mbox{\bf Y}}$,
exist for all $t\ge 0$, ($t\in R$) and are unique.

A much studied class of examples of $a\in{\mbox{\bf Y}}$ is that of 
vortex patches: the initial vorticity
$a_0(x)$ is a simple function
$$
a_0 = \sum_{j=1}^{N}\omega_0^{(j)}\chi_{D_j}
$$
where $\omega_0^{(j)}$ are real constants and 
$\chi_{D_j}$ are characteristic functions of bounded, 
simply connected domains in $ R^2$.

 
We associate to any $a \in {\mbox{\bf Y}}$ certain basic objects: two 
functions and four numbers. The functions are a stream function
$\psi_a$ and a velocity field $u_a$:
$$
\psi_a (x) = \frac{1}{2\pi}\int \log(|x-y|)a(y)dy,
$$
and
$$
u_a = \nabla^{\perp}\psi_a.
$$
The numbers are a length scale $L_a$, a
time scale $T_a$, a velocity scale $U_a$, and a kinetic energy
$E_a$:
$$
L_a =\sqrt{ \frac{\|a\|_{L^1(R^2)}}{\|a\|_{L^{\infty}(R^2)}}},
$$
$$
T_a = \frac{1}{\|a\|_{L^{\infty }(R^2)}},
$$
$$
U_a = \sqrt{\|a\|_{L^1(R^2)}\|a\|_{L^{\infty}(R^2)}},
$$
and
$$
E_a = -\frac{1}{2}\int \psi_a(y) a(y)dy.
$$
We also associate to $a\in{\mbox{\bf Y}}$ a distribution $\pi_a(dy)$
defined by
$$
\int f(y)\pi_a(dy) = \int_{spt \, a}f(a(x))dx.
$$

If the initial vorticity is in ${\mbox{\bf Y}}$ then the fundamental
existence result, due to Yudovich (\cite{yu}) is 
\begin{thm} For every $a\in{\mbox{\bf Y}}$ there exists a unique
weak solution
$$
\omega^E(x,t) = S^E(t)a
$$
of the Euler equations that satisfies $\omega^E(x,0) = a(x)$. 
\end{thm}

The quantities $L_a$, $T_a$, $U_a$, $E_a$ and the distribution $\pi_a$
are conserved by the Eulerian flow, i.e.
$$
C_{S^E(t)a} = C_a
$$
if $C_a$ stands for any of these quantities. The velocity
$$
u^E(x,t) = u_{S^E(t)a}
$$
satisfies
$$
\|u^E(\cdot ,t)\|_{L^{\infty}} \le U_a
$$
for all $t\in R$. We denote by $S$ the
strain matrix -- the
symmetric part of the gradient of velocity:
$$
S(x,t) = \frac{1}{2} \left((\nabla u)+ (\nabla u)^*\right ).
$$ 

If the initial vorticity is
smooth then the solution is a classical solution. The following
precise estimates will be used in the sequel:

\begin{thm}
Let $a\in Y\cap W^{1,\infty}$ be a smooth initial vorticity. Then the
strain matrix
$$
S(x,t) = \frac{1}{2} \left((\nabla u^E)+ (\nabla u^E)^*\right )
$$
satisfies
$$
\|S(\cdot ,t)\|_{L^{\infty}} \le
\|a\|_{L^{\infty}}\left [\left (2+\frac{1}{\pi}\right ) +
2\log_+{\left(L_a\frac{\|\nabla a\|_{L^{\infty
        }}}{\|a\|_{L^{\infty}}}\right )}\right
]\exp{\left (2\|a\|_{L^{\infty }}t\right )}.
$$

The gradient of the vorticity $\omega^E = S^E(t)a$ satisfies
$$
\|\nabla \omega^E(\cdot ,t)\|_{L^p} \le \|\nabla
a\|_{L^p}\exp{(\int_0^t\|S(\cdot ,s)\|_{L^{\infty}}ds)}.
$$
for all time $t\in R$ and all $p$ including infinity: $1\le p\le
\infty $. 

The Lagrangian trajectory map $X(x,t)$ defined by
$$
\frac{d}{dt}(X(x,t)) = u^E(X(x,t),t), \quad X(x,0) = x
$$
satisfies
$$
\|\nabla X(\cdot ,t)\|_{L^{\infty }} \le \exp{(\int_0^t\|S(\cdot
  ,s)\|_{L^{\infty}}ds)}.
$$
\end{thm}
\vspace{1cm}

The quantity
$$
{\cal A}(t) = \int_0^t\|S(\cdot ,s)\|_{L^{\infty}}ds
$$
plays an important role. It controls not only the growth of the
Lipschitz
norm of particle trajectories and of the $L^p$ norms of gradients of
vorticity but also the $L^2$ operator norm of the Gateaux derivative of the
velocity solution map. The class of initial vorticities for
which the quantity ${\cal A}(t)$ is finite for all time is therefore
included in the class of initial vorticities for which the velocity
map $u_a \mapsto u^E (\cdot ,t)$ is continuous in $L^2$. The only
class of non-smooth functions $a\in {\mbox{\bf Y}}$ that are known to 
have ${\cal  A}(t)$ finite for all time are vortex patches with {\it smooth}
boundaries (\cite{bc}) or minor variations thereof. 


We start by estimating the difference between velocities of solutions 
of the  Navier-Stokes equations and Euler equations. Assume that
$a\in{\mbox{\bf Y}}$ and $b\in{\mbox{\bf Y}}$ are initial vorticities
for the Euler and respectively Navier-Stokes equation. The difference
$$
u(x,t) = u_{S^{NS}(t)b} - u_{S^E(t)a}
$$
satisfies
$$
\left (\partial_t + u^{NS}\cdot\nabla - \nu\Delta \right )u + \nabla q
= \nu\Delta u^E - u\cdot\nabla u^E.
$$
Using the method of \cite{cw1} one obtains
\begin{thm} Let $a\in{\mbox{\bf Y}}$ be the initial vorticity for
a solution of the Euler equations and $b\in{\mbox{\bf Y}}$ the
initial vorticity for a solution of the Navier-Stokes equations with
kinematic viscosity $\nu$. If the corresponding velocities, $u_a$ and
$u_b$ are such that $u_b- u_a$ is square integrable then
$$
\|u^{NS}(\cdot ,t) - u^{E}(\cdot ,t)\|_{L^2(R^2)} \le \left (
\|u_b - u_a\|_{L^2(R^2)} + \|a\|_{L^2(R^2)}\sqrt{\nu t}\right )
\exp{\left ({\cal A}(t)\right )}
$$
holds for all $t\ge 0$ with
$$
{\cal A}(t) = \int_0^t\|\frac{1}{2}\left (\left(\nabla u^E\right ) + 
\left (\nabla u^E\right )^*\right )\|_{L^{\infty}}ds.
$$
\end{thm}

For general $a,\, b\in{\mbox{\bf Y}}$, $u_a-u_b$ is not square 
integrable. We have, however, quite obviously
\begin{prop} Assume that $b\in{\mbox{\bf Y}}$ and that $a =
b_{\delta}$ where
$$
b_{\delta } = b*\phi_{\delta}
$$
with $\phi_{\delta}(x) = \delta^{-2}\phi (\frac{x}{\delta})$ a
standard mollifier. Then
$$
\|u_a-u_b\|_{L^2(R^2)} \le C \delta\|b\|_{L^2(R^2)}.
$$
\end{prop}
\vspace{1cm}

\section{Further Results}


If $a = b_{\delta}$ and $b\in{\mbox{\bf Y}}$ we have so far
a $L^2$ bound
$$
\|u^{NS}(\cdot ,t) - u^E(\cdot ,t)\|_{L^2(R^2)}\le C\|b\|_{L^2(R^2)}
\left [\delta + \sqrt{\nu t}\right ]\exp{\left ({\cal A}_{\delta}(t)\right )}.
$$
where ${\cal A}_{\delta}$ is computed on the Euler solution
$S^E(t)b_{\delta}$. 
We will keep the notation $b$ for the initial vorticity for
the Navier-Stokes evolution and $a$ for that of the Euler evolution.
A direct consequence of Theorem 2 is:
\begin{lemma}
Let $b\in{\mbox{\bf Y}}$ and let $a = b_{\delta}$. Then there exists
a nondimensional constant $C$ depending only on the mollifier $\phi$
such that
$$ 
{\cal A}_{\delta}(t) \le \left [C +
 \log_+{\left (\frac{L_b}{\delta}\right )}\right ]\left [\exp{\left
   (2\|b\|_{L^{\infty}(R^2)}t\right )} - 1\right ]
$$
\end{lemma}
As a consequence of this inequality it follows that the exponential
$\exp{\left ({\cal A}_{\delta }(t)\right )}$ is bounded by a small
power of $\delta^{-1}$ for times that are short compared to $T_b$.
In order to continue the estimates we will make an additional assumption
regarding $b$: we will assume a certain degree of regularity:
$$
b\in {\mbox{\bf Y}}\cap \left (\cup_{0<s<1}B^{s, \infty}_2(R^2)\right) 
$$
where $B^{s, \infty}_2(R^2)$ is the Besov space (see for instance,
\cite{pet}) of vorticities in
$L^2(R^2)$ that have an $L^2$ modulus of continuity that is H\"{o}lder
continuous of order $s$. If $b\in B^{s, \infty}_2(R^2)$ then
$$
\delta \|\nabla b_{\delta }\|_{L^2(R^2)} \le C\left (\frac{\delta}{\rho _b}
\right )^s\|b\|_{L^2(R^2)}
$$
where
$$
\rho_b = \left (\frac{\|b\|_{L^2(R^2)}}{N_s(b)}\right )^{\frac{1}{s}}
$$
and
$$
N_s(b) = \sup_{y\in R^2}\frac{\|b(\cdot + y) - b(\cdot
  )\|_{L^2(R^2)}}{|y|^s}.
$$

Note that, if $b$ is a vortex patch initial datum and if the
boundary of the patch is smooth then $b\in B^{\frac{1}{2}, \infty}_2$.
More generally,

\begin{lemma} Let $b = \chi_D$ be 
the characteristic function of a bounded domain whose boundary
has box-counting (fractal) dimension not larger than $d<2$:
$$
d_F\left (\partial D\right )\le d.
$$
Then 
$$ 
b \in B^{\frac{2-d}{p}, \infty}_p(R^2)
$$
for $1\le p<\infty$.
\end{lemma}

We start with a short time result:
\begin{thm}
Assume that $b\in{\mbox{\bf Y}}\cap B^{s,\infty}_2$ for some $0<s<1$. 
Consider
$$
\omega^{NS}(\cdot , t) = S^{NS}(t)b,
$$
and
$$
\omega^E(\cdot ,t) = S^E(t)b_{\delta}
$$
with $\delta = \sqrt{\nu T_b}$.

For every $\epsilon > 0$, there exists an absolute constant $\gamma $
depending only on $s, \epsilon$ such
that, if
$$
0 \le \frac{t}{T_b}\le \gamma
$$
then, for every $p\ge 2$ there exists a constant $K_b$ depending on
$p$
and $b$ alone, such that
$$
\|\omega^{NS}(\cdot ,t) - \omega^E(\cdot ,t)\|_{L^p(R^2)} \le
K_b\nu ^{\frac{s-\epsilon}{2p}}
$$
holds for all $\nu$ small enough. 
\end{thm}

In order to obtain a long time result we need to know that
$$
\lim\sup_{\delta\to 0}{\cal A}_{\delta}(t) < \infty .
$$
Recall that this quantity is computed by solving a family of Euler
equations. 
The map $\delta\mapsto {\cal A}_{\delta}(t)$  is not known to be
upper semicontinuous. In other words, even in the class of vortex patches with
smooth boundaries, we can not
rule out the possibility that there exists $b$, a time $t$ and a sequence
$\delta\to 0$ such that ${\cal A}(t)<\infty$ for the solution starting from
$b$ but $\lim_{\delta\to 0}{\cal A}_{\delta}(t) = \infty$. If
this does not happen then the result 
above holds without loss of $\epsilon$ and without restriction on time.
Remarkably though, the global estimates can be obtained
if one reverses the order of operations and, instead of
mollifying the initial datum and then solving the Euler equations, one
rather solves first the Euler equations and then mollifies.
Let us consider thus $b\in{\mbox{\bf Y}}\cap B^{s, \infty}_2$ and assume
that
$$
{\cal A}(t) < \infty
$$
for $0\le t\le T$. In view of the results of \cite{bc} this is the case
if $b$ represents a vortex patch with smooth boundaries. Now we consider
$$
\omega^E(x,t) = S^E(t)b
$$
and mollify it, i.e. we consider the function
$$
\omega _{\delta}(x,t) = \left (S^E(t)b\right )*\phi_{\delta}.
$$
The equation obeyed by the mollified vorticity $\omega_{\delta }= 
\left (S^E(t)b\right )_{\delta}$ is
$$
\left (\partial_t + u_{\delta }\cdot\nabla \right )\omega_{\delta }
= \nabla\cdot\tau_{\delta }(u^E, \omega^E).
$$
where
$$
\tau_{\delta}(v,w) = \left (v-v_{\delta}\right )\left (w-w_{\delta}
\right ) - r_{\delta }(v,w)
$$
with
$$
r_{\delta}(v,w)(x) = \int\phi (y)\left (v(x-\delta y)-v(x)\right)
\left (w(x-\delta y)-w(x)\right )dy.
$$ 
The three dimensional analogues of these formulae were first used
in a proof of the Onsager conjecture (\cite{cet}).



We will choose $\delta = \sqrt{\nu T_b}$ and compare $\omega_{\delta}$
to $\omega^{NS}(x,t) = S^{NS}(t)b$.
\begin{thm}
Let $b\in{\mbox{\bf Y}}\cap B^{s, \infty}_2\cap B^{\frac{s}{2}, \infty}_4$
with $0<s<1$. Consider the solution
$$
\omega ^E(\cdot, t) = S^E(t)b
$$
and assume that
$$
{\cal A}(T) = \int_0^T\|\frac{1}{2}\left ((\nabla u^E ) + (\nabla u^E )^*
\right )\|_{L^{\infty}}dt < \infty
$$
and that the solution satisfies
$$
\omega^E \in L^2(0,T; B^{\frac{s}{2}, \infty}_4\cap 
 B^{s, \infty}_2).
$$
Consider $\delta = \sqrt{\nu T_b}$ and set
$$
\omega_{\delta } = \omega^E * \phi_{\delta}
$$
and
$$
u_{\delta} = u^E * \phi_{\delta}
$$
where $\phi_{\delta }$ is an appropriate mollifier. Consider the solution
$$
\omega^{NS} (\cdot ,t) = S^{NS}(t)b.
$$
Then there exists $K$ (depending on $b$ only) such that
$$
\|u^{NS} -u_{\delta}\|_{L^2} \le K\sqrt{\nu}
$$
holds for all $0\le t\le T$ and $\nu$ small. Moreover, for every $p\ge 2$ there exists
a constant $K_p$ (depending only on $p$ and and the solution
$S^E(t)b$ for $0\le t\le T$ ) such that
$$
\|\omega^{NS}(\cdot ,t) - \omega_{\delta }(\cdot ,t)\|_{L^p}
\le K_p\nu^{\frac {s}{2p}}
$$
holds for all $0\le t\le T$. In particular, if $b$ is a vortex patch
with smooth $\left ({\cal C} ^{1,\gamma}\right )$ boundaries then
$$\|S^{NS}(t)b - \left(S^E(t)b\right)_{\sqrt{\nu T_b}}\|_{L^p} \le
K_p\nu^{\frac{1}{4p}}
$$
holds for all $0\le t\le T$.
\end{thm}
As a consequence of the {\it non-pathwise, uniform} results one can obtain 
{\it pathwise, non-uniform}  results. We recall a theorem from
\cite{cw1}:
\begin{thm}
Assume that the initial vorticity $b\in L^1\cap L^{\infty }$
has compact support, included in the disk
$$
\{x; |x|\le L\}.
$$
Then
$$
\|\omega^{NS}(\cdot , t)\|_{L^{\infty}(Q_t)} \le \|b\|_{L^{\infty }}e^{-\frac{U_bL}{\nu}}.
$$
where 
$$
Q_t = \{x; |x| \ge C\left (L + \frac{\nu}{U_b} + U_bt \right )\}
$$
and $C$ is an absolute constant. Moreover, if we set
$$
[[\frac{x}{\delta}]] = \sqrt{1 + \frac{|x|^2}{\delta ^2}},
$$
then, for any $\delta > 0$
$$
\|\omega ^{NS}(\cdot , t)  e^{[[\frac{\cdot }{\delta }]]}\|_{L^p} \le
\|b(\cdot )e^{[[\frac{\cdot }{\delta }]]}\|_{L^p}e^{\left
(\frac{U_bt}{\delta} + \frac{ 7\nu t}{\delta ^2}\right )}
$$
holds for any $p$, $1\le p\le\infty $.
\end{thm}
We will now state the pathwise results:
\begin{thm} Let $b\in{\mbox{\bf Y}}\cap B^{s, \infty}_2$, $0<s<1$, be
an initial vorticity and consider $\omega^{NS} = S^{NS}(t)b $. 
There exists an absolute constant $\gamma $ such that, for all 
$t\in [0,\gamma T_b]$ and all 
Lipschitz functions $f$ that vanish at the origin
$$
\lim_{\nu\to 0}\int f(\omega^{NS}(x,t)dx = \int f(b(x))dx
$$
holds. Consequently the weak limit of distributions is
$$
\lim_{\nu\to 0}\pi_{\left [S^{NS}(t)b \right ]}(dy) = \pi_{b}(dy).
$$
The same result holds for arbitrary any time interval $[0, T]$
provided the solution of the Euler equation $\omega^E = S^E(t)b$
satisfies the assumptions
$$
b\in{\mbox{\bf Y}}\cap B^{s, \infty}_2\cap B^{\frac{s}{2}, \infty}_4,
$$
$$
{\cal A}(T) = \int_0^T\|\frac{1}{2}\left ((\nabla u^E ) + (\nabla u^E )^*
\right )\|_{L^{\infty}}dt < \infty
$$
and
$$
\omega^E \in L^2(0,T; B^{\frac{s}{2}, \infty}_4\cap 
 B^{s, \infty}_2).
$$
As a consequence, the strong limit
$$
\lim_{\nu\to 0}S^{NS}(t)b = S^E(t)b
$$
holds in the $L^p$ norm for vorticities, for all $1<p<\infty $ and the
time intervals corresponding to the two situations considered above.
\end{thm}






\vspace{1cm}

\section{Proofs}


The proof of Theorem 2 follows from a few well-known observations.
First a classical inequality for Calderon-Zygmund singular integrals
yields in this case
$$
\|S\|_{L^{\infty}} \le \|\omega^E\|_{L^{\infty}}\left [\left
(2+\frac{1}{\pi}\right ) + 2\log_{+}\left
(L_{\omega^E}\frac{\|\nabla\omega^E\|_{L^{\infty}}}{\|\omega^E\|_{L^{\infty
      }}}\right )\right ].
$$
Using the conservation laws this becomes
$$
\|S\|_{L^{\infty}} \le \|a\|_{L^{\infty}}\left [\left
(2+\frac{1}{\pi}\right ) + 2\log_{+}\left
(L_{a}\frac{\|\nabla\omega^E\|_{L^{\infty}}}{\|a\|_{L^{\infty
      }}}\right )\right ].
$$
The equation obeyed by $\nabla^{\perp}\omega^E$  is
$$
\left (\partial_t + u^E\cdot\nabla \right )\nabla^{\perp}\omega^E =
(\nabla u^E)\nabla^{\perp}\omega^E.
$$
Consequently
$$
\|\nabla^{\perp}\omega^E\|_{L^{\infty}} \le
\|\nabla^{\perp}a\|_{L^{\infty}}\exp{\left(\int_0^t\|S\|_{L^{\infty}}ds\right )}
$$
and the theorem follows by combining these facts and a straightforward
Gronwall inequality argument.

The proof of Theorem 3 follows closely the proof of the similar
pathwise result in \cite{cw1} and will not be repeated
here. Proposition 1 is elementary and Lemma 1 is just a direct
application of Theorem 2 for the case of $a=b_{\delta}$. 


We sketch the proof
of Lemma 2. Consider $b =\chi_D$ and consider a vector $y$ of small length
$\delta $. The function $b(\cdot + y) - b(\cdot)$ is supported in a
thin open neighborhood of $\partial D$ of width $\delta$. The two
dimensional area of this open set vanishes as  $\delta^{2-d}$: Indeed
one can cover $\partial D$ with $N(\delta )\sim \delta^{-d}$ balls of
radii less or equal to $\delta$. Any point that is at
distance
at most $\delta $ from $\partial D$ is at distance at most  $2\delta $
from the center of at least one of these balls. So the union of the balls with the
same centers but with radii $2\delta $ covers the $\delta $
neighborhood
of the boundary and has the advertised area. The $L^p$ norm of the
difference
is bounded by the $\frac{1}{p}$ power of this area. Note that the
argument fails if we replace box-counting dimension by Hausdorff
dimension.







Now we turn to the ideas for the proof of Theorem 4. We note first
that, in view of the fact that $b\in
{\mbox{\bf Y}}\cap B^{s,\infty}_2$,  it follows that
$$
\|b-b_{\delta}\|_{L^p} \le C\|b\|_{L^\infty}^{1-\frac
  {2}{p}}\|b\|_{L^2}^{\frac{2}{p}}\left
(\frac{\delta}{\rho_b}\right)^{\frac
{2s}{p}}.
$$

The equation obeyed by the difference of vorticities
$$
\omega = \omega ^{NS} - \omega ^{E}
$$
between solutions of the Navier-Stokes equations with initial data $b$
and solutions of the Euler equation with initial data $a= b_{\delta}$
is
$$
\left (\partial_t + u^{NS}\cdot\nabla - \nu \Delta \right )\omega
= \nu\Delta \omega^{E} - u\cdot\nabla\omega^{E}
$$
where
$$
u = u_{\omega}
$$
is the corresponding velocity. We take $p\ge 2$, multiply by $|\omega
|^{p-2}\omega $ and integrate. The first term on the right hand side is
integrated by parts and a straightforward Holder inequality is
applied.


Using the bounds in Theorem 2 and Theorem 3 one can estimate
$$
\int|u||\nabla \omega^E||\omega |^{p-1}dx \le
C\|b\|_{L^{\infty}}^{p-1}\|b\|_{L^2}\left [\delta + \sqrt{\nu
  t}\right]\|\nabla b_{\delta}\|_{L^2}\exp{\left (2{\cal
    A}_{\delta}(t)\right )}.
$$
and
$$
\nu\int |\nabla \omega^E|^2 |\omega |^{p-2} dx \le
C\nu\|b\|_{L^{\infty}}^{p-2}\|\nabla b_{\delta}\|_{L^2}^2\exp{\left (2{\cal
    A}_{\delta}(t)\right )}.
$$
Now, in view of the fact that $b\in B^{s,\infty}_2$ these would be
respectively $\delta^s$ and  $\nu \delta^{-2+2s}$ estimates if
$\exp{\left (2{\cal
    A}_{\delta}(t)\right )}$ would be bounded uniformly as $\delta\to
0$. For general $b\in{\mbox{\bf Y}}\cap B^{s,\infty}_2$ we can use
Lemma 1  to estimate this exponential by a small power
$\delta^{-\epsilon}$ for times that are short by comparison to $T_b$.
We omit further details of the proof of Theorem 4.

For the long time estimates of Theorem 5 we mollify the Euler solution. If
$u^E$ solves the Euler equation then $u_{\delta} = \left (u^E\right)_{\delta}$ 
solves (\cite{cet})
$$
\left (\partial_t + u_{\delta}\cdot\nabla \right )u_{\delta} +\nabla
p_{\delta} = \nabla\cdot \tau_{\delta}(u^E,u^E)
$$
where
$$
\tau_{\delta}(v,w) = \left (v-v_{\delta}\right )\otimes\left (w-w_{\delta}
\right ) - r_{\delta }(v,w)
$$
with
$$
r_{\delta}(v,w)(x) = \int\phi (y)\left (v(x-\delta y)-v(x)\right)\otimes
\left (w(x-\delta y)-w(x)\right )dy.
$$

If $u^{NS}$ solves the Navier-Stokes equations with initial velocity
$u_b$ for $b\in{\mbox{\bf Y}}$ then the difference
$u = u^{NS} - u_{\delta}$ solves
$$
\left (\partial_t +u^{NS}\cdot -\nu\Delta \right )u +\nabla q
= \nu\Delta u_{\delta } - u\cdot\nabla u_{\delta} - \nabla\cdot\tau_{\delta}
(u^E,u^E).
$$
Note that, because we assumed (or proved) that 
$$
\int_0^T\|\frac{1}{2}\left (\left (\nabla u^E\right ) + \left (\nabla
u^E\right )^*\right )\|_{L^\infty }ds
$$
is finite then it follows immediately that
the same is true for 
$$
\int_0^T\|\frac{1}{2}\left (\left (\nabla u^E_{\delta}\right ) + \left (\nabla
u^E_{\delta}\right )^*\right )\|_{L^\infty }ds. 
$$
The term involving $\tau_{\delta }$ is handled in the following
manner. One integrates by parts and after using the viscosity one has
to estimate
$$
\frac{1}{\nu}\|\tau_{\delta}(u^E,u^E)\|_{L^2}^2.
$$
In view of the fact that
$$
\|u_{\omega}(\cdot - y) - u_{\omega}(\cdot )\|_{L^4} \le C|y|\|\omega\|_
{L^4}
$$
it follows that
$$
\|\tau_{\delta}(u^E, u^E)\|_{L^2} \le C\delta^2\|b\|_{L^4}^2.
$$
But $\nu\sim\delta^2$ so we conclude that
$$
\|u\|_{L^2} = O(\delta).
$$
The equation obeyed by the mollified vorticity $\omega_{\delta }= 
\left (S^E(t)b\right )_{\delta}$ is
$$
\left (\partial_t + u_{\delta }\cdot\nabla \right )\omega_{\delta }
= \nabla\cdot\tau_{\delta }(u^E, \omega^E).
$$
Consequently the equation for the difference
$$
w = S^{NS}(t)_b - \left (S^{E}(t)b\right )_{\delta}
$$
is 
$$
\left (\partial_t + u^{NS}\cdot\nabla - \nu\Delta \right )w =
\nu\Delta \omega_{\delta } - u\cdot\nabla\omega_{\delta} - \nabla\cdot
\tau_{\delta }(u^E,\omega^E).
$$
Using the fact that $\omega_{\delta}\in B^{s,\infty }_2$ we
obtain
$$
\|\nabla\omega_{\delta }\|_{L^2} = O(\delta^{-1+s})
$$
and together with the estimate $\|u\|_{L^2} = O(\delta )$
it follows that the estimates for the first two terms on the right hand side
of the equation obeyed by $\int|w|^pdx$ are similar to the
corresponding ones in the proof of Theorem 4.
The estimate for the term involving $\tau_{\delta }$ uses the
viscosity, integration by parts and the estimate
$$
\frac{1}{\nu }\|\tau_{\delta }(u^E, \omega ^E)\|_{L^2}^2
\le K\nu^{-1}\delta^{2+s}
$$
where
$$
K = C\|b\|_{L^4}^2\|S^E(t)b\|_{B^{\frac{s}{2}, \infty}_4}^2.
$$
One obtains thus estimates of the type 
$\|w\|_{L^p} = O(\delta^{\frac{s}{p}}) = O(\nu^{\frac{s}{2p}})$,
and this concludes our presentation of the ideas for the proof of
Theorem 5. 

We show now hints for the proof of Theorem 7. In view of the
result of Theorem 6 that was proved in \cite{cw1} we have
$$
\int _{R^2} f(\omega^{NS}(x,t))dx \simeq \int_{R^2\setminus Q_t} 
f(\omega^{NS}(x,t))dx.
$$
(We use $\simeq$ to denote quantities that become equal in the limit
$\nu\to 0$.) In view of the fact that
$$
\int_{R^2\setminus Q_t} f\left(\left (\omega^{E}(x,t)\right
)_{\delta}\right )dx
\simeq \int_{R^2\setminus Q_t} f(\omega^{E}(x,t))dx = 
\int_{R^2} f(\omega^{E}(x,t))dx
$$
one needs to show that
$$
\int_{R^2\setminus Q_t} f\left (\left (\omega^{E}(x,t)\right
)_{\delta} \right )dx\simeq\int_{R^2\setminus Q_t} 
f(\omega^{NS}(x,t))dx.
$$
Using the Lipschitz condition for $f$ and the fact that $R^2\setminus
Q_t$ is compact it is enough to have
$$
\int_{R^2\setminus Q_t}\left |\omega^{NS}(x,t) -\left (\omega^{E}(x,t)\right
)_{\delta} \right |^2dx\simeq 0.
$$ 
But this follows from the non-pathwise estimates 
$$
\|\omega^{NS} -\omega_{\delta }\|_{L^2} \le K\nu^{\frac {s}{4}}.
$$
The proof of the $L^p$ convergence follows from the fact that the
norms converge ($f(y) =|y|^p$)
$$
\lim_{\nu\to 0}\|S^{NS}(t)b\|_{L^p} = \|S^E(t)b\|_{L^p}
$$
and the strong convergence of velocities in $L^2$ (Theorem 3). This
last fact implies that any weak limit as $\nu\to 0$ of a sequence $S^{NS}(t)b$
for fixed $b$ and $t$ must be $S^E(t)b$. Strong convergence in
$L^p$ for $1<p<\infty$ follows from weak convergence and convergence
of the norms because these spaces are uniformly convex.

\newpage
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