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:
$$
b\in {\mbox{\bf Y}}\cap \left (\cup_{0~~ 0$, there exists an absolute constant $\gamma $
depending only on $s, \epsilon$ such
that, if
$$
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.
$$
The three dimensional analogues of these formulae were first used
in a proof of the Onsager conjecture (\cite{cet}).
We will choose $\delta = \sqrt{\nu T_b}$ and compare $\omega_{\delta}$
to $\omega^{NS}(x,t) = S^{NS}(t)b$.
\begin{thm}
Let $b\in{\mbox{\bf Y}}\cap B^{s, \infty}_2\cap B^{\frac{s}{2}, \infty}_4$
with $0~~~~ 0$
$$
\|\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~~