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\author{D. C\'ordoba}
\address{IMAFF,
Consejo Superior de Investigaciones Cient\'{\i}ficas, Madrid,
28006} \email[DC]{dcg@imaff.cfmac.csic.es}
\author{C. Fefferman}
\address{Dept. of Mathematics,
Princeton University, Princeton NJ 08540}
\email[CF]{cf@math.princeton.edu}
\author{R. de la Llave}
\address{Dept. of Mathematics,
Univ. of Texas at Austin, Austin, TX 78712-1802}
\title{On squirt singularities}
\email[R.L]{llave@math.utexas.edu}
\begin{document}
\begin{abstract}
We consider certain singularities of hydrodynamic equations that
have been proposed in the literature.
We present a kinematic argument that shows that, if a volume
preserving field presents these singularities, certain integrals
related to the vector field have to diverge.
We also show that, if the vector fields satisfy certain partial
differential equations (Navier Stokes, Boussinesq) then the
integrals have to be finite.
As a consequence, these singularities are absent in the solutions
of the equations. This answers a question posed by K. Moffat in
\cite{Moffatt}.
\end{abstract}
\maketitle
\section{Introduction}
One way to make progress towards settling the question of
existence of singularities in incompressible fluid motion is to
conjecture plausible scenarios for the formation of singularities
supported by numerical evidence. Then, it becomes a natural
object to develop mathematically rigorous arguments that derive
quantitative consequences of the different scenarios and, possibly
show that these singularities cannot occur in solutions of
hydrodynamic equations.
In this note we introduce some classes of singularities which we
call \emph{``squirt''} singularities in which some portion of
material is ejected from a set of positive measure. These squirt
singularities include as particular cases several other
singularities that had been considered in the literature (for
example, the \emph{``potato chip''} singularities, the
\emph{``saddle collapse''}, the \emph{``tube collapse''}. See
Section \ref{definitions} for precise definitions. )
In Section \ref{kinematic} we present a very simple argument that
shows that if a volume preserving vector field $u$ presents a
squirt singularity at time $T$, then
\begin{equation}\label{divergence}
\int_0^T || u||_{L^\infty} \, dt = \infty.
\end{equation}
In Section \ref{apriori} we show that in the vector field
satisfies certain partial differential equations (e.g.
Navier-Stokes in 2,3 dimensions, Boussinesq equations in 2, 3
dimensions with positive viscosity), then
\begin{equation}\label{convergence}
\int_0^T || u||_{L^\infty} \, dt < \infty.
\end{equation}
As a consequence of the results in Sections \ref{kinematic} and
\ref{apriori}, we conclude that volume preserving vector fields
satisfying the partial differential equations considered in
Section \ref{apriori} do not experience any of the squirt
singularities.
The above results include as a particular case an answer to a
question proposed by K. Moffatt in \cite{Moffatt}. We show that if
a two dimensional fluid satisfies the Boussinesq equation
describing a fluid moving under buoyancy forces with positive
fluid viscosity $\nu > 0$, but possibly with zero molecular
diffusivity $\kappa = 0$, then, it cannot have a saddle collapse.
Even if the arguments presented here exclude that the
singularities happen, they give little information on how fast,
the singular terms may grow. In some of the cases discussed here,
these more quantitative arguments are available in the literature
(see \cite{Cordoba98}).
They, of
course, require using more heavily the details of the equation and
the singularity.
\section{Squirt singularities} \label{definitions}
In this section we collect the definitions of the different types
of singularities that we will be considering in this paper.
\subsection{Notation}
We denote the Lebesgue measure of a set $A$ by $\abs{A}$ and the
ball centered at $\cx$ with radius $r$ by $B_r(\cx)$.
Let $\Omega \subset \real^n$ be an open set. We consider a $C^1$
time dependent vector field $u: \Omega \bigtimes [0,T)\rightarrow
\real^n$.
This vector field defines an evolution for trajectories
$\Phi_t(x)$, where $\Phi_t(x)$ denotes the position at time $t$ of
the trajectory with initial condition $x$ at time $t = 0$. More
generally, we denote by $\Phi_{t,a}(x)$ the position at time $t$
of the trajectory which at time $t =a$ is in $x$. Note that, when
both sides of the formulas make sense, $\Phi_t(x) =
\Phi_{t,0}(x)$, $\Phi_{t,a} = \Phi_{t} \circ \Phi_a^{-1}(x)$,
$\Phi_{t,a} \circ \Phi_{a,b}(x) = \Phi_{t,b}(x)$.
For $\cS \subset \Omega$, we denote by $\Phi_{t,a}^\Omega \cS = \{
x \in \Omega \mid x = \Phi_t(y),\ y \in \cS,\ \Phi_s(y) \in
\Omega,\ 0 \leq s \leq t\}$. That is, $\Phi_{t,a}^\Omega$ is the
evolution of the set $\cS$, starting at time $a$, after we
eliminate the trajectories which step out of $\Omega$.
We will, henceforth assume that $u$ is divergence free. Given the
fact that $u$ has zero divergence, we have that $\abs{\Phi_{t,s}
\cS}$ is independent of $t$ and $\abs{\Phi^\omega_{t,a} \cS}$ is
non-increasing in $t$.
\subsection{Definition of singularities}
\subsubsection{Squirt singularities}
The following definition will be the hypothesis of the main
kinematic result of this paper, Theorem \ref{main}.
\begin{defin} \label{squirt}
Let $\Omega_-, \Omega_+$ be open and bounded sets. $\overline{
\Omega_-} \subset \Omega_+$. (Therefore, $\dist( \Omega_-, \real^d
- \Omega_+ \ge r > 0$.)
We say that $u$ experiences a squirt singularity in $\Omega_-$, at
time $T > 0$
when for every $0 \le s < T$.
we can find a set $\cS_s \subset \Omega_+ $ such that
\begin{itemize}
\item $\cS_s \cap \Omega_-$ has positive
measure, $0 \leq s < T$
\item $\lim_{t \to T} \abs{\Phi_{t,s}^{\Omega_+}\cS_s} = 0$
\end{itemize}
\end{defin}
The physical intuition is that there is a region of positive
volume so that all the fluid occupying it gets ejected from a slightly
bigger region in a finite time.
As we see in the following subsections, Definition \ref{squirt}
includes as particular cases, other singularities that have been
considered in the literature. Of course, the conclusions of
Theorem \ref{kinematic}, which uses
only Definition \ref{squirt} as hypothesis, are a fortiori valid
when we use as hypothesis the existence of the other singularities
that we now formulate.
\subsubsection{Potato chip singularities}\label{potatochip}
\begin{defin} \label{def:potatochip}
We say that $u$ experiences a potato chip singularity when we can
find continuous functions
\begin{equation*}
\begin{split}
f_\pm & : \real^{n-1} \times [0,T) \to \real \\
f_+(\ccx,t) & \geq f_-(\ccx,t) \quad t \in [0,T], \ccx\in B_r(\Pi \cx)\\
f_+(\ccx,0) & > f_-(\ccx,0) \quad \ccx\in B_{\frac r2}(\Pi \cx)\\
\lim_{t\rightarrow T^-}(f_+(\ccx,t) & - f_-(\ccx,t)) = 0 \quad
\text{for all} \quad \ccx\in B_r(\Pi \cx) \\
\end{split}
\end{equation*}
The surfaces
$$
\Sigma_{\pm,t} = \{x_n = f_\pm(x_1, \ldots, x_{n-1}, t)\}
\subset\Omega
$$
are transformed into each other by the flow
$$
\Phi_t(\Sigma_{\pm,0}) \supset \Sigma_{\pm, t}\ .
$$
\end{defin}
Note that in definition \ref{potatochip} we are not requiring that
the functions are $C^1$ as it was done in \cite{CordobaF}. For us,
it suffices that $f_\pm$ are continuous. That is, we allow the
singularities to be ruffled potato chips. Since the arguments we
will present in Section \ref{kinematic} do not depend on calculus
identities, there is no need for the boundaries of the sets to be
differentiable.
If we denote by $\cS^{(f)}_t = \{x \mid f_-(x_1,t) \leq x_n \leq
f_+(x_1,t)\}$, we have, by the intermediate value theorem,
and the continuity of the trajectories
\begin{equation}
\Phi^{B_r(\cx)}_{t,s} (\cS^{(f)}_s) \subset \cS_t.
\end{equation} In particular,
if $f_\pm$ verify definition \ref{potatochip}, then $\cS^{(f)}_s$
verifies the assumptions of Definition \ref{squirt}.
Hence, if a system satisfies Definition \ref{potatochip}, it also
satisfies Definition \ref{squirt}.
Potato chips singularities were introduced as a conjectural
mechanism (see \cite{GrauerM} and \cite{KerrB})
of singularities for a 3D ideal Magnetohydrodymamic flow,
in which two linked flux rings approach each other forming
two-dimensional current sheets. In two dimensional cases, similar
singularities were proposed by \cite{Parker}, \cite{PriestTitov}.
The two dimensional potato chips singularities were considered in
\cite{Corfef1}, where they were called \emph{``sharp fronts''}.
Using calculus identities and the fact that the fluid admits a
stream function representation, it was shown that if a sharp
front exits, then \eqref{divergence} holds. This result was
generalized to three dimensions in \cite{CordobaF}. Both of these
results follow from Theorem \ref{main}.
\subsubsection{Tube collapse singularities}
The following definition appears for the case $n = 3 $ in
\cite{Cordoba1}. In the case $d = 2$ the concept was introduced in
\cite{Corfef1}, \cite{Corfef2}.
Let $I_i \subset \real$, $i = 1,\ldots n$ be bounded intervals.
Let $Q = \times_{i} I_i \subset \real^n$ be a cube.
\begin{defin}\label{tube}
A regular tube is a relatively open set $\cS \subset Q$.
characterized as
\begin{equation}
\cS = \{ x \in Q \, | \, f(x) < 0\}
\end{equation}
where $f: Q \rightarrow \real$ is a $C^1$ function that satisfies
\begin{equation*}
f(x)= 0 \implies \nabla_{x_1,\ldots, x_{n-1}}f \ne 0
\end{equation*}
For every $x_n \in I_n$, the set
\begin{equation*}
\cS(x_n) = \cS \cap I_1 \times \cdots \times I_{n-1} \times
\{x_n\}
\end{equation*}
is non empty and its closure is contained in the interior of
$I_1 \times \cdots \times I_{n-1} \times \{x_n\}$.
\end{defin}
We will also consider the situation when $f_t$ is a family of
functions indexed by time, $t \in [0, T)$.
\begin{defin}\label{tubecollapse}.
We say that the vector field $u$ experiences a tube collapse
singularity at time $T$ when the boundaries of the tube evolve
with the velocity field $u$ and $\liminf_{t \to T} | \cS_t | = 0$.
\end{defin}
An example that is worth keeping in mind is when $f_t(x) =
\dist(x, \gamma) + r(t)$, where $\gamma$ is a curve, $\dist$
denotes the distance and $r(t) \to 0 $ as $t \to 0$. $S_t$ is the
set of points which are at a distance less than $r(t)$ from the
curve $\gamma$. (Of course, we could let the curve $\gamma$
depend on time provided that it does not become too pathological.)
Again, we point that Definition \ref{tubecollapse} implies
Definition \ref{squirt}. We can take $\Omega_- = \times_{i =
1,\ldots n - 1} I_i \times J $, $\Omega_- = \times_{i = 1,\ldots n
- 1} I_i \times I_d $ where $J \subset I_d$ is an interval
contained in the interior of $I_d$.
\subsubsection{Saddle collapse singularity}
This singularity is specific of two dimensional flows. We follow
the definition in \cite{Cordoba98}. We refer to that paper for a
comparison with alternative definitions in the literature.
\begin{defin}\label{saddlecollapse}
We consider foliations of a neighborhood of the origin (with
coordinates $x_1, x_2$) whose leaves are given by equations of the
form
\begin{equation}
\rho \equiv (y_1 \beta(t) + y_2)\cdot (y_1 \delta(t) + y_2) = \cte
\end{equation}
and $(y_1,y_2) = F_t(x_1, x_2)$, where $\beta, \delta: [0, T)
\rightarrow \real^+$, are $C^1$ functions and $F$ is a $C^2$
fuction of $x,t$, for a fixed $t$, $F_t$ is an orientation
preserving diffeormorphism.
We say that the foliation experiences a saddle collapse when
$\liminf_{ t \to T} \beta(t) + \delta(t) = 0$.
If the leaves of the foliation are transported by a vector field
$u$, we say that the vector field $u$ experiences a saddle
collapse.
\end{defin}
If we take as $\Omega_\pm$ balls centered at $F(0,0,T)$, and as
the set $\cS_s$ a connected component of the set $\rho < 0$, we
see that Definition \ref{saddlecollapse} implies Definition
\ref{squirt}.
\section{Kinematic arguments} \label{kinematic}
The main result of this section is:
\begin{thm} \label{main}
If $u$ as before has a squirt singularity, then
\begin{equation} \label{conclusion1}
\int_s^T \sup_x \abs{u(x,t)}\, dt = \infty \quad \forall s \in
(0,T)
\end{equation}
Moreover, if $u$ has a potato chip singularity, then
\begin{equation} \label{conclusion2}
\int_s^T \sup_x \abs{\Pi u(x,t)}\, dt = \infty
\end{equation}
\end{thm}
\begin{remark} \label{generalizations}
We note that in the argument for Theorem \ref{main}, some of the
hypothesis can be somewhat weakened
For example, using the theory of \cite{DiPernaL}, the hypothesis
that $u \in C^1$ can be weakened to $u \in H^1$.
We also note that strict volume preservation is not needed. It
suffices that the volume contraction remains bounded. That is, for
some constant $C \ge 1$ and all $M \subset \real^n$ measurable, $
C^{-1} |M| \le |\Phi_t(M)| \le C |M| $.
\end{remark}
\begin{remark}
We note that if $u(x,t)$ experiences a squirt singularity at $t =
T$ and $\Gamma:[0,T) \hookleftarrow [0,T)$ is a
reparameterization, then
$$
\tilde{u}(x,t) = u(x, \Gamma(t))\Gamma'(t)
$$
also has a potato chip singularity.
It is reassuring to note that the conclusions of Theorem
\ref{main} remain true for $\tilde{u}$. But the observation that
the existence of potato chip singularities is invariant under time
reparameterizations shows that, with the present assumptions,
one cannot obtain more precise rates of the blow-up of
$\sup_x\abs{u(x,t)}$ than \eqref{conclusion1}.
\end{remark}
In case that we assume that singularities are somewhat more
uniform, it is possible to develop more quantitative information
about the rate at which happen.
Roughly speaking, we just need to assume that the exit area of the
set $\cS_s$ controls the volume of the set.
For example, in potato chip singularities (Definition
\ref{def:potatochip}), we say that the collapse is uniform when
\[
\max_{x_1, x_2} (f_+(x_1, x_2,t) - f_-(x_1, x_2,t)) \le M
\min_{x_1, x_2} (f_+(x_1, x_2,t) - f_-(x_1, x_2,t))
\]
where $M$ is a constant independent of time.
In tube collapse singularities (Definitions~\ref{tubecollapse},
\ref{tube}
we say that the collapse is uniform when
\[
\max |S(x_n) |_{n-1} \le M \min |S( x_n)|
\]
where $M$ is a constant independent of time and $| \cdot |_{n-1}$
denotes the $n-1$ dimensional area.
Given a $S_t$ a $C^1$ set, we denote by $\tilde \partial S_t$
the portion of the boundary which is not evolving with the fluid.
We note that by zero divergence of the fluid, the change of volume
is the integral of $u$ over $\tilde \partial S_t$. Hence, we
always have
\[
\frac{d}{dt} |\cS_t| \ge - ||u||_{L^\infty} | \tilde \partial
\cS_t|_{n-1}
\]
In the uniform cases, we have:
\[
\frac{d}{dt} |\cS_t| \ge - M ||u||_{L^\infty} |\cS_t|
\]
Integrating the above equation we have
\[
|\cS_t | \ge |\cS_0| \exp( - M \int_0^t ||u(s)||_{L^\infty} \, ds )
\]
\subsection{Proof of Theorem \ref{main}}
{From} the assumption that $\abs{\Phi_{T,s}^{\Omega_+}\cS_s} \to
0$,
we conclude that almost all
the trajectories starting in $\cS_s $ at time
$s$ leave the set $\Omega_+$ at a time in $(s,T)$.
Therefore we conclude that for a trajectory $x(t)$ starting in $
\Omega_- \cap \cS_s$ at time $s$
we have,
\begin{equation}\label{displacement}
\Big|\int_s^Tu(\Phi_t(x), t)\, dt\Big| \geq r > 0 \ .
\end{equation}
Therefore,
\begin{equation}\label{integral}
\int_s^T \sup_x\abs{u(x,t)}\, dt \geq r > 0 \ .
\end{equation}
Since \eqref{integral} holds for every $s \in (0,T)$ we conclude
that \eqref{conclusion1} holds.
To establish \eqref{conclusion2} we observe that in the classical
potato chip singularity, since the escape can only happen by
increasing the $n-1$ first components we can sharpen
\eqref{displacement} to
$$
\Big|\int_s^T \Pi u(\Phi_t(x),t)\, dt\Big| \geq r/2\ .
$$
again for all $s \in [0,T)$.
\section{A priori bounds} \label{apriori}
In this Section, we show how if the vector field $u$ satisfies
certain partial differential equations, then, \eqref{convergence}
holds. By Theorem \ref{main}, we conclude immediately that these
equations do not exhibit any of the singularities considered in
Definition \ref{squirt}.
We consider, two and three dimensional Boussinesq equations and
Navier-Stokes equations in three dimensions.
We note that the results on two dimensional Boussinesq equations
solve the problem proposed by K. Moffatt \cite{Moffatt}:
\\
\underline{XXI Century Problem 3:} {\it The problem is to examine the
evolution of the $\theta$-field for Boussinesq equations (see
\cite{Batchelor}) in the neighborhood of its
saddle points, to determine whether singularities of $\nabla
\theta$ can develop, and to examine the influence of weak
molecular diffusivity k in controlling the approach to such
singularities.}
The case of Navier-Stokes equations has been in the literature for
a long time. See, for example \cite{FoiasGT81} and the exposition
in \cite{DoeringG}, where it is called the \emph{``second $F_N$
ladder''}.
We point out
that the proofs are based on very elementary arguments. Basically,
integration by parts, Sobolev and interpolation inequalities.
Hence, they remain valid for all the boundary conditions that
allow to carry out these operations. These includes problems
defined in the whole space, in a bounded domain with periodic,
Neuman and Dirichlet conditions. We will, therefore, not state
this very explicitly in the calculations.
\subsection{ Two dimensional Boussinesq Equations} \label{sec:2dbouss}
The Boussinesq equations are:
\begin{eqnarray}
\frac{\partial u}{\partial t} + u\cdot \nabla u = -\nabla p + \nu
\Delta u + (0,\theta) \label{vel}\\
\nabla\cdot u = 0 \label{inc} \\
\left (\partial_t + u\cdot \nabla \right )\theta = \kappa \Delta
\theta \label{tet}
\end{eqnarray}
with $u= (u_1,u_2)$, $x=(x_1,x_2) \in R^2$ or $R^2/Z^2$ and finite
energy at initial time.
The Cauchy problem for the system (10), (11) and (12) have been
extensively studied in the literature, see \cite{canon},
\cite{guo1} and \cite{teman}. In the case $\kappa
> 0$ it is known that the equation does not develop
singularities in finite time. But in order to study the evolution
of the level sets of $\theta $ it is reasonable to take $\kappa
=0$, where the collapse of the saddle would produce a singularity
on $\nabla\theta$. This is a 2-dimensional potato
chip singularity, for more details see \cite{Corfef1}.
In \cite{chae1}, \cite{chae2} and \cite{e} the 2-dimensional
Boussinesq convection in the absence of viscous effects were
studied numerically and analytically.
\begin{thm}\label{2dbouss}
If $u$ satisfies the two-dimensional Boussinesq equation with $\nu
> 0$ and $|| \theta(0)||_{L^2} \le A < \infty$, then
\eqref{convergence} holds.
In particular, using Theorem \ref{main}, $u$ does not exhibit
any singularity satisfying Definition \ref{squirt}.
\end{thm}
{\bf Proof} We denote by $C_1, C_2$ constants that depend only on
$\nu, A$ and the initial conditions. In particular, they can
change the meaning from line to line.
{From} equation \eqref{tet} we obtain that the $L^p$ norms $p \ge 1$ are
nonincreasing -- they are conserved if $\kappa = 0$.
\[
\|\theta(\cdot,t)\|_{L^p} \le \|\theta( \cdot, 0) \|_{L^p} \quad\mbox{for
$1\leq p\leq \infty $} \quad\mbox{for all $t\ge 0$}.
\]
Taking the curl of the equations \eqref{vel} of the velocity
field we get
\begin{eqnarray}\label{vort}
\left (\partial_t + u\cdot \nabla \right )\omega = \theta_{x_1} +
\nu \Delta \omega
\end{eqnarray}
where $\omega = \curl (u)$.
We multiply \eqref{vort} by $\omega$ and integrate by parts to
obtain:
\begin{equation}\label{intermediate}
\frac{1}{2}\frac{d}{dt} \int |\omega|^2\, dx + \nu \int |\nabla
\omega|^2 dx = \int \omega \theta_{x_1} dx.
\end{equation}
Integration by parts and H\"older inequality gives
\[
\frac{1}{2}\frac{d}{dt} \int |\omega|^2\, dx +
\nu ||\nabla \omega||_{L^2}^2
\le || \nabla \omega||_{L^2} ||\theta||_{L^2}.
\]
This implies that
\[
\frac{d}{dt} \int |\omega|^2\, dx \le C_1
\]
and, therefore $|| \omega||_{L^2} \le C_1 t + C_2$.
Substituting this into \eqref{intermediate} gives
\[
\nu || \nabla \omega||_{L^2}^2 \le A ||\nabla \omega||_{L^2} -
\frac{1}{2}\frac{d}{dt} || \omega||_{L^2}
\]
An integration with respect to time and a H\"older inequality for
the integration with respect to time yields
\[
\begin{split}
\nu \int_0^t ds\, || \nabla \omega(s)||_{L^2}^2 &
\le A \int_0^t
ds\, || \nabla \omega(s)||_{L^2} - \frac{1}{2} ||
\omega(t)||_{L^2}^2
+ \frac{1}{2} || \omega(0)||_{L^2}^2 \\
& \le A t^{1/2} \left( \int_0^t ds\,
|| \nabla \omega(s)||_{L^2}^2 \right)^{1/2} +
\frac{1}{2} || \omega(0)||_{L^2}^2
\end{split}
\]
This yields
\begin{equation}\label{est1}
\int_0^t ds \, || \nabla \omega(s) ||_{L^2}^2 \le C_1 t + C_2
\end{equation}
and using H\"older inequality again,
\begin{equation}\label{est2}
\int_0^t ds \, || \nabla \omega(s) ||_{L^2} \le C_1 t + C_2
\end{equation}
The well known Biot-Savart law, recovers the velocity field from
the vorticity by the integral operator
$$
u(x,t) = \frac{1}{2\pi}\int K(x - y) \omega (y, t) dy
$$
with $K(x) =(-\frac{x_2}{x_1^2 + x_2^2}, \frac{x_1}{x_1^2 +
x_2^2})$ for $x\in R^2$ and a similar formula holds for $R^2/Z^2$.
Furthermore, $\nabla u$ is a singular integral operator of
$\omega$ and $\Delta u$ is a singular integral operator of
$\nabla\omega$ (for details see \cite{Bertozzi}).
{From} the classical Calderon-Zygmund theory we have
\begin{eqnarray}\label{est3}
\|\nabla u\|_{L^2} \leq C \|\omega\|_{L^2}, \quad\quad \ \|\Delta
u\|_{L^2} \leq C \|\nabla\omega\|_{L^2}
\end{eqnarray}
Combining estimates \eqref{est1}, \eqref{est2}, and \eqref{est3}
and using Sobolev inequalities we finally get
\[
\begin{split}
\int_{0}^{t} \|u\|_{L^{\infty}} ds &\leq
C \int_{0}^{t} ( \|u\|_{L^2} + \|\Delta u\|_{L^2}) ds \\
&\leq C_1 t + C_2
\end{split}
\]
\qed
\begin{remark}
The argument above works for $\nu >0$. For $\nu =0$ we do not have
control on any norm of the derivatives of the vorticity.
\end{remark}
\subsection{Three dimensional Boussinesq equations} \label{sec3dbouss}
In this section we adapt Theorem \ref{2dbouss} to three
dimensions.
\begin{thm}\label{3dbouss}
If $u$ satisfies the three-dimensional Boussinesq equation with
$\nu > 0$ and $||\theta_0||_{L^2} < \infty$, then \eqref{convergence} holds.
In particular, using Theorem \ref{main}, $u$ does not exhibit any
singularity satisfying Definition \ref{squirt}.
\end{thm}
Compared with the proof of Theorem \ref{2dbouss}, the proof of
Theorem \ref{3dbouss} requires an extra estimate on the nonlinear
term that appears on the vorticity equation. Below we give the
argument which is based on the argument in \cite{FoiasGT81} for
Navier-Stokes.
By the usual integration by parts
\[
\frac12\frac{d}{dt} \int |u|^2\, dx + \nu \int |\nabla u|^2 dx
\leq C \int |u \theta | dx.
\]
therefore, proceeding as before
\begin{eqnarray}\label{est4}
\|u\|^{2}_{L^2} \leq C_1 t + C_2\nonumber\\
\int_{0}^{t} \|\nabla u\|^2_{L^2} ds \leq \tilde{C_1} t +
\tilde{C_2} \label{est5}
\end{eqnarray}
The vorticity equation is
\begin{eqnarray*}\label{vortbis}
\left (\partial_t + u\cdot \nabla \right )\omega =
\omega\cdot\nabla u + \theta_{x_1} - \theta_{x_2} + \nu \Delta
\omega
\end{eqnarray*}
Multiply the vorticity equation by $\omega$ and integrate by parts
\[
\frac12\frac{d}{dt} \int |\omega|^2\, dx + \nu \int |\nabla
\omega|^2 dx < \int |(\omega\cdot\nabla u)\omega| dx +
\frac{1}{2\nu}\int |\theta|^2\, dx + \frac{\nu}{2}\int |\nabla
\omega|^2 dx
\]
The nonlinear term can be bounded by (see \cite{FoiasGT81})
\begin{eqnarray*}
\int |(\omega\cdot\nabla u)\omega| dx &\leq&
C\|\omega\|^{\frac32}_{L^2} \|\nabla \omega\|^{\frac32}_{L^2}\\
&\leq& \tilde{C}\|\omega\|^{6}_{L^2} + \frac{\nu}{4}\|\nabla
\omega\|^{2}_{L^2}
\end{eqnarray*}
then
\[
\frac12\frac{d}{dt} \int |\omega|^2\, dx + \frac{\nu}{4} \int
|\nabla \omega|^2 dx \leq \tilde{C}( 1 + \|\omega\|^{6}_{L^2})
\]
and
\[
\frac{\frac12\frac{d}{dt} \|\omega\|^2_{L^2}}{(1 +
\|\omega\|^{2}_{L^2})^2} + \nu \frac{\|\nabla \omega\|^2_{L^2}}{(
1 + \|\omega\|^{2}_{L^2})^2} \leq \tilde{C}( 1 +
\|\omega\|^{2}_{L^2})
\]
we get
\begin{eqnarray}\label{est6}
\int_{0}^{t} \frac{\|\nabla \omega\|^2_{L^2}}{( 1 +
\|\omega\|^{2}_{L^2})^2} ds \leq \tilde{\tilde{C}}(1 + t)
\end{eqnarray}
Finally we estimate $\int_{0}^{t} \|u\|_{L^{\infty}} ds$ applying
Sobolev inequalities, Calderon-Zygmund theory, \eqref{est5} and
\eqref{est6}
\begin{eqnarray*}
\int_{0}^{t} \|u\|_{L^{\infty}} ds &\leq & C \int_{0}^{t} \|\nabla
u\|^{\frac12}_{L^2} \|\Delta u\|^{\frac12}_{L^2} ds\\
&\leq& C \left(\int_{0}^{t} \|\nabla u\|^2_{L^2} ds + \int_{0}^{t}
\|\Delta u\|^{\frac23}_{L^2} ds\right)\\
&\leq& C \left[\int_{0}^{t} \|\omega\|^2_{L^2} ds +
\left(\int_{0}^{t} \frac{\|\nabla \omega\|^2_{L^2}}{( 1 +
\|\omega\|^{2}_{L^2})^2} ds \right)^{\frac13}\left(\int_{0}^{t} (
1 + \|\omega\|^{2}_{L^2}) ds\right)^{\frac23} \right] \\
&\leq& C (1 + t)
\end{eqnarray*}
where C depends on the initial data and on the viscosity.
\section{Acknowledgments}
D.C was partially
supported by Ministerio de Ciencia y Tecnolog\'{\i}a,
BFM2002-02042. C.F. and R.L have been supported by NSF.
%\bibliographystyle{alpha}
%\bibliography{chip}
\begin{thebibliography}{99}
\bibitem{Batchelor}G.K. Batchelor, V.M. Canuto and J.R. Chasnov.
Homogeneous buoyancy generated turbulance. J. Fluid Mech.
\textbf{212}(1990), 337-363.
\bibitem{BKM}J.~T. Beale, T.~Kato, and A.~Majda. Remarks on the breakdown
of smooth solutions for the {3D} {E}uler
equations. Comm. Math. Phys., 94:61--64, (1984).
\bibitem{Bertozzi} A.L. Bertozzi and A.J. Majda. Vorticity and
incompressible flow. Cambridge University Press. (2002).
\bibitem{CaffarelliKN}L. Caffarelli, R. Kohn and L. Nirenberg. Partial
regularity of suitable weak solutions of the Navier-Stokes
equations. Comm. Pure Appl. Math. \textbf{35}(1982) no.6, 711-831.
\bibitem{canon}J.R. Cannon and E. Dibenedetto. The initial problem
for the Boussinesq equation with data in $L^p$, Lecture Notes in
Math., Springer, \textbf{771} (1980), 129-144.
\bibitem{chae1}D. Chae, O. Y. Imanuvilov. Generic solvability of the
axisymmetric 3-D Euler equations and the 2-D Boussinesq equations.
J. Differential Equations \textbf{156} (1999), no. 1, 1--17.
\bibitem{chae2}D. Chae, S-K. Kim and H-S. Nam. Local existence and
blow-up criterion of Holder continuous solutions of the Boussinesq
equations. Nagoya Math. J. \textbf{155} (1999), 55-80.
\bibitem{Cordoba98} D. Cordoba, ``Nonexistence of simple hyperbolic
blow-up for the quasi-geostrophic equation,'' {\it Ann. of Math.}
148 (1998), 1135--1152.
\bibitem{Corfef1}D. Cordoba and C. Fefferman, ``Scalars convected by a 2D
incompressible flow,'' {\it Comm. Pure Appl. Math.} 55 (2002),
no. 2, 255--260.
\bibitem{Corfef2}D. Cordoba and C. Fefferman, ``Behavior of
several 2D fluid equations in singular scenarios,'' {\it Proc.
Nat. Acad. Sci.} 98 (2001), 4311--4312.
\bibitem{Cordoba1}D. C\'{o}rdoba and C. Fefferman. On the collapse
of tubes carried by 3D incompressible flows. Comm. Math. Phys.
\textbf{222}(2001) no.2, 293-298.
\bibitem{CordobaF}D. C\'{o}rdoba and C. Fefferman. Potato chip
singularities of 3D flows. SIAM J. Math. Anal. \textbf{33}(2001)
no.4, 786-789
\bibitem{DiPernaL}R. J. Diperna and P.L. Lions. Ordinary differential
equations, transport theory and Sobolev spaces. Invent. Math.
\textbf{98}(1989) no.3, 511-547.
\bibitem{DoeringG} C.~R. Doering, J.~D. Gibbon, Applied analysis of
the Navier Stokes equations, Cambridge U. Press. (1995).
\bibitem{e} W. E. and C-W. Shu. Small-scale structures in
Boussinesq convection. Phys. Fluids. \textbf{6} (1994) no. 1,
49-58.
\bibitem{FoiasGT81}C. Foias, C. Guillope and R. Temam. New a priori
estimates for {N}avier-{S}tokes equations in dimension 3. Comm.
Partial Differential Equations. \textbf{6}(1981) no. 3, 329-359.
\bibitem{GrauerM}R. Grauer and C. Marliani, ``Current sheet formation in
3D ideal incompressible magnetohydrodynamics,'' {\it Phys. Rev.
Lett.} {\bf 84} , 4850-4853, (2000).
\bibitem{guo1}B. Guo. Spectral method for solving two-dimensional
Newton-Boussinesq equation. {\it Acta Math. Appl. Sin.} {\bf 5},
(1989), 208-218.
\bibitem{KerrB}R. Kerr and A. Brandenburg, ``Evidence for a Singularity
in Ideal Magnetohydrodynamics: Implications for Fast
Reconnection'' {\it Phys. Rev. Lett.} {\bf 83} , 1155-1158,
(1999).
%\bibitem{Li} Y. Li. Global regularity for the viscous Boussinesq
%equations. Preprint.
\bibitem{Moffatt} H.K. Moffatt. Some remarks on topological fluid
mechanics. R.L. Ricca(ed.) An introduction to the geometry and
topology of fluid flows. Kluwer Academic Publ. (2001), 3-10.
\bibitem{Parker}E.N.Parker. Spontaneous current sheets in magnetic
fields. Oxford University Press, New York (1994).
\bibitem{PriestTitov} E. R. Priest and V. S. Titov. Magnetic reconnection
at three-dimensional null points. {\it Philos. Trans. Roy. Soc.
London Ser. A} {\bf 354} (1996), no. 1721, 2951--2992.
\bibitem{Tartar}L. Tartar. Topics in nonlinear analysis,
Publications Mat. D'Orsay. (1978)
\bibitem{teman}R. Temam. Navier-Stokes equations, theory and
numerical analysis. North-Holland, Amsterdam (1984).
\end{thebibliography}
\end{document}