\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 89, pp. 1--19.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/89\hfil LANS equation in Besov spaces]
{Local and global existence for the Lagrangian Averaged Navier-Stokes 
equations in Besov spaces}

\author[N. Pennington\hfil EJDE-2012/89\hfilneg]
{Nathan Pennington}

\address{Nathan Pennington \newline
Department of Mathematics, Kansas State University, 138 Cardwell Hall,
Manhattan, KS 66506, USA}
\email{npenning@math.ksu.edu}

\thanks{Submitted February 22, 2012. Published June 5, 2012.}
\subjclass[2000]{76D05, 35A02, 35K58}
\keywords{Navier-Stokes; Lagrangian averaging; global existence; Besov spaces}

\begin{abstract}
 Through the use of a non-standard Leibntiz rule estimate, we prove the
 existence of unique short time solutions to the incompressible,
 iso\-tropic Lagrangian Averaged Navier-Stokes equation with initial
 data in the Besov space $B^{r}_{p,q}(\mathbb{R}^n)$, $r>0$,
 for $p>n$ and $n\geq 3$.  When $p=2$, we obtain unique local
 solutions with initial data in the Besov space $B^{n/2-1}_{2,q}(\mathbb{R}^n)$,
 again with $n\geq 3$, which recovers the optimal regularity available
 by these methods for the Navier-Stokes equation.  Also, when $p=2$ and $n=3$,
 the local solution can be extended to a global solution for all
 $1\leq q\leq \infty$.  For $p=2$ and $n=4$, the local solution can be extended
 to a global solution for $2\leq q\leq \infty$.
 Since $B^s_{2,2}(\mathbb{R}^n)$ can be identified with the Sobolev space
 $H^s(\mathbb{R}^n)$, this improves previous Sobolev space results,
 which only held for initial data in $H^{3/4}(\mathbb{R}^3)$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

The Lagrangian Averaged Navier-Stokes (LANS) equation is a recently derived 
approximation to the Navier-Stokes equation.  The equation is obtained via 
an averaging process applied at the Lagrangian level, resulting in a modified 
energy functional.  The geodesics of this energy functional satisfy the 
Lagrangian Averaged Euler (LAE) equation, and the LANS equation is derived 
from the LAE equation in an analogous fashion to the derivation of the 
Navier-Stokes equation from the Euler equation.  For an exhaustive treatment
 of this process, see \cite{Shkoller}, \cite{SK}, \cite{MRS} and \cite{MS2}. 
In \cite{MKSM} and \cite{CHMZ}, the authors discuss the numerical improvements 
that use of the LANS equation provides over more common approximation 
techniques of the Navier-Stokes equation.

On a region without boundary, the isotropic, incompressible form of the LANS 
equation is given by
\begin{equation}\label{LANS}
\begin{gathered}
 \partial_t u+(u\cdot\nabla)u+\operatorname{div}
\tau^\alpha u=-(1-\alpha^2\Delta)^{-1}\nabla p+\nu\Delta u \\ 
u=u(t,x),\quad \operatorname{div} u=0,\quad u(0,x)=u_0(x),
\end{gathered} 
\end{equation}
with the terms defined as follows.  First,
 $u:I\times \mathbb{R}^n\to \mathbb{R}^n$ for some time strip $I=[0,T)$
denotes the velocity of the fluid, $\alpha>0$ is a constant, 
$p:I\times\mathbb{R}^n\to \mathbb{R}^n$ denotes the fluid pressure,
 $\nu>0$ is a constant due to the viscosity of the fluid, and 
$u_0:\mathbb{R}^n\to\mathbb{R}^n$, with $\operatorname{div} u_0=0$.
Next, the differential operators $\nabla, \Delta,$ and $\operatorname{div}$ 
are spatial differential operators with their standard definitions.  
The term $(v\cdot \nabla)w$, also denoted $\nabla_v w$, is the vector field
with $j^{th}$ component $\sum_{i=1}^n v_i\partial_i w_j$.  The Reynolds
stress $\tau^\alpha$ is given by 
\[
\tau^\alpha u=\alpha^2(1-\alpha^2\Delta)^{-1}[Def(u)\cdot Rot(u)],
\]
where $Rot(u)=(\nabla u-\nabla u^T)/2$ and $Def(u)=(\nabla u+\nabla u^T)/2$.
 We remark that setting $\alpha=0$ in equation \eqref{LANS} recovers the
Navier-Stokes equation.

There is a wide variety of local existence results for the LANS equation in
 various settings, including \cite{Shkoller,MRS,MS,sobpaper}. 
In \cite{MS}, Marsden and Shkoller proved the existence of global solutions 
to the LANS equation with initial data in the Sobolev space 
$H^{3,2}(\mathbb{R}^3)$.  In \cite{sobpaper}, this result was improved, 
achieving global existence for data in the space $H^{3/4,2}(\mathbb{R}^3)$ 
and local existence for initial data in the space $H^{n/2p,p}(\mathbb{R}^n)$.

The most significant obstacle to lowering the initial data regularity necessary 
to obtain these results is the nonlinear terms.  
These terms are typically controlled by the Leibnitz rule type estimate 
(see \cite{CW} for the original reference or Proposition 1.1 in\cite{tt}):
\begin{equation}\label{old product rule}
\|fg\|_{H^{s,p}}\leq \|f\|_{H^{s,p_1}}\|g\|_{L^{p_2}}+\|f\|_{L^{q_1}}\|g\|_{H^{s,q_2}},
\end{equation}
where $1/p=1/p_1+1/p_2=1/q_1+1/q_2$, $s>0$, and $\|\cdot\|_{H^{s,p}}$ denotes 
the Sobolev space norm. In this article, we obtain better regularity results 
by changing to the Besov space $B^s_{p,q}(\mathbb{R}^n)$ setting, where we have
 access to the following, non-standard Leibnitz rule type result:
\begin{equation}\label{new product eqn}
\|fg\|_{B^s_{p,q}}\leq \|f\|_{B^{s_1}_{p_1,q}}\|g\|_{B^{s_2}_{p_2,q}},
\end{equation}
provided $s_1<n/p_1$, $s_2<n/p_2$, $s_1+s_2>0$, $1/p\leq 1/p_1+1/p_2$, and 
$s=s_1+s_2-n(1/p_1+1/p_2-1/p)$.  This is Proposition 
\ref{new product estimate} below, and can be found in \cite{chemin}.  
This result has two advantages over equation \eqref{old product rule}. 
 First, equation \eqref{new product eqn} allows for ``spreading" the regularity $s$ 
between the two terms.  This is not of particular value here, since in the LANS 
equation the nonlinearity is of quadratic type, but it is useful when estimating 
products of functions with varying degrees of regularity 
(see, for example, \cite{galplanme}).  The second advantage 
(and the the one used in this article) equation \eqref{new product eqn}
has over \eqref{old product rule} is that there is no requirement that $s>0$ and, 
by allowing $s_1+s_2>s$, $p_1$, $p_2$ and $p$ are no longer required to 
satisfy the Holder condition.

This is particularly helpful when dealing with negative regularity operators, 
like $\operatorname{div}(1-\alpha^2\Delta)^{-1}$.  Specifically,
\[
\|\operatorname{div}(\tau^\alpha(u))\|_{B^r_{p,q}}\leq \|Def(u)\cdot
Rot(u)\|_{B^{r-1}_{p,q}}.
\]
For $r<1$, further estimating of this term using equation \eqref{old product rule}
 would require first embedding back to $B^s_{p,q}(\mathbb{R}^n)$, $s>0$, and then
applying the equation, which ``wastes" $r-1$ derivatives.  Using equation
\eqref{new product eqn}, we manage to make some (though not full) use of these
 $r-1$ derivatives.  In the statement of our local existence results below,
we will further elaborate on the benefits of equation \eqref{new product eqn}.

The paper is organized as follows.  We devote the rest of this section to 
defining solution spaces and stating our main theorems.  In Section
\ref{Function Space definitions and basic Besov space results}
 we outline some fundamental, known Besov space results.  
In Sections \ref{local continuous solutions} and \ref{local integral solutions},
 we prove Theorems \ref{local lp thm} and \ref{local l2 thm}, respectively,
 stated below.  In Section \ref{extension} we prove
 Theorem \ref{global extension thm}, which extends some of the local solutions 
from Theorem \ref{local l2 thm} to global solutions.  
Section \ref{Extending the local result: Higher regularity} contains a 
technical result necessary for the proof of Theorem \ref{global extension thm}.

As mentioned above, we denote Besov spaces by $B^s_{p,q}(\mathbb{R}^n)$,
 with norm denoted by $\|\cdot\|_{B^s_{p,q}}=\|\cdot\|_{s,p,q}$ 
(a complete definition of these spaces can be found in Section $2$).  
We define the space
\[
C^T_{a;s,p,q}=\{f\in
C((0,T):B^s_{p,q}(\mathbb{R}^n)):\|f\|_{a;s,p,q}<\infty\},
\]
where
\[
\|f\|_{a;s,p,q}=\sup\{t^a\|f(t)\|_{s,p,q}:t\in (0,T)\},
\]
$T>0$, $a\geq 0$, and $C(A:B)$ is the space of continuous functions from $A$ to $B$.
 We let ${\dot{C}}^T_{a;s,p,q}$ denote the subspace of $C^T_{a;s,p,q}$ consisting
of $f$ such that
\[
\lim_{t\to 0^+}t^a f(t)=0 \quad \text{(in }B^s_{p,q}(\mathbb{R}^n)).
\]
Note that while the norm $\|\cdot\|_{a;s,p,q}$ lacks an explicit reference to $T$,
there is an implicit $T$ dependence. We also say $u\in BC(A:B)$ if $u\in C(A:B)$
and $\sup_{a\in A}\|u(a)\|_{B}<\infty$.  Lastly, setting
 $\mathbb{M}((0,T):\mathbb{E})$ to be the set of measurable
functions defined on $(0,T)$ with values in the space $\mathbb{E}$, we define
\[
L^a((0,T):B^s_{p,q}(\mathbb{R}^n))=
\big\{f\in \mathbb{M}((0,T):B^s_{p,q}(\mathbb{R}^n)):\Big(\int_0^T
\|f(t)\|^a_{s,p,q}dt\Big)^{1/a}<\infty\big\}.
\]

Finally, because the Navier-Stokes equation is globally well-posed with initial 
data in $L^2(\mathbb{R}^2)$ (see, for example, Chapter $17$ in \cite{T3}),
 we will restrict ourselves to the case where $n\geq 3$.  We are now ready to 
state our two local existence theorems.

\begin{theorem}\label{local lp thm}
Let $0<r_1<n/p$, with $p>n$, and let $u_0\in B^{r_1}_{p,q}(\mathbb{R}^n)$ 
be divergence free.  Then there exists a unique local solution $u$ to the LANS 
equation \eqref{LANS}, where
\begin{equation}
u\in BC([0,T):B^{r_1}_{p,q}(\mathbb{R}^n))\cap \dot{C}^T_{(r_2-r_1)/2;r_2,p,q},
\end{equation}
$1<r_2<r_1+1$, and $T$ is a non-increasing function of $\|u_0\|_{B^{r_1}_{p,q}}$,
 with $T=\infty$ if $\|u_0\|_{s^+,2,q}$ is sufficiently small.

Similarly, with $0<r_1<n/p$, $p>n$, and $u_0\in B^{r_1}_{p,q}(\mathbb{R}^n)$ 
divergence free, there exists a unique local solution $u$ to the LANS equation 
\eqref{LANS}, where
\begin{equation}
u\in BC([0,T):B^{r_1}_{p,q}(\mathbb{R}^n))\cap L^a((0,T):B^{r_2}_{p,q}(\mathbb{R}^n)),
\end{equation}
$a=2/(r_2-r_1)$, $1<r_2<r_1+1$, and $T$ is a non-increasing function of 
$\|u_0\|_{B^{r_1}_{p,q}}$, with $T=\infty$ if $\|u_0\|_{s^+,2,q}$ is sufficiently 
small.
\end{theorem}

\begin{theorem}\label{local l2 thm}
Let $u_0\in B^{n/2-1}_{2,q}(\mathbb{R}^n)$ be divergence free. 
 Then there exists a unique local solution $u$ to the LANS equation \eqref{LANS},
 where
\begin{equation}\label{local l2 thm eqn}
u\in BC([0,T):B^{n/2-1}_{2,q}(\mathbb{R}^n))\cap \dot{C}^T_{(r-n/2+1)/2;r,2,q},
\end{equation}
$\max (1,n/2-1)<r<n/2$ and $T$ is a non-increasing function of
 $\|u_0\|_{B^{n/2-1}_{2,q}}$, with $T=\infty$ if $\|u_0\|_{s^+,2,q}$ is 
sufficiently small.

Similarly, with $u_0\in B^{n/2-1}_{2,q}(\mathbb{R}^n)$ divergence free,
 there exists a unique local solution $u$ to the LANS equation \eqref{LANS}, where
\begin{equation}
u\in BC([0,T):B^{n/2-1}_{2,q}(\mathbb{R}^n))\cap L^a((0,T):B^{r}_{2,q}
(\mathbb{R}^n)),
\end{equation}
$a=2/(r-n/2+1)$, $\max (1,n/2-1)<r<n/2$, and $T$ is a non-increasing function
 of $\|u_0\|_{B^{n/2-1}_{2,q}}$, with $T=\infty$ if $\|u_0\|_{s^+,2,q}$ 
is sufficiently small.
\end{theorem}

We pause here to address the distinction between these two results. 
 Using techniques like those in \cite{KP}, the nonlinear term from the
 Navier-Stokes equation $\operatorname{div}(u\otimes u)$ can be controlled
provided the initial data has regularity at least $n/p-1$. 
 Using Proposition \ref{new product estimate}, we are able to control 
the LANS specific nonlinear term, $\operatorname{div}(\tau^\alpha(u))$, 
provided the initial data has strictly positive regularity.  Thus, when $p>n$, 
the limiting factor will be LANS specific term $\operatorname{div}(\tau^\alpha(u))$, 
and we obtain local existence provided the initial data has strictly positive regularity.  For $p=2$, the limiting factor is the Navier-Stokes nonlinear term $\operatorname{div}(u\otimes u)$, and we obtain existence provided the data has regularity $n/2-1$.  We remark that this means, for $p=2$, the additional nonlinear term in the LANS equation is no longer limiting the existence result.

Finally, we state our global existence extension.

\begin{theorem}\label{global extension thm}
When $n=3$, the local solutions with initial data 
$u_0\in B^{1/2}_{2,q}(\mathbb{R}^3)$ from Theorem \ref{local l2 thm}
 can be extended to global solutions.  When $n=4$, the local solutions 
with initial data $u_0\in B^{1}_{2,q}(\mathbb{R}^4)$, with $2\leq q\leq \infty$, 
can be extended to global solutions.  In particular, the local solutions 
from Theorem \ref{local l2 thm} can be extended to global solutions when 
$u_0\in B^{n/2-1}_{2,2}(\mathbb{R}^n)=H^{n/2-1,2}(\mathbb{R}^n)$ for $n=3,4$.
\end{theorem}
We remark that this last statement improves the result from \cite{sobpaper},
 which only gave global existence for initial data in $H^{3/4,2}(\mathbb{R}^3)$.

\section{Besov spaces}\label{Function Space definitions and basic Besov space results}

We begin by defining the Besov spaces $B^s_{p,q}(\mathbb{R}^n)$.  
Let $\psi_0\in\mathcal{S}$ be an even, radial function with Fourier 
transform $\hat{\psi_0}$ that has the following properties:
\begin{gather*} 
 \hat{\psi_0}(x)\geq 0\\  
\operatorname{support}\hat{\psi_0}\subset A_0:=\{\xi\in \mathbb{R}^n:2^{-1}<|\xi|<2\}
\\ 
\sum_{j\in\mathbb{Z}} \hat{\psi_0}(2^{-j}\xi)=1, ~\quad \text{for all } \xi\neq 0.
\end{gather*}

We then define
$\hat{\psi_j}(\xi)=\hat{\psi}_0(2^{-j}\xi)$ (from Fourier inversion, this also means
 $\psi_j(x)=2^{jn}\psi_0(2^jx)$), and remark that $\hat{\psi_j}$ is supported 
in $A_j:=\{\xi\in\mathbb{R}^n:2^{j-1}<|\xi|<2^{j+1}\}$.  We also define $\Psi$ by
\begin{equation}\label{low freq part}
\hat{\Psi}(\xi)=1-\sum_{k=0}^\infty \hat{\psi}_k(\xi).
\end{equation}

We define the Littlewood Paley operators $\Delta_j$ and $S_j$ by
\[
\Delta_j f=\psi_j\ast f, \quad S_jf=\sum_{k=-\infty}^{j}\Delta_k f,
\]
and record some properties of these operators.  Applying the Fourier Transform and
recalling that $\hat{\psi}_j$ is supported on
$2^{j-1}\leq |\xi|\leq2^{j+1}$, it follows that
\begin{equation} \label{besovlemma1}
\begin{gathered}  \Delta_j\Delta_k f= 0, \quad |j-k|\geq 2 \\
\Delta_j (S_{k-3}f\Delta_{k}g)= 0 \quad |j-k|\geq 4,
\end{gathered}
\end{equation}
and, if $|i-k|\leq 2$, then
\begin{equation}\label{besovpieces67}
\Delta_j(\Delta_kf\Delta_i g)=0 \quad j>k+4.
\end{equation}

For $s\in\mathbb{R}$ and $1\leq p,q\leq \infty$ we define
the space $\tilde{B}^s_{p,q}(\mathbb{R}^n)$ to be the set of distributions such that
\[
\|u\|_{\tilde{B}^s_{p,q}}=\Big(\sum_{j=0}^\infty (2^{js}\|\Delta_j
u\|_{L^p})^q\Big)^{1/q}<\infty,
\]
with the usual modification when $q=\infty$.  Finally, we define the Besov
spaces $B^s_{p,q}(\mathbb{R}^n)$ by the norm
\[
\|f\|_{B^s_{p,q}}=\|\Psi*f\|_p+\|f\|_{\tilde{B}^s_{p,q}},
\]
for $s>0$.  For $s>0$, we define $B^{-s}_{p',q'}$ to be the dual
of the space $B^s_{p,q}$, where $p',q'$ are the Holder-conjugates to
$p,q$.

These Littlewood-Paley operators are also used to define Bony's paraproduct. 
 We have
\begin{equation}\label{lp start}
fg=\sum_{k} S_{k-3}f\Delta_k g + \sum_{k}S_{k-3}g\Delta_k f
+ \sum_{k}\Delta_k f\sum_{l=-2}^2 \Delta_{k+l} g.
\end{equation}

The estimates \eqref{besovlemma1} and \eqref{besovpieces67} imply that
\begin{equation}  \label{bony256}
\begin{aligned}
\Delta_j (fg)&\leq \sum_{k=-3}^3 \Delta_j (S_{j+k-3}f\Delta_{j+k} g)
 + \sum_{k=-3}^3 \Delta_j (S_{j+k-3}g\Delta_{j+k} f)\\ 
&\quad + \sum_{k>j-4}\Delta_j \Big(\Delta_k f\sum_{l=-2}^2 \Delta_{k+l}g\Big).
\end{aligned}
\end{equation}
This calculation will be very useful in Section \ref{A Modified Product Estimate}.


Now we turn our attention to establishing some basic Besov space estimates. 
 First, we let $1\leq q_1\leq q_2\leq \infty$, $\beta_1\leq \beta_2$, 
$1\leq p_1\leq p_2\leq\infty$, $\gamma_1=\gamma_2+n(1/p_1-1/p_2)$, and $r>s>0$.  
Then we have the following:
\begin{equation}\label{besov embedding}
\begin{gathered} 
\|f\|_{B^{\beta_1}_{p,q_2}}\leq C\|f\|_{B^{\beta_2}_{p,q_1}},\\ 
\|f\|_{B^{\gamma_2}_{p_2,q}}\leq C\|f\|_{B^{\gamma_1}_{p_1,q}},\\ 
\|f\|_{H^{s,p}}\leq \|f\|_{B^r_{p,q}},\\ 
\|f\|_{H^{s,2}}=\|f\|_{B^s_{2,2}}\leq \|f\|_{B^{r}_{2,q}}.
\end{gathered}
\end{equation}
These will be referred to as the Besov embedding results.  
Next, we record a Leibnitz-rule type estimate.  
This can be found in \cite{chemin}, and for the reader's convenience, 
the proof can be found in Section \ref{A Modified Product Estimate}.
\begin{proposition}\label{new product estimate} 
Let $f\in B^{s_1}_{p_1,q}(\mathbb{R}^n)$ and let $g\in B^{s_2}_{p_2,q}(\mathbb{R}^n)$.  Then, for any $p$ such that $1/p\leq 1/p_1+1/p_2$ and with $s=s_1+s_2-n(1/p_1+1/p_2-1/p)$,  we have
\[
\|fg\|_{B^s_{p,q}}\leq \|f\|_{B^{s_1}_{p_1,q}}\|g\|_{B^{s_2}_{p_2,q}},
\]
provided $s_1<n/p_1$, $s_2<n/p_2$, and $s_1+s_2>0$.
\end{proposition}

Our third result is the Bernstein inequalities (see Appendix A in \cite{taonde}). 
 We let $A=(-\Delta)$, $\alpha\geq 0$, and $1\leq p\leq q\leq\infty$.  
If $\operatorname{supp}\hat{f}\subset\{\xi\in\mathbb{R}^n:|\xi|\leq
2^jK\}$ and 
$\operatorname{supp}\hat{g}\subset\{\xi\in\mathbb{R}^n:2^jK_1\leq |\xi|\leq
2^jK_2\}$ for some $K, K_1, K_2>0$ and some integer $j$, then
\begin{equation}\label{Bernstein}
\begin{gathered}
\tilde{C}2^{j\alpha+jn(1/p-1/q)}\|g\|_p \leq \|A^{\alpha/2}g\|_q\leq
C2^{j\alpha+jn(1/p-1/q)}\|g\|_p.
\\ \|A^{\alpha/2} f\|_q \leq C2^{j\alpha+jn(1/p-1/q)}\|f\|_p
\end{gathered}
\end{equation}

Next, we establish estimates for the heat kernel on Besov spaces.

\begin{proposition}\label{hkb} 
Let $1\leq p_1\leq p_2<\infty$,
$-\infty<s_1\leq s_2<\infty$, and let $0<q<\infty$.  Then
\[
\|e^{t\Delta} f\|_{B^{s_2}_{p_2,q}}\leq
Ct^{-(s_2-s_1+n/p_1-n/p_2)/2}\|f\|_{B^{s_1}_{p_1,q}},
\]
provided $0<t<1$.
\end{proposition}

Using the Sobolev space heat kernel estimate, we obtain, for $0<t<1$,
\begin{align*} 
 \|e^{t\Delta}f\|_{B^{s_2}_{p_2,q}}
&=\|\Psi*e^{t\Delta} f\|_{L^{p_2}}+\Big(\sum
(2^{js_1}\|2^{j(s_2-s_1)}\Delta_j e^{t\Delta}f\|_{L^{p_2}})^q\Big)^{1/q}\\
&\leq t^{(n/p_1-n/p_2)/2}\|\Psi*f\|_{L^{p_1}}+\Big(\sum
(2^{js_1}\|e^{t\Delta}\Delta_j f\|_{H^{s_2-s_1,p_2}})^q\Big)^{1/q}\\
&\leq t^{-(n/p_1-n/p_2)/2}\|\Psi * f\|_{L^{p_1}}+t^{\sigma}\Big(\sum
(2^{js_1}\|\Delta_j * f\|_{L^{p_1}})^q\Big)^{1/q}\\
&\leq t^{\sigma}\|f\|_{B^{s_1}_{p_1,q}}.
\end{align*}
where $\sigma=-(s_2-s_1+n/p_1-n/p_2)/2$, and we made liberal use of the fact 
that $e^{t\Delta}$ commutes with convolution operators.
We remark that a straightforward density argument can be used to show that, 
for any $\varepsilon$,
\begin{equation}\label{hkb ext}
\sup_{0\leq t<T}t^{(s_2-s_1+n/p_1-n/p_2)/2}\|e^{t\Delta} f\|_{B^{s_2}_{p_2,q}}
<\varepsilon,
\end{equation}
where $T$ depends only on $\|f\|_{B^{s_1}_{p_1,q}}$.

\subsection{Integral-in-time results}

In this subsection we establish integral-in-time results for Besov space. 
 The proofs are similar to those in \cite{sobpaper} used for the analogous 
operators in Sobolev spaces.  In this section, the operators $\Gamma$ and $G$
 are defined by
\begin{gather*}
  \Gamma f=e^{t\Delta}f,\\
 G(f)(t)=\int_0^t e^{(t-s)\Delta}f(s) ds.
\end{gather*}
We start with a result for $\Gamma$.

\begin{proposition}\label{gammaprop3}
Let $1<p_0\leq p_1<\infty$,
$1\leq q<\infty$, $-\infty<s_0\leq s_1<\infty$, and assume
$0<(s_1-s_0+n/p_0-n/p_1)/2=1/\sigma$.  Then $\Gamma$ maps
$B^{s_0}_{p_0,q_0}$ continuously into
$L^\sigma((0,\infty):B^{s_1}_{p_1,q_1})$ with the estimate
\[
\|\Gamma f\|_{L^\sigma((0,\infty):B^{s_1}_{p_1,q_1})}
\leq C\|f\|_{B^{s_0}_{p_0,q_0}}.
\]
Also, for any $\varepsilon>0$,
\[
\|\Gamma f\|_{L^\sigma((0,T):B^{s_1}_{p_1,q_1})} \leq \varepsilon
\]
provided $T$ is sufficiently small.  The necessary $T$ depends only
 on $\|f\|_{B^{s_0}_{p_0,q_0}}$.
\end{proposition}

The proof is similar to \cite[Prop. 4]{sobpaper}, with two main distinctions,
 both due to the differences in interpolation theory between Sobolev and Besov spaces.  The first is that we interpolate using $s_0$ instead of $p_0$.  The second difference is that we do not require $p_0\leq \sigma$, as we did in Proposition $4$ of \cite{sobpaper}.

The remaining results in this section are for the operator $G$.

\begin{proposition}\label{Gprop3}
Given $1\leq p_0\leq p_1<\infty$, $1\leq q<\infty$,
$-\infty<s_0\leq s_1<\infty$, $1<\sigma_0<\sigma_1<\infty$ and
$1/\sigma_0-1/\sigma_1=1-(s_1-s_0+n/p_0-n/p_1)/2$, for any $T\in
(0,\infty]$, $G$ sends $L^{\sigma_0}((0,T):B^{s_0}_{p_0,q_0})$ into
$L^{\sigma_1}((0,T):B^{s_1}_{p_1,q_1})$ with the estimate
\[
\|G(f)\|_{L^{\sigma_1}((0,T):B^{s_1}_{p_1,q_1})}
\leq C\|f\|_{L^{\sigma_0}((0,T):B^{s_0}_{p_0,q_0})}.
\]
\end{proposition}

\begin{proposition}\label{Gprop4}
$1<p_0\leq p_1<\infty$, $1\leq q<\infty$,
$-\infty<s_0\leq s_1<\infty$, and assume $1/p_1\leq
1/\sigma=1-(s_1-s_0+n/p_0-n/p_1)/2=$.  Then $G$ maps
$L^\sigma((0,T):B^{s_0}_{p_0,q_0})$ continuously into
$BC([0,T):B^{s_1}_{p_1,q_1})$ with the estimate
\[\sup_{t\in[0,T)}\|G(f)(t)\|_{B^{s_1}_{p_1,q_1}}
\leq C\|f\|_{L^{\sigma}((0,T):B^{s_0}_{p_0,q_0})}.
\]
\end{proposition}


\section{Local solutions in $\dot{C}^T_{a;s,p,q}$}\label{local continuous solutions}

We begin by re-writing the LANS equation as
\begin{equation}\label{LANS2}
 \partial_t u-Au+P^\alpha (\operatorname{div}\cdot(u\otimes
u)+\operatorname{div}\tau^\alpha u)=0,
\end{equation}
where the recurring terms are as in \eqref{LANS}, with the exception 
that we set $\nu=1$.  For the new terms, we set $A=P^\alpha\Delta$, $u\otimes u$ 
is the tensor with
$jk$-component $u_ju_k$ and $\text{div}\cdot(u\otimes u)$ is the
vector with $j$-component $\sum_k\partial_k(u_ju_k)$.  $P^\alpha$ is
the Stokes Projector, defined as
\[
P^\alpha(w)=w-(1-\alpha^2\Delta)^{-1}\nabla f
\]
where $f$ is a solution of the Stokes problem: Given $w$, there is a
unique divergence-free $v$ and a unique (up to additive constants) function $f$ such
that
\[
(1-\alpha^2\Delta)v+\nabla f=(1-\alpha^2\Delta)w.
\]
For a more explicit treatment of the Stokes
Projector, see \cite[Theorem 4]{SK}.

Using Duhamel's principle, we write \eqref{LANS2} as the
integral equation
\begin{equation}\label{intversion}
u=\Gamma\varphi-G\cdot P^\alpha(\operatorname{div}(u\otimes u+\tau^\alpha(u)))
\end{equation}
with
\[
(\Gamma\varphi)(t)=e^{tA}\varphi,
\]
where $A$ agrees with $\Delta$ when restricted to $P^\alpha H^{r,p}$, and
\[
G\cdot g(t)=\int_0^t e^{(t-s)A}\cdot g(s)ds.
\]

We prove local existence using the standard contraction mapping method and
 heavy use of the results from Section 
\ref{Function Space definitions and basic Besov space results}. 
 We begin by defining the nonlinear operator $\Phi$ by
\[
\Phi(u)=e^{t\Delta}u_0+\Psi(u),
\]
where
\[
  \Psi(u)=\int_0^t e^{(t-s)\Delta}(V(u))ds
\]
with $V$ (essentially) given by
\[
 V(u)=\operatorname{div}(u\otimes u)+\operatorname{div}
(1-\Delta)^{-1}(\nabla u\nabla u),
\]
where the full definition of $V$ involves additional terms whose behavior
is controlled by the terms shown.

The proofs of local existence in $\dot{C}_{a;r,p,q}$ for the two cases $p=2$ 
and $p>n$ are sufficiently similar that we only present the $p=2$ case here. 
 In Section \ref{local integral solutions} we address the Integral in time case,
 and there we provide the details for the $p>n$ case.

Having set $p=2$, we seek a fixed point of $\Phi$ in the space
\begin{align*} 
 E&=\big\{f\in BC([0,T):B^{n/2-1}_{2,q}(\mathbb{R}^n))\cap 
 \dot{C}^T_{\frac{r-n/2+1}{2};r,2,q}:\\ 
 &\quad \sup_{t\in [0,T)} \|f-e^{t\Delta}u_0\|_{n/2-1,2,q}
 +\|f\|_{(r-n/2+1)/2;r,2,q}<M\big\},
\end{align*} 
for some $T$, $M$, to be determined below.  First, we show that
 $\Phi:E\to E$, and we have
\[
\|\Phi(u)\|_{E}= I+J+K,
\]
where
\begin{equation}  \label{dirk1}
\begin{gathered}
I=\|e^{t\Delta}u_0\|_{(r-n/2+1)/2; r,2,q}\\
J=\sup_{t\in [0,T)}\|\Psi(u)\|_{n/2-1,2,q}\\
K=\|\Psi(u)\|_{(r-n/2+1)/2;r,2,q}.
\end{gathered}
\end{equation}

For $I$, Proposition \ref{hkb} and equation \eqref{hkb ext} give that
\begin{equation}\label{iest}
\|e^{t\Delta}u_0\|_{r-n/2+1;r,2,q}<M/3,
\end{equation}
provided $T$ is sufficiently small.  Estimating $J$ and $K$ is significantly 
more work, and is the focus of the next two subsections.

\subsection{Estimating $J$}

We begin by writing $J\leq J_1+J_2$ where
\begin{gather*} 
 J_1=\sup_{t\in [0,T)}\| \int_0^t e^{(t-s)\Delta}\operatorname{div}(u(s)
\otimes u(s))ds\|_{B^{n/2-1}_{2,q}},\\ 
J_2=\sup_{t\in [0,T)}\|\int_0^t e^{(t-s)\Delta}\operatorname{div}
(1-\Delta)^{-1}(\nabla u(s)\nabla u(s))ds\|_{B^{n/2-1}_{2,q}},
\end{gather*}
and for notational convenience we set $a=(r-n/2+1)/2$.  
Starting with $J_1$, we use Minkowski's inequality and then the heat kernel 
estimate to get
\begin{equation}\label{j11}
J_1\leq \sup_{t\in [0,T)} \int_0^t |t-s|^{(n/2-1-(r-1)+n/p-n/2)/2}\|u(s)
\otimes u(s)\|_{B^{r}_{p,q}}ds,
\end{equation}
where $1/p=1-r/n$.  By Proposition \ref{new product estimate}, we have
\[
\|u(s)\otimes u(s)\|_{B^{r}_{p,q}}\leq \|u(s)\|_{B^r_{2,q}}
\|u(s)\|_{B^0_{\tilde{p},q}}
\leq \|u(s)\|_{B^r_{2,q}}^2,
\]
where $1/p=1/2+1/\tilde{p}$ (which, combined with the definition of $p$,
implies $1/\tilde{p}=1/2-r/n$) and the second inequality used equation
\eqref{besov embedding}.  Substituting back into equation \eqref{j11}
 above, we obtain
\begin{equation}  \label{j12}
\begin{aligned}
J_1&\leq \sup_{t\in [0,T)} \int_0^t |t-s|^{(-r+n-r)/2}\|u(s)
 \otimes u(s)\|_{B^{r}_{p,q}}ds
\\ &\leq C\sup_{t\in [0,T)} \int_0^t |t-s|^{-(n/2-r)}s^{-2a}s^{2a}
 \|u(s)\|_{B^{r}_{2,q}}^2ds
\\ &\leq C\sup_{t\in [0,T)} \|u\|_{a;r,2,q}^2t^{-(n/2-r)-(r-n/2+1)+1}
 \leq C\|u\|_{a;r,2,q}^2.
\end{aligned}
\end{equation}
We remark that this calculation required $n/2-r<1$ and $2a=r-n/2+1<1$,
 which are both satisfied for $n/2-1<r<n/2$.

For $J_2$, with $1/p=1-(r-1)/n$, we have that
\begin{equation}\label{j21}
J_2\leq \sup_{t\in [0,T)}\int_0^t |t-s|^{-(n/p-n/2)/2}\|
\operatorname{div}(1-\Delta)^{-1}(\nabla u(s)\nabla u(s))\|_{B^{n/2-1}_{p,q}}ds.
\end{equation}
By Proposition \ref{new product estimate}, we have
\begin{align*} 
 \|\operatorname{div}(1-\Delta)^{-1}(\nabla u(s)\nabla u(s))\|_{B^{n/2-1}_{p,q}}
&\leq \|\nabla(u(s))\nabla(u(s))\|_{B^{n/2-2}_{p,q}}\\ 
&\leq \|\nabla u(s)\|_{B^0_{2,q}}\|\nabla u(s)\|_{B^{r-1}_{2,q}}\\
&\leq \|u(s)\|_{B^r_{2,q}}^2,
\end{align*}
provided $n/2-2\leq 0+(r-1)-n/2-n/2+n/p$.  Recalling the definition of $p$, 
this simplifies to $n/2-2\leq r-1-n+n-(r-1)=0$, which holds for $n\leq 4$. 
 We pause here to remark that this would not follow from a more standard Leibnitz 
rule estimate, since $n/2+n/2\neq n/p$.  Returning to equation \eqref{j21}, 
we have
 \begin{equation}\label{j22}
\begin{aligned}  
J_2&\leq \sup_{t\in [0,T)}\int_0^t |t-s|^{-(n/2-(r-1))/2}s^{-2a}s^{2a}
 \|u\|^2_{B^{r}_{p,q}}ds
\\ &\leq C\sup_{t\in [0,T)}\|u\|^2_{a;r,2,q}t^{-(n/2-r+1)/2-(r-n/2+1)+1}
 \leq C\|u\|_{a;r,2,q}^2,
\end{aligned}
\end{equation}
again provided $n/2-1<r<n/2$.  Combining equations \eqref{j12} and \eqref{j22}, 
we obtain
\begin{equation}\label{jest}
J\leq C\|u\|_{a;r,2,q}^2\leq CM^2.
\end{equation}
Now we turn to $K$.

\subsection{Estimating $K$}

As with $J$, we write $K$ as $K\leq K_1+K_2$, where
\begin{gather*}
K_1=\sup_{t\in [0,T)}t^a\| \int_0^t e^{(t-s)\Delta}\operatorname{div}
 (u(s)\otimes u(s))ds\|_{B^{r}_{2,q}},\\
K_2=\sup_{t\in [0,T)}t^a\|\int_0^t e^{(t-s)\Delta}\operatorname{div}
(1-\Delta)^{-1}(\nabla u(s)\nabla u(s))ds\|_{B^{r}_{2,q}},
\end{gather*}
where again $a=(r-(n/2-1))/2$.  For $K_1$, we have
\begin{equation}\label{k11}
\begin{aligned}  K_1&\leq \sup_{t\in [0,T)}t^a 
 \int_0^t |t-s|^{-(r-(r-1)+n/p-n/2)/2}\|\operatorname{div}(u(s)
 \otimes u(s))\|_{B^{r-1}_{2,q}}ds \\ 
&\leq C\sup_{t\in [0,T)}t^a \int_0^t |t-s|^{-(1+n/2-r)/2}\|u(s)\|_{B^{r}_{2,q}}^2ds\\ 
&\leq C\sup_{t\in [0,T)} \|u\|_{a;r,2,q}t^at^{-(1+n/2-r)/2-(r-n/2+1)+1}
\leq C\|u\|_{a;r,2,q},
\end{aligned}
\end{equation}
where $p$ is as in the estimate of $J_1$ and we again used
 Proposition \ref{new product estimate}.

For $K_2$, using an argument similar to that used for $J_2$, we have with 
$1/p=1-(r-1)/n$,
\begin{equation}\label{k21}
\begin{aligned} 
 K_2&\leq \sup_{t\in [0,T)}t^a \int_0^t |t-s|^{-(n/p-n/2)/2}\|
 \operatorname{div}(1-\Delta)^{-1}(\nabla(u(s))\nabla (u(s)))\|_{B^{r}_{2,q}}
\\ &\leq \sup_{t\in [0,T)}t^a \int_0^t |t-s|^{-(n/2-r+1)/2}
 \|\nabla(u(s))\nabla (u(s))\|_{B^{r-1}_{2,q}}ds
\\ &\leq C\sup_{t\in [0,T)}t^a \int_0^t |t-s|^{-(n/2-r+1)/2}
 \|u(s)\|_{B^{r}_{2,q}}^2ds
\\ &\leq C\sup_{t\in [0,T)} \|u\|^2_{a;r,2,q}t^at^{-(1+n/2-r)/2-(r-n/2+1)+1}
\leq C\|u\|^2_{a;r,2,q},
\end{aligned}
\end{equation}
where this time the use of Proposition \ref{new product estimate} required
 $n/2-1\leq r$.  Combining equations \eqref{k11} and \eqref{k21}, we obtain
\begin{equation}\label{kest}
K\leq C\|u\|^2_{a;r,2,q}\leq CM^2.
\end{equation}

\subsection{Finishing Theorem \ref{local l2 thm}}

From equations \eqref{iest}, \eqref{jest} and \eqref{kest}, we have that
\[
\|\Phi(u)\|_E\leq I+J+K <M/3+CM^2<M,
\]
provided $T$ and $M$ are sufficiently small, and thus $\Phi:E\to E$.
To show that $\Phi$ is a contraction, we observe that
\begin{gather*}
 u\otimes u-v\otimes v=(u-v)\otimes u+v\otimes (u-v), \\
\nabla u\nabla u-\nabla v\nabla v=\nabla(u-v)\nabla u+v\nabla (u-v),
\end{gather*}
and so, using a slight modification of equations \eqref{jest} and \eqref{kest},
 we have
\[
\|\Phi(u)-\Phi(v)\|_E\leq CM\|u-v\|_E,
\]
which proves that $\Phi$ is a contraction for a sufficiently small choice of $M$.
This completes the proof of the first part of Theorem \ref{local l2 thm}.
  Adapting the above argument to the $p>n$ case proves the first part of
Theorem \ref{local lp thm}.  The details necessary for this adaptation
are similar to those found in the next section.

\section{Local solutions in $L^a((0,T):B^s_{p,q}(\mathbb{R}^n))$}\label{local integral solutions}

As in Section \ref{local continuous solutions}, we seek a fixed point of the map
\[
\Phi(u)=e^{t\Delta}u_0+\Psi(u),
\]
where
\[
\Psi(u)=\int_0^t e^{(t-s)\Delta}(V(u))ds
\]
with $V$ (essentially) given by
\[
 V(u)=\operatorname{div}(u\otimes u)
+\operatorname{div}(1-\Delta)^{-1}(\nabla u\nabla u).
\]
We present the details for the $p>n$ case.  The $p=2$ case is handled
by a combination of the arguments presented here and the arguments used
in Section \ref{local continuous solutions}.

We begin by defining $F$, for a $T$ and $M$ to be chosen later, as
\begin{align*}  
F&=\{f\in BC([0,T):B^{r_1}_{p,q}(\mathbb{R}^n))\cap L^a((0,T):B^{r_2}_{p,q}):
\\ &\quad \sup_{t\in [0,T)} \|f-e^{t\Delta}u_0\|_{B^{r_1}_{p,q}}
+\|f\|_{L^a(B^{r_2}_{p,q})}<M\},
\end{align*}  
where $a=2/(r_2-r_1)$, $r_1$ is an arbitrarily small positive number, 
and $1<r_2<1+r_1$.  As in the previous section, we first show that $\Phi:F\to F$, 
and we have
\[
\|\Phi(u)\|_{F}= I+J+K,
\]
where
\begin{gather*}
I=\|e^{t\Delta}u_0\|_{L^a(B^{r_2}_{p,q})}
\\ J=\sup_{t\in [0,T)}\|\Psi(u)\|_{B^{r_1}_{p,q}}
\\ K=\|\Psi(u)\|_{L^a(B^{r_2}_{p,q})}.
\end{gather*}

For $I$, Proposition \ref{Gprop3} gives that
\begin{equation}\label{2iest}
\|e^{t\Delta}u_0\|_{L^a(B^{r_2}_{p,q})}<M/3,
\end{equation}
provided $T$ is sufficiently small.  As in the previous section,
 estimating $J$ and $K$ is the focus of the next two subsections.

\subsection{Estimating $J$}

We write $J\leq J_1+J_2$, where
\begin{gather*}
  J_1=\sup_{t\in [0,T)}\|G(\operatorname{div}(u\otimes u))(t))\|_{B^{r_1-1}_{p,q}},
\\ J_2=\sup_{t\in [0,T)}\|G(\operatorname{div}(1-\Delta)^{-1}(\nabla u\nabla u))
(t)\|_{B^{r_1}_{p,q}}.
\end{gather*}

For $J_1$, we use Proposition \ref{Gprop4} and get
\begin{equation}\label{2j11}
J_1\leq \|\operatorname{div}(u\otimes u)\|_{L^\sigma(B^{r-1}_{p/2,q})}
\leq \|u\otimes u\|_{L^\sigma(B^{r}_{\tilde{p},q})},
\end{equation}
where $1/\sigma=1-(r_1-r_2+1+2n/p-n/p)/2=(r_2-r_1+1-n/p)/2$.  
Using Proposition \ref{new product estimate}, we have
\[
\|u(s)\otimes u(s)\|_{B^{r_2}_{p/2,q}}\leq \|u(s)\|_{B^{r_2}_{p,q}}
\|u(s)\|_{B^0_{p,q}}.
\]
Plugging back into equation \eqref{2j11}, we obtain
\begin{equation}\label{2j12}
\begin{aligned}
J_1&\leq \|u\otimes u\|_{L^\sigma(B^{r}_{\tilde{p},q})}
 \leq \Big(\int_0^T (\|u(s)\|_{B^{r_2}_{p,q}}\|u(s)\|_{B^{r_1}_{p,q}}
 )^\sigma ds\Big)^{1/\sigma}
\\ &\leq C\sup_{t\in [0,T)}\|u(t)\|_{B^{r_1}_{p,q}}\|u\|_{L^a(B^{r_2}_{p,q})}
\leq CM^2,
\end{aligned}
\end{equation}
where we used that $\|\cdot\|_{L^\sigma}\leq \|\cdot\|_{L^a}$, since $\sigma\leq a$.

For $J_2$, again using Proposition \ref{Gprop4}, we have
\begin{equation}\label{2j21}
  J_2\leq \|\operatorname{div}(1-\Delta)^{-1}(\nabla u\nabla u)\|_{L^1(B^{r_1}_{p,q})}
\leq \|\nabla u\nabla u\|_{L^1(B^{r_1-1}_{p,q})}.
\end{equation}
Using Proposition \ref{new product estimate}, we have
\begin{equation}\label{multi34}
\|\nabla u(s)\nabla u(s)\|_{B^{r_1-1}_{p,q}}\leq \|\nabla u\|_{B^{r_2-1}_{p,q}}^2
\leq \|u(s)\|_{B^{r_2}_{p,q}}^2,
\end{equation}
provided $r_1-1\leq 2(r_2-1)-n/p$ (recall $r_2>1$, so $r_2-1>0$).  
This condition is equivalent to $n/p\leq 2r_2-1-r_1$, and since $r_1<n/p-1$,
 equation \eqref{multi34} holds.  Using equation \eqref{multi34}
 in equation \eqref{2j21}, we have
\begin{equation}\label{2j22}
 J_2\leq \|\nabla u\nabla u\|_{L^1(B^{r_1-1}_{p,q})}
\leq C\|u\|_{L^2(B^{r_2}_{p,q})}^2\leq C\|u\|_{L^a(B^{r_2}_{p,q})}^2\leq CM^2,
\end{equation}
since $2<a=2/(r_2-r_1)$.  So using equations \eqref{2j12} and \eqref{2j22},
 we have
\begin{equation}\label{2jest}
J\leq CM^2.
\end{equation}

\subsection{Estimating $K$}

We have that $K\leq K_1+K_2$ with
\begin{gather*}
  K_1=\|G(\operatorname{div}(u\otimes u))\|_{L^a(B^{r_2}_{p,q})},
\\ K_2= \|G(\operatorname{div}(1-\Delta)^{-1}(\nabla u\nabla u))\|_{L^a(B^{r_2}_{p,q})}.
\end{gather*}

Using Proposition \ref{Gprop3}, for $K_1$, we have
\begin{equation}\label{2k11}
K_1=\|G(\operatorname{div}(u\otimes u))\|_{L^a(B^{r_2}_{p,q})}
\leq \|\operatorname{div}(u\otimes u)\|_{L^\sigma(B^{r_2-1}_{p/2,q})}
\leq \|u\otimes u\|_{L^\sigma(B^{r_2}_{p/2,q})},
\end{equation}
where $1/\sigma-1/a=1-(r_2-(r_2-1)+2p/n-n/p)/2$, which can be rewritten as 
$1/\sigma=(r_2-r_1)/2+(1-n/p)/2$.  Using Proposition \ref{new product estimate}, 
we have
\[
\|u(s)\otimes u(s)\|_{B^{r_2}_{p/2,q}}\leq \|u(s)\|_{B^{r_2}_{p,q}}\|u(s)\|_{B^0_{p,q}}.
\]
Applying this to equation \eqref{2k11}, we have
\begin{equation}\label{2k12}
\begin{aligned}
K_1&\leq C\Big(\int_0^T (\|u(s)\|_{B^0_{p,q}}\|u(s)\|_{B^{r_2}_{p,q}})^{\sigma}ds\Big)
 ^{1/\sigma}
\\ &\leq C\sup_{t\in [0,T)}\|u(t)\|_{B^{r_1}_{p,q}}\|u\|_{L^\sigma(B^{r_2}_{p,q})}
 \leq CM\|u\|_{L^a(B^{r_2}_{p,q})}\leq CM^2,
\end{aligned}
\end{equation}
which required $1/\sigma>1/a$, which holds since $p>n$.  Now we turn to $K_2$,
where we have
\begin{equation}\label{2k21}
\begin{aligned}
K_2&=\|G(\operatorname{div}(1-\Delta)^{-1}(\nabla u\nabla u))\|_{L^a(B^{r_2}_{p,q})}
\\ &\leq \|\operatorname{div}(1-\Delta)^{-1}\nabla u\nabla u\|_{L^\sigma(B^{r_1}_{p,q})}\leq \|\nabla u\nabla u\|_{L^\sigma(B^{r_1-1}_{p,q})},
\end{aligned}
\end{equation}
provided $1/\sigma-1/a=1-(r_2-r_1)/2$, which implies $\sigma=1$.
Then, by equation \eqref{2k21} above, we have
\begin{equation}\label{2k22}
K_2\leq \|\nabla u\nabla u\|_{L^\sigma(B^{r_1-1}_{p,q})}\leq CM^2.
\end{equation}
Combining equations \eqref{2j12} and \eqref{2j22}, we obtain
\begin{equation}\label{2kest}
K\leq CM^2.
\end{equation}

Given equations \eqref{2iest}, \eqref{2jest}, and \eqref{2kest}, we have that
\[
\Phi(u)\leq M/3+CM^2<M,
\]
provided $M$ is sufficiently small.  From here, local existence follows
from the standard method.

\section{Proof of Theorem \ref{global extension thm}}\label{extension}

In this section we prove Theorem \ref{global extension thm}, and we start
 by proving the following \emph{a priori} estimate.

\begin{lemma}\label{a priori} 
Let $f$ be a solution to the LANS equation such that
 $f(t)\in H^{2,2}(\mathbb{R}^n)$ for all $t\in [a,T)$ for some $a\geq 0$.  Then
\[
\sup_{t\in [a,T)} \|f(t)\|_{H^{1,2}}\leq \|f(a)\|_{H^{1,2}}.
\]
\end{lemma}

We begin the proof of the Lemma by stating the following equivalent form of the
 LANS equation (see \cite[Section 3]{MS}):
\begin{equation}\label{LANSglobalpi}
\begin{aligned}  
&\partial_t (1-\alpha^2\Delta)f(t) -(1-\alpha^2\Delta)\Delta f(t)\\
&=-\nabla p -\alpha^2(\nabla {f(t)})^T\cdot (-\Delta){f(t)}
 -\nabla_{f(t)}[(1-\alpha^2\Delta){f(t)}]
\end{aligned}
\end{equation}
Taking the $L^2$ product of the equation with ${f(t)}$, we obtain
\begin{equation} \label{monticeto}
\partial_t (\|f(t)\|_{L^2}^2+\alpha^2\|\nabla f(t)\|_{L^2}^2) 
 +\|\nabla f(t)\|_{L^2}^2+\alpha^2\|\Delta f(t)\|_{L^2}^2
\leq I_1+I_2+I_3,
\end{equation}
where
\begin{gather*}
  I_1=(\nabla_{f(t)} {f(t)},{f(t)}),
\\ I_2=\alpha^2\Big((\nabla_{f(t)} \Delta {f(t)},{f(t)})+((\nabla {f(t)})^T\cdot (-\Delta){f(t)}, {f(t)})\Big),
\\ I_3=(\nabla p,{f(t)}).
\end{gather*}

An application of integration by parts and recalling that 
$\operatorname{div} {f(t)}=0$ gives that $I_3=0$.  For $I_1$, writing the
 expression in its coordinate form gives
\begin{align*} 
 I_1&=(\nabla_{f(t)} f(t), f(t))=\sum_{i,j=1}^n 
 \int f_i(t)(\partial_{x_i}f_j(t))f_j(t)
\\ &=\sum_{i,j=1}^n \frac{1}{2}\int f_i(t)(\partial_{x_i}(f_j(t))^2)
=-\frac{1}{2}\int \operatorname{div} (f(t))|f(t)|^2 =0.
\end{align*}

For $I_2$, writing it in coordinates (and temporarily suppressing the time 
dependence), we see that
\begin{align*}
  I_2&=\sum_{i,j=1}^n\alpha^2\int f_i(\partial_{x_i} \Delta f_j)f_j
 +(\Delta f_i)(\partial_{x_j} f_i)f_j \\
 &=\sum_{i,j=1}^3\alpha^2\int -(f_i(\Delta f_j)(\partial_{x_i} f_j))
+(\Delta f_i)(\partial_{x_j} f_i)f_j= 0,
\end{align*}
where we again used integration by parts and exploited the divergence free condition.
  We remark here that it is these cancellations which make it easier to control the 
long time behavior of the LANS equations.  Returning to equation \eqref{monticeto}, 
we have
\[
\partial_t (\|f(t)\|_{L^2}^2+\alpha^2\|\nabla f(t)\|_{L^2}^2)
\leq -(\|\nabla f(t)\|_{L^2}^2+\alpha^2 \|\Delta f(t)\|_{L^2}^2),
\]
which, combined with Gronwall's inequality, completes the Lemma.
Note that, if $\alpha=0$, this reduces to the well-known $L^2$
control of the solution.

Now we are ready to prove Theorem \ref{global extension thm}. 
 The extension arguments for the two different local existence results 
from Theorem \ref{local l2 thm} are similar, and we present here the argument 
for the local solution $u$ given in equation \eqref{local l2 thm eqn}.  
First, because the time interval of the local solution given by 
Theorem \ref{local l2 thm} depends only on $\|u_0\|_{B^{n/2-1}_{2,q}}$, 
global existence will follow from a standard bootstrapping argument once 
we have a uniform in time bound on $\|u(t)\|_{B^{n/2-1}_{p,q}}$.

Because $u\in BC([0,T)^{n/2-1}_{2,q}(\mathbb{R}^n))$, there exists an $a<T$ such that
\begin{equation}\label{chavez}
\sup_{t\in [0,T)}\|u(t)\|_{B^{n/2-1}_{2,q}}\leq 2\|u_0\|_{B^{n/2-1}_{2,q}}
+\sup_{t\in[a,T)}\|u(t)\|_{B^{n/2-1}_{2,q}}.
\end{equation}
So our remaining task is to bound the second term, and this will follow from 
Lemma \ref{a priori}.  First, from Lemma \ref{higher reg} in the next section, 
we have that $u(t)\in B^{n/2+1}_{2,q}(\mathbb{R}^n)$ for all $t>0$.  
From the Besov embedding results in equation \eqref{besov embedding}, 
this means $u(t)\in H^{2,2}(\mathbb{R}^n)$ for all $t>0$, and thus 
Lemma \ref{a priori} can be applied to our solution $u$.  
Using Lemma \ref{a priori}, when $n=3$, we have
\[
\sup_{t\in[a,T)}\|u(t)\|_{B^{n/2-1}_{2,q}}
\leq \sup_{t\in[a,T)}\|u(t)\|_{H^{1,2}}\leq \|u(a)\|_{H^{1,2}}.
\]
Plugging this back into  \eqref{chavez} gives the desired uniform
 bound on $\|u(t)\|_{B^{3/2-1}_{2,q}}$.  For $n=4$, $n/2-1=1$,
and Lemma \ref{a priori} provides the desired bound when
$\|u(t)\|_{B^{1}_{2,q}}\leq \|u(t)\|_{H^{1,2}}=\|u(t)\|_{B^1_{2,2}}$,
 which holds for $2\leq q\leq \infty$.

For the integrable in time spaces, the only distinction in the argument
 is that Lemma \ref{a priori} only provides a bound almost everywhere,
 since Lemma \ref{higher reg} gives that $u(t)\in B^{2}_{2,q}(\mathbb{R}^n)$
 for almost every $t>0$.  So, in this case, Lemma \ref{a priori}
 and the Besov embedding results only give that $\|u(t)\|_{B^{n/2-1}_{2,q}}$ 
is uniformly bounded for almost all $t$.  However, since 
$u\in BC([0,T):B^{n/2-1}_{2,q}(\mathbb{R}^n))$, continuity extends the bound to
 all time.

\section{Higher regularity for the local existence result}\label{Extending the local result: Higher regularity}

In this section we quantify the smoothing effect of the heat kernel on our 
local solutions.  The proof is an induction argument, similar to the 
one in \cite{sobpaper} applied to the LANS equation (which was in turn 
inspired by the argument in \cite{katoinduction} for the Navier-Stokes equation).


\begin{lemma}\label{higher reg}
Let $u_0\in B^{s}_{p,q}(\mathbb{R}^n)$ and let $u$ be an associated solution
 to the LANS equation with initial data $u_0$ such that
\[
u\in BC([0,T):B^{r}_{p,q}(\mathbb{R}^n))\cap \dot{C}^T_{(s-r)/2;{s},p,q},
\]
where $0<s-r<1$ and $s>1$.  Then for all $k\geq s$, we have
that $u\in \dot{C}^T_{(k-s)/2;k,p,q}$.
\end{lemma}

We have an analogous result for the integral in time case.

\begin{lemma}\label{higher regularity theorem2}
Let $k>s_2>s_1$, with $s_2\geq 1$, and let $\varepsilon$ be a small positive number.  
Then, for $k-s_2=s_2-s_1=\varepsilon$, for any solution $u$ to the 
LANS equation \eqref{LANS} where
\[
u\in BC([0,T):B^{s_1}_{p,q}(\mathbb{R}^n)\cap L^{2/(s_2-s_1)}((0,T):
B^{s_2}_{p,q}(\mathbb{R}^n)),
\]
we have that $u\in L^1((0,T):B^{k}_{p,q}(\mathbb{R}^n))$.
\end{lemma}

The proofs of the two Lemmas are similar.  
The rest of the section is devoted to the proof of Lemma \ref{higher reg}.

\begin{proof}
We start with a solution to the LANS equation $u$.  Then let $\delta>0$
 be arbitrary, and let $w=t^\delta u$.  We note that $w(0)=0$.  Then
\begin{align*}
\partial_t w&=\delta t^{\delta-1} u+t^\delta \partial_t u
\\ &=\delta t^{-1} w+t^\delta (\Delta u-\operatorname{div}(u\otimes u
+\tau^\alpha (u,u)))
\\ &=\delta t^{-1} w+ \Delta w-t^{-\delta}\operatorname{div} (w\otimes w
+\tau^\alpha (w,w)).
\end{align*}
Applying Duhamel's principle, we obtain
\begin{align*}
w&=e^{t\Delta}w_0+\int_0^t e^{(t-s)\Delta}s^{-1}w(s)ds\\
&\quad +\int_0^t e^{(t-s)\Delta}s^{-\delta}(\operatorname{div}(w(s)\otimes w(s)
+\tau^\alpha(w(s),w(s))))ds.
\end{align*}
Recalling that $w(0)=w_0=0$, and substituting $w=t^\delta u$, we obtain
\[
 u=t^{-\delta}\int_0^t e^{(t-s)\Delta}s^{\delta-1}u(s)ds
+t^{-\delta}\!\int_0^t e^{(t-s)\Delta}s^\delta(\operatorname{div}(u(s)\otimes u(s)
+\tau^\alpha(u(s),u(s)))) ds.
\]

Now we are ready to apply the induction.  We have by assumption that $u$ 
is in $\dot{C}^T_{(r-s)/2;r,p,q}$, where $r> 1$.  For induction, 
we assume this solution $u$ is also in $\dot{C}^T_{(k-r)/2;k,p,q}$, 
and seek to show that $u$ is in $\dot{C}^T_{(k+h-r)/2;k+h,p,q}$, 
where $0<h<1$ is fixed and will be chosen later.  We have
\[
  \|u\|_{B^{k+h}_{p,q}}\leq I+J_1+J_2,
\]
with $I$, $J_1$, and $J_2$ defined by
\begin{gather*}
I=t^{-\delta}\int_0^t \|e^{(t-s)\Delta} s^{\delta-1}u(s)\|_{B^{k+h}_{p,q}}ds
\\ J_1=t^{-\delta}\int_0^t \|e^{(t-s)\delta} s^{\delta} (\operatorname{div}(1-\alpha^2\Delta)^{-1}(\nabla u(s)\nabla u(s)))\|_{B^{k+h}_{p,q}}ds
\\ J_2=t^{-\delta}\int_0^t \|e^{(t-s)\delta} s^{\delta}(\operatorname{div}(u(s)\otimes u(s)))\|_{B^{k+h}_{p,q}}ds
\end{gather*}
where, as usual, we have suppressed terms from $\tau^\alpha$ that are
controlled by the terms we included.

\subsection{Bounding $I$, $J_1$, and $J_2$}

Starting with $I$, we have
\begin{equation}\label{fsu1}
\begin{aligned}  
I&\leq t^{-\delta}\int_0^t |t-s|^{-h/2}s^{\delta-1}\|u(s)\|_{B^{k}_{p,q}}
\\ &\leq t^{-\delta}\|u\|_{(k-r)/2;k,p,q}\int_0^t |t-s|^{-h/2}s^{\delta-1-(k-r)/2}ds
\\ &\leq C\|u\|_{(k-r)/2;k,p,q}t^{-\delta}t^{-h/2}t^{\delta-1-(k-{n/2})/2+1}
\\ &\leq Ct^{-(k+h-r)/2}\|u\|_{(k-r)/2;k,2,q},
\end{aligned}
\end{equation}
provided
\[
1>h/2, \quad -1 <\delta-1-(k-r)/2,
\]
which clearly holds for sufficiently large $\delta$.
We observe that, without modifying the PDE to include these $t^\delta$ terms,
 we would need $(k-r)/2$ to be less than $1$, which does not hold for large $k$.

For $J_1$, we have
\begin{align} 
J_1&\leq t^{-\delta}\int_0^t|t-s|^{-(h+2n/p-n/p)/2}s^{\delta} \|\operatorname{div}(1-\Delta)^{-1}(\nabla u \nabla u)\|_{B^{k}_{p/2,q}}ds
\nonumber\\ 
&\leq t^{-\delta}\int_0^t|t-s|^{-(h+n/p)/2}s^{\delta} \|(\nabla u \nabla u)\|_{B^{k-1}_{p/2,q}}ds
\nonumber \\ 
&\leq t^{-\delta}\int_0^t|t-s|^{-(h+n/p)/2}s^{\delta} \| u \|_{B^{k}_{p,q}}\|\nabla u\|_{B^{1}_{p,q}}ds
\nonumber\\ 
&\leq t^{-\delta}\|u\|_{(k-r)/2;k,p,q}\|u\|_{(1-r)/2;1,p,q} \int_0^t|t-s|^{-(h+n/p)/2}s^{\delta-(k-r)/2-(1-r)/2}ds
\nonumber\\ 
&\leq t^{-\delta-(h+n/p)/2-(k-r)/2-(1-r)/2+1+\delta}\|u\|_{(k-r)/2;k,p,q}^2
\nonumber\\ 
&\leq t^{-(k+h-r))/2-(n/p-1-r)/2}\|u\|_{(k-r)/2;k,p,q}^2\leq t^{-(k+h-r)/2
\|u\|_{(k-r)/2;k,p,q}^2}, \label{fsu2}
\end{align}
provided
\[
 \delta>(k-r)/2+(1-r)/2,\quad 
 2>h+{n/p}, \quad 
 r\geq n/p-1,
\]
and we again see that this is easily satisfied by choosing $\delta$ large and 
$h$ small.  For $J_2$, we handle the cases $p=2$ and $p>n$ separately.  
For $p>n$, we have
\begin{equation}\label{fsu3}
\begin{aligned}  
J_2 &\leq t^{-\delta}\int_0^t|t-s|^{-(h+1+2n/p-{n/p})/2}s^{\delta} 
 \|u\otimes u\|_{B^{k}_{p/2,q}}ds
\\ &\leq t^{-\delta}\int_0^t|t-s|^{-(h+1+n/p)/2}s^{\delta} \|u\|_{B^{k}_{p,q}}
 \|u\|_{B^{s}_{p,q}}ds
\\ &\leq t^{-\delta}\|u\|_{(k-{r})/2;k,p,q}\|u\|_{0;{s},p,q} 
 \int_0^t|t-s|^{-(h+1+n/p)/2}s^{\delta-(k-r)/2}ds
\\ &\leq t^{-(h+k-r)/2-(1+n/p-2)/2}\|u\|_{(k-{n/2})/2;k,2,q}\|u\|_{0;{n/2},2,q}\\
&\leq t^{-(h+k-r)/2}\|u\|_{(k-{n/2})/2;k,2,q}\|u\|_{0;{n/2},2,q},
\end{aligned}
\end{equation}
provided
\[
  1>h+n/p, \quad -1<\delta-(k-r)/2.
\]

For the $p=2$ case, we specialize to the case $r=n/2-1$, which is the minimal 
$s$ allowed by our local existence theorem.  The argument for larger
 $s$ is a straightforward generalization of the one presented here. 
 Defining $1/\tilde{p}=1-1/n$, we have
\begin{equation}\label{fsu4}
\begin{aligned}  
J_2&\leq t^{-\delta}\int_0^t|t-s|^{-(h+1+n/\tilde{p}-n/2)/2}s^{\delta} \|u\otimes u\|_{B^{k}_{\tilde{p},q}}ds
\\ &\leq t^{-\delta}\int_0^t|t-s|^{-(h+1+n/2-1)/2}s^{\delta} \|u\|_{B^{k}_{2,q}}L^{2n/(n-2)}ds
\\ &\leq t^{-\delta}\|u\|_{(k-{r})/2;k,2,q}\|u\|_{(1-r)/2;1,2,q} \int_0^t|t-s|^{-(h+n/2)/2}s^{\delta-(k-r)/2-(1-r)/2}ds
\\ &\leq t^{-(h+k-r)/2-(n/p-r-1)/2}\|u\|_{(k-{n/2})/2;k,2,q}\|u\|_{(1-r)/2;1,2,q}
\\ &\leq t^{-(h+k-r)/2}\|u\|_{(k-{n/2})/2;k,2,q}\|u\|_{(1-r)/2;1,2,q},
\end{aligned}
\end{equation}
provided
\[
  2>h+n/2, \quad
 -1<\delta-(k-r)/2-(1-r)/2, \quad
 r\geq n/2-1,
\]
which, again, are easily satisfied.

Combining equations \eqref{fsu1}, \eqref{fsu2} and \eqref{fsu3} for $p>n$ 
(or \eqref{fsu4} if $p=2$), we have that, for $h$ small enough and $\delta$ 
large enough,
\[
I+J_1+J_2\leq Ct^{-(h+k-n/2)/2}\|u\|_{(k-{n/2})/2;k,2,q}^2
\]
This in turn gives
\[
  \|u\|_{B^{k+h}_{p,q}}\leq Ct^{(k+h-r)/2}\|u\|^2_{(k-n/2)/2;k,2,q}
\]
which proves the desired result.  We remark that $\delta$ is chosen after
 beginning the induction step, while the appropriate value of $h$ is
fixed by the choices of the parameters.
\end{proof}

\section{Appendix: A Modified Product Estimate}\label{A Modified Product Estimate}

In this appendix we prove Proposition \ref{new product estimate},
 which can be found in Corollary $1.3.1$ in \cite{chemin}.  
Before beginning, we establish another result for the Littlewood-Paley operators 
and make a slight notational change.  First, we observe that, by changing variables,
\begin{equation}\label{omega results}
\|\psi_j\|_{L^p}\leq 2^{jn/p'}\|\psi_0\|_{L^p}\leq C2^{jn/p'},
\end{equation}
where $p'$ is the Holder' conjugate to $p$; i.e., $1=1/p+1/p'$.
Next, we make a slight notational change.  For $j>0$, we leave $\psi_j$ 
as defined in Section \ref{Function Space definitions and basic Besov space results}. 
 For $j=0$, we set $\psi_0=\Psi$, so $\hat{\psi_0}$ is now supported on the
 ball centered at the origin of radius $1/2$ and $\Delta_0 f=\psi_0 *f=\Psi *f$.  
Then the Besov norm can be defined by
\[
\|f\|_{B^r_{p,q}}=\Big(\sum_{j=0}^\infty 2^{rjq}\|\Delta_j u\|_{L^p}^q\Big)^{1/q}.
\]
We are now ready to prove Proposition \ref{new product estimate}.


\begin{proof}[Proof of Proposition \ref{new product estimate}]
We start by taking the $L^p$ norm of equation \eqref{bony256}, and get:
\begin{align*}
 \|\Delta_j (fg)\|_{L^p}
&\leq \sum_{k=-3}^3 \|\Delta_j (S_{j+k-3}f\Delta_{j+k} g)
\|_{L^p}+ \sum_{k=-3}^3 \|\Delta_j (S_{j+k-3}g\Delta_{j+k} f)\|_{L^p}
\\ 
&\quad + \sum_{k>j-4}\|\Delta_j \Big(\Delta_k f\sum_{l=-2}^2 \Delta_{k+l}g\Big)\|_{L^p}.
\end{align*}
We first observe that, without loss of generality, we can set $k=l=0$ in the
 finite sums and replace $k>j-4$ with $k>j$.  Doing so, we obtain
\[
\|\Delta_j (fg)\|_{L^p}
\leq \|\Delta_j (S_{j-3}f\Delta_{j} g)\|_{L^p}
+ \|\Delta_j (S_{j-3}g\Delta_{j} f)\|_{L^p}
+ \sum_{k>j}\|\Delta_j \Big(\Delta_k f\Delta_{k}g\Big)\|_{L^p}.
\]

Starting with the first term, and defining $\tilde{p}$ by
 $1+1/p=1/\tilde{p}+1/p_2$, we have
\begin{align*}
 \|\Delta_j (S_{j-3}f\Delta_{j} g)\|_{L^p}
 &\leq \|\psi_j\|_{L^{\tilde{p}}}\|\Delta_j f S_{j-3} g\|_{L^{p_2}}\\
&\leq C2^{jn/\tilde{p}'}\|\Delta_j g\|_{L^{p_2}}\|S_{j-3}f\|_{L^\infty}
\\ &\leq C2^{jn/\tilde{p}'} \|\Delta_j g\|_{L^{p_2}}\sum_{m<j-3}\|\Delta_m f\|_{L^\infty}
\\ &\leq C2^{jn(1/p_2-1/p)/\tilde{p}'} \|\Delta_j g\|_{L^{p_2}}\sum_{m<j-3}2^{mn/p_1}\|\Delta_m f\|_{L^{p_1}},
\end{align*}
where we used Young's inequality, equation \eqref{omega results},
 Holder's inequality, and finally Bernstein's inequality.

A similar calculation for the second term yields
\[
\|\Delta_j (S_{j-3}g\Delta_{j} f)\|_{L^p}\leq C2^{jn(1/p_1-1/p)}
 \|\Delta_j f\|_{L^{p_2}}\sum_{m<j-3}2^{mn/p_2}\|\Delta_m g\|_{L^{p_1}}.
\]

For the third term, we have
\begin{align*} 
 \sum_{k>j} \|\Delta_j(\Delta_k f \Delta_k g\|_p)
&\leq \|\psi_j \|_{\tilde{q}}\sum_{k>j}\|\Delta_k u \Delta_k v\|_{L^q}
\\ &\leq 2^{jn/\tilde{p}'}\sum_{k>j}\|\Delta_k f\|_{p_1}\|\Delta_k g\|_{p_2}
\\ &\leq 2^{jn(1/p-1/p_1-1/p_2)}\sum_{k>j} \|\Delta_k f\|_{p_1}\|\Delta_k g\|_{p_2},
\end{align*}
where $1+1/p=1/\tilde{q}+1/q$ and $1/q=1/p_1+1/p_2$.

So we have that
\begin{equation}\label{chuck1}
\begin{aligned}  
\|\Delta_j (fg)\|_{L^p}
&\leq 2^{jn(1/p_2-1/p)} \|\Delta_j g\|_{L^{p_2}}
 \sum_{m<j-3}2^{jn/p_1}\|\Delta_m f\|_{L^{p_1}}\\
 &\quad + 2^{jn(1/p_1-1/p)} \|\Delta_j f\|_{L^{p_1}}
 \sum_{m<j-3}2^{jn/p_2}\|\Delta_m g\|_{L^{p_2}}\\ 
&\quad + 2^{jn(1/p-1/p_1-1/p_2)}\sum_{k>j} \|\Delta_k f\|_{p_1}\|\Delta_k g\|_{p_2}
\end{aligned}
\end{equation}

Multiplying \eqref{chuck1} by $2^{j(s_1+s_2-n(1/p_2+1/p_1-1/p))}$ and taking 
the $l^q$ norm in $j$, we obtain
\[
\|fg\|_{B^s_{p,q}}\leq I+J+K,
\]
where
\begin{gather*}
 I=\Big(\sum_j 2^{(s_1+s_2-n/p_1)jq} \|\Delta_j g\|_{L^{p_2}}^q
\Big(\sum_{m<j-3}2^{mn/p_1}\|\Delta_m f\|_{L^{p_1}}\Big)^q\Big)^{1/q},
\\ J=\Big(\sum_j 2^{(s_1+s_2-n/p_2)jq} \|\Delta_j f\|^q_{L^{p_1}}
\Big(\sum_{m<j-3}2^{mn/p_2}\|\Delta_m g\|_{L^{p_2}}\Big)^q\Big)^{1/q},
\\ K=\Big(\sum_j (2^{j(s_1+s_2)}\sum_{k>j} \|\Delta_k f\|_{p_1}
 \|\Delta_k g\|_{p_2})^q\Big)^{1/q}.
\end{gather*}
For $I$, we have
\begin{align*}
I&\leq \Big(\sum_j 2^{(s_1+s_2-n/p_1)jq} \|\Delta_j g\|_{L^{p_2}}^q
 (\sum_{m<j-3}2^{jn/p_1}\|\Delta_m f\|_{L^{p_1}})^q\Big)^{1/q}\\
&\leq \Big(\sum_j (2^{js_2}\|\Delta_j g\|_{L^{p_2}})^q
 (\sum_{m<j-3}2^{m(n/p_1+s_1-n/p_1)}2^{(j-m)(s_1-n/p_1)}
 \|\Delta_m f\|_{L^{p_1}})^q\Big)^{1/q}\\
&\leq \|f\|_{B^{s_1}_{p_1,\infty}}\sum_k 2^{-(s_1-n/p_2)}
 \Big(\sum_j (2^{js_2}\|\Delta_j g\|_{L^{p_2}})\Big)^{1/q}\\
&\leq \|f\|_{B^{s_1}_{p,q}}\|g\|_{B^{s_2}_{s_2,q}},
\end{align*}
provided $s_1<n/p_1$.  A similar calculation for $J$ yields
\[
J\leq \|f\|_{B^{s_1}_{p,q}}\|g\|_{B^{s_2}_{s_2,q}},
\]
provided $s_2<n/p_2$.  For $K$, we have, using Young's inequality for sums,
\begin{align*}
K&=\Big(\sum_j (\sum_{k>j}2^{(j-k)(s_1+s_2)} 2^{ks_1}
 \|\Delta_k f\|_{p_1}2^{ks_2}\|\Delta_k g\|_{p_2})^q\Big)^{1/q}
\\ &\leq \|g\|_{B^{s_2}_{p_2,\infty}}\Big(\sum_j
 (\sum_{k>j}2^{(j-k)(s_1+s_2)} 2^{ks_1}\|\Delta_k f\|_{p_1})^q\Big)^{1/q}
\\ &\leq \|g\|_{B^{s_2}_{p_2,\infty}}\sum_{k}2^{-k(s_1+s_2)}
 \Big(\sum_k (2^{ks_1}\|\Delta_k f\|_{p_1})^q\Big)^{1/q}
\\ &\leq C\|f\|_{B^{s_1}_{p_1,q}}\|g\|+{B^{s_2}_{p_2,q}},
\end{align*}
provided $s_1+s_2>0$.  This completes the proof.
\end{proof}


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\end{document}