\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 $$\label{LANS} \begin{gathered} \partial_t u+(u\cdot\nabla)u+\operatorname{div} \tau^\alpha u=-(1-\alpha^2\Delta)^{-1}\nabla p+\nu\Delta u \\ u=u(t,x),\quad \operatorname{div} u=0,\quad u(0,x)=u_0(x), \end{gathered}$$ 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}): $$\label{old product rule} \|fg\|_{H^{s,p}}\leq \|f\|_{H^{s,p_1}}\|g\|_{L^{p_2}}+\|f\|_{L^{q_1}}\|g\|_{H^{s,q_2}},$$ 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: $$\label{new product eqn} \|fg\|_{B^s_{p,q}}\leq \|f\|_{B^{s_1}_{p_1,q}}\|g\|_{B^{s_2}_{p_2,q}},$$ provided $s_10$, $1/p\leq 1/p_1+1/p_2$, 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 $0n$, 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 $$u\in BC([0,T):B^{r_1}_{p,q}(\mathbb{R}^n))\cap \dot{C}^T_{(r_2-r_1)/2;r_2,p,q},$$ $1n$, 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 $$u\in BC([0,T):B^{r_1}_{p,q}(\mathbb{R}^n))\cap L^a((0,T):B^{r_2}_{p,q}(\mathbb{R}^n)),$$ $a=2/(r_2-r_1)$, $1n$, 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 $$\label{low freq part} \hat{\Psi}(\xi)=1-\sum_{k=0}^\infty \hat{\psi}_k(\xi).$$ 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 $$\label{besovlemma1} \begin{gathered} \Delta_j\Delta_k f= 0, \quad |j-k|\geq 2 \\ \Delta_j (S_{k-3}f\Delta_{k}g)= 0 \quad |j-k|\geq 4, \end{gathered}$$ and, if $|i-k|\leq 2$, then $$\label{besovpieces67} \Delta_j(\Delta_kf\Delta_i g)=0 \quad j>k+4.$$ 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 $$\label{lp start} fg=\sum_{k} S_{k-3}f\Delta_k g + \sum_{k}S_{k-3}g\Delta_k f + \sum_{k}\Delta_k f\sum_{l=-2}^2 \Delta_{k+l} g.$$ The estimates \eqref{besovlemma1} and \eqref{besovpieces67} imply that \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} 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: $$\label{besov embedding} \begin{gathered} \|f\|_{B^{\beta_1}_{p,q_2}}\leq C\|f\|_{B^{\beta_2}_{p,q_1}},\\ \|f\|_{B^{\gamma_2}_{p_2,q}}\leq C\|f\|_{B^{\gamma_1}_{p_1,q}},\\ \|f\|_{H^{s,p}}\leq \|f\|_{B^r_{p,q}},\\ \|f\|_{H^{s,2}}=\|f\|_{B^s_{2,2}}\leq \|f\|_{B^{r}_{2,q}}. \end{gathered}$$ 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 $s_10$. \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 $$\label{Bernstein} \begin{gathered} \tilde{C}2^{j\alpha+jn(1/p-1/q)}\|g\|_p \leq \|A^{\alpha/2}g\|_q\leq C2^{j\alpha+jn(1/p-1/q)}\|g\|_p. \\ \|A^{\alpha/2} f\|_q \leq C2^{j\alpha+jn(1/p-1/q)}\|f\|_p \end{gathered}$$ Next, we establish estimates for the heat kernel on Besov spaces. \begin{proposition}\label{hkb} Let $1\leq p_1\leq p_2<\infty$, $-\infty0$, $\|\Gamma f\|_{L^\sigma((0,T):B^{s_1}_{p_1,q_1})} \leq \varepsilon$ 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 $1\leq p_0\leq p_1<\infty$, $1\leq q<\infty$, $-\inftyn$ 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}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$Mto 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})}1, so $r_2-1>0$). This condition is equivalent to $n/p\leq 2r_2-1-r_1$, and since $r_11/a$, which holds since $p>n$. Now we turn to $K_2$, where we have \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} provided $1/\sigma-1/a=1-(r_2-r_1)/2$, which implies $\sigma=1$. Then, by equation \eqref{2k21} above, we have $$\label{2k22} K_2\leq \|\nabla u\nabla u\|_{L^\sigma(B^{r_1-1}_{p,q})}\leq CM^2.$$ Combining equations \eqref{2j12} and \eqref{2j22}, we obtain $$\label{2kest} K\leq CM^2.$$ Given equations \eqref{2iest}, \eqref{2jest}, and \eqref{2kest}, we have that $\Phi(u)\leq M/3+CM^20. 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 $01$. 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 $0h/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 \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} provided $1>h+n/p, \quad -1<\delta-(k-r)/2.$ For thep=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 \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} 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} forp>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, $$\label{omega results} \|\psi_j\|_{L^p}\leq 2^{jn/p'}\|\psi_0\|_{L^p}\leq C2^{jn/p'},$$ 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$L^pnorm of equation \eqref{bony256}, and get: \begin{align*} \|\Delta_j (fg)\|_{L^p} &\leq \sum_{k=-3}^3 \|\Delta_j (S_{j+k-3}f\Delta_{j+k} g) \|_{L^p}+ \sum_{k=-3}^3 \|\Delta_j (S_{j+k-3}g\Delta_{j+k} f)\|_{L^p} \\ &\quad + \sum_{k>j-4}\|\Delta_j \Big(\Delta_k f\sum_{l=-2}^2 \Delta_{k+l}g\Big)\|_{L^p}. \end{align*} We first observe that, without loss of generality, we can setk=l=0$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$1+1/p=1/\tilde{p}+1/p_2, we have \begin{align*} \|\Delta_j (S_{j-3}f\Delta_{j} g)\|_{L^p} &\leq \|\psi_j\|_{L^{\tilde{p}}}\|\Delta_j f S_{j-3} g\|_{L^{p_2}}\\ &\leq C2^{jn/\tilde{p}'}\|\Delta_j g\|_{L^{p_2}}\|S_{j-3}f\|_{L^\infty} \\ &\leq C2^{jn/\tilde{p}'} \|\Delta_j g\|_{L^{p_2}}\sum_{mj} \|\Delta_j(\Delta_k f \Delta_k g\|_p) &\leq \|\psi_j \|_{\tilde{q}}\sum_{k>j}\|\Delta_k u \Delta_k v\|_{L^q} \\ &\leq 2^{jn/\tilde{p}'}\sum_{k>j}\|\Delta_k f\|_{p_1}\|\Delta_k g\|_{p_2} \\ &\leq 2^{jn(1/p-1/p_1-1/p_2)}\sum_{k>j} \|\Delta_k f\|_{p_1}\|\Delta_k g\|_{p_2}, \end{align*} where1+1/p=1/\tilde{q}+1/q$and$1/q=1/p_1+1/p_2. So we have that \label{chuck1} \begin{aligned} \|\Delta_j (fg)\|_{L^p} &\leq 2^{jn(1/p_2-1/p)} \|\Delta_j g\|_{L^{p_2}} \sum_{mj} \|\Delta_k f\|_{p_1}\|\Delta_k g\|_{p_2} \end{aligned} Multiplying \eqref{chuck1} by2^{j(s_1+s_2-n(1/p_2+1/p_1-1/p))}$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_{mj} \|\Delta_k f\|_{p_1} \|\Delta_k g\|_{p_2})^q\Big)^{1/q}. \end{gather*} For$I, we have \begin{align*} I&\leq \Big(\sum_j 2^{(s_1+s_2-n/p_1)jq} \|\Delta_j g\|_{L^{p_2}}^q (\sum_{mj}2^{(j-k)(s_1+s_2)} 2^{ks_1} \|\Delta_k f\|_{p_1}2^{ks_2}\|\Delta_k g\|_{p_2})^q\Big)^{1/q} \\ &\leq \|g\|_{B^{s_2}_{p_2,\infty}}\Big(\sum_j (\sum_{k>j}2^{(j-k)(s_1+s_2)} 2^{ks_1}\|\Delta_k f\|_{p_1})^q\Big)^{1/q} \\ &\leq \|g\|_{B^{s_2}_{p_2,\infty}}\sum_{k}2^{-k(s_1+s_2)} \Big(\sum_k (2^{ks_1}\|\Delta_k f\|_{p_1})^q\Big)^{1/q} \\ &\leq C\|f\|_{B^{s_1}_{p_1,q}}\|g\|+{B^{s_2}_{p_2,q}}, \end{align*} provideds_1+s_2>0$. This completes the proof. \end{proof} \begin{thebibliography}{10} \bibitem{chemin} J. Y. Chemin, \emph{About the navier-stokes system}, Publications du Laboratoire d'analyse numerique (1996). \bibitem{CHMZ} S.~Chen, D.~D. Holm, L.~Margolin, and R.~Zhang, \emph{Direct numerical simulations of the {N}avier-{S}tokes alpha model}, Phys. D \textbf{133} (1999), no.~1-4, 66--83, Predictability: quantifying uncertainty in models of complex phenomena (Los Alamos, NM, 1998). \bibitem{CW} F.~Christ and M.~Weinstein, \emph{Dispersion of {S}mall {A}mplitude {S}olutions of the {G}eneralized {K}orteweg-de {V}ries {E}quation}, Journal of {F}unctional {A}nalysis \textbf{100} (1991), 87--109. \bibitem{katoinduction} T.~Kato, \emph{The {N}avier-{S}tokes equation for an incompressible fluid in {${\bf R}^2\$} with a measure as the initial vorticity}, Differential Integral Equations \textbf{7} (1994), no.~3-4, 949--966. \bibitem{KP} T.~Kato and G.~Ponce, \emph{The {N}avier-{S}tokes equation with weak initial data}, Internat. Math. Res. Notices (1994), no.~10. \bibitem{MRS} J.~Marsden, T.~Ratiu, and S.~Shkoller, \emph{The geometry and analysis of the averaged {E}uler equations and a new diffeomorphism group}, Geom. Funct. Anal. \textbf{10} (2000), no.~3, 582--599. \bibitem{MS} J.~Marsden and S.~Shkoller, \emph{Global well-posedness for the {L}agrangian averaged {N}avier-{S}tokes equations on bounded domains}, {P}hil. {T}rans. {R}. {S}oc. {L}ond. (2001), no.~359, 1449--1468. \bibitem{MS2} J.~Marsden and S.~Shkoller, \emph{The {A}nisotropic {L}agrangian {A}veraged {E}uler and {N}avier-{S}tokes equations}, {A}rch. {R}ational {M}ech. {A}nal. (2003), no.~166, 27--46. \bibitem{MKSM} K.~Mohseni, B.~Kosovi{\'c}, S.~Shkoller, and J.~Marsden, \emph{Numerical simulations of the {L}agrangian averaged {N}avier-{S}tokes equations for homogeneous isotropic turbulence}, Phys. Fluids \textbf{15} (2003), no.~2, 524--544. \bibitem{galplanme} N.~Pennington, \emph{Global solutions to the lagrangian averaged navier-stokes equation in low regularity besov spaces}, Advances in Differential Equations, to appear. \bibitem{sobpaper} N.~Pennington, \emph{Lagrangian {A}veraged {N}avier-{S}tokes equation with rough data in {S}obolev space}, http://arxiv.org/abs/1011.1856v2. \bibitem{Shkoller} S.~Shkoller, \emph{On incompressible averaged Lagrangian hydrodynamics}, E-print, (1999), http: //xyz.lanl.gov/abs/math.AP/9908109. \bibitem{SK} S.~Shkoller, \emph{Analysis on groups of diffeomorphisms of manifolds with boundary and the averaged motion of a fluid}, Journal of {D}ifferential {G}eometry (2000), no.~55, 145--191. \bibitem{taonde} T.~Tao, \emph{Nonlinear {D}ispersive {E}quations}, American Mathematical Society, 2006. \bibitem{T3} M.~Taylor, \emph{Partial {D}ifferential {E}quations}, Springer-{V}erlag {N}ew {Y}ork, {I}nc., 1996. \bibitem{tt} M.~Taylor, \emph{Tools for {P}{D}{E}}, {M}athematical {S}urveys and {M}onographs, {V}ol. 81, {A}merican {M}athematical {S}ociety, Providence {R}{I}, 2000. \end{thebibliography} \end{document}