\documentstyle[12pt,amsfonts]{article} \begin{document} \title{Quasi-geostrophic Type Equations \\with Initial Data in Morrey Spaces} \author{Jiahong Wu \\School of Mathematics\\The Institute for Advanced Study \\Princeton, NJ 08540 } \date{} \maketitle \newtheorem{thm}{Theorem}[section] \newtheorem{cor}[thm]{Corollary} \newtheorem{prop}[thm]{Proposition} \newtheorem{define}[thm]{Definition} \newtheorem{rem}[thm]{Remark} \newtheorem{example}[thm]{Example} \newtheorem{lemma}[thm]{Lemma} \def\theequation{\thesection.\arabic{equation}} \begin{abstract} This paper studies the well-posedness of the initial value problem for the quasi-geostrophic type equations $$\frac{\partial \theta}{\partial t}+u\cdot\nabla\theta + (-\Delta)^{\gamma}\theta=0,\quad \mbox{on}\quad {\Bbb R}^n \times (0,\infty),$$ $$\theta(x,0)=\theta_0(x), \quad x\in {\Bbb R}^n$$ where $\gamma(0\le \gamma\le 1)$ is a fixed parameter and $u=(u_j)$ is divergence free and determined from $\theta$ through the Riesz transform $u_j=\pm {\cal R}_{\pi(j)}\theta$ ($\pi(j)$ being a permutation of $j$, $j=1,2,\cdots,n)$. The initial data $\theta_0$ is taken in certain Morrey spaces ${\cal M} _{p,\lambda}({\Bbb R}^n)$ (see text for the definition). The local well-posedness is proved for $$\frac{1}{2}<\gamma \le 1, \quad 1\frac{1}{2}, the existence, uniqueness and regularity are obtained, but the same questions remain open for \gamma\le \frac{1}{2}. Now the important question is if the index \gamma=\frac{1}{2} is actually sharp and how this index is related to the singularity of the integral operator determining the velocity field. The answer to this question may provide a clue to the solution of the outstanding problem concerning the existence of smooth solutions for the Navier-Stokes equations. \vspace{.14in} This paper is organized as follows. In Section \ref{sec:lin} the definition and some properties of Morrey spaces are given as well as the solution operator of the linear equation and its properties. The well-posedness results are stated and proved in Section \ref{sec:main} while the smoothness of the solution is established in Section \ref{sec:reg}. \vspace{.14in} It is a great pleasure to thank Professor Peter Constantin for his constant encouragement and suggestions. This work is supported by the NSF grant \#DMS 9304580 at the Institute for Advanced Study. \newpage \section{Morrey spaces and the linear equation } \setcounter{equation}{0} \label{sec:lin} In this section we shall list some basic properties of Morrey spaces and then study the the solution operator for the linear equation over Morrey spaces. \begin{define} \label{morrey} For 1\le p< \infty, 0\le \lambda< n, the Morrey space {\cal M}_{p, \lambda}({\Bbb R}^n) is defined as$$ {\cal M}_{p,\lambda}\equiv \{f\in L_{loc}^{p}({\Bbb R}^n): \|f\|_{p, \lambda}<\infty\} $$where the norm is given by$$ \|f\|_{p,\lambda} =sup_{\{x\in {\Bbb R}^n, R>0\}} R^{-\frac{\lambda}{p}} \left(\int_{|y-x|\le R}|f|^p(y) dy\right)^{\frac{1}{p}} $$\ddot{\cal M}_{p,\lambda} is defined to be a subspace of {\cal M}_{p, \lambda}:$$ \ddot{\cal M}_{p,\lambda}=\{f\in {\cal M}_{p,\lambda}: \quad \|f(\cdot-y)-f(\cdot)\|_{p,\lambda}\to 0,\quad \mbox{as $y\to 0$}\} $$\end{define} {\cal M}_{p,\lambda} is a Banach space and \ddot{\cal M}_{p,\lambda} is a closed subspace of {\cal M}_{p,\lambda}. For p>1, {\cal M}_{p,0}=L^p and {\cal M}_{1,0}={\cal M}, where {\cal M} is the space of finite measures. We shall also use the following properties of Morrey spaces (see e.g., \cite{cam},\cite{Pe},\cite{kat1}). When not specified, the indices in the notation {\cal M}_{p,\lambda} will be restricted in 1\le p<\infty, 0\le \lambda1, \quad \mu> \lambda}$$ where $L_{s,p}$ is the weighted Lebesgue space consisting of functions $f$ such that $(-\Delta)^{s/2}f\in L^p$. \end{description} \end{lemma} \vspace{.13in} We now consider the solution operator for the linear equation $$\partial_t \theta + \Lambda^{2\gamma}\theta=0, \qquad \Lambda =(-\Delta)^{\frac{1}{2}}$$ on the whole space ${\Bbb R}^n$. For a given initial data $\theta_0$, the solution of this equation is given by $\theta=K(t)\theta_0=e^{-\Lambda^{2\gamma}t} \theta_0$, where $K(t)$ is a convolution operator with the kernel $k_t(x)=k_{t}^{(\gamma)}(x)$ being defined through its Fourier transform \begin{equation}\label{fo} \widehat{k}_{t}^{(\gamma)}(\xi) =e^{-|\xi|^{2\gamma}t}, \end{equation} In particular, $k_t$ is the heat kernel for $\gamma=1$ and the Poisson kernel for $\gamma=\frac{1}{2}$. \vspace{.1in} As observed in \cite{Re}, $k_{t}^{(\gamma)}$ can be expressed as an average of the heat kernel, \begin{equation}\label{av} k_{1}^{(\gamma)}(x)=\int_{0}^{\infty}k_{s}^{(1)}(x)dP_{\gamma}(s), \qquad k_{t}^{(\gamma)}=t^{-\frac{n}{2\gamma}}k^{(\gamma)}_{1} \left(\frac{x}{t^\frac{1}{2\gamma}}\right) \end{equation} where $P_{\gamma}$ is some probability distribution. Furthermore, \begin{lemma}\label{lin} For $0<\gamma\le 1$, \begin{description} \item[(a)] $k_{t}^{(\gamma)}\ge 0$, and is a non-increasing radial function on ${\Bbb R}^{n}$ with $$|x|^{l}k_{t}^{(\gamma)} (|x|) \longmapsto 0,\quad \mbox{as\quad |x|\to \infty}$$ for any real power $l$. \item[(b)] For $t>0$, $\|k_{t}^{(\gamma)} \|_{L^1}=1$ and thus for $f_t=k_{t}^{(\gamma)}*f$, $$|f_t|^p \le k_{t}^{(\gamma)}* |f|^p, \qquad 1\le p<\infty$$ \item[(c)] For $s\in {\Bbb R}$ and $q\in [1,\infty)$, $$k_{t}^{(\gamma)}*f \longmapsto f, \quad\mbox{in L_{s,q} \quad as t\to 0}$$ for any $f\in L{s,q}$. \end{description} \end{lemma} {\bf Proof.} The proof of (a) is an easy consequence of equation (\ref{av}) and the corresponding properties of the heat kernel. (b) can be seen from the Fourier transform (\ref{fo}) of $k_{t}^{(\gamma)}$. The proof of (c) involves merely the definition of the norm in $L_{s,q}$ and the dominated convergence theorem. \vspace{.16in} We now establish estimates for the operator $K(t)$ between Morrey spaces. The following proposition is a straightforward generalization of Lemma 2.1 of Kato \cite{kat1}. \begin{prop}\label{prop1} Let $1\le q_1\le q_2\le \infty$ and $0\le \lambda_1=\lambda_20$, the operators $K(t)$, $W(t)=\nabla K(t)$ and $\partial_t K(t)$ are bounded operators from ${\cal M}_{q_1,\lambda_1}$ to $\ddot{\cal M}_{q_2, \lambda_2}$ and depend on $t$ continuously , where $\nabla$ denotes the space derivarive. Furthermore, we have for $f\in {\cal M}_{q_1,\lambda_1}$, \begin{equation}\label{k1} t^{\frac{1}{2\gamma}\left(\alpha_1-\alpha_2\right)} \|K(t) f\|_{q_2,\lambda_2}\le C\|f\|_{q_1,\lambda_1}, \end{equation} \begin{equation}\label{k2} t^{\frac{1}{2\gamma}+\frac{1}{2\gamma}\left(\alpha_1-\alpha_2\right)} \|W(t) f\|_{q_2,\lambda_2}\le C\|f\|_{q_1,\lambda_1}, \end{equation} \begin{equation}\label{k3} t^{1+\frac{1}{2\gamma}\left(\alpha_1-\alpha_2\right)} \|\partial_t K(t) f\|_{q_2,\lambda_2}\le C\|f\|_{q_1,\lambda_1}, \end{equation} where $\alpha_i=\frac{n-\lambda_i}{p_i}(i=1,2)$ and constants $C$ depend on $\gamma, q_1,q_2, \lambda_1, \lambda_2$. \end{prop} The proof of this proposition is a modification of that for Lemma 2.1 in \cite{kat1}, but for reader's convenience, it will be given in the Appendix. \newpage \section{Well-posedness in Morrey spaces} \setcounter{equation}{0} \label{sec:main} In this section we deal with the IVP (\ref{eq1}),(\ref{ndu}), (\ref{eq2}) for the QGS equations with initial data in Morrey spaces. The solutions are found to be in the spaces of weighted continuous functions in time, which we now introduce. Kato and collaborators first define such spaces in solving the IVP for the Navier-Stokes equations (\cite{kat1},\cite{k2},\cite{kp1},\cite{kp2}). \begin{define} Let $00$ such that if $\|\theta_0\|_{p,\lambda}<\delta$, the IVP (\ref{eq1}), (\ref{ndu}), (\ref{eq2}) has a solution $\theta$ on $(0,T)$ for some $T>0$ satisfying \begin{equation}\label{re1} \theta\in BC((0,T);\ddot{\cal M}_{p,\lambda})\cap (\cap_{ 1n-(2\gamma-1)q}BC([0,T);L_{-k/q,q})) \end{equation} Furthermore, for any $p0$ and $b>0$, we obtain that $$\|G(f,g)\|_{m,q,\lambda}\le C\|f\|_{h,q_1,\lambda_1}\|g\|_{l,q_2,\lambda_2}$$ under the assumptions of this proposition. The estimate (\ref{est}) is thus proved. The continuity of $G(f,g)$ in $t$ comes from applying dominated convergence theorem to the quantity $$t^mG(f,g)(t)=t^m\int_{0}^{t}W(t-\tau)(fg)(\tau)d\tau$$ \vspace{.14in} We will need a result concerning the Calderon-Zygmund type singular integral operators on Morrey spaces. The Riesz transform is a particular example of these types of singular integral operators. \begin{lemma}\label{sing} Let ${\cal Z}$ be a Calderon-Zygmund type singular integral operator, i.e., ${\cal Z}: {\Bbb R}^n\setminus \{0\}\to {\Bbb R}$ is a homogeneous continuous function of degree $-n$ and integral on unit sphere vanishes. Let ${\cal M}_{q,\mu}$ with $10$ and $T=\infty$ if $\|\theta_0\|_{p,\lambda}$ is small enough). \vspace{.15in} To show that $\theta$ satisfies (\ref{re1}), we first note that $$\theta={\cal A}(\theta)\equiv K\theta_0 -G(u\theta)$$ For $\theta_0\in {\cal M}_{p,\lambda}$ with $\lambda= n-(2\gamma-1)p$, $$K\theta_0\in BC((0,T); \ddot{\cal M}_{p, \lambda})$$ as implied by Proposition \ref{prop1}. Furthermore, because of the continuous embedding for $1n-(2\gamma-1)q$ $${\cal M}_{p,\lambda}\subset {\cal M}_{q,n-(2\gamma-1)q} \subset L_{-\frac{k}{q},q}$$ and the fact that $K(t)$ forms a $C_0$ semigroup on $L_{-k/q,q}$ for $11$ and $k>n-(2\gamma-1)q$. \vspace{.1in} For the term $G(u\theta)$, we first apply Proposition \ref{prop2} with $$\lambda_1=\lambda_2=\lambda, \quad q_1=q_2=2p,\quad q=p,\quad h=l=\frac{1}{2}-\frac{1}{4\gamma}, \quad m=0$$ to show that \begin{equation}\label{111} G(u\theta)\in BC((0,T); \ddot{\cal M}_{p,\lambda}) \end{equation} Now we use (\ref{111}) and apply Proposition \ref{prop2} with $$\lambda_1=\lambda_2=\lambda,\quad \quad q_1=q_2=p, \quad q=\frac{p}{1+\eta},$$ $$h=l=0,\quad m=-\eta\left(1-\frac{1}{2\gamma}\right)\quad\mbox{\eta>0 is small}$$ to obtain that $$G(u\theta) \in {C}_{-\eta\left(1-\frac{1}{2\gamma}\right) }((0,T); {\cal M}_{q, \lambda}), \quad \mbox{for q=\frac{p}{1+\eta}n-(2\gamma-1)q\ge\lambda,$$ G(u\theta) \to 0, \quad \mbox{in $L_{-k/q,q}$},\quad \mbox{as}\quad t\to 0, $$Therefore G(u\theta)\in BC([0,T);L_{-\frac{k}{q},q}) for 1n-(2\gamma-1)q. Summing up, we've shown that \theta=K\theta_0 - G(u\theta) is exactly in the class defined by (\ref{re1}). \vspace{.17in} We now prove that \theta satisfies (\ref{re}). K\theta_0 satisfying (\ref{re}) is an easy consequence of Proposition \ref {prop1}. We apply Proposition \ref{prop2} to G(u\theta) with$$ h=l=\left(\frac{1}{2}-\frac{1}{4\gamma}\right) , \quad m=m'  q=q,\quad q_1=q_2=2p, \quad \lambda=\lambda_1=\lambda_2 $$and use the fact that \theta\in X (\ref{xx}) to show that G(u\theta) is in the class defined by (\ref{re}). This argument, combined with the uniqueness of \theta in (\ref{xx}), indicates the uniqueness of \theta in (\ref{re}). \vspace{.13in} The proof of the Lipschitz property is routine (see i.e., \cite{wu2}) and is therefore omitted. \newpage \section{Further regularity} \setcounter{equation}{0} \label{sec:reg} In this section we prove that the solution \theta obtained in Theorem \ref{main} is actually smooth. More precisely, we have \begin{thm} \label{reg} Let \theta be the solution obtained in theorem \ref{main}, Then for any p\le q<\infty and k,j=0,1,2,\cdots \begin{equation}\label{more} \partial_{t}^{k}\nabla^j \theta \in C((0,T); \ddot{\cal M}_{q, \lambda}), \end{equation} where \lambda=n-(2\gamma-1)p, \nabla denotes the space derivative and C((0,T);X) is the space of X-valued continuous functions on (0,T). \end{thm} {\bf Proof} \quad The smoothness of \theta is proved by standard schemes. First we consider the case when k=0. For j=0, (\ref{more}) can be seen from (\ref{re1}), (\ref{re}) in Theorem \ref{main} . We now prove that (\ref{more}) is true for j=1. We take any t_1>0 and prove the results for t>t_1. \vspace{.12in} We take \nabla of G and apply the Leibnitz rule to obtain$$ \nabla G(u,\theta) = G_1(\nabla u,\theta) + G_2(u,\nabla\theta) $$where G_1 and G_2 are integral operators on (t_1,T) with the same properties as G. First let q>p and X be the space consisting of functions \theta such that \begin{equation}\label{X_R} \theta\in C([t_1,T); \ddot{\cal M}_{q,\lambda}),\qquad \nabla\theta\in C_{1-\frac{1}{2\gamma}}((t_1,T); \ddot{\cal M}_{q, \lambda}) \end{equation} and X_R be the closed ball of radius R in X. The idea is to apply contraction mapping arguments to {\cal A} on X_R with T and R to be determined. First we choose R appropriately such that K(t)\theta_1\in X_R, where \theta_1=\theta(t_1) is the value of \theta at t_1. As in the proof of Theorem \ref{main}, we apply Proposition \ref{prop2} to G, G_1 and G_2 to show that for \theta\in X_R \begin{equation}\label{G1} G(u,\theta)\in C_{\left(-\left(1-\frac{1}{2\gamma}\right) \left(1-\frac{p}{q}\right)\right)} ((0,T); \ddot{\cal M}_{q,\lambda}) \end{equation} \begin{equation}\label{G2} \nabla G(u,\theta) \in C_{\left(1-\frac{1}{2\gamma}\right)\left(1-\frac{p}{q}\right)} ((0,T); \ddot{\cal M}_{q,\lambda}) \end{equation} and Lemma \ref{sing} is used in estimating u and \nabla u as usual. (\ref{G1}) and (\ref{G2}) imply not only G\in X_R, but also that the norm of G(u,\theta) in X_R has a small factor (T-t_1)^\varrho if T-t_1 is small, where$$ \varrho=\min\{\left(1-\frac{1}{2\gamma}\right)\left(1-\frac{p}{q}\right), \left(1-\frac{1}{2\gamma}\right)\frac{p}{q}\}$$If T-t_1 is chosen small and R taken as above, then {\cal A} maps X_R to itself. Furthermore , it can be shown in the same spirit as in the proof of Theorem \ref{main} that {\cal A} is a contraction map on X_R. Therefore {\cal A} has a fixed point \theta in X_R, which solves \ref{int}. The uniqueness result in Theorem \ref{main} indicates that this \theta is just the original \theta obtained in Theorem \ref{main}. Thus we've shown that \theta\in C((t_1,T); \ddot{\cal M}_{q,\lambda}), which implies that \theta\in C((0,T); \ddot{\cal M}_{q,\lambda}) because of the arbitrariness of t_1. \vspace{.12in} For the case q=p, the result \nabla\theta\in C((t_1,T);\ddot{\cal M}_{p, \lambda}) can be established by applying Proposition \ref{prop2} to \nabla G again and using the relation in (\ref{X_R}). \vspace{.13in} Repeating the same argument for higher spatial derivatives of \theta, we obtain the result \nabla^j\theta\in C((0,T); \ddot{\cal M}_{q, \lambda}). This finishes the proof for the case k=0. \vspace{.17in} We now prove (\ref{more}) for k=1. It is easy to see from the regularity result we've just obtained that$$ \nabla^j\theta,\quad \nabla^j u,\quad (-\Delta)^{\lambda}\nabla^j\theta \in C((0,T);\ddot{\cal M}_{q,\lambda}),\quad j=0,1,2,\cdots $$for any p\le q<\infty. Using Equation \ref{eq1}$$ \partial_t \theta =-u\cdot\nabla\theta -(-\Delta)^{\gamma}\theta $$and the H\"{o}lder inequatlity for the Morrey spaces (i.e., (ii) of Lemma \ref{MP}), we obtain for j=0,1,2,\cdots$$ \partial_t\nabla^j\theta\in C((0,T);\ddot{\cal M}_{q,\lambda}) $$The result for general k can be established by induction. This concludes the proof of Theorem \ref{more}. \newpage \section{Appendix} \setcounter{equation}{0} \label{sec:app} We prove Proposition \ref{prop1} in this appendix. For simplicity of notation, we'll drop \gamma when writing the kernel k_{t}^{(\gamma)} and use f_t for k_t*f. \vspace{.14in} We first prove estimate (\ref{k1}). For q_1=q_2=\infty, (\ref{k1}) is obvious. For q_1=q_2<\infty, we use (b) of Lemma \ref{lin}, i.e., \begin{equation}\label{ker} \|k_t\|_{L^1}=1,\qquad |f_t|^p\le k_t*|f|^p, \end{equation} to obtain for q_2=q_1, \lambda_2=\lambda_1 \begin{equation}\label{q12} \|f_t\|_{q_2,\lambda_2}\le\|f\|_{q_1,\lambda_1}. \end{equation} \vspace{.12in} To prove (\ref{k1}) in the general case, we first estimate \|f_t\|_{L^ \infty}. By (\ref{ker}),$$ |f_t(0)|^{q_1}\le \int_{{\Bbb R}^n} k_{t}(|x|)|f(x)|^{q_1}dx=\int_{0}^{\infty} k_t(\rho)dr(\rho)  \le \int_{0}^{\infty}|k_{t}'(\rho)|r(\rho)d\rho\le \|f\|_{q_1,\lambda_1}^{q_1} \int_{0}^{\infty}|k_t'(\rho)|\rho^{\lambda_1}d\rho $$where r(\rho)=\int_{|y|<\rho}|f(y)|^{q_1}dy and we've used the fact that r(\rho)\le \|f\|_{q_1,\lambda_1}^{q_1}\rho^{\lambda_1}. The decay properties of k_t (see Lemma \ref{lin}) are also used here. Thus, \begin{equation}\label{big} \|f_t\|_{L^\infty}^{q_1}\le c \|f\|_{q_1,\lambda_1}^{q_1} t^{-\frac{n+1}{2\gamma}}\int_{0}^{\infty} |k_{1}'(\rho t^{-\frac{1}{2\gamma}})|\rho^{\lambda_1}d\rho \le c\|f\|_{q_1,\lambda_1}^{q_1} t^{\frac{\lambda_1-n}{2\gamma}} \end{equation} which also proves (\ref{k1}) for q_10 and x\in {\Bbb R}^n,$$ R^{-\lambda_2}\int_{|x-y|