ABSTRACT

In this paper, using the Littlewood-Paley decomposition and commutator estimates, we establish the local well-posedness for the quasi-geostrophic equation without viscosity in Besov-Herz spaces, which improves the results in [1] and [4].

Keywords:Local Well-Posedness, Besov-Herz Spaces, Quasi-Geostrophic Equation

准地转方程在Besov-Herz空间上的局部适定性

徐艳霞，陈晓莉^*

江西师范大学数学与信息科学学院，江西 南昌

收稿日期：2020年4月22日；录用日期：2020年5月14日；发布日期：2020年5月25日

摘 要

本文利用Littlewood-Paley分解和交换子估计，建立了无粘准地转方程在Besov-Herz空间上的局部适定性，推广了 [1] 和 [4] 的结果。

关键词 :局部适定性，Besov-Herz空间，准地转方程

Copyright © 2020 by author(s) and Hans Publishers Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/

1. 引言

本文研究带有非线性项的热方程

$\{\begin{array}{l}{\partial }_t\theta -\kappa \Delta \theta =-u\cdot \nabla \theta ,\left(x,t\right)\in {ℝ}^2\times \left[0,T\right),\hfill \\ \text{div}u=0,\hfill \\ \theta \left(0,x\right)={\theta }_0\left(x\right),x\in {ℝ}^2,\hfill \end{array}$ (1.1)

即标准的准地转(quasi-geostrophic)方程，其中 $\kappa >0$ 是粘性系数， $\theta \left(t,x\right)$ 是关于t和x的实值函数。函数 $\theta$ 表示位势温度，流速u与流函数 $\Psi$ 和 $\theta$ 的关系如下

$u=\left(-{\partial }_{x_2}\Psi ,{\partial }_{x_1}\Psi \right),{\left(-\Delta \right)}^{\frac{1}{2}}\Psi =-\theta .$ (1.2)

分数阶Laplace ${\left(-\Delta \right)}^{-\alpha }$ 按Fourier定义为 $\widehat{{\left(-\Delta \right)}^{\alpha }f}\left(\xi \right)={\left|\xi \right|}^{2\alpha }\hat{f}\left(\xi \right)$，这里 $\hat{f}\left(\xi \right)$ 表示f的Fourier变换。

关于第j个分量的Riesz变换 ${\mathcal{R}}_j$，其Fourier变换 $\widehat{{\mathcal{R}}_{j}f}\left(\xi \right)=-\frac{i{\xi }_j}{\left|\xi \right|}\hat{f}\left(\xi \right),j=1,\cdots ,N$，其中N表示空间维数。

因此利用Riesz变换的定义，可将(1.2)式改写成

$u=\left({\partial }_{x_2}{\left(-\Delta \right)}^{-\frac{1}{2}}\theta ,{\partial }_{x_1}{\left(-\Delta \right)}^{-\frac{1}{2}}\theta \right)=\left(-{\mathcal{R}}_2\theta ,{\mathcal{R}}_1\theta \right).$

除了本身有地球流体力学的相关背景外，二维准地转方程可以看成是三维Navier-Stokes方程在二维空间的模型。Constantin，Majda和Tabak在文献 [1] 首先研究了方程(1.1)的无粘情形。他们建立了方程(1.1)在Sobolev空间中的局部适定性和爆破判别准则。此后，准地转方程在各种函数空间上的适定性和爆破准则得以建立。例如Wu [2] 得到了准地转方程在Morrey空间中的适定性和爆破判别准则；Chae [3] 在Triebel-Lizorkin中建立相似的结论；Wang和Jia [4] 将文 [1] 中结果推广到Besov空间，Xu和Tan [5] 进一步将他们推广到Besov-Morrey空间。

为了得到一类Lipschitz函数做Fourier变换后的像函数的Bernsttein型定理，Herz在 [6] 中引入Herz空间 $K_{p,q}^{\alpha }\left({ℝ}^N\right)$，它在调和分析的算子理论上有重要的作用。下面给出Herz空间和Besov-Herz空间的定义。

定义1.1 [6] 对整数 $k,k\ge -1$，$A_k$ 定义如下

$A_{-1}=B\left(0,\frac{1}{2}\right),A_k=\left\{x\in R^N:2^{k-1}\le \left|x\right|<2^k\right\},k\ge 0,$

其中 $B\left(x_0,r\right)=\left\{x\in {ℝ}^N:\left|x-x_0\right|<r\right\}$。设 $1\le p,q\le \infty ,\alpha \in ℝ$。Herz空间 $K_{p,q}^{\alpha }=K_{p,q}^{\alpha }\left({ℝ}^N\right)$ 定义为

$K_{p,q}^{\alpha }:=\{\begin{array}{l}{\left({\displaystyle \underset{k\ge -1}{\sum }{2}^{k\alpha q}{\Vert f\Vert }_{L^p\left(A_k\right)}^q}\right)}^{\frac{1}{q}},q<\infty ,\hfill \\ \underset{k\ge -1}{\sup }{2}^{k\alpha }{\Vert f\Vert }_{L^p\left(A_k\right)},q={\infty }^{\frac{1}{q}}.\hfill \end{array}$

定义1.2 [7] 假设 $1\le p,q,r\le \infty ,\alpha ,s\in ℝ$。空间 $B{K}_{p,q,r}^{\alpha ,s}=B{K}_{p,q,r}^{\alpha ,s}\left({ℝ}^N\right)$ 定义为

$B{K}_{p,q,r}^{\alpha ,s}=\left\{f\in {\mathcal{S}}^{\prime }\left({ℝ}^n\right):{\Vert f\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}<\infty \right\},$

其中

${\Vert f\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}:=\{\begin{array}{l}{\left({\displaystyle \underset{j\ge -1}{\sum }{2}^{jsr}}{\Vert {\Delta }_{j}f\Vert }_{K_{p,q}^{\alpha }}^r\right)}^{\frac{1}{r}},若r<\infty ,\hfill \\ \underset{j\ge -1}{\text{sup}}{2}^{js}{\Vert {\Delta }_{j}f\Vert }_{K_{p,q}^{\alpha }},若r=\infty .\hfill \end{array}$

根据Besov-Herz空间的定义可知对任意 ，这里 $B_{p,r}^s\left({ℝ}^N\right)$ 就是标准的Besov空间。

这一节的主要目的是在Besov-Herz空间建立无粘准地转方程(1.1)的局部适定性。下面给出当 $\kappa =0$ 时准地转方程的局部适定性结果。

定理1 假设 $\kappa =0,1\le p<\infty ,1\le r,q\le \infty ,s\ge \frac{2}{p}+1,0\le \alpha <2\left(1-\frac{1}{p}\right)$。若 ${\theta }_0\in B{K}_{p,q,r}^{\alpha ,s}$，则存在时间 $T=T\left({\Vert {\theta }_0\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}\right)$ 使得方程(1.1)有一个唯一的解 $\theta \in C\left(\left[0,T\right);B{K}_{p,q,r}^{\alpha ,s}\right)$。

注 由于 $K_{p,p}^0\left({ℝ}^n\right)=L^p\left({ℝ}^n\right)$，空间 $B{K}_{p,q,r}^{\alpha ,s}\left({ℝ}^N\right)$ 包含经典的Besov空间。因此定理改进了文 [1] 和 [4] 中的局部适定性结论。

2. 预备知识和相关引理

首先给出Fourier变换和Fourier逆变换的定义。设 $f\in S^{\prime }\left({ℝ}^N\right)$，其Fourier变换为

$\hat{f}\left(\xi \right)=\mathcal{F}f\left(\xi \right)={\displaystyle {\int }_{{ℝ}^N}{\text{e}}^{-ix\cdot \xi }}f\left(x\right)\text{d}x.$

$f\in S^{\prime }\left({ℝ}^N\right)$，其Fourier逆变换为

$\stackrel{⌣}{f}\left(x\right)={\mathcal{F}}^{-1}f\left(x\right)={\left(2\pi \right)}^{-\frac{N}{2}}{\displaystyle {\int }_{{ℝ}^N}{\text{e}}^{ix\cdot \xi }}f\left(\xi \right)\text{d}\xi .$

下面回顾一下Littlewood-Paley分解。假设 $\phi \in S\left({ℝ}^N\right)$ 满足

$0\le \phi \le 1,\text{supp}\left(\phi \right)\subset \left\{\xi \in {ℝ}^N\text{}:\frac{3}{4}\le \left|\xi \right|\le \frac{8}{3}\right\}$ 且当 $\xi \ne 0$ 时 ${\displaystyle \underset{j\in \mathbb{Z}}{\sum }\phi }\left(2^{-j}\xi \right)=1.$

记

${\phi }_j\left(\xi \right)=\phi \left(2^{-j}\xi \right),{\psi }_j\left(\xi \right)={\displaystyle \underset{k\le j-1}{\sum }{\phi }_k}\left(\xi \right),{\psi }_j\left(\xi \right)={\displaystyle \underset{k\le j-1}{\sum }{\phi }_k}\left(\xi \right),$

$h\left(x\right)={\mathcal{F}}^{-1}\phi \left(x\right),g\left(x\right)={\mathcal{F}}^{-1}{\psi }_0\left(x\right).$

下面给出几个频率局部化算子

${\stackrel{\dot{}}{\Delta }}_{j}f={\phi }_j\left(D\right)f=2^{Nj}{\displaystyle \int h}\left(2^{j}y\right)f\left(x-y\right)\text{d}y$

和

${\stackrel{\dot{}}{S}}_{j}f={\displaystyle \underset{k\le j-1}{\sum }{\stackrel{\dot{}}{\Delta }}_k}f={\psi }_j\left(D\right)f=2^{Nj}{\displaystyle {\int }_{{ℝ}^d}g}\left(2^{j}y\right)f\left(x-y\right)\text{d}y.$

根据定义，可推得

${\stackrel{\dot{}}{\Delta }}_j{\stackrel{\dot{}}{\Delta }}_{k}f=0,若\left|j-k\right|\ge 2,$

${\stackrel{\dot{}}{\Delta }}_j\left({\stackrel{\dot{}}{S}}_{k-1}f{\stackrel{\dot{}}{\Delta }}_{k}f\right)=0,若\left|j-k\right|\ge 5.$

下面给出Bony仿积分解的定义

$uv={\stackrel{\dot{}}{T}}_{u}v+{\stackrel{\dot{}}{T}}_{v}u+R\left(u,v\right),$

其中

${\stackrel{\dot{}}{T}}_{u}v={\displaystyle \underset{j\in \mathbb{Z}}{\sum }{\stackrel{\dot{}}{S}}_{j-1}}u{\stackrel{\dot{}}{\Delta }}_{j}v,\stackrel{\dot{}}{R}\left(u,v\right)={\displaystyle \underset{j\in \mathbb{Z}}{\sum }{\stackrel{\dot{}}{\Delta }}_j}u{\widetilde{\stackrel{\dot{}}{\Delta }}}_{j}v,{\widetilde{\stackrel{\dot{}}{\Delta }}}_{j}v={\displaystyle \underset{\left|j^{\prime }-j\right|\le 1}{\sum }{\stackrel{\dot{}}{\Delta }}_{j^{\prime }}}v.$

引理2.1 设 $1\le p<\infty$，则有 $L^p\left({ℝ}^N\right)\subset K_{p,\infty }^0\left({ℝ}^N\right)$。

下面几个引理在定理的证明过程中起到关键的作用(见 [8]、 [9])。

引理2.2 [8] 设 $1<p<\infty ,-\frac{N}{p}<\alpha <N\left(1-\frac{1}{p}\right)$。若T是在 $L^p\left({ℝ}^N\right)$ 上有界的次线性算子且对任意具有

紧支集的函数f满足

$\left|Tf\left(x\right)\right|\le C{\displaystyle {\int }_{R^n}\frac{\left|f\left(y\right)\right|}{{\left|x-y\right|}^n}\text{d}y},x\in \text{supp}f.$

则算子T一定在Herz空间 $K_{p,q}^{\alpha }\left({ℝ}^N\right)$ 上有界。

显然算子T包含Hardy-Littlewood极大函数M和Riesz变换 ${\mathcal{R}}_j,j=1,\cdots ,N$。

引理2.3 [9] 设 $\phi$ 是 ${ℝ}^N$ 上可积函数，令 ${\phi }_{ϵ}\left(x\right)=\frac{1}{{ϵ}^N}\phi \left(\frac{x}{ϵ}\right),ϵ>0$。假设 $\phi$ 的严格递减的最小径向控制

函数 $\Psi$ 是可积的，即

$\Psi \left(x\right)=\underset{\left|y\right|\ge \left|x\right|}{\text{sup}}\left|\phi \left(y\right)\right|且{\displaystyle {\int }_{{ℝ}^d}\Psi }\left(x\right)\text{d}x=A<\infty .$

则对任意 $f\in L^p\left({ℝ}^N\right),1\le p\le \infty$，存在常数 $C>0$ 使得

其中M为Hardy-Littlewood极大函数。

3. 主要引理的证明

证明定理1.1之前，先给出几个重要引理的证明。

引理3.1 设 $1\le p<\infty ,1\le r,q\le \infty ,\alpha \ge 0$ 以及f是一螺线向量场. 则当 $s\ge 0$ 时，有

${\Vert 2^{ks}{\Vert \left[f,{\Delta }_k\right]\cdot \nabla g\Vert }_{K_{p,q}^{\alpha }\left(A^k\right)}\Vert }_{l^r}\lesssim {\Vert \nabla f\Vert }_{\infty }{\Vert g\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}+{\Vert \nabla g\Vert }_{\infty }{\Vert f\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}.$ (3.1)

证明 根据Einstein关于哑指标 $i\in \left[1,N\right]$ 求和的习惯以及Bony仿积分解可得

$\begin{array}{c}\left[f,{\Delta }_k\right]\cdot \nabla g=\left[f_i,{\Delta }_k\right]{\partial }_{i}g=f_i{\Delta }_k{\partial }_{i}g-{\Delta }_k\left(f_i{\partial }_{i}g\right)\\ =\left[T_{f_i},{\Delta }_k\right]{\partial }_{i}g+{T^{\prime }}_{{\Delta }_k{\partial }_{i}g}{f}_i-{\Delta }_k\left(T_{{\partial }_{i}g}{f}_i\right)-{\Delta }_k\left(R\left(f_i,{\partial }_{i}g\right)\right)\\ :=I+II+III+IV,\end{array}$

其中 ${T^{\prime }}_{u}v$ 表示 $T_{u}v+R\left(u,v\right)$。结合支集条件，分部积分和散度条件 $\text{div}f=0$，可得

$\begin{array}{c}\left|I\right|=\left|{\displaystyle \underset{\left|k^{\prime }-k\right|\le 4}{\sum }\left[S_{k^{\prime }-1}{f}_i,{\Delta }_k\right]{\partial }_i{\Delta }_{k^{\prime }}g}\right|\\ =\left|{\displaystyle \underset{\left|k^{\prime }-k\right|\le 4}{\sum }{\displaystyle {\int }_{{ℝ}^d}\left(S_{k^{\prime }-1}{f}_i\left(x\right)-S_{k^{\prime }-1}{f}_i\left(y\right)\right)2^{Nk}}}h\left(2^k\left(x-y\right)\right){\partial }_i{\Delta }_{k^{\prime }}g\left(y\right)\text{d}y\right|\\ =\left|{\displaystyle \underset{\left|k^{\prime }-k\right|\le 4}{\sum }{\displaystyle {\int }_{{ℝ}^d}\left(S_{k^{\prime }-1}{f}_i\left(x\right)-S_{k^{\prime }-1}{f}_i\left(y\right)\right)2^{k\left(N+1\right)}}}\left({\partial }_i\right)h\left(2^k\left(x-y\right)\right){\Delta }_{k^{\prime }}g\left(y\right)\text{d}y\right|\\ \le C{\displaystyle \underset{\left|k^{\prime }-k\right|\le 4}{\sum }{\Vert \nabla S_{k^{\prime }-1}f\Vert }_{\infty }}{\displaystyle {\int }_{{ℝ}^d}{2}^k}\left|x-y\right|2^{Nk}\left|\nabla h\left(2^k\left(x-y\right)\right){\Delta }_{k^{\prime }}g\left(y\right)\right|\text{d}y.\end{array}$ (3.2)

注意到 $h\left(x\right)\in \mathcal{S}\left({ℝ}^d\right)$，容易验证 $\left|x\nabla h\left(x\right)\right|$ 满足引理2.3的条件，因此

$\begin{array}{c}\left|I\right|\le C{\displaystyle \underset{\left|k^{\prime }-k\right|\le 4}{\sum }{\Vert \nabla S_{k^{\prime }-1}f\Vert }_{\infty }}M\left(\left|{\Delta }_{k^{\prime }}g(\cdot )\right|\right)\left(x\right)\end{array}.$ (3.3)

在(3.3)式两边取 $K_{p,q}^{\alpha }$ 范数并用引理2.2可得

(3.4)

在(3.4)式两边同时乘以 $2^{ks}$，然后取 $l^r\left(k\ge -1\right)$ 范数可得

$\begin{array}{c}{\Vert 2^{ks}{\Vert I\Vert }_{K_{p,q}^{\alpha }}\Vert }_{l^r\left(k\ge -1\right)}\lesssim {\Vert \nabla f\Vert }_{\infty }{\Vert {\displaystyle \underset{\left|k^{\prime }-k\right|\le 4}{\sum }{2}^{\left(k-k^{\prime }\right)s}}{2}^{k^{\prime }s}{\Vert {\Delta }_{k^{\prime }}g\Vert }_{K_{p,q}^{\alpha }}\Vert }_{l^r}\\ \lesssim {\Vert \nabla f\Vert }_{\infty }{\Vert 2^{k^{\prime }s}{\Vert {\Delta }_{k^{\prime }}g\Vert }_{K_{p,q}^{\alpha }}\Vert }_{l^r}\\ \lesssim {\Vert \nabla f\Vert }_{\infty }{\Vert 2^{ks}{\Vert {\Delta }_{k}g\Vert }_{K_{p,q}^{\alpha }}\Vert }_{l^r}\\ \lesssim {\Vert \nabla f\Vert }_{\infty }{\Vert g\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}.\end{array}$ (3.5)

下面估计II项。根据II的定义有

$\left|II\right|=\left|{\displaystyle \underset{k^{\prime }\ge k-2}{\sum }{S}_{k^{\prime }+2}}{\partial }_i{\Delta }_{k}g{\Delta }_{k^{\prime }}{f}_i\left(x\right)\right|\le {\displaystyle \underset{k^{\prime }\ge k-2}{\sum }{\Vert \nabla {\Delta }_{k}g\Vert }_{\infty }}\left|{\Delta }_{k^{\prime }}f\left(x\right)\right|.$ (3.6)

当 $s>0$ 时，利用离散形式的卷积不等式可得

$\begin{array}{c}{\Vert 2^{ks}{\Vert II\Vert }_{K_{p,q}^{\alpha }}\Vert }_{l^r\left(k\ge -1\right)}\lesssim {\Vert \nabla g\Vert }_{\infty }{\Vert {\displaystyle \underset{k^{\prime }\ge k-2}{\sum }{2}^{\left(k-k^{\prime }\right)s}}{2}^{k^{\prime }s}{\Vert {\Delta }_{k^{\prime }}f\Vert }_{K_{p,q}^{\alpha }}\Vert }_{l^r}\\ \lesssim {\Vert \nabla g\Vert }_{\infty }{\Vert 2^{-ks}{\chi }_{\left\{k\ge -2\right\}}\Vert }_{l^1}{\Vert 2^{k^{\prime }s}{\Vert {\Delta }_{k^{\prime }}f\Vert }_{K_{p,q}^{\alpha }}\Vert }_{l^r}\\ \lesssim {\Vert \nabla f\Vert }_{\infty }{\Vert 2^{ks}{\Vert {\Delta }_{k}g\Vert }_{K_{p,q}^{\alpha }}\Vert }_{l^r}\lesssim {\Vert \nabla g\Vert }_{\infty }{\Vert f\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}.\end{array}$ (3.7)

对于III项，

$\begin{array}{c}{\Vert III\Vert }_{K_{p,q}^{\alpha }}={\Vert {\displaystyle \underset{\left|k^{\prime }-k\right|\le 4}{\sum }{\Delta }_k}\left(S_{k^{\prime }-1}{\partial }_{i}g{\Delta }_{k^{\prime }}{f}_i\right)\left(x\right)\Vert }_{K_{p,q}^{\alpha }}\\ \lesssim {\Vert {\displaystyle \underset{\left|k^{\prime }-k\right|\le 4}{\sum }M}\left(S_{k^{\prime }-1}{\partial }_{i}g{\Delta }_{k^{\prime }}{f}_i\right)\Vert }_{K_{p,q}^{\alpha }}\\ \lesssim {\displaystyle \underset{\left|k^{\prime }-k\right|\le 4}{\sum }{\Vert M\left(\left|{\Delta }_{k^{\prime }}f\right|\right)\Vert }_{K_{p,q}^{\alpha }}{\Vert S_{k^{\prime }-1}\nabla g\Vert }_{\infty }}\\ \lesssim {\Vert \nabla g\Vert }_{\infty }{\displaystyle \underset{\left|k^{\prime }-k\right|\le 4}{\sum }{\Vert {\Delta }_{k^{\prime }}f\Vert }_{K_{p,q}^{\alpha }}}.\end{array}$ (3.8)

因此类似于(3.5)可得

(3.9)

利用 和分部积分，可得

$\begin{array}{c}\left|IV\right|=\left|{\displaystyle \underset{k^{\prime }\ge k-3}{\sum }{\Delta }_k}\left({\Delta }_{k^{\prime }}{f}_i{\partial }_i{\tilde{\Delta }}_{k^{\prime }}g\right)\right|=\left|{\displaystyle \underset{k^{\prime }\ge k-3}{\sum }{\displaystyle {\int }_{{ℝ}^d}{2}^{kd}}}h\left(2^k\left(x-y\right)\right){\Delta }_{k^{\prime }}{f}_i\left(y\right){\partial }_i{\tilde{\Delta }}_{k^{\prime }}g\left(y\right)\text{d}y\right|\\ =\left|{\displaystyle \underset{k^{\prime }\ge k-3}{\sum }{\displaystyle {\int }_{{ℝ}^d}{2}^{kd+k}}}\left({\partial }_{i}h\right)\left(2^k\left(x-y\right)\right){\Delta }_{k^{\prime }}{f}_i\left(y\right){\partial }_i{\tilde{\Delta }}_{k^{\prime }}g\left(y\right)\text{d}y\right|\\ \lesssim {\displaystyle \underset{k^{\prime }\ge k-3}{\sum }{2}^k}M\left({\Delta }_{k^{\prime }}f{\tilde{\Delta }}_{k^{\prime }}g\right)\left(x\right)\lesssim {\displaystyle \underset{k^{\prime }\ge k-3}{\sum }{2}^k}M\left({\tilde{\Delta }}_{k^{\prime }}g\right)\left(x\right){\Vert {\Delta }_{k^{\prime }}f\Vert }_{\infty }\\ \lesssim {\displaystyle \underset{k^{\prime }\ge k-3}{\sum }{2}^k}M\left({\tilde{\Delta }}_{k^{\prime }}g\right)\left(x\right)2^{-k^{\prime }}{\Vert \nabla {\Delta }_{k^{\prime }}f\Vert }_{\infty }\\ \lesssim {\Vert \nabla f\Vert }_{\infty }{\displaystyle \underset{k^{\prime }\ge k-3}{\sum }{2}^{k-k^{\prime }}}M\left({\tilde{\Delta }}_{k^{\prime }}g\right)\left(x\right),\end{array}$ (3.10)

利用离散形式的卷积不等式和引理2.2，可得当 $s+1>0$ 时有

$\begin{array}{c}{\Vert 2^{ks}{\Vert IV\Vert }_{K_{p,q}^{\alpha }}\Vert }_{l^r}\lesssim {\Vert \nabla f\Vert }_{\infty }{\Vert {\displaystyle \underset{k^{\prime }\ge k-3}{\sum }{2}^{k-k^{\prime }}}{2}^{ks}{\Vert M\left({\tilde{\Delta }}_{k^{\prime }}g\right)\Vert }_{K_{p,q}^{\alpha }}\Vert }_{l^r}\\ \lesssim {\Vert \nabla f\Vert }_{\infty }{\Vert 2^{k-k^{\prime }}{2}^{ks}{2}^{-k^{\prime }s}{2}^{k^{\prime }s}{\Vert M\left({\tilde{\Delta }}_{k^{\prime }}g\right)\Vert }_{K_{p,q}^{\alpha }}\Vert }_{l^r}\\ \lesssim {\Vert \nabla f\Vert }_{\infty }{\Vert 2^{\left(k-k^{\prime }\right)\left(s+1\right)2^{k^{\prime }s}}{\Vert {\Delta }_{k}g\Vert }_{K_{p,q}^{\alpha }}\Vert }_{l^r}\\ \lesssim {\Vert \nabla f\Vert }_{\infty }{\Vert g\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}.\end{array}$ (3.11)

结合不等式(3.5)、(3.7)、(3.9)和(3.11)可得不等式(3.1)。

引理3.2 设 $1\le p<\infty ,1\le r,q\le \infty ,\alpha \ge 0$ 以及f是一螺线向量场。设 $s\ge \frac{n}{p}+1$ 且当 $s=\frac{n}{p}+1$ 时 $\mathrm{r=1}$，则

存在常数 $C>0$ 使得

${\Vert 2^{ks}{\Vert \left[f,{\Delta }_k\right]\cdot \nabla g\Vert }_{K_{p,q}^{\alpha }\left(A^k\right)}\Vert }_{l^r}\lesssim {\Vert f\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}{\Vert g\Vert }_{B{K}_{p,q,r}^{\alpha ,s}},$ (3.12)

${\Vert 2^{ks}{\Vert \left[f,{\Delta }_k\right]\cdot \nabla g\Vert }_{K_{p,q}^{\alpha }\left(A^k\right)}\Vert }_{l^r}\lesssim {\Vert f\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}{\Vert g\Vert }_{B{K}_{p,q,r}^{\alpha ,s}},$ (3.13)

${\Vert 2^{k\left(s-1\right)}{\Vert \left[f,{\Delta }_k\right]\cdot \nabla g\Vert }_{K_{p,q}^{\alpha }\left(A^k\right)}\Vert }_{l^r}\lesssim {\Vert f\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}{\Vert g\Vert }_{B{K}_{p,q,r}^{\alpha ,s-1}}.$ (3.14)

当 $r=\infty$ 时，按通常的定义做相应的改动即可。

证明 根据Ein stein关于哑指标 $i\in \left[1,N\right]$ 求和的习惯以及Bony仿积分解可得

$\begin{array}{l}\left[f,{\Delta }_k\right]\cdot \nabla g=\left[f_i,{\Delta }_k\right]{\partial }_{i}g=f_i{\Delta }_k{\partial }_{i}g-{\Delta }_k\left(f_i{\partial }_{i}g\right)\\ =\left[T_{f_i},{\Delta }_k\right]{\partial }_{i}g+{T^{\prime }}_{{\Delta }_k{\partial }_{i}g}{f}_i-{\Delta }_k\left(T_{{\partial }_{i}g}{f}_i\right)-{\Delta }_k\left(R\left(f_i,{\partial }_{i}g\right)\right)\\ :=I+II+III+IV,\end{array}$

类似引理3.1中的(3.3)可得

$\begin{array}{c}\left|I\right|\lesssim {\displaystyle \underset{\left|k^{\prime }-k\right|\le 4}{\sum }{\Vert \nabla S_{k^{\prime }-1}f\Vert }_{\infty }}M\left(\left|{\Delta }_{k^{\prime }}g(\cdot )\right|\right)\left(x\right)\\ \lesssim {\displaystyle \underset{\left|k^{\prime }-k\right|\le 4}{\sum }{\displaystyle \underset{m\le k^{\prime }-2}{\sum }{\Vert \nabla {\Delta }_{m}f\Vert }_{\infty }}}M\left(\left|{\Delta }_{k^{\prime }}g(\cdot )\right|\right)\left(x\right),\end{array}$ (3.15)

取 $K_{p,q}^{\alpha }$ 范数得

$\begin{array}{c}{\Vert I\Vert }_{K_{p,q}^{\alpha }}\lesssim {\displaystyle \underset{\left|k^{\prime }-k\right|\le 4}{\sum }{\displaystyle \underset{m\le k^{\prime }-2}{\sum }{\Vert \nabla {\Delta }_{m}f\Vert }_{\infty }{\Vert M\left(\left|{\Delta }_{k^{\prime }}g\right|\right)\Vert }_{K_{p,q}^{\alpha }}}}\\ \lesssim {\displaystyle \underset{\left|k^{\prime }-k\right|\le 4}{\sum }{\displaystyle \underset{m\le k^{\prime }-2}{\sum }{\Vert \nabla {\Delta }_{m}f\Vert }_{\infty }}}{\Vert {\Delta }_{k^{\prime }}g\Vert }_{K_{p,q}^{\alpha }}.\end{array}$ (3.16)

对任意 $\rho$，按照文献 [7] 引理4.4估计 $R_j^1\left(u,v\right)$ 的方法可得

${\Vert 2^{k\rho }{\Vert I\Vert }_{K_{p,q}^{\alpha }}\Vert }_{l^r}\lesssim {\Vert f\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}{\Vert g\Vert }_{B{K}_{p,q,r}^{\alpha ,\rho }}.$ (3.17)

当 $r=\infty$ 时，做相应的改动即可。特别地，在(3.17)中取 $\rho =s$ 和 $\rho =s-1$ 可得，对I可得

${\Vert 2^{ks}{\Vert I\Vert }_{K_{p,q}^{\alpha }}\Vert }_{l^r}\lesssim {\Vert f\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}{\Vert g\Vert }_{B{K}_{p,q,r}^{\alpha ,s}},$ (3.18)

${\Vert 2^{k\left(s-1\right)}{\Vert I\Vert }_{K_{p,q}^{\alpha }}\Vert }_{l^r}\lesssim {\Vert f\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}{\Vert g\Vert }_{B{K}_{p,q,r}^{\alpha ,s-1}}.$ (3.19)

利用(3.3)和(3.16)，按照文献 [7] 引理4.4(ii)估计(4.27)式的方法可得

${\Vert 2^{k\left(s-1\right)}{\Vert I\Vert }_{K_{p,q}^{\alpha }}\Vert }_{l^r}\lesssim {\Vert f\Vert }_{B{K}_{p,q,r}^{\alpha ,s-1}}{\Vert g\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}.$ (3.20)

下面，对II利用(3.6)和文献 [7] 引理4.4(ii)估计 $R_j^{4,1}\left(u,v\right)$ 的方法可得

${\Vert 2^{ks}{\Vert II\Vert }_{K_{p,q}^{\alpha }}\Vert }_{l^r}\lesssim {\Vert f\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}{\Vert g\Vert }_{B{K}_{p,q,r}^{\alpha ,\rho }},$ (3.21)

${\Vert 2^{k\left(s-1\right)}{\Vert II\Vert }_{K_{p,q}^{\alpha }}\Vert }_{l^r}\lesssim {\Vert f\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}{\Vert g\Vert }_{B{K}_{p,q,r}^{\alpha ,s-1}}.$ (3.22)

此外，类似(3.20)可得

${\Vert 2^{k\left(s-1\right)}{\Vert II\Vert }_{K_{p,q}^{\alpha }}\Vert }_{l^r}\lesssim {\Vert f\Vert }_{B{K}_{p,q,r}^{\alpha ,s-1}}{\Vert g\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}.$ (3.23)

按照(3.17)的方法，对III，(3.12)~(3.14)右端的不等式也一样成立。对IV，利用(3.10)可得

$\begin{array}{c}{\Vert IV\Vert }_{K_{p,q}^{\alpha }}\lesssim {\displaystyle \underset{k^{\prime }\ge k-3}{\sum }{\Vert 2^{k}M\left({\Delta }_{k^{\prime }}f{\tilde{\Delta }}_{k^{\prime }}g\right)\left(x\right)\Vert }_{K_{p,q}^{\alpha }}}\\ \lesssim {\displaystyle \underset{k^{\prime }\ge k-3}{\sum }{\Vert 2^{k}M\left({\Delta }_{k^{\prime }}f\right)\Vert }_{K_{p,q}^{\alpha }}}{\Vert {\tilde{\Delta }}_{k^{\prime }}g\Vert }_{\infty }\\ \lesssim {\displaystyle \underset{k^{\prime }\ge k-3}{\sum }{2}^{k-k^{\prime }}}{\Vert {\Delta }_{k^{\prime }}f\Vert }_{K_{p,q}^{\alpha }}{2}^{k^{\prime }\left(\frac{n}{p}+1\right)}{\Vert {\Delta }_{k^{\prime }}g\Vert }_{K_{p,q}^{\alpha }}\\ \lesssim {\Vert f\Vert }_{B{K}_{p,q,\infty }^{\alpha ,s}}{\displaystyle \underset{k^{\prime }\ge k-3}{\sum }{2}^{k-k^{\prime }}}{2}^{k^{\prime }\left(\frac{n}{p}+1-s\right)}{\Vert {\Delta }_{k^{\prime }}g\Vert }_{K_{p,q}^{\alpha }}.\end{array}$ (3.24)

因此，利用(3.6)和文献 [7] 引理4.4(ii)估计 $R_j^{4,1}\left(u,v\right)$ 的方法可得，对IV，(3.12)~(3.13)右端的不等式也一样成立。由于 ${\tilde{\Delta }}_{k^{\prime }}~{\Delta }_{k^{\prime }}$，根据对称性对IV，(3.14)右端的不等式也成立。

4. 定理1.1的证明

在这一节，考虑方程

$\{\begin{array}{l}{\partial }_t\theta +u\cdot \nabla \theta =0,\left(x,t\right)\in {ℝ}^2\times \left[0,T\right),\hfill \\ \text{div}u=0,\hfill \\ \theta \left(0,x\right)={\theta }_0\left(x\right),x\in {ℝ}^2.\hfill \end{array}$ (4.1)

定理1.1的证明分成下面的五步。

第一步 先验界估计。用算子 ${\Delta }_k$ 作用在方程(4.1)的两边得

${\partial }_t\Delta \theta +u\cdot \nabla \theta =\left[u,{\Delta }_k\right]\cdot \nabla \theta .$ (4.2)

假设 $X_t\left(\alpha \right)$ 是下面的常微分方程的解

$\{\begin{array}{l}{\partial }_t{X}_t\left(\alpha \right)=u\left(X_t\left(\alpha \right),t\right),\hfill \\ {X_t\left(\alpha \right)|}_{t=0}=\alpha .\hfill \end{array}$ (4.3)

则从(4.2)可得

$\frac{\text{d}}{\text{d}t}{\Delta }_k\theta \left(X_t\left(\alpha \right),t\right)=\left[u,{\Delta }_k\right]\cdot \nabla \theta \left(X_t\left(\alpha \right),t\right),$

因此可推得

$\left|{\Delta }_k\theta \left(X_t\left(\alpha \right),t\right)\right|\le \left|{\Delta }_k{\theta }_0\left(\alpha \right)\right|+{\displaystyle {\int }_0^t\left|\left[u,{\Delta }_k\right]\cdot \nabla \theta \left(X_t\left(\alpha \right),t\right)\right|\text{d}\tau }.$ (4.4)

在(4.4)式两边先取 范数，再乘以 $2^{ks}$ 并取 $l^r$ 范数，则利用Minkowski不等式可得

$\begin{array}{l}{\Vert 2^{ks}{\Vert \Delta \theta \left(X_t\left(\alpha \right),t\right)\Vert }_{K_{p.q}^{\alpha }}\Vert }_{l^r}\\ \le {\Vert {\theta }_0\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}+{\displaystyle {\int }_0^t{\Vert 2^{ks}{\Vert \left[u,{\Delta }_k\right]\cdot \nabla \theta \left(X_t\left(\alpha \right),t\right)\Vert }_{K_{p.q}^{\alpha }}\Vert }_{l^r}}\text{d}\tau .\end{array}$

因为 $\text{div}u=0$，故 $X_t\left(\alpha \right)$ 是保体积的微分同胚。再结合引理3.2，从上面的不等式可得

$\begin{array}{c}{\Vert \theta \Vert }_{B{K}_{p,q,r}^{\alpha ,s}}\le {\Vert {\theta }_0\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}+{\displaystyle {\int }_0^t{\Vert 2^{ks}{\Vert \left[u,{\Delta }_k\right]\cdot \nabla \theta \Vert }_{K_{p.q}^{\alpha }}\Vert }_{l^r}\text{d}\tau }\\ \le {\Vert {\theta }_0\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}+{\displaystyle {\int }_0^t{\Vert u\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}{\Vert \theta \Vert }_{B{K}_{p,q,r}^{\alpha ,s}}\text{d}\tau }\\ \le {\Vert {\theta }_0\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}+{\displaystyle {\int }_0^t{\Vert \theta \Vert }_{B{K}_{p,q,r}^{\alpha ,s}}^2}\text{d}\tau ,\end{array}$ (4.5)

这里用到了

${\Vert u\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}={\Vert {\mathcal{R}}^{\perp }\theta \Vert }_{B{K}_{p,q,r}^{\alpha ,s}}\le C{\Vert \theta \Vert }_{B{K}_{p,q,r}^{\alpha ,s}}.$

因此利用Gronwall不等式，存在 $T_0=T\left({\Vert {\theta }_0\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}\right)$，当 $t\in \left[0,T_0\right)$ 时有

${\Vert \theta \Vert }_{B{K}_{p,q,r}^{\alpha ,s}}\lesssim {\Vert {\theta }_0\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}.$ (4.6)

第二步 逼近解和一致界估计。先构造问题(4.1)的逼近解，通过解方程组

$\{\begin{array}{l}{\partial }_t{\theta }^{\left(n+1\right)}+u^{\left(n\right)}\cdot \nabla {\theta }^{\left(n+1\right)}=0,\left(x,t\right)\in {ℝ}^2\times \left[0,T\right),\hfill \\ \text{div}{u}^{\left(n\right)}=0,\hfill \\ {\theta }^{\left(n+1\right)}\left(0,x\right)=S_{n+2}{\theta }_0\left(x\right),x\in {ℝ}^2\hfill \end{array}$ (4.7)

来定义序列 ${\left\{{\theta }^{\left(n\right)},u^{\left(n\right)}\right\}}_{N_0=N\cup \left\{0\right\}}$，其中。类似于(4.5)式的证明可得

$\begin{array}{c}{\Vert {\theta }^{\left(n+1\right)}\left(t\right)\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}\lesssim {\Vert {\theta }_0\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}+{\displaystyle {\int }_0^t{\Vert u^{\left(n\right)}\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}}{\Vert {\theta }^{\left(n+1\right)}\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}\text{d}\tau \\ \lesssim {\Vert {\theta }_0\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}+{\displaystyle {\int }_0^t{\Vert {\theta }^{\left(n\right)}\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}}{\Vert {\theta }^{\left(n+1\right)}\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}\text{d}\tau ,\end{array}$ (4.8)

其中用到事实

${\Vert S_{n+2}{\theta }_0\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}\lesssim {\Vert {\theta }_0\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}和u=R^{\perp }\theta .$

利用Gronwall不等式可知存在 $T_0=T\left({\Vert {\theta }_0\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}\right)$ 使得对任意 $n,t\in \left[0,T_0\right]$，

${\Vert {\theta }^{\left(n\right)}\left(t\right)\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}\lesssim 2{\Vert {\theta }_0\Vert }_{B{K}_{p,q,r}^{\alpha ,s}}.$

第三步 存在性。现在将证明存在与n无关的时间 $T_1\left(\le T_0\right)$ 使得 $\left\{{\theta }^{\left(n\right)}\right\}$ 是 $X_T^{s-1}\triangleq \mathcal{C}\left(\left[0,T_1\right];B{K}_{p,q,r}^{\alpha ,s-1}\right)$ 中的柯西序列。为此令

$\delta {\theta }^{\left(n+1\right)}={\theta }^{\left(n+1\right)}-{\theta }^{\left(n\right)},\delta u^{\left(n+1\right)}=u^{\left(n+1\right)}-u^{\left(n\right)}.$

则不难验证差分 满足方程

(4.9)

用 ${\Delta }_k$ 作用在(4.9)的第一个方程可得

${\partial }_t{\Delta }_k\delta {\theta }^{\left(n+1\right)}+u^{\left(n\right)}\cdot \nabla {\Delta }_k\delta {\theta }^{\left(n+1\right)}=\left[u^{\left(n\right)},{\Delta }_k\right]\cdot \nabla \delta {\theta }^{\left(n+1\right)}-{\Delta }_k\left(\delta u^{\left(n\right)}\cdot \nabla {\theta }^{\left(n\right)}\right).$

类似(4.5)可得

$\begin{array}{c}{\Vert \delta {\theta }^{\left(n+1\right)}\Vert }_{B{K}_{p,q,r}^{\alpha ,s-1}}\le {\Vert {\Delta }_{n+1}{\theta }_0\Vert }_{B{K}_{p,q,r}^{\alpha ,s-1}}+{\displaystyle {\int }_0^t{\Vert 2^{k\left(s-1\right)}{\Vert \left[u^{\left(n\right)},{\Delta }_k\right]\cdot \nabla \delta {\theta }^{\left(n+1\right)}\Vert }_{K_{p,q}^{\alpha }}\Vert }_{l^r}\text{d}\tau }\\ +{\displaystyle {\int }_0^t{\Vert \delta u^{\left(n\right)}\cdot \nabla {\theta }^{\left(n\right)}\Vert }_{B{K}_{p,q,r}^{\alpha ,s-1}}}\text{d}\tau .\end{array}$ (4.10)