西北师范大学，甘肃 兰州

收稿日期：2021年12月13日；录用日期：2022年1月3日；发布日期：2022年1月19日

摘要

本文研究了带有旋转效应的分数阶Navier-Stokes方程在给定周期外力作用下周期mild解的存在唯一性，并且建立了Besov空间中分数阶热半群的线性估计。首先，给出符号及函数空间的定义。其次，采用分数阶热半群的 $L^p-L^q$ 估计，分别对分数阶不可压缩Navier-Stokes-Coriolis方程的线性项及非线性项进行了估计。最后证明了给定一个具有周期ω的外力f，分数阶不可压缩Navier-Stokes-Coriolis方程周期mild解是唯一存在的，且周期也为ω。

关键词

分数阶不可压缩Navier-Stokes-Coriolis方程，周期解，半群算子估计

Existence of Periodic Solutions to the Fractional Navier-Stokes-Coriolis Equation

Gangjing He, Xiaochun Sun^*

Northwest Normal University, Lanzhou Gansu

Received: Dec. 13^th, 2021; accepted: Jan. 3^rd, 2022; published: Jan. 19^th, 2022

ABSTRACT

In this paper, we study the existence and uniqueness of periodic mild solution for fractional incompressible Navier-Stokes equations in the rotational framework and establish the linear estimation of fractional heat semigroups in Besov space. Firstly, the definition of symbol and function space is given. Secondly, the linear and nonlinear terms of the fractional incompressible Navier-Stokes-Coriolis equations were estimated by using the $L^p-L^q$ estimates of fractional heat semigroups. Finally, we proved that given an external force with periodic ω, the periodic mild solution of the fractional incompressible Navier-Stokes-Coriolis equation is uniqueness and its period is also ω.

Keywords:Fractional Incompressible Navier-Stokes-Coriolis Equation, Periodic Solution, Semigroup Operator Estimation

Copyright © 2022 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

考虑分数阶不可压缩Navier-Stokes-Coriolis方程

$\{\begin{array}{l}\frac{\partial u}{\partial t}+{\left(-\Delta \right)}^{\alpha }u+\Omega e_3\times u+\left(u\cdot \nabla \right)u+\nabla p=f,\\ div\begin{array}{c}u=0\end{array},\end{array}$ (1.1)

的周期解。其中，向量函数 $u=u\left(x,t\right)=\left(u^1\left(x,t\right),u^2\left(x,t\right),u^3\left(x,t\right)\right)$ 表示流体在点 $\left(x,t\right)\in {ℝ}^3\times ℝ$ 的未知速度，标量函数 $p=p\left(x,t\right)$ 是流体在 $\left(x,t\right)\in {ℝ}^3\times ℝ$ 的未知压力。 $\Omega \in ℝ$ 是围绕垂直单位向量 $e_3=\left(0,0,1\right)$ 的旋转速度。 $f=f\left(x,t\right)=\left(f^1\left(x,t\right),f^2\left(x,t\right),f^3\left(x,t\right)\right)$ 表示给定具有时间周期 $\omega$ 的外力， $\Delta ={\displaystyle {\sum }_{i=1}^3{\partial }_{xi}^2}$ 是关于空间变量x的Laplacian微分算子。设 $\alpha >0$，分数阶Laplacian微分算子 ${\left(-\Delta \right)}^{\alpha }$ 通过Fourier变换定义为：

.

不可压缩Navier-Stokes方程在气象学、海洋学和地球物理学等众多领域中有着广泛的应用。由于地球自转对空气流及海洋流产生着旋转效应与周期效应，因此研究带有旋转效应的不可压缩Navier-Stokes方程周期解的适定性问题具有一定的理论意义。

当 $\alpha =1$ 时，(1.1)式为经典的不可压缩Navier-Stokes-Coriolis方程。Hieber, M.和Shibata, Y.在文献 [1]

中证明了初值为 $u_0$ 的 $H_{\sigma }^{\frac{1}{2}}$ 空间范数充分小时，不可压缩Navier-Stokes-Coriolis方程初值问题存在唯一的整

体mild解。Zhao, H.和Wang, Y. X.在文献 [2] 中证明了对于一类特殊大初值 $u_0\left(x\right)=u_{01}\left(x\right)+u_{02}\left(x\right)$，不可压缩Navier-Stokes-Coriolis方程初值问题整体解的存在性，其中 $\nabla \cdot u_{01}=0$ 而 $u_{02}\left(x\right)$ 满足其它更多条件。Wang, W. H.和Wu, G.在文献 [3] 中证明了空间 ${\mathcal{X}}^{1-2\alpha }$ 中三维不可压缩广义Navier-Stokes-Coriolis方程解的

整体适定性，其中 $\frac{1}{2}\le \alpha \le 1$。Iwabuchi, T.和Takada, R.在文献 [4] 中证明了当 $\Omega$ 足够大时，具有时间周期

外力Navier-Stokes-Coriolis方程的周期解。

当 $\alpha >0$ 时，(1.1)式为分数阶不可压缩Navier-Stokes-Coriolis方程。Ding, Y.和Sun, X. C.在文献 [5] 中证明了分数阶不可压缩Navier-Stokes-Coriolis方程的色散效应和解的局部适定性。Kiahimoto, N.和Yoneda, T.在文献 [6] 中研究了Ω足够大时在周期坐标系内三维旋转流体方程整体解的存在性。Wang, W. H.和Wu, G. [7] 证明了在空间 $F{\stackrel{\dot{}}{B}}_{b,r}^{1-2\alpha +3/n}$ 中分数阶随机不可压缩Navier-Stokes-Coriolis方程解的全局存在性，

其中 $2\le b\le \infty$，$\frac{2}{3}<\alpha <\frac{5}{3}-\frac{1}{b}$，$1\le r\le \infty$。

根据Duhamel定理，(1.1)式等价于下列积分方程：

, (IP)

其中， $\mathbb{P}=\left({\delta }_{ij}+R_i{R}_j\right)\left(1\le i,j\le 3\right)$ 为散度自由向量域上的Helmholtz投影，(IP)中的 $T_{\Omega }(\cdot )$ 为(1.1)式的线性方程的解半群：

$T_{\Omega }\left(t\right)u={\mathcal{F}}^{-1}\left[\cos \left(\Omega \frac{{\xi }_3}{\left|\xi \right|}t\right){\text{e}}^{-t{\left|\xi \right|}^{2\alpha }}I\hat{u}\left(\xi \right)+\sin \left(\Omega \frac{{\xi }_3}{\left|\xi \right|}t\right){\text{e}}^{-t{\left|\xi \right|}^{2\alpha }}R\left(\xi \right)\hat{u}\left(\xi \right)\right]$,

其中 $t\in ℝ$，$divu=\text{0}$。I表示单位矩阵， $R\left(\xi \right)$ 为Riesz变换： $\widehat{Rf}\left(\xi \right):=\frac{i\xi }{\left|\xi \right|}\hat{f}\left(\xi \right)$。

本文考虑了分数阶不可压缩Navier-Stokes-Coriolis方程的周期性解。主要结论为：

定理1.1设 $s>0$，$1<p\le 2$，$\frac{3}{4}<\alpha \le 1$，$2\le r<3$，$2\le q\le \frac{12}{5}$，并且 $\left(r,q\right)\ne \left(2,2\right)$，且

$\frac{s}{3}+\frac{1}{p}>\frac{1}{r}+\frac{2}{3}$,$\frac{1}{2}\le \frac{1}{l}<\frac{1}{q}+\frac{1}{6}$. (1.2)

存在正常数 $\varsigma =\varsigma \left(r,q,s,p,l\right)$，$K=K\left(r,q,s,p,l\right)$ 及 $\omega$，使得 $\forall f\in BC\left(ℝ;{\stackrel{\dot{}}{B}}_{p,2}^{-s}\left({ℝ}^3\right)\right)\cap BC\left(ℝ;L^l\left({ℝ}^3\right)\right)$，对 $\forall \text{}\text{}\text{}\text{}\text{}\text{}\text{}t\in ℝ$，$f\left(t\right)=f\left(t+\omega \right)$ 且

$\underset{t\in ℝ}{\sup }{\Vert f\left(t\right)\Vert }_{{\stackrel{\dot{}}{B}}_{p,2}^{-s}}+\underset{t\in ℝ}{\sup }{\Vert f\left(t\right)\Vert }_{L^l}\le \varsigma$,

(1.1)式存在唯一的周期mild解 $u\in X_K^{r,q}$，并且 $u\left(t\right)=u\left(t+\omega \right)$，其中

$X_K^{r,q}:=\left\{u\in BC\left(ℝ;L_{\sigma }^r\left({ℝ}^3\right)\right)\cap BC\left(ℝ;{\stackrel{\dot{}}{W}}^{1,q}\left({ℝ}^3\right)\right)|{\Vert u\Vert }_{X^{r,q}}\le K\right\}$,

${\Vert u\Vert }_{X^{r,q}}:=\underset{t\in ℝ}{\sup }{\Vert u\left(t\right)\Vert }_{L^r}+\underset{t\in ℝ}{\sup }{\Vert \nabla u\left(t\right)\Vert }_{L^q}$.

注记1.2 Kozono, H.和Nakao, M.在文献 [8] 中证明了三维不可压缩Navier-Stokes方程在条件 $2<r<3$，$\frac{3}{2}<q<3$，$s>0$，$1<p<\min \left\{r,q\right\}$ 及 $\frac{s}{3}+\frac{1}{p}>\frac{1}{r}+\frac{2}{3}$，$\frac{1}{q}\le \frac{1}{l}<\frac{1}{q}+\frac{1}{3}$ 下存在唯一的周期解。Iwabuchi, T.和Takada, R.在文献 [4] 中证明了外力满足与参数 $\left|\Omega \right|$ 相关的尺寸条件下不可压缩Navier-Stoke-Coriolis方程周期解的存在性。受以上两篇文献启发，本文定理1.1将结论推广至分数阶不可压缩Navier-Stoke-Coriolis方程，其中 $\frac{3}{4}<\alpha \le 1$。

2. 预备知识

首先给出符号说明及相关函数空间的定义。

$C_{0,\sigma }^{\infty }\left({ℝ}^3\right)$ 表示具有紧支集的所有实值 $C^{\infty }$ 向量函数 $\varphi =\left({\varphi }^1,{\varphi }^2,{\varphi }^3\right)$ 的集合，且 $div\varphi =0$。 $L_{\sigma }^r\left({ℝ}^3\right)$ 为 $C_{0,\sigma }^{\infty }\left({ℝ}^3\right)$ 关于范数 ${\Vert \begin{array}{c}\cdot \end{array}\Vert }_{L^r}$ 的闭包，其中 $1<r<\infty$。 $L^r\left({ℝ}^3\right)$ 表示 ${ℝ}^3$ 上r次Lebesgue可积函数空间， ${\stackrel{\dot{}}{W}}^{1,r}(\; ℝ\; 3\; )$

表示齐次Sobolev空间。 $L_{\sigma }^{r^{\prime }}\left({ℝ}^3\right)$ 表示为 $L_{\sigma }^r\left({ℝ}^3\right)$ 的对偶空间， $1<r<\infty$，$\frac{1}{r}+\frac{1}{r^{\prime }}=1$。

其次介绍Littlewood-Paley分解.设 $\mathfrak{S}\left({ℝ}^3\right)$ 为Schwartz函数空间， ${\mathfrak{S}}^{\prime }\left({ℝ}^3\right)$ 为缓增广义函数空间.设 $\phi \in \mathfrak{S}\left({ℝ}^3\right)$ 为径向函数，且 $\hat{\phi }$ 满足下列性质

$\text{supp}\hat{\phi }\subset \mathcal{C}:=\left\{\xi \in {ℝ}^3:2^{-1}<\left|\xi \right|<2\right\}$ ; ${\displaystyle \underset{j\in \mathbb{Z}}{\sum }\hat{\phi }}\left(2^{-j}\xi \right)=1$,$\forall \xi \in {ℝ}^3\backslash \left\{0\right\}$.

令 ${\phi }_j\left(x\right):=2^{3j}\phi \left(2^{j}x\right)$，定义频率局部化算子 ${\Delta }_j$ 为

${\Delta }_{j}f:={\phi }_j\ast f$,$\forall j\in \mathbb{Z}$,$f\in {\mathfrak{S}}^{\prime }\left({ℝ}^3\right)$.

令 $\tilde{\mathfrak{S}}\left({ℝ}^3\right):={\mathfrak{S}}^{\prime }\left({ℝ}^3\right)/\mathcal{P}\left[{ℝ}^3\right]$，其中 $\mathcal{P}\left[{ℝ}^3\right]$ 为定义在 ${ℝ}^3$ 上的全体多项式所构成的线性空间。

设 $s\in ℝ$，$1\le p$，$q\le \infty$，则齐次Besov空间 ${\stackrel{\dot{}}{B}}_{p,q}^s\left({ℝ}^3\right)$ 定义为

$\left\{f\in \tilde{\mathfrak{S}}\left({ℝ}^3\right):{\Vert f\Vert }_{{\stackrel{\dot{}}{B}}_{p,q}^s\left({ℝ}^3\right)}<\infty \right\}$,

其中

${\Vert f\Vert }_{{\stackrel{\dot{}}{B}}_{p,q}^s\begin{array}{c}\left({ℝ}^3\right)\end{array}}=\{\begin{array}{l}{\left({\displaystyle \underset{j\in \mathbb{Z}}{\sum }{2}^{sqj}{\Vert {\Delta }_{j}f\Vert }_{L^p\left({ℝ}^3\right)}^q}\right)}^{\frac{1}{q}},1\le q<\infty ,\\ \underset{j\in \mathbb{Z}}{\sup }{2}^{sj}{\Vert {\Delta }_{j}f\Vert }_{L^p\left({ℝ}^3\right)},q=\infty .\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\end{array}$

3. 非线性估计

对(IP)中的非线性项进行估计。设

$N\left(u,v\right)\left(t\right)={\displaystyle {\int }_{-\infty }^t{T}_{\Omega }\left(t-\tau \right)\mathbb{P}\left[\left(u\left(\tau \right)\cdot \nabla \right)u\left(\tau \right)\right]\text{d}\tau }$. (3.1)

Hieber, M.和Shibata, Y.在文献 [1] 中给出了热半群 $T_{\Omega }$ 的 $L^p-L^q$ 估计。在此基础上，Liu, J.和Sun, X. C. [9] 及Miao, C. X.，Yuan, B. Q.，Zhang, B.在文献 [10] 中给出了分数阶热半群 $T_{\Omega }$ 的 $L^p-L^q$ 估计。

引理3.1 [9] 设 $1<p_0\le \text{2}\le p_1<\infty$，且 $\beta =\left({\beta }_1,{\beta }_2,{\beta }_3\right)\in {\left(\mathbb{N}\cup \left\{0\right\}\right)}^3$，则存在正常数 $C=C\left(\alpha ,p_0,p_1,\beta \right)$，使得 $\forall t>0$，$\alpha >0$ 及 $f\in L^{p_0}\left({ℝ}^3\right)$，有

${\Vert {\partial }_x^{\beta }{T}_{\Omega }\left(t\right)f\Vert }_{L^{p_1}}\le C{t}^{-\frac{\beta }{2\alpha }-\frac{3}{2\alpha }\left(\frac{1}{p_0}-\frac{1}{p_1}\right)}{\Vert f\Vert }_{L^{p_0}}$. (3.2)

引理3.2 [10] 设 $1<p_0\le \text{2}\le p_1<\infty$，且 $s>0$，则存在正常数 $C=C\left(\alpha ,p_0,p_1,s\right)$，使得 $\forall t>0$，$\alpha >0$ 及 $f\in L^{p_0}\left({ℝ}^3\right)$，有

${\Vert {\left(-\Delta \right)}^{\frac{s}{2}}{T}_{\Omega }\left(t\right)f\Vert }_{L^{p_1}}\le C{t}^{-\frac{s}{2\alpha }-\frac{3}{2\alpha }\left(\frac{1}{p_0}-\frac{1}{p_1}\right)}{\Vert f\Vert }_{L^{p_0}}$. (3.3)

引理3.3设 $\frac{3}{4}<\alpha \le 1$，$2\le r<3$，$2\le q\le \frac{12}{5}$，并且 $\left(r,q\right)\ne \left(2,2\right)$。对

$\forall u,v\in BC\left(ℝ;L_{\sigma }^r\left({ℝ}^3\right)\right)\cap BC\left(ℝ;{\stackrel{\dot{}}{W}}^{1,q}\left({ℝ}^3\right)\right)$，$\forall C=C\left(q,r\right)>0$，使得 $N\left(u,v\right)\in BC\left(ℝ;L_{\sigma }^r\left({ℝ}^3\right)\right)\cap BC\left(ℝ;{\stackrel{\dot{}}{W}}^{1,q}\left({ℝ}^3\right)\right)$，且

$\underset{t\in ℝ}{\sup }{\Vert N\left(u,v\right)\left(t\right)\Vert }_{L^r}\le C\underset{t\in ℝ}{\sup }{\Vert u\left(t\right)\Vert }_{L^r}\left\{\underset{t\in ℝ}{\sup }{\Vert v\left(t\right)\Vert }_{L^r}+\underset{t\in ℝ}{\sup }{\Vert \nabla v\left(t\right)\Vert }_{L^q}\right\}$, (3.4)

$\underset{t\in ℝ}{\sup }{\Vert \nabla N\left(u,v\right)\left(t\right)\Vert }_{L^q}\le C\underset{t\in ℝ}{\sup }{\Vert \nabla v\left(t\right)\Vert }_{L^q}\left\{\underset{t\in ℝ}{\sup }{\Vert u\left(t\right)\Vert }_{L^r}+\underset{t\in ℝ}{\sup }{\Vert \nabla u\left(t\right)\Vert }_{L^q}\right\}$. (3.5)

证明：将 $N\left(u,v\right)\left(t\right)$ 分为两个部分

$N\left(u,v\right)\left(t\right)=N_1\left(u,v\right)\left(t\right)+N_2\left(u,v\right)\left(t\right)$,

其中

$N_1\left(u,v\right)\left(t\right)={\displaystyle {\int }_{-\infty }^{t-1}{T}_{\Omega }\left(t-\tau \right)\mathbb{P}\left[\left(u\left(\tau \right)\cdot \nabla \right)u\left(\tau \right)\right]\text{d}\tau }$,

$N_2\left(u,v\right)\left(t\right)={\displaystyle {\int }_{t-1}^t{T}_{\Omega }\left(t-\tau \right)\mathbb{P}\left[\left(u\left(\tau \right)\cdot \nabla \right)u\left(\tau \right)\right]\text{d}\tau }$.

首先证明(3.4)式。

$N_1\left(u,v\right)\left(t\right)$ 的估计：由于 $1\le \frac{r}{2}<\frac{3}{2}$，$3<{\left(\frac{r}{2}\right)}^{\prime }\le \infty$，$\frac{3}{2}<r^{\prime }\le 2$，$\frac{r}{2}$ 与 ${\left(\frac{r}{2}\right)}^{\prime }$，r与 $r^{\prime }$ 分别为共轭关系， $\varphi \in C_{0,\sigma }^{\infty }\left({ℝ}^3\right)$。由引理3.1及Hölder不等式可得

$\begin{array}{c}\left|\left(N_1\left(u,v\right)\left(t\right),\varphi \right)\right|=\left|{\displaystyle {\int }_{-\infty }^{t-1}\left(\left(u\left(\tau \right)\cdot \nabla \right)T_{\Omega }\left(t-\tau \right)\varphi ,v\left(\tau \right)\right)\text{d}\tau }\right|\\ \le {{\displaystyle {\int }_{-\infty }^{t-1}\Vert \nabla T_{\Omega }\left(t-\tau \right)\varphi \Vert }}_{L^{{\left(\frac{r}{2}\right)}^{\prime }}}{\Vert u\left(\tau \right)\otimes v\left(\tau \right)\Vert }_{L^{\frac{r}{2}}}\text{d}\tau \\ \le C{\displaystyle {\int }_{-\infty }^{t-1}{\left(t-\tau \right)}^{-\frac{1}{2\alpha }-\frac{3}{2\alpha }\left(\frac{1}{r^{\prime }}-\frac{1}{{\left(r/2\right)}^{\prime }}\right)}{\Vert \varphi \Vert }_{L^{r^{\prime }}}}{\Vert u\left(\tau \right)\Vert }_{L^r}{\Vert v\left(\tau \right)\Vert }_{L^r}\text{d}\tau \\ \le C\underset{t\in ℝ}{\sup }{\Vert u\left(t\right)\Vert }_{L^r}\underset{t\in ℝ}{\sup }{\Vert v\left(t\right)\Vert }_{L^r}{\Vert \varphi \Vert }_{L^{r^{\prime }}}{\displaystyle {\int }_1^{\infty }{\tau }^{-\frac{1}{2\alpha }-\frac{3}{2\alpha r}}\text{d}\tau }\\ \le C\underset{t\in ℝ}{\sup }{\Vert u\left(t\right)\Vert }_{L^r}\underset{t\in ℝ}{\sup }{\Vert v\left(t\right)\Vert }_{L^r}{\Vert \varphi \Vert }_{L^{r^{\prime }}}\end{array}$

由对偶性可得

${\Vert N_1\left(u,v\right)\left(t\right)\Vert }_{L^r}\le C\underset{t\in ℝ}{\sup }{\Vert u\left(t\right)\Vert }_{L^r}\underset{t\in ℝ}{\sup }{\Vert v\left(t\right)\Vert }_{L^r}$. (3.6)

$N_2\left(u,v\right)\left(t\right)$ 的估计：对 $\forall t\in ℝ$，$1<k<\frac{4}{3}$，

$\frac{1}{k}=\frac{1}{r}+\frac{1}{q}$, (3.7)

$\begin{array}{c}{\Vert N_2\left(u,v\right)\left(t\right)\Vert }_{L^r}\le {{\displaystyle {\int }_{t-1}^t\Vert T_{\Omega }\left(t-\tau \right)\mathbb{P}\left[\left(u\left(\tau \right)\cdot \nabla \right)v\left(\tau \right)\right]\Vert }}_{L^r}\text{d}\tau \\ \le C{{\displaystyle {\int }_{t-1}^t{\left(t-\tau \right)}^{-\frac{3}{2\alpha }\left(\frac{1}{k}-\frac{1}{r}\right)}\Vert \left(u\left(\tau \right)\cdot \nabla \right)v\left(\tau \right)\Vert }}_{L^k}\text{d}\tau \\ \le C{\displaystyle {\int }_{t-1}^t{\left(t-\tau \right)}^{-\frac{3}{2\alpha q}}}{\Vert u\left(\tau \right)\Vert }_{L^r}{\Vert \nabla v\left(\tau \right)\Vert }_{L^q}\text{d}\tau \\ \le C\underset{t\in ℝ}{\sup }{\Vert u\left(t\right)\Vert }_{L^r}\underset{t\in ℝ}{\sup }{\Vert \nabla v\left(t\right)\Vert }_{L^q}{\displaystyle {\int }_0^1{\tau }^{-\frac{3}{2\alpha q}}}\text{d}\tau \\ \le C\underset{t\in ℝ}{\sup }{\Vert u\left(t\right)\Vert }_{L^r}\underset{t\in ℝ}{\sup }{\Vert \nabla v\left(t\right)\Vert }_{L^q}\end{array}$ (3.8)

故由(3.6)式和(3.8)式可得

${\Vert N\left(u,v\right)\left(t\right)\Vert }_{L^r}\le C\underset{t\in ℝ}{\sup }{\Vert u\left(t\right)\Vert }_{L^r}\left\{\underset{t\in ℝ}{\sup }{\Vert v\left(t\right)\Vert }_{L^r}+\underset{t\in ℝ}{\sup }{\Vert \nabla v\left(t\right)\Vert }_{L^q}\right\}$.

其次，证明(3.5)式。

$\nabla N_1\left(u,v\right)\left(t\right)$ 的估计：由(3.7)式，引理3.1及Hölder不等式可得

$\begin{array}{c}{\Vert \nabla N_1\left(u,v\right)\left(t\right)\Vert }_{L^q}\le {{\displaystyle {\int }_{-\infty }^{t-1}\Vert \nabla T_{\Omega }\left(t-\tau \right)\mathbb{P}\left[\left(u\left(\tau \right)\cdot \nabla \right)v\left(\tau \right)\right]\Vert }}_{L^q}\text{d}\tau \\ \le C{{\displaystyle {\int }_{-\infty }^{t-1}{\left(t-\tau \right)}^{-\frac{1}{2\alpha }-\frac{3}{2\alpha }\left(\frac{1}{k}-\frac{1}{q}\right)}\Vert \left(u\left(\tau \right)\cdot \nabla \right)v\left(\tau \right)\Vert }}_{L^k}\text{d}\tau \\ \le C{{\displaystyle {\int }_{-\infty }^{t-1}{\left(t-\tau \right)}^{-\frac{1}{2\alpha }-\frac{3}{2\alpha r}}\Vert u\left(\tau \right)\Vert }}_{L^r}{\Vert \nabla v\left(\tau \right)\Vert }_{L^q}\text{d}\tau \\ \le C\underset{t\in ℝ}{\sup }{\Vert u\left(t\right)\Vert }_{L^r}\underset{t\in ℝ}{\sup }{\Vert \nabla v\left(t\right)\Vert }_{L^q}{\displaystyle {\int }_1^{\infty }{\tau }^{-\frac{1}{2\alpha }-\frac{3}{2\alpha r}}\text{d}\tau }\\ \le C\underset{t\in ℝ}{\sup }{\Vert u\left(t\right)\Vert }_{L^r}\underset{t\in ℝ}{\sup }{\Vert \nabla v\left(t\right)\Vert }_{L^q}\end{array}$ (3.9)

$\nabla N_2\left(u,v\right)\left(t\right)$ 的估计：引入参数 $q^{\ast }$ 和l， $2\le q\le \frac{12}{5}$，$\frac{3}{2}\le l\le 2$，$\frac{1}{q^{\ast }}=\frac{1}{q}-\frac{1}{3}$，$\frac{1}{l}=\frac{1}{q^{\ast }}+\frac{1}{q}$。

根据嵌入关系 ${\stackrel{\dot{}}{W}}^{1,q}\left({ℝ}^3\right)$ ↪ $L^{q^{\ast }}\left({ℝ}^3\right)$ 和引理3.1可得

$\begin{array}{c}{\Vert \nabla N_2\left(u,v\right)\left(t\right)\Vert }_{L^q}\le {{\displaystyle {\int }_{t-1}^t\Vert \nabla T_{\Omega }\left(t-\tau \right)\mathbb{P}\left[\left(u\left(\tau \right)\cdot \nabla \right)v\left(\tau \right)\right]\Vert }}_{L^q}\text{d}\tau \\ \le C{{\displaystyle {\int }_{t-1}^t{\left(t-\tau \right)}^{-\frac{1}{2\alpha }-\frac{3}{2\alpha }\left(\frac{1}{l}-\frac{1}{q}\right)}\Vert \left(u\left(\tau \right)\cdot \nabla \right)v\left(\tau \right)\Vert }}_{L^l}\text{d}\tau \\ \le C{{\displaystyle {\int }_{t-1}^t{\left(t-\tau \right)}^{-\frac{1}{2\alpha }-\frac{3}{2\alpha q^{\ast }}}{\Vert u\left(\tau \right)\Vert }_{L^{q^{\ast }}}\Vert \nabla v\left(\tau \right)\Vert }}_{L^q}\text{d}\tau \\ \le C{{\displaystyle {\int }_{t-1}^t{\left(t-\tau \right)}^{-\frac{3}{2\alpha q}}{\Vert \nabla u\left(\tau \right)\Vert }_{L^q}\Vert \nabla v\left(\tau \right)\Vert }}_{L^q}\text{d}\tau \\ \le C\underset{t\in ℝ}{\sup }{\Vert \nabla u\left(t\right)\Vert }_{L^q}\underset{t\in ℝ}{\sup }{\Vert \nabla v\left(t\right)\Vert }_{L^q}{\displaystyle {\int }_0^1{\tau }^{-\frac{3}{2\alpha q}}}\text{d}\tau \\ \le C\underset{t\in ℝ}{\sup }{\Vert \nabla u\left(t\right)\Vert }_{L^q}\underset{t\in ℝ}{\sup }{\Vert \nabla v\left(t\right)\Vert }_{L^q}\end{array}$ (3.10)

故由(3.9)式和(3.10)式可得

$\underset{t\in ℝ}{\sup }{\Vert \nabla N\left(u,v\right)\left(t\right)\Vert }_{L^r}\le C\underset{t\in ℝ}{\sup }{\Vert \nabla v\left(t\right)\Vert }_{L^q}\left\{\underset{t\in ℝ}{\sup }{\Vert u\left(t\right)\Vert }_{L^q}+\underset{t\in ℝ}{\sup }{\Vert \nabla u\left(t\right)\Vert }_{L^q}\right\}$.

4. 线性估计

对(IP)中的线性项进行估计。设

$F\left(t\right)={\displaystyle {\int }_{-\infty }^t{T}_{\Omega }\left(t-\tau \right)\mathbb{P}f\left(\tau \right)\text{d}\tau }$.

Kozono, H.，Ogawa, T.和Taniuchi, Y. [11] 在齐次Besov空间中对热半群 ${\text{e}}^{\Delta t}$ 做估计。Miao, C. X.，Yuan, B. Q.和Zhang, B. [10] 对分数阶的热半群 ${\text{e}}^{-t{\left(-\Delta \right)}^{\alpha }}$ 做了 $L^p-L^q$ 的估计。Sun, X. C.和Ding, Y. [12] 推广到齐次Besov空间。

引理4.1 [12] 设 $-\infty <s_0\le s_1<\infty$，$1\le p\le q\le \infty$，则存在正常数 $C=C\left(s_0,s_1\right)$，使得对 $\forall t>0$，$\alpha >0$ 及 $f\in {\stackrel{\dot{}}{B}}_{p,q}^{s_0}\left({ℝ}^3\right)$，有

${\Vert {\text{e}}^{-t{\left(-\Delta \right)}^{\alpha }}f\Vert }_{{\stackrel{\dot{}}{B}}_{p,q}^{s_1}}\le C{t}^{-\frac{1}{2\alpha }\left(s_1-s_0\right)}{\Vert f\Vert }_{{\stackrel{\dot{}}{B}}_{p,q}^{s_0}}$.

引理4.2设 $-\infty <s_0\le s_1<\infty$，$1\le p_0\le p_1\le \infty$，$1\le q\le \infty$，则存在正常数 $C=C(s_0,s_1,p_0,p_1)$，使得 $\forall t>0$，$\alpha >0$ 及 $f\in {\stackrel{\dot{}}{B}}_{p,q}^{s_0}\left({ℝ}^3\right)$，有

${\Vert {\text{e}}^{-t{\left(-\Delta \right)}^{\alpha }}f\Vert }_{{\stackrel{\dot{}}{B}}_{p_1,q}^{s_1}}\le C{t}^{-\frac{1}{2\alpha }\left(s_1-s_0\right)-\frac{3}{2\alpha }\left(\frac{1}{p_0}-\frac{1}{p_1}\right)}{\Vert f\Vert }_{{\stackrel{\dot{}}{B}}_{p_0,q}^{s_0}}$.

证明：设 $G_t\left(x\right)={\text{e}}^{-t{\left(-\Delta \right)}^{\alpha }}$，记 $G_t\left(x\right)\ast f={\text{e}}^{-t{\left(-\Delta \right)}^{\alpha }}f$。

令 ${G^{\prime }}_t\left(x\right)={\text{e}}^{-\frac{t}{2}{\left(-\Delta \right)}^{\alpha }}$，则 ${G^{\prime }}_t\left(x\right)\ast {G^{\prime }}_t\left(x\right)\ast f={\text{e}}^{-\frac{t}{2}{\left(-\Delta \right)}^{\alpha }}{\text{e}}^{-\frac{t}{2}{\left(-\Delta \right)}^{\alpha }}f$。

下证

$2^{s_{1}j}{\Vert {\phi }_j\ast {G^{\prime }}_t\ast {G^{\prime }}_t\ast a\Vert }_{L^{p_1}}\le C{t}^{-\frac{1}{2\alpha }\left(s_1-s_0\right)-\frac{3}{2\alpha }\left(\frac{1}{p_0}-\frac{1}{p_1}\right)}{2}^{s_{0}j}{\Vert {\phi }_j\ast a\Vert }_{L^{p_0}}$,

$\text{supp}\widehat{{\phi }_j}=\left\{\xi \in {ℝ}^n;2^{j-1}\le \left|\xi \right|\le 2^{j+1}\right\}$，存在正常数 $C=C\left(s_0,s_1\right)$，$j\in \mathbb{Z}$，$1<P\le \infty$。

有 ${\phi }_j\ast {G^{\prime }}_t\ast {G^{\prime }}_t\ast f=\widetilde{{\phi }_j}\ast {G^{\prime }}_t\ast {\phi }_j\ast {G^{\prime }}_t\ast a$，$\widetilde{{\phi }_j}={\phi }_{j-1}+{\phi }_j+{\phi }_{j+1}$。

$\begin{array}{c}{2}^{s_{1}j}{\Vert {\phi }_j\ast {G^{\prime }}_t\ast {G^{\prime }}_t\ast a\Vert }_{L^{p_1}}=2^{s^{\prime }j}{2}^{s_{0}j}{\Vert {\left(-\Delta \right)}^{-\frac{s^{\prime }}{2}}\widetilde{{\phi }_j}\ast {\left(-\Delta \right)}^{\frac{s^{\prime }}{2}}{G^{\prime }}_t\ast {\phi }_j\ast {G^{\prime }}_t\ast a\Vert }_{L^{p_1}}\\ \le 2^{s^{\prime }j}{\Vert {\left(-\Delta \right)}^{-\frac{s^{\prime }}{2}}\widetilde{{\phi }_j}\Vert }_{L^1}{\Vert {\left(-\Delta \right)}^{\frac{s^{\prime }}{2}}{G^{\prime }}_t\Vert }_{L^1}\left(2^{s_{0}j}{\Vert {\phi }_j\ast {G^{\prime }}_t\ast a\Vert }_{L^{p_1}}\right)\\ \le C{t}^{-\frac{1}{2\alpha }\left(s_1-s_0\right)}\left(2^{s_{0}j}{\Vert {\phi }_j\ast {G^{\prime }}_t\ast a\Vert }_{L^{p_1}}\right)\end{array}$

其中 $s^{\prime }=s_1-s_0$，${\Vert {\left(-\Delta \right)}^{-\frac{s^{\prime }}{2}}\widetilde{{\phi }_j}\Vert }_{L^1}=C{2}^{-s^{\prime }j}$，${\Vert {\left(-\Delta \right)}^{\frac{s^{\prime }}{2}}{G^{\prime }}_t\Vert }_{L^1}=C{t}^{-\frac{1}{2\alpha }\left(s_1-s_0\right)}$。

再根据引理3.1可得

${\Vert {\phi }_j\ast {G^{\prime }}_t\ast a\Vert }_{L^{p_1}}\le C{t}^{-\frac{3}{2\alpha }\left(\frac{1}{p_0}-\frac{1}{p_1}\right)}{\Vert {\phi }_j\ast a\Vert }_{L^{p_0}}$,

可得

$2^{s_{1}j}{\Vert {\phi }_j\ast {G^{\prime }}_t\ast {G^{\prime }}_t\ast a\Vert }_{L^{p_1}}\le C{t}^{-\frac{1}{2\alpha }\left(s_1-s_0\right)-\frac{3}{2\alpha }\left(\frac{1}{p_0}-\frac{1}{p_1}\right)}{2}^{s_{0}j}{\Vert {\phi }_j\ast a\Vert }_{L^{p_0}}$.

引理4.3设 $\frac{3}{4}\le \alpha \le 1$，$2\le r<3$，$2\le q<3$，并且 $\left(r,q\right)\ne \left(2,2\right)$。指标s, p, l满足

$s>0$,$1<p\le 2$,$\frac{s}{3}+\frac{1}{p}>\frac{1}{r}+\frac{2}{3}$,$\frac{1}{2}\le \frac{1}{l}<\frac{1}{q}+\frac{1}{6}$. (4.1)

对 $\forall f\in BC\left(ℝ;{\stackrel{\dot{}}{B}}_{p,2}^{-s}\left({ℝ}^3\right)\right)\cap BC\left(ℝ;L^l\left({ℝ}^3\right)\right)$，使得 $F\in BC\left(ℝ;L_{\sigma }^r\left({ℝ}^3\right)\right)\cap BC\left(ℝ;{\stackrel{\dot{}}{W}}^{1,q}\left({ℝ}^3\right)\right)$，并存在正常数 $C=C\left(r,q,s,p,l\right)$ 有

$\underset{t\in ℝ}{\sup }{\Vert F\left(t\right)\Vert }_{L^r}\le C\left(\underset{t\in ℝ}{\sup }{\Vert f\left(t\right)\Vert }_{{\stackrel{\dot{}}{B}}_{p,2}^{-s}}+\underset{t\in ℝ}{\sup }{\Vert f\left(t\right)\Vert }_{L^l}\right)$, (4.2)

$\underset{t\in ℝ}{\sup }{\Vert \nabla F\left(t\right)\Vert }_{L^q}\le C\left(\underset{t\in ℝ}{\sup }{\Vert f\left(t\right)\Vert }_{{\stackrel{\dot{}}{B}}_{p,2}^{-s}}+\underset{t\in ℝ}{\sup }{\Vert f\left(t\right)\Vert }_{L^l}\right)$. (4.3)

证明：将 $F\left(t\right)$ 分为两个部分

$F\left(t\right)=F_1\left(t\right)+F_2\left(t\right)$,

其中

$F_1\left(t\right)={\displaystyle {\int }_{-\infty }^{t-1}{T}_{\Omega }\left(t-\tau \right)\mathbb{P}f\left(\tau \right)\text{d}\tau }$,

$F_2\left(t\right)={\displaystyle {\int }_{t-1}^t{T}_{\Omega }\left(t-\tau \right)\mathbb{P}f\left(\tau \right)\text{d}\tau }$.

首先估计(4.2)式。

$F_1\left(t\right)$ 的估计： $1<p\le 2$，$2\le r<3$。由引理4.3，Plancherel定理及参考文献 [13] 连续嵌入关系 $L^p\left({ℝ}^3\right)$ ↪ ${\stackrel{\dot{}}{B}}_{p,2}^0\left({ℝ}^3\right)$ $\left(1<p\le 2\right)$ 和 ${\stackrel{\dot{}}{B}}_{p,2}^0\left({ℝ}^3\right)$ ↪ $L^p\left({ℝ}^3\right)$ $\left(2\le p<\infty \right)$。

$\begin{array}{c}{\Vert F_1\left(t\right)\Vert }_{L^r}\le {{\displaystyle {\int }_{-\infty }^{t-1}\left(t-\tau \right)}}^{-\frac{3}{2\alpha }\left(\frac{1}{2}-\frac{1}{r}\right)}{\Vert T_{\Omega }\left(\frac{t-\tau }{2}\Delta \right)\mathbb{P}f\left(\tau \right)\Vert }_{L^2}\text{d}\tau \\ \le C{{\displaystyle {\int }_{-\infty }^{t-1}\left(t-\tau \right)}}^{-\frac{3}{2\alpha }\left(\frac{1}{2}-\frac{1}{r}\right)}{\Vert {\text{e}}^{\frac{t-\tau }{2}\Delta }\mathbb{P}f\left(\tau \right)\Vert }_{L^2}\text{d}\tau \\ \le C{{\displaystyle {\int }_{-\infty }^{t-1}\left(t-\tau \right)}}^{-\frac{3}{2\alpha }\left(\frac{1}{2}-\frac{1}{r}\right)-\frac{s}{2\alpha }}{\Vert {\text{e}}^{\frac{t-\tau }{4}\Delta }\mathbb{P}f\left(\tau \right)\Vert }_{{\stackrel{\dot{}}{B}}_{2,2}^{-s}}\text{d}\tau \\ \le C{{\displaystyle {\int }_{-\infty }^{t-1}\left(t-\tau \right)}}^{-\frac{3}{2\alpha }\left(\frac{1}{2}-\frac{1}{r}\right)-\frac{s}{2\alpha }-\frac{3}{2\alpha }\left(\frac{1}{p}-\frac{1}{2}\right)}{\Vert f\left(\tau \right)\Vert }_{{\stackrel{\dot{}}{B}}_{p,2}^{-s}}\text{d}\tau \\ \le C\underset{t\in ℝ}{\sup }{\Vert f\left(t\right)\Vert }_{{\stackrel{\dot{}}{B}}_{p,2}^{-s}}{\displaystyle {\int }_1^{\infty }{\tau }^{-\frac{3}{2\alpha }\left(\frac{s}{3}+\frac{1}{p}-\frac{1}{r}\right)}}\text{d}\tau \\ \le C\underset{t\in ℝ}{\sup }{\Vert f\left(t\right)\Vert }_{{\stackrel{\dot{}}{B}}_{p,2}^{-s}}\end{array}$ (4.4)

$F_2\left(t\right)$ 的估计： $\frac{3}{2}<l\le 2$，$2\le r<3$。由引理3.1可得

$\begin{array}{c}{\Vert F_2\left(t\right)\Vert }_{L^r}\le C{{\displaystyle {\int }_{t-1}^t\left(t-\tau \right)}}^{-\frac{3}{2\alpha }\left(\frac{1}{l}-\frac{1}{r}\right)}{\Vert f\left(\tau \right)\Vert }_{L^l}\text{d}\tau \\ \le C\underset{t\in ℝ}{\sup }{\Vert f\left(t\right)\Vert }_{L^l}{{\displaystyle {\int }_0^1\tau }}^{-\frac{3}{2\alpha }\left(\frac{1}{l}-\frac{1}{r}\right)}\text{d}\tau \\ \le C\underset{t\in ℝ}{\sup }{\Vert f\left(t\right)\Vert }_{L^l}\end{array}$ (4.5)

故由(4.4)式和(4.5)式可证得

$\underset{t\in ℝ}{\sup }{\Vert F\left(t\right)\Vert }_{L^r}\le C\left(\underset{t\in ℝ}{\sup }{\Vert f\left(t\right)\Vert }_{{\stackrel{\dot{}}{B}}_{p,2}^{-s}}+\underset{t\in ℝ}{\sup }{\Vert f\left(t\right)\Vert }_{L^l}\right)$.

其次，证明(4.3)式。

$\nabla F_1\left(t\right)$ 的估计： $1<p\le 2$，$2\le q<3$。由引理3.1，引理4.2，Plancherel定理和(4.1)式可得

$\begin{array}{c}{\Vert \nabla F_1\left(t\right)\Vert }_{L^q}\le C{{\displaystyle {\int }_{-\infty }^{t-1}\left(t-\tau \right)}}^{-\frac{3}{2\alpha }\left(\frac{1}{2}-\frac{1}{q}\right)-\frac{1}{2\alpha }}{\Vert T_{\Omega }\left(\frac{t-\tau }{2}\right)\mathbb{P}f\left(\tau \right)\Vert }_{L^2}\text{d}\tau \\ \le C{{\displaystyle {\int }_{-\infty }^{t-1}\left(t-\tau \right)}}^{-\frac{3}{2\alpha }\left(\frac{1}{2}-\frac{1}{q}\right)-\frac{1}{2\alpha }}{\Vert {\text{e}}^{\frac{t-\tau }{2}\Delta }\mathbb{P}f\left(\tau \right)\Vert }_{L^2}\text{d}\tau \\ \le C{{\displaystyle {\int }_{-\infty }^{t-1}\left(t-\tau \right)}}^{-\frac{3}{2\alpha }\left(\frac{1}{2}-\frac{1}{q}\right)-\frac{1}{2\alpha }-\frac{s}{2\alpha }}{\Vert {\text{e}}^{\frac{t-\tau }{4}\Delta }\mathbb{P}f\left(\tau \right)\Vert }_{{\stackrel{\dot{}}{B}}_{2,2}^{-s}}\text{d}\tau \\ \le C{{\displaystyle {\int }_{-\infty }^{t-1}\left(t-\tau \right)}}^{-\frac{3}{2\alpha }\left(\frac{1}{2}-\frac{1}{q}\right)-\frac{1}{2\alpha }-\frac{s}{2\alpha }-\frac{3}{2\alpha }\left(\frac{1}{p}-\frac{1}{2}\right)}{\Vert f\left(\tau \right)\Vert }_{{\stackrel{\dot{}}{B}}_{p,2}^{-s}}\text{d}\tau \\ \le C\underset{t\in ℝ}{\sup }{\Vert f\left(t\right)\Vert }_{{\stackrel{\dot{}}{B}}_{p,2}^{-s}}{\displaystyle {\int }_1^{\infty }{\tau }^{-\frac{3}{2\alpha }\left(\frac{s}{3}+\frac{1}{p}+\frac{1}{3}-\frac{1}{q}\right)}}\text{d}\tau \\ \le C\underset{t\in ℝ}{\sup }{\Vert f\left(t\right)\Vert }_{{\stackrel{\dot{}}{B}}_{p,2}^{-s}}\end{array}$ (4.6)