Time periodic solutions to the 2D quasi-geostrophic equation with the supercritical dissipation

Abstract

We consider the 2D dissipative quasi-geostrophic equation with the time periodic external force and prove the existence of a unique time periodic solution in the case of the supercritical dissipation. In this case, the smoothing effect of the semigroup generated by the dissipation term is too weak to control the nonlinearity in the Duhamel term of the corresponding integral equation. In this paper, we give a new approach which does not depend on the contraction mapping principle for the integral equation.

1 Introduction

We consider the 2D dissipative quasi-geostrophic equation with the time periodic external force:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t\theta +(-\Delta )^{\frac{\alpha }{2}}\theta +u \cdot \nabla \theta =F, &{}\qquad t\in {\mathbb {R}}, x\in {\mathbb {R}}^2,\\ u={\mathcal {R}}^{\perp }\theta =(-{\mathcal {R}}_2\theta ,{\mathcal {R}}_1\theta ), &{}\qquad t\in {\mathbb {R}}, x\in {\mathbb {R}}^2, \end{array}\right. } \end{aligned}$$

(1.1)

where $\theta =\theta (t,x)$ and $u=(u_1(t,x),u_2(t,x))$ represent the unknown potential temperature of the fluid and the unknown velocity field of the fluid, respectively. The given external force $F=F(t,x)$ is T-time periodic, that is F satisfies $F(t+T)=F(t)$ $(t\in {\mathbb {R}})$ for some $T>0$. The two operators $(-\Delta )^{\frac{\alpha }{2}}$ $(0<\alpha \leqslant 2)$ and ${\mathcal {R}}_k$ $(k=1,2)$ denote the nonlocal differential operators so-called the fractional Laplacian and the Riesz transforms on ${\mathbb {R}}^2$, respectively, and they are defined by

$$\begin{aligned} (-\Delta )^{\frac{\alpha }{2}}f ={\mathscr {F}}^{-1}\left[ |\xi |^{\alpha } {\widehat{f}}(\xi ) \right] , \qquad {\mathcal {R}}_kf =\partial _{x_k}(-\Delta )^{-\frac{1}{2}}f ={\mathscr {F}}^{-1}\left[ \frac{i\xi _k}{|\xi |} {\widehat{f}}(\xi ) \right] . \end{aligned}$$

In this paper, we prove the existence of a unique T-time periodic solution of (1.1) with the supercritical dissipation if the given T-time periodic external force is sufficiently small.

Before we state the main result precisely, we recall some known results for the initial value problem of the 2D dissipative quasi-geostrophic equation with the case $F=0$:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t\theta +(-\Delta )^{\frac{\alpha }{2}}\theta +u\cdot \nabla \theta =0, \qquad &{} t>0,x\in {\mathbb {R}}^2,\\ u={\mathcal {R}}^{\perp }\theta =(-{\mathcal {R}}_2\theta ,{\mathcal {R}}_1\theta ), \qquad &{} t\geqslant 0,x\in {\mathbb {R}}^2,\\ \theta (0,x)=\theta _0(x), \qquad &{} x\in {\mathbb {R}}^2. \end{array}\right. } \end{aligned}$$

(1.2)

Based on the scaling transform and the $L^{\infty }({\mathbb {R}}^2)$-conservation, the dissipative quasi-geostrophic equation is divided into the subcritical case $1<\alpha \leqslant 2$, critical case $\alpha =1$ and supercritical case $0<\alpha <1$. In the subcritical case, Constantin-Wu [6] proved the existence of a weak solution and decay estimates with respect to $L^2$ norm for the initial data $\theta _0\in L^2({\mathbb {R}}^2)$. Wu [14] proved the global well-posedness for small data in the scaling subcritical setting $\theta _0\in L^p({\mathbb {R}}^2)$ ($p>2/(\alpha -1)$) via the contraction mapping principle for the corresponding integral equation. In the critical case, the order of the spatial derivative in the dissipation term coincides with that in the nonlinear term. Zhang [16] used this property and proved the existence of the global in time mild solution in the scaling critical Besov space ${\dot{B}}_{p,1}^{\frac{2}{p}}({\mathbb {R}}^2)$ ($1\leqslant p \leqslant \infty$). Global well-posedness in the Triebel–Lizorkin spaces $F_{p,q}^s({\mathbb {R}}^2)$ ($s>2/p$, $1<p,q<\infty$) is proved by Chen-Zhang [5]. In the supercritical case, the order of the spatial derivative in the dissipation term is less than that in the nonlinear term. Therefore, the smoothing effect of the fractional heat kernel $e^{-t(-\Delta )^{\frac{\alpha }{2}}}$ is too weak to control the spatial derivative in the nonlinear term. This implies that it seems to be impossible to construct a solution of (1.2). It is able to overcome this and the local well-posedness for large data and the global well-posedness for small data in the scaling critical Sobolev $H^{2-\alpha }({\mathbb {R}}^2)$ by Miura [12] and Besov spaces ${B}_{p,q}^{1+\frac{2}{p}-\alpha }({\mathbb {R}}^2)$ ($2\leqslant p<\infty$, $1\leqslant q<\infty$) by Chae-Lee [3] and Chen-Miao-Zhang [4]. Their method is based on the energy estimates for the iteration of the transport-diffusion-type equation, and they control the nonlinear term by the divergence free condition $\nabla \cdot u=0$ and the commutator estimates.

On the other hand, despite the large number of previous studies on the well-posedness of the initial value problem (1.2), the study on the existence of time periodic solutions to the 2D quasi-geostrophic equation is hardly known.

In this manuscript, we consider the supercritical case and prove the existence of a unique time periodic solution to (1.1) in the scaling critical Besov space if the given time periodic external force is sufficiently small. More precisely, our main result of this paper reads as follows:

Theorem 1.1

Let $T>0$ and $2/3<\alpha <1$. Let exponents p, q and r satisfy

$$\begin{aligned} \frac{2}{2\alpha -1}<r\leqslant p<\frac{4}{\alpha }, \qquad 1\leqslant q<\infty . \end{aligned}$$

(1.3)

Then, there exist positive constants $\delta =\delta (\alpha ,p,q,r,T)$ and $K=K(\alpha ,p,q,r,T)$ such that if the given T-time periodic external force $F\in BC({\mathbb {R}}; {\dot{B}}_{r,\infty }^0({\mathbb {R}}^2))$ satisfies

$$\begin{aligned} \sup _{t\in {\mathbb {R}}} \Vert F(t) \Vert _{{\dot{B}}_{r,\infty }^0}\leqslant \delta , \end{aligned}$$

then there exist a unique T-time periodic solution $\theta$ to (1.1) satisfying

$$\begin{aligned} \theta \in BC({\mathbb {R}};B_{p,q}^{1+\frac{2}{p}-\alpha }({\mathbb {R}}^2)), \qquad \Vert \theta \Vert _{{\widetilde{L}}^{\infty }({\mathbb {R}};{\dot{B}}_{p,q}^{1+\frac{2}{p}-\alpha })}\leqslant K. \end{aligned}$$

(1.4)

Remark 1.2

• (1) If $\theta$ and F satisfy (1.1), then

  $$\begin{aligned} \theta _{\lambda }(t,x)=\lambda ^{\alpha -1}\theta (\lambda ^{\alpha }t,\lambda x),\quad F_{\lambda }(t,x)=\lambda ^{2\alpha -1}F(\lambda ^{\alpha }t,\lambda x) \end{aligned}$$

  also satisfy (1.1) for all $\lambda >0$. Since it holds

  $$\begin{aligned} \begin{aligned} \sup _{t\in {\mathbb {R}}}\Vert \theta _{\lambda }(t)\Vert _{{\dot{B}}_{p,q}^{1+\frac{2}{p}-\alpha }}&=\sup _{t\in {\mathbb {R}}}\Vert \theta (t)\Vert _{{\dot{B}}_{p,q}^{1+\frac{2}{p}-\alpha }},\\ \sup _{t\in {\mathbb {R}}}\Vert F_{\lambda }(t)\Vert _{{\dot{B}}_{2/(2\alpha -1),\infty }^0}&=\sup _{t\in {\mathbb {R}}}\Vert F(t)\Vert _{{\dot{B}}_{2/(2\alpha -1),\infty }^0} \end{aligned} \end{aligned}$$

  for all dyadic numbers $\lambda >0$, the function spaces $BC({\mathbb {R}};B_{p,q}^{1+\frac{2}{p}-\alpha }({\mathbb {R}}^2))$ and $BC({\mathbb {R}};{\dot{B}}_{r,\infty }^0({\mathbb {R}}^2))$ in Theorem 1.1 are scaling critical and subcritical setting, respectively.

• (2) The assumption $2/3<\alpha$ in Theorem 1.1 ensures the existence of p and r satisfying (1.3).

Our approach can be applied to the time periodic problem on the half time line $(0,\infty )$:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t\theta +(-\Delta )^{\frac{\alpha }{2}}\theta +u \cdot \nabla \theta =F, &{}\qquad t>0, x\in {\mathbb {R}}^2,\\ u={\mathcal {R}}^{\perp }\theta =(-{\mathcal {R}}_2\theta ,{\mathcal {R}}_1\theta ), &{}\qquad t\geqslant 0, x\in {\mathbb {R}}^2,\\ \theta (0,x)=\theta _0(x), &{}\qquad x\in {\mathbb {R}}^2, \end{array}\right. } \end{aligned}$$

(1.5)

Here, we note that F is a given T-time periodic external force and the initial data $\theta _0$ is unknown. In this setting, we prove the existence of a unique suitable initial data $\theta _0$ such that (1.5) possesses a unique T-time periodic solution $\theta$. More precisely, the following theorem is obtained:

Theorem 1.3

Let $T>0$ and $2/3<\alpha <1$. Let exponents p, q and r satisfy

$$\begin{aligned} \frac{2}{2\alpha -1}<r\leqslant p<\frac{4}{\alpha }, \qquad 1\leqslant q<\infty . \end{aligned}$$

(1.6)

Then, there exist positive constants $\delta =\delta (\alpha ,p,q,r,T)$ and $K=K(\alpha ,p,q,r,T)$ such that if the given T-time periodic external force $F\in BC((0,\infty ); {\dot{B}}_{r,\infty }^0({\mathbb {R}}^2))$ satisfies

$$\begin{aligned} \sup _{t>0} \Vert F(t) \Vert _{{\dot{B}}_{r,\infty }^0}\leqslant \delta , \end{aligned}$$

then there exist a unique initial data $\theta _0 \in B_{p,q}^{1+\frac{2}{p}-\alpha }({\mathbb {R}}^2)$ and a unique T-time periodic solution $\theta$ to (1.5) satisfying

$$\begin{aligned} \theta \in BC([0,\infty );B_{p,q}^{1+\frac{2}{p}-\alpha }({\mathbb {R}}^2)), \qquad \Vert \theta \Vert _{{\widetilde{L}}^{\infty }(0,\infty ;{\dot{B}}_{p,q}^{1+\frac{2}{p}-\alpha })}\leqslant K. \end{aligned}$$

(1.7)

In the study of the Navier–Stokes equation, the existence of time periodic solutions is often proved by applying the contraction mapping principle to the corresponding integral equation. (It was in [10] that first used this idea.) However, in the case of our problem, the supercritical dissipation prevents us from using this scheme. Indeed, when we apply the idea of [10] to (1.1), we meet the difficulty that the smoothing effect of the fractional heat kernel $e^{-t(-\Delta )^{\frac{\alpha }{2}}}$ is too weak to control the first-order spatial derivative of the nonlinear term and it is pretty difficult to find a Banach space X satisfying

$$\begin{aligned} \sup _{t\in {\mathbb {R}}} \left\| \int _{-\infty }^t e^{-(t-\tau )(-\Delta )^{\frac{\alpha }{2}}} ({\mathcal {R}}^{\perp }\theta (\tau ) \cdot \nabla \theta (\tau )) d\tau \right\| _X \leqslant C\left( \sup _{t\in {\mathbb {R}}}\Vert \theta (t)\Vert _X\right) ^2. \end{aligned}$$

As another approach, let us consider the successive approximation defined by the transport diffusion-type equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \theta ^{(n+1)} +(-\Delta )^{\frac{\alpha }{2}} \theta ^{(n+1)} +u^{(n)} \cdot \nabla \theta ^{(n+1)} =F, &{}\qquad t\in {\mathbb {R}},x\in {\mathbb {R}}^2,\\ u^{(n)}={\mathcal {R}}^{\perp } \theta ^{(n)}, &{}\qquad t\in {\mathbb {R}}, x\in {\mathbb {R}}^2.\\ \end{array}\right. } \end{aligned}$$

(1.8)

Then, we can obtain the a priori estimates for the approximation solutions by the energy method, with some commutator estimates, which is the similar method to the analysis of the initial value problem. However, it seems to be difficult to construct a time periodic linear solution $\theta ^{(n+1)}$ of (1.8) when $\theta ^{(n)}$ is determined. Therefore, we are not able to proceed in parallel with the energy method of the initial value problem for the supercritical case.

We now introduce an idea to overcome these difficulties and get a time periodic solution. Once Theorem 1.3 is proved, then we can get the time periodic solution on ${\mathbb {R}}$ by extending the time periodic solution on $[0,\infty )$ to the other half time line $(-\infty ,0)$ periodically. Thus, in the following, we only consider the time periodic problem (1.5) and Theorem 1.3.

Outline of our idea is to construct a specific initial data $\theta _0$ and the corresponding local in time solution $\theta$ on [0, T] satisfying $\theta (T)=\theta (0)=\theta _0$. Then, we get a T-time periodic solution by extending $\theta$ periodically in time. In order to construct such a solution on [0, T], we define a new approximation sequences.

Let us explain the new approximation system more precisely. If a solution $\theta$ to (1.5) satisfies $\theta (T)=\theta (0)=\theta _0$, then we have

$$\begin{aligned} (1-e^{-T(-\Delta )^{\frac{\alpha }{2}}})\theta _0 =\theta (T)-e^{-T(-\Delta )^{\frac{\alpha }{2}}}\theta _0. \end{aligned}$$

(1.9)

Therefore, from this observation, we approximate (1.5) together with (1.9) successively and define the iteration scheme $\{\theta _0^{(n)}\}_{n=0}^{\infty }$ and $\{\theta ^{(n)}\}_{n=0}^{\infty }$ as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \theta ^{(n+1)} +(-\Delta )^{\frac{\alpha }{2}} \theta ^{(n+1)} +u^{(n)} \cdot \nabla \theta ^{(n+1)} =F,\\ u^{(n)}={\mathcal {R}}^{\perp } \theta ^{(n)},\\ \theta ^{(n+1)}(0,x)=\theta _0^{(n+1)};\\ (1-e^{-T(-\Delta )^{\frac{\alpha }{2}}})\theta _0^{(n+1)} =\theta ^{(n)}(T)-e^{-T(-\Delta )^{\frac{\alpha }{2}}}\theta ^{(n)}(0). \end{array}\right. } \end{aligned}$$

(1.10)

((1.10) is the formal definition. See Sect. 3 for the accurate definition of the approximation sequences.)

Finally, we remark the estimates for the approximation sequences. It is well known that the uniform boundedness is a key property of the convergence of sequences. The uniform boundedness of $\{\theta ^{(n)}\}_{n=0}^{\infty }$ can be obtained by the classical energy estimates (see, for instance, [1]). However, the new idea is required for the uniform boundedness of $\{\theta _0^{(n)}\}_{n=0}^{\infty }$ since $\{\theta _0^{(n)}\}_{n=0}^{\infty }$ is defined by solving the following time-independent equation:

$$\begin{aligned} (1-e^{-T(-\Delta )^{\frac{\alpha }{2}}})u=f. \end{aligned}$$

To obtain estimates for $\{\theta _0^{(n)}\}_{n=0}^{\infty }$, we establish Lemmas 2.6 and 2.7 in Sect. 2 which are inspired by the idea of [8, Theorem 2.4].

This paper is organized as follows. In Sect. 2, we summarize some notations and introduce lemmas which are used in the proof of the main results. In Sect. 3, we prove Theorem 1.1.

Throughout this paper, we denote by C the constant, which may differ in each line. In particular, $C=C(a_1,...,a_n)$ means that C depends only on $a_1,...,a_n$. We define a commutator for two operators A and B as $[A,B]=AB-BA$.

2 Preliminaries

Let ${\mathscr {S}}({\mathbb {R}}^2)$ be the set of all Schwartz functions on ${\mathbb {R}}^2$, and let ${\mathscr {S}}'({\mathbb {R}}^2)$ be the set of all tempered distributions on ${\mathbb {R}}^2$. For $f\in {\mathscr {S}}({\mathbb {R}}^2)$, we define the Fourier transform and the inverse Fourier transform of f by

$$\begin{aligned} \begin{array}{cc} {\mathscr {F}}[f](\xi )={\widehat{f}}(\xi ):=\displaystyle \int _{{\mathbb {R}}^2}e^{-i\xi \cdot x}f(x)\ dx,&{\mathscr {F}}^{-1}[f](x):=\dfrac{1}{(2\pi )^2}\displaystyle \int _{{\mathbb {R}}^2}e^{i\xi \cdot x}f(\xi )\ d\xi , \end{array} \end{aligned}$$

respectively. $\{\varphi _j\}_{j\in {\mathbb {Z}}}$ is called the homogeneous Littlewood–Paley decomposition if $\varphi _0\in {\mathscr {S}}({\mathbb {R}}^2)$ satisfy $\mathrm{supp}\ \widehat{\varphi _0} \subset \{2^{-1}\leqslant |\xi |\leqslant 2\}$, $0\leqslant \widehat{\varphi _0}\leqslant 1$ and

$$\begin{aligned} \sum _{j\in {\mathbb {Z}}}\widehat{\varphi _j}(\xi )=1, \quad \quad \xi \in {\mathbb {R}}^2\setminus \{0\}, \end{aligned}$$

where $\widehat{\varphi _j}(\xi )=\widehat{\varphi _0}(2^{-j}\xi )$. Let us write

$$\begin{aligned} \Delta _jf:= \varphi _j*f \end{aligned}$$

for $j\in {\mathbb {Z}}$ and $f\in {\mathscr {S}}'({\mathbb {R}}^2)$. Using the homogeneous Littlewood–Paley decomposition, we define the Besov spaces. For $1\leqslant p,q\leqslant \infty$ and $s\in {\mathbb {R}}$, the homogeneous Besov space ${\dot{B}}^s_{p,q}({\mathbb {R}}^2)$ is defined by

$$\begin{aligned} \begin{aligned} {\dot{B}}^s_{p,q}({\mathbb {R}}^2)&:=\left\{ f\in {\mathscr {S}}_0'({\mathbb {R}}^2)\ ;\ \Vert f\Vert _{{\dot{B}}^s_{p,q}}<\infty .\right\} ,\\ \Vert f\Vert _{{\dot{B}}^s_{p,q}}&:=\left\| \left\{ 2^{js}\Vert \Delta _j f\Vert _{L^p}\right\} _{j\in {\mathbb {Z}}} \right\| _{l^q({\mathbb {Z}})}, \end{aligned} \end{aligned}$$

where ${\mathscr {S}}_0'({\mathbb {R}}^2)$ is the dual space of

$$\begin{aligned} {\mathscr {S}}_0({\mathbb {R}}^2):=\left\{ f\in {\mathscr {S}}({\mathbb {R}}^2)\ ;\ \int _{{\mathbb {R}}^2}x^{\gamma }f(x) dx=0\mathrm{\ for\ all\ }\gamma \in ({\mathbb {N}}\cup \{0\})^2. \right\} . \end{aligned}$$

Note that ${\dot{B}}_{p,q}^s({\mathbb {R}}^2)$ is a Banach space with respect to the norm $\Vert \cdot \Vert _{{\dot{B}}_{p,q}^s}$. It is well known that if $1\leqslant p,q\leqslant \infty$ and $s<2/p$, then we can identify ${\dot{B}}_{p,q}^s({\mathbb {R}}^2)$ as:

$$\begin{aligned} \left\{ f\in {\mathscr {S}}'({\mathbb {R}}^2)\ ;\ f=\sum _{j\in {\mathbb {Z}}}\Delta _jf\ \mathrm{in}\ {\mathscr {S}}'({\mathbb {R}}^2)\ \mathrm{and}\ \Vert f\Vert _{{\dot{B}}_{p,q}^s}<\infty .\right\} . \end{aligned}$$

(See for the detail in [11] and [13].) For $s>0$ and $1\leqslant p,q\leqslant \infty$, the inhomogeneous Besov space $B_{p,q}^s({\mathbb {R}}^2)$ is defined by

$$\begin{aligned} \begin{aligned} B_{p,q}^s({\mathbb {R}}^2)&:={\dot{B}}_{p,q}^s({\mathbb {R}}^2)\cap L^p({\mathbb {R}}^2),\\ \Vert f\Vert _{B_{p,q}^s}&:=\Vert f\Vert _{{\dot{B}}_{p,q}^s}+\Vert f\Vert _{L^p}. \end{aligned} \end{aligned}$$

In this paper, we also use the space-time Besov spaces defined by

$$\begin{aligned} \begin{aligned} {\widetilde{L}}^r(0,T;{\dot{B}}_{p,q}^s({\mathbb {R}}^2))&:=\left\{ F:(0,T)\rightarrow {\mathscr {S}}_0'({\mathbb {R}}^2)\ ;\ \Vert F\Vert _{{\widetilde{L}}^r(0,T;{\dot{B}}_{p,q}^s)}<\infty .\right\} ,\\ \Vert F\Vert _{{\widetilde{L}}^r(0,T;{\dot{B}}_{p,q}^s)}&:=\left\| \left\{ 2^{sj}\Vert \Delta _jF\Vert _{L^r(0,T;L^p)}\right\} _{j\in {\mathbb {Z}}}\right\| _{l^{q}({\mathbb {Z}})} \end{aligned} \end{aligned}$$

for $1\leqslant p,q,r\leqslant \infty ,s\in {\mathbb {R}}$ and $0<T\leqslant \infty$.

Next, we introduce the semigroup generated by the fractional Laplacian $(-\Delta )^{\frac{\alpha }{2}}$. It is given explicitly by using the Fourier transform:

$$\begin{aligned} e^{-t(-\Delta )^{\frac{\alpha }{2}}}f ={\mathscr {F}}^{-1}\left[ e^{-t|\xi |^{\alpha }} {\widehat{f}}(\xi ) \right] . \end{aligned}$$

Then, this semigroup possesses the following properties:

Lemma 2.1

Let $\alpha >0$ and $1\leqslant p,q \leqslant \infty$. Then, the followings hold:

• (1) There exists a positive constant $C=C(\alpha )$ such that

  $$\begin{aligned} \left\| e^{-t(-\Delta )^{\frac{\alpha }{2}}} \Delta _j f \right\| _{L^p} \leqslant C e^{-C^{-1}2^{\alpha j}t} \Vert \Delta _jf\Vert _{L^p} \end{aligned}$$

  holds for all $t>0$, $j\in {\mathbb {Z}}$ and $f\in {\mathscr {S}}'_0({\mathbb {R}}^2)$ with $\Delta _j f\in L^p({\mathbb {R}}^2)$.

• (2) Let $s_1,s_2\in {\mathbb {R}}$ satisfy $s_1\leqslant s_2$. Then, there exists a positive constant $C=C(\alpha ,s_1,s_2)$ such that

  $$\begin{aligned} 2^{s_2 j}\left\| e^{-t(-\Delta )^{\frac{\alpha }{2}}}\Delta _j f\right\| _{L^p} \leqslant C t^{-\frac{s_2-s_1}{\alpha }}2^{s_1 j}\Vert \Delta _j f\Vert _{L^p} \end{aligned}$$

  holds for all $t>0$, $j\in {\mathbb {Z}}$ and $f\in {\mathscr {S}}'_0({\mathbb {R}}^2)$ with $\Delta _j f\in L^p({\mathbb {R}}^2)$. In particular, it holds

  $$\begin{aligned} \left\| e^{-t(-\Delta )^{\frac{\alpha }{2}}}f\right\| _{{\dot{B}}_{p,q}^{s_2}} \leqslant C t^{-\frac{s_2-s_1}{\alpha }}\Vert f\Vert _{{\dot{B}}_{p,q}^{s_1}} \end{aligned}$$

  for all $t>0$ and $f\in {\dot{B}}_{p,q}^{s_1}({\mathbb {R}}^2)$.

• (3) Let $s\in {\mathbb {R}}$. Then, for each $f\in {\dot{B}}_{p,q}^s({\mathbb {R}}^2)$, it holds

  $$\begin{aligned} \lim _{t\rightarrow \infty } \left\| e^{-t(-\Delta )^{\frac{\alpha }{2}}}f\right\| _{{\dot{B}}_{p,q}^s}=0. \end{aligned}$$

Proof

(1) is proved in [9] and [16]. (2) is immediately obtained by (1) and

$$\begin{aligned} 2^{s_2j}e^{-C^{-1}2^{\alpha j}t}\leqslant Ct^{-\frac{s_2-s_1}{\alpha }}2^{s_1j}. \end{aligned}$$

Let us prove (3). The density property yields that for any $\varepsilon >0$, there exists a $f_{\varepsilon }\in {\mathscr {S}}_0({\mathbb {R}}^2)$ such that $\Vert f_{\epsilon }-f\Vert _{{\dot{B}}_{p,q}^s}<\varepsilon$. Then, we see that

$$\begin{aligned} \begin{aligned} \left\| e^{-t(-\Delta )^{\frac{\alpha }{2}}}f\right\| _{{\dot{B}}_{p,q}^s}&\leqslant \left\| e^{-t(-\Delta )^{\frac{\alpha }{2}}}(f-f_{\varepsilon })\right\| _{{\dot{B}}_{p,q}^s} +\left\| e^{-t(-\Delta )^{\frac{\alpha }{2}}}f_{\varepsilon }\right\| _{{\dot{B}}_{p,q}^s}\\&\leqslant C\Vert f-f_{\epsilon }\Vert _{{\dot{B}}_{p,q}^s}+Ct^{-\frac{1}{\alpha }}\Vert f_{\varepsilon }\Vert _{{\dot{B}}_{p,q}^{s-1}}\\&<C\varepsilon +Ct^{-\frac{1}{\alpha }}\Vert f_{\varepsilon }\Vert _{{\dot{B}}_{p,q}^{s-1}}, \end{aligned} \end{aligned}$$

which implies

$$\begin{aligned} \limsup _{t\rightarrow \infty }\left\| e^{-t(-\Delta )^{\frac{\alpha }{2}}}f\right\| _{{\dot{B}}_{p,q}^s} \leqslant C\varepsilon . \end{aligned}$$

Since $\varepsilon >0$ is arbitrary, the proof is completed. $\square$

Next, we derive some bilinear estimates. We first recall the definition and basic properties of the Bony paraproduct formula. For $f,g\in {\mathscr {S}}_0({\mathbb {R}}^2)$, we decompose the product fg as:

$$\begin{aligned} fg=T_fg+R(f,g)+T_gf, \end{aligned}$$

where

$$\begin{aligned} T_fg:=\sum _{l\in {\mathbb {Z}}}S_lf\Delta _lg, \qquad R(f,g):=\sum _{l\in {\mathbb {Z}}}\sum _{|k-l|\leqslant 2}\Delta _kf\Delta _lg. \end{aligned}$$

Here, $S_lf$ is defined by

$$\begin{aligned} S_lf:=\sum _{k\leqslant l-3}\Delta _kf,\qquad l\in {\mathbb {Z}}. \end{aligned}$$

Then, considering the supports of the functions of the Fourier side, we have

$$\begin{aligned} \Delta _jT_{f}g=\sum _{l:|l-j|\leqslant 3}\Delta _j(S_lf\Delta _lg), \qquad \Delta _jR(f,g)=\sum _{(k,l):\begin{array}{c} \max \{j,k\}\geqslant l-3,\\ |k-l|\leqslant 2 \end{array}}\Delta _j(\Delta _kf\Delta _lg). \end{aligned}$$

Using them, we have for $T>0$, $1\leqslant p,q \leqslant \infty$ and $s_1,s_2\in {\mathbb {R}}$ with $s_1<0$ (if $q=1$, then $s_1\leqslant 0$) that

$$\begin{aligned} 2^{(s_1+s_2-\frac{2}{p})j}\Vert \Delta _jT_{f}g\Vert _{L^{\infty }(0,T;L^p)} \leqslant C\Vert f\Vert _{{\widetilde{L}}^{\infty }(0,T{\dot{B}}_{p,q}^{s_1})}\nonumber \\ \sum _{|l-j|\leqslant 3}2^{s_2l}\Vert \Delta _lg\Vert _{L^{\infty }(0,T;L^p)} \end{aligned}$$

(2.1)

and it also holds for $1\leqslant p,q \leqslant \infty$ and $s_1,s_2\in {\mathbb {R}}$ with $s_1+s_2>0$

$$\begin{aligned} \Vert R(f,g)\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_1+s_2})} \leqslant C\Vert f\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_1})} \Vert g\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_2})}. \end{aligned}$$

(2.2)

See [2] for the idea of the proof of these estimates. From easy applications of (2.1) and (2.2), we obtain the following lemma:

Lemma 2.2

Let $2\leqslant p\leqslant \infty$ and $1\leqslant q \leqslant \infty$. Let $s_1,s_2\in {\mathbb {R}}$ satisfy $s_1+s_2>0$ and $s_1,s_2<2/p$. Then, there exists a positive constant $C=C(p,q,s_1,s_2)$ such that

$$\begin{aligned} \Vert fg\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_1+s_2-\frac{2}{p}})} \leqslant C \Vert f\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_1})} \Vert g\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_2})} \end{aligned}$$

holds for all $T>0$, $f\in {\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_1}({\mathbb {R}}^2))$ and $g\in {\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_2}({\mathbb {R}}^2))$.

By the standard argument of the proof of commutator estimates (see, for instance, [2, 12]), we get the following lemma:

Lemma 2.3

Let $2\leqslant p\leqslant \infty$ and $1\leqslant q \leqslant \infty$. Let $s_1,s_2\in {\mathbb {R}}$ satisfy $s_1+s_2>0$, $0<s_1<1+2/p$ and $s_2<2/p$. Then, there exists a positive constant $C=C(p,q,s_1,s_2)$ such that

$$\begin{aligned} \left\| \left\{ 2^{(s_1+s_2-\frac{2}{p})j}\Vert [f,\Delta _j]g\Vert _{L^{\infty }(0,T;L^p)}\right\} _{j\in {\mathbb {Z}}}\right\| _{l^{q}({\mathbb {Z}})} \leqslant C \Vert f\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_1})} \Vert g\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_2})} \end{aligned}$$

holds for all $T>0$, $f\in {\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_1}({\mathbb {R}}^2))$ and $g\in {\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_2}({\mathbb {R}}^2))$.

The next lemma helps us to control the product term which will appear in equations of the perturbation such as (3.5), (3.22) and (3.32).

Lemma 2.4

Let $\lambda >0$, $\alpha >0$, $\beta \leqslant \alpha$, $2\leqslant p\leqslant \infty$ and $1\leqslant q \leqslant \infty$. Let $s_1,s_2\in {\mathbb {R}}$ satisfy $s_1+s_2>0$, $2/p<s_1<2/p+\alpha$ and $s_2<2/p$. Then, there exists a positive constant $C=C(\lambda ,\alpha ,\beta ,p,q,s_1,s_2)$ such that

$$\begin{aligned}&\left\| \left\{ \int _0^T2^{\beta j} e^{-\lambda 2^{\alpha j}(T-\tau )}2^{(s_1+s_2-\frac{2}{p})j} \Vert \Delta _j(f(\tau ) e^{-\tau (-\Delta )^{\frac{\alpha }{2}}}g)\Vert _{L^p} d\tau \right\} _{j\in {\mathbb {Z}}}\right\| _{l^q({\mathbb {Z}})}\nonumber \\&\leqslant CT^{1-\frac{\beta }{\alpha }-\frac{1}{\alpha }(s_1-\frac{2}{p})} \left\{ \Vert f\Vert _{L^{\infty }(0,T;L^p)} +\left( 1+T^{\frac{1}{\alpha }(s_1-\frac{2}{p})}\right) \Vert f\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_1})} \right\} \Vert g\Vert _{{\dot{B}}_{p,q}^{s_2}}\nonumber \\ \end{aligned}$$

(2.3)

holds for all $T>0$, $f\in L^{\infty }(0,T;L^p({\mathbb {R}}^2))\cap {\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_1}({\mathbb {R}}^2))$ and $g\in {\dot{B}}_{p,q}^{s_2}({\mathbb {R}}^2)$.

In particular, if $\beta <\alpha$, then the following estimate holds:

$$\begin{aligned}&\sum _{j\in {\mathbb {Z}}} \int _0^T2^{\beta j} e^{-\lambda 2^{\alpha j}(T-\tau )} \Vert f(\tau ) e^{-\tau (-\Delta )^{\frac{\alpha }{2}}}g\Vert _{{\dot{B}}_{p,q}^{s_1+s_2-\frac{2}{p}}} d\tau \nonumber \\&\qquad \leqslant CT^{1-\frac{\beta }{\alpha }-\frac{1}{\alpha }(s_1-\frac{2}{p})} \left\{ \Vert f\Vert _{L^{\infty }(0,T;L^p)} +\left( 1+T^{\frac{1}{\alpha }(s_1-\frac{2}{p})}\right) \Vert f\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_1})} \right\} \Vert g\Vert _{{\dot{B}}_{p,q}^{s_2}}.\nonumber \\ \end{aligned}$$

(2.4)

Remark 2.5

Let $1/2<\alpha <1$ and $2\leqslant p < 4/(2\alpha -1)$. Then, it immediately follows from (2.3) with $s_1=s_\mathrm{c}:=1+2/p-\alpha$ and the continuous embedding ${\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,1}^0({\mathbb {R}}^2))\hookrightarrow L^{\infty }(0,T;L^p({\mathbb {R}}^2))$ that

$$\begin{aligned}&\left\| \left\{ \int _0^T2^{\beta j} e^{-\lambda 2^{\alpha j}(T-\tau )}2^{(s_\mathrm{c}+s_2-\frac{2}{p})j} \Vert \Delta _j(f(\tau ) e^{-\tau (-\Delta )^{\frac{\alpha }{2}}}g)\Vert _{L^p} d\tau \right\} _{j\in {\mathbb {Z}}}\right\| _{l^q({\mathbb {Z}})}\nonumber \\&\qquad \leqslant CT^{1-\frac{\beta }{\alpha }} (T^{-\frac{1-\alpha }{\alpha }}+1) \Vert f\Vert _{X_T^{p,q}} \Vert g\Vert _{{\dot{B}}_{p,q}^{s_2}}, \end{aligned}$$

(2.5)

where $X_T^{p,q}:={\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,1}^0({\mathbb {R}}^2)) \cap {\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_\mathrm{c}}({\mathbb {R}}^2))$. If $\beta <\alpha$, then (2.4) yields that

$$\begin{aligned}&\sum _{j\in {\mathbb {Z}}} \int _0^T2^{\beta j} e^{-\lambda 2^{\alpha j}(T-\tau )} \Vert f(\tau ) e^{-\tau (-\Delta )^{\frac{\alpha }{2}}}g\Vert _{{\dot{B}}_{p,q}^{s_1+s_2-\frac{2}{p}}} d\tau \nonumber \\&\qquad \leqslant CT^{1-\frac{\beta }{\alpha }} (T^{-\frac{1-\alpha }{\alpha }}+1) \Vert f\Vert _{X_T^{p,q}} \Vert g\Vert _{{\dot{B}}_{p,q}^{s_2}}. \end{aligned}$$

(2.6)

Proof of Lemma 2.4

First, we prove (2.3). It follows from an inequality of (2.1) type and (2) of Lemma 2.1 that

$$\begin{aligned}&\int _0^T2^{\beta j} e^{-\lambda 2^{\alpha j}(T-\tau )}2^{(s_1+s_2-\frac{2}{p})j} \Vert \Delta _jT_{f(\tau )} e^{-\tau (-\Delta )^{\frac{\alpha }{2}}}g\Vert _{L^p} d\tau \nonumber \\&\qquad \leqslant C\int _0^T2^{\beta j} e^{-\lambda 2^{\alpha j}(T-\tau )} \Vert f(\tau )\Vert _{{\dot{B}}_{p,1}^{\frac{2}{p}}} \sum _{|l-j|\leqslant 3}2^{((s_1-\frac{2}{p})+s_2)l}\Vert e^{-\tau (-\Delta )^{\frac{\alpha }{2}}}\Delta _lg\Vert _{L^p} d\tau \nonumber \\&\qquad \leqslant C\int _0^T2^{\beta j} e^{-\lambda 2^{\alpha j}(T-\tau )} \tau ^{-\frac{1}{\alpha }(s_1-\frac{2}{p})} d\tau \Vert f\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,1}^{\frac{2}{p}})} \sum _{|l-j|\leqslant 3}2^{s_2l}\Vert \Delta _lg\Vert _{L^p}. \end{aligned}$$

(2.7)

By virtue of $s_1-2/p<\alpha$ and $\beta \leqslant \alpha$, it is easy to see that

$$\begin{aligned} \sup _{j\in {\mathbb {Z}}} \int _0^T2^{\beta j} e^{-\lambda 2^{\alpha j}(T-\tau )} \tau ^{-\frac{1}{\alpha }(s_1- \frac{2}{p})} d\tau \leqslant CT^{1-\frac{\beta }{\alpha }-\frac{1}{\alpha }(s_1- \frac{2}{p})}. \end{aligned}$$

Hence, taking $l^q({\mathbb {Z}})$-norm of (2.7) and using

$$\begin{aligned} \Vert f\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,1}^{\frac{2}{p}})}\leqslant & {} \sum _{j\leqslant 0}2^{\frac{2}{p}j}\Vert \Delta _jf\Vert _{L^{\infty }(0,T;L^p)} +\left\| \left\{ 2^{(\frac{2}{p}-s_1)j}\right\} \right\| _{l^{\frac{q}{q-1}}({\mathbb {N}})} \Vert f\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_1})}\nonumber \\\leqslant & {} C\left( \Vert f\Vert _{L^{\infty }(0,T;L^p)}+\Vert f\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_1})}\right) , \end{aligned}$$

(2.8)

we obtain that

$$\begin{aligned}&\left\| \left\{ \int _0^T2^{\beta j} e^{-\lambda 2^{\alpha j}(T-\tau )}2^{(s_1+s_2-\frac{2}{p})j} \Vert \Delta _jT_{f(\tau )} e^{-\tau (-\Delta )^{\frac{\alpha }{2}}}g\Vert _{L^p} d\tau \right\} _{j\in {\mathbb {Z}}}\right\| _{l^q({\mathbb {Z}})}\nonumber \\&\quad \leqslant CT^{1-\frac{\beta }{\alpha }-\frac{1}{\alpha }(s_1-\frac{2}{p})} \left( \Vert f\Vert _{L^{\infty }(0,T;L^p)}+\Vert f\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_1})}\right) \left\| g\right\| _{{\dot{B}}_{p,q}^{s_2}}. \end{aligned}$$

(2.9)

Since it holds

$$\begin{aligned}&\int _0^T2^{\beta j} e^{-\lambda 2^{\alpha j}(T-\tau )}2^{(s_1+s_2-\frac{2}{p})j} \left\| \Delta _jR(f(\tau ),e^{-\tau (-\Delta )^{\frac{\alpha }{2}}}g)\right\| _{L^p}d\tau \nonumber \\&\leqslant \int _0^T2^{\beta j} e^{-\lambda 2^{\alpha j}(T-\tau )} d\tau 2^{(s_1+s_2-\frac{2}{p})j} \left\| \Delta _jR(f,e^{-t(-\Delta )^{\frac{\alpha }{2}}}g)\right\| _{L^{\infty }_t(0,T;L^p)}, \end{aligned}$$

(2.10)

taking $l^q({\mathbb {Z}})$-norm of (2.10), we see by (2.2) and (2) of Lemma 2.1 that

$$\begin{aligned}&\left\| \left\{ \int _0^T2^{\beta j} e^{-\lambda 2^{\alpha j}(T-\tau )}2^{(s_1+s_2-\frac{2}{p})j} \left\| \Delta _jR(f(\tau ),e^{-\tau (-\Delta )^{\frac{\alpha }{2}}}g)\right\| _{L^p} d\tau \right\} _{j\in {\mathbb {Z}}}\right\| _{l^q({\mathbb {Z}})}\nonumber \\&\quad \leqslant \sup _{j\in {\mathbb {Z}}}\int _0^T2^{\beta j} e^{-\lambda 2^{\alpha j}(T-\tau )} d\tau \left\| R(f,e^{-t(-\Delta )^{\frac{\alpha }{2}}}g)\right\| _{{\widetilde{L}}^{\infty }_t(0,T;{\dot{B}}_{p,q}^{s_1+s_2-\frac{2}{p}})}\nonumber \\&\quad \leqslant CT^{1-\frac{\beta }{\alpha }} \Vert f\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_1})} \left\| g\right\| _{{\dot{B}}_{p,q}^{s_2}}. \end{aligned}$$

(2.11)

Here, we have used

$$\begin{aligned} \sup _{j\in {\mathbb {Z}}}\int _0^T 2^{\beta j}e^{\lambda 2^{\alpha j}(T-\tau )} d\tau \leqslant CT^{1-\frac{\beta }{\alpha }}, \qquad \beta \leqslant \alpha . \end{aligned}$$

(2.12)

Similarly, it follows from (2.1) and (2) of Lemma 2.1 that

$$\begin{aligned}&\left\| \left\{ \int _0^T2^{\beta j} e^{-\lambda 2^{\alpha j}(T-\tau )}2^{(s_1+s_2-\frac{2}{p})j} \left\| \Delta _jT_{e^{-\tau (-\Delta )^{\frac{\alpha }{2}}}g}f(\tau )\right\| _{L^p} d\tau \right\} _{j\in {\mathbb {Z}}}\right\| _{l^q({\mathbb {Z}})}\nonumber \\&\quad \leqslant \sup _{j\in {\mathbb {Z}}}\int _0^T2^{\beta j} e^{-\lambda 2^{\alpha j}(T-\tau )} d\tau \left\| T_{e^{-t(-\Delta )^{\frac{\alpha }{2}}}g}f(t)\right\| _ {{\widetilde{L}}^{\infty }_t(0,T;{\dot{B}}_{p,q}^{s_1+s_2-\frac{2}{p}})}\nonumber \\&\quad \leqslant CT^{1-\frac{\beta }{\alpha }} \Vert f\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_1})} \left\| e^{-t(-\Delta )^{\frac{\alpha }{2}}}g\right\| _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_2})}\nonumber \\&\quad \leqslant CT^{1-\frac{\beta }{\alpha }} \Vert f\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_1})} \left\| g\right\| _{{\dot{B}}_{p,q}^{s_2}}. \end{aligned}$$

(2.13)

Combining (2.9), (2.11) and (2.13), we complete the proof of (2.5). Next, we show (2.4). By similar inequality to (2.1) and (2) of Lemma 2.1, we see that

$$\begin{aligned}&\sum _{j\in {\mathbb {Z}}} \int _0^T2^{\beta j} e^{-\lambda 2^{\alpha j}(T-\tau )} \Vert T_{f(\tau )} e^{-\tau (-\Delta )^{\frac{\alpha }{2}}}g\Vert _{{\dot{B}}_{p,q}^{s_1+s_2-\frac{2}{p}}} d\tau \nonumber \\&\quad \leqslant C\sum _{j\in {\mathbb {Z}}} \int _0^T2^{\beta j} e^{-\lambda 2^{\alpha j}(T-\tau )}\tau ^{-\frac{1}{\alpha }(s_1-\frac{2}{p})}d\tau \Vert f\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,1}^{\frac{2}{p}})} \Vert g\Vert _{{\dot{B}}_{p,q}^{s_2}}\nonumber \\&\quad \leqslant CT^{1-\frac{\beta }{\alpha }-\frac{1}{\alpha }(s_1-\frac{2}{p})} \left( \Vert f\Vert _{L^{\infty }(0,T;L^p)}+\Vert f\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_1})}\right) \left\| g\right\| _{{\dot{B}}_{p,q}^{s_2}}. \end{aligned}$$

(2.14)

We also obtain from (2.2) and (2) of Lemma 2.1 that

$$\begin{aligned}&\sum _{j\in {\mathbb {Z}}}\int _0^T2^{\beta j} e^{-\lambda 2^{\alpha j}(T-\tau )} \left\| R(f(\tau ),e^{-\tau (-\Delta )^{\frac{\alpha }{2}}}g)\right\| _{{\dot{B}}_{p,q}^{s_1+s_2-\frac{2}{p}}}d\tau \nonumber \\&\leqslant \sum _{j\in {\mathbb {Z}}}\int _0^T2^{\beta j} e^{-\lambda 2^{\alpha j}(T-\tau )} d\tau \left\| R(f,e^{-\tau (-\Delta )^{\frac{\alpha }{2}}}g)\right\| _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_1+s_2-\frac{2}{p}})}\nonumber \\&\leqslant CT^{1-\frac{\beta }{\alpha }} \Vert f\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_1})} \left\| g\right\| _{{\dot{B}}_{p,q}^{s_2}}. \end{aligned}$$

(2.15)

Here, we have used the following inequalities in (2.14) and (2.15):

$$\begin{aligned}&\sum _{j\in {\mathbb {Z}}}\int _0^T2^{\beta j}e^{-\lambda 2^{\alpha j}(T-\tau )}\tau ^{-\frac{\gamma }{\alpha }} d\tau \leqslant CT^{1-\frac{\beta }{\alpha }-\frac{\gamma }{\alpha }}, \qquad \beta<\alpha ,\ \gamma<\alpha ,\nonumber \\&\sum _{j\in {\mathbb {Z}}}\int _0^T2^{\beta j}e^{-\lambda 2^{\alpha j}(T-\tau )} d\tau \leqslant CT^{1-\frac{\beta }{\alpha }}, \qquad \beta <\alpha . \end{aligned}$$

(2.16)

Similarly, we have

$$\begin{aligned}&\sum _{j\in {\mathbb {Z}}} \int _0^T2^{\beta j} e^{-\lambda 2^{\alpha j}(T-\tau )}2^{(s_1+s_2-\frac{2}{p})j} \left\| T_{e^{-\tau (-\Delta )^{\frac{\alpha }{2}}}g}f(\tau )\right\| _{{\dot{B}}_{p,q}^{s_1+s_2-\frac{2}{p}}} d\tau \nonumber \\&\quad \leqslant CT^{1-\frac{\beta }{\alpha }} \Vert f\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_1})} \left\| g\right\| _{{\dot{B}}_{p,q}^{s_2}}. \end{aligned}$$

(2.17)

Hence, we complete the proof by combining (2.14), (2.15) and (2.17).

To derive some estimates for initial data related to a time periodic solution in the proof of the main results, we introduce the following two lemmas whose ideas are inspired by [8].

Lemma 2.6

Let $\alpha >0$, $1\leqslant p \leqslant \infty$, $1\leqslant q< \infty$ and $s\in {\mathbb {R}}$. Then, for any $f\in {\dot{B}}_{p,q}^s({\mathbb {R}}^2)\cap {\dot{B}}_{p,q}^{s-\alpha }({\mathbb {R}}^2)$, the series

$$\begin{aligned} u=\sum _{k=0}^{\infty } e^{-Tk(-\Delta )^{\frac{\alpha }{2}}}f \end{aligned}$$

(2.18)

converges in ${\dot{B}}_{p,q}^s({\mathbb {R}}^2)$ and u satisfies

$$\begin{aligned} (1-e^{-T(-\Delta )^{\frac{\alpha }{2}}})u=f \qquad \mathrm{in}\ {\dot{B}}_{p,q}^s({\mathbb {R}}^2). \end{aligned}$$

(2.19)

Moreover, there exists a positive constant $C=C(\alpha )$ such that

$$\begin{aligned} \Vert u\Vert _{{\dot{B}}_{p,q}^s} \leqslant C \left( T^{-1}\Vert f\Vert _{{\dot{B}}_{p,q}^{s-\alpha }} +\Vert f\Vert _{{\dot{B}}_{p,q}^s} \right) . \end{aligned}$$

(2.20)

Proof

Let $m,n\in {\mathbb {N}}$ satisfy $m<n$. Then, it follows from (1) of Lemma 2.1 that

$$\begin{aligned} \left\| \sum _{k=m}^n e^{-Tk(-\Delta )^{\frac{\alpha }{2}}}f \right\| _{{\dot{B}}_{p,q}^s}^q= & {} \sum _{j\in {\mathbb {Z}}} \left( 2^{sj} \left\| \sum _{k=m}^n e^{-Tk(-\Delta )^{\frac{\alpha }{2}}}\Delta _jf \right\| _{L^p} \right) ^q\nonumber \\\leqslant & {} \sum _{j\in {\mathbb {Z}}} \left( 2^{sj} \Vert \Delta _jf\Vert _{L^p} \sum _{k=m}^n Ce^{-C^{-1}T2^{\alpha j}k} \right) ^q, \end{aligned}$$

(2.21)

where C is the same constant as in (1) of Lemma 2.1. Since it holds

$$\begin{aligned}&\sum _{j\in {\mathbb {Z}}} \left( 2^{sj} \Vert \Delta _jf\Vert _{L^p} \frac{C}{1-e^{-C^{-1}2^{\alpha j}T}} \right) ^q\nonumber \\&\quad = \sum _{j:2^{\alpha j}T<1} \left( 2^{sj} \Vert \Delta _jf\Vert _{L^p} \frac{C^{-1}2^{\alpha j}T}{1-e^{-C^{-1}2^{\alpha j}T}}\cdot C^2 2^{-\alpha j}T^{-1} \right) ^q\nonumber \\&\qquad + \sum _{j:2^{\alpha j}T\geqslant 1} \left( 2^{sj} \Vert \Delta _jf\Vert _{L^p} \frac{C}{1-e^{-C^{-1}2^{\alpha j}T}} \right) ^q \nonumber \\&\quad \leqslant \left( \frac{C}{1-e^{-C^{-1}}} \right) ^q \left( T^{-q}\Vert f\Vert _{{\dot{B}}_{p,q}^{s-\alpha }}^q+\Vert f\Vert _{{\dot{B}}_{p,q}^s}^q \right) \nonumber \\&\quad \leqslant \left( \frac{2C}{1-e^{-C^{-1}}} \right) ^q \left( T^{-1}\Vert f\Vert _{{\dot{B}}_{p,q}^{s-\alpha }}+\Vert f\Vert _{{\dot{B}}_{p,q}^s}\right) ^q <\infty , \end{aligned}$$

(2.22)

we have

$$\begin{aligned} \begin{aligned} 2^{sj} \Vert \Delta _jf\Vert _{L^p} \sum _{k=m}^n Ce^{-C^{-1}T2^{\alpha j}k}&\leqslant 2^{sj} \Vert \Delta _jf\Vert _{L^p} \sum _{k=0}^{\infty } Ce^{-C^{-1}T2^{\alpha j}k}\\&= 2^{sj} \Vert \Delta _jf\Vert _{L^p} \frac{C}{1-e^{-C^{-1}2^{\alpha j}T}}\in l^q({\mathbb {Z}}). \end{aligned} \end{aligned}$$

Hence, it follows from (2.21) and the dominated convergence theorem that

$$\begin{aligned} \limsup _{n,m\rightarrow \infty }\left\| \sum _{k=m}^n e^{-Tk(-\Delta )^{\frac{\alpha }{2}}}f \right\| ^q_{{\dot{B}}_{p,q}^s} \leqslant \sum _{j\in {\mathbb {Z}}} \left( 2^{sj} \Vert \Delta _jf\Vert _{L^p} \lim _{m,n\rightarrow \infty } \sum _{k=m}^n Ce^{-C^{-1}T2^{\alpha j}k} \right) ^q =0. \end{aligned}$$

Thus, the series (2.18) converges in ${\dot{B}}_{p,q}^s({\mathbb {R}}^2)$, and we find that u satisfies (2.20) by

$$\begin{aligned} \begin{aligned} \Vert u\Vert _{{\dot{B}}_{p,q}^s}^q&=\sum _{j\in {\mathbb {Z}}} \left( 2^{sj} \left\| \sum _{k=0}^{\infty } e^{-Tk(-\Delta )^{\frac{\alpha }{2}}}\Delta _jf \right\| _{L^p} \right) ^q\\&\leqslant \sum _{j\in {\mathbb {Z}}} \left( 2^{sj} \sum _{k=0}^{\infty } \left\| e^{-Tk(-\Delta )^{\frac{\alpha }{2}}}\Delta _jf \right\| _{L^p} \right) ^q\\&\leqslant \sum _{j\in {\mathbb {Z}}} \left( 2^{sj} \Vert \Delta _jf\Vert _{L^p} \sum _{k=0}^{\infty } Ce^{-C^{-1}T2^{\alpha j}k} \right) ^q\\&\leqslant \left( \frac{2C}{1-e^{-C^{-1}}} \right) ^q \left( T^{-1}\Vert f\Vert _{{\dot{B}}_{p,q}^{s-\alpha }}+\Vert f\Vert _{{\dot{B}}_{p,q}^s} \right) ^q, \end{aligned} \end{aligned}$$

where we have used (2.22). Finally, we show (2.19). Let

$$\begin{aligned} u_N:=\sum _{k=0}^{N-1}e^{-Tk(-\Delta )^{\frac{\alpha }{2}}}f, \qquad N\in {\mathbb {N}}. \end{aligned}$$

Note that $u_N$ converges to u in ${\dot{B}}_{p,q}^s({\mathbb {R}}^2)$ as $N\rightarrow \infty$. By a simple calculation, we see that

$$\begin{aligned} (1-e^{-T(-\Delta )^{\frac{\alpha }{2}}})u_N=f-e^{-TN(-\Delta )^{\frac{\alpha }{2}}}f. \end{aligned}$$

(2.23)

Here, it follows from (2) of Lemma 2.1 that

$$\begin{aligned}&\left\| (1-e^{-T(-\Delta )^{\frac{\alpha }{2}}})u_N -(1-e^{-T(-\Delta )^{\frac{\alpha }{2}}})u\right\| _{{\dot{B}}_{p,q}^s}\nonumber \\&\qquad \leqslant \Vert u-u_N\Vert _{{\dot{B}}_{p,q}^s} +\left\| e^{-T(-\Delta )^{\frac{\alpha }{2}}}(u-u_N)\right\| _{{\dot{B}}_{p,q}^s}\nonumber \\&\qquad \leqslant C\Vert u-u_N\Vert _{{\dot{B}}_{p,q}^s}\nonumber \\&\qquad \rightarrow 0 \end{aligned}$$

(2.24)

as $N\rightarrow \infty$ and it holds by (3) of Lemma 2.1 that

$$\begin{aligned} \left\| e^{-TN(-\Delta )^{\frac{\alpha }{2}}}f\right\| _{{\dot{B}}_{p,q}^s}\rightarrow 0 \end{aligned}$$

(2.25)

as $N\rightarrow \infty$. Hence, letting $N\rightarrow \infty$ in (2.23) by (2.24) and (2.25), we find that u satisfies (2.19). This completes the proof. $\square$

Lemma 2.7

Let $T>0$, $\alpha >0$, $1\leqslant p\leqslant \infty$, $1\leqslant q <\infty$ and $s<2/p$. Then, for any T-time periodic function F satisfying

$$\begin{aligned} f(t):=\int _0^t e^{-(t-\tau )(-\Delta )^{\frac{\alpha }{2}}}F(\tau ) d\tau \in BC((0,\infty ); {\dot{B}}_{p,q}^{s-\alpha }({\mathbb {R}}^2)\cap {\dot{B}}_{p,q}^s({\mathbb {R}}^2)), \end{aligned}$$

there exists a unique element $u_0\in {\dot{B}}_{p,q}^s({\mathbb {R}}^2)$ such that the function

$$\begin{aligned} u(t)=e^{-t(-\Delta )^{\frac{\alpha }{2}}}u_0+f(t),\qquad t\geqslant 0 \end{aligned}$$

is T-time periodic. Moreover, $u_0$ satisfies

$$\begin{aligned} \Vert u_0\Vert _{{\dot{B}}_{p,q}^s} \leqslant C\left( T^{-1}\Vert f(T)\Vert _{{\dot{B}}_{p,q}^{s-\alpha }} +\Vert f(T)\Vert _{{\dot{B}}_{p,q}^{s}} \right) , \end{aligned}$$

(2.26)

where C is the same constant as in Lemma 2.6.

Proof

By Lemma 2.6, the series

$$\begin{aligned} u_0:=\sum _{k=0}^{\infty } e^{-Tk(-\Delta )^{\frac{\alpha }{2}}}f(T) \end{aligned}$$

converges in ${\dot{B}}_{p,q}^s({\mathbb {R}}^2)$ and $u_0$ satisfies (2.26) and

$$\begin{aligned} (1-e^{-T(-\Delta )^{\frac{\alpha }{2}}})u_0=f(T). \end{aligned}$$

(2.27)

By the periodicity of F, we have

$$\begin{aligned} f(t+T)=f(t)+e^{-t(-\Delta )^{\frac{\alpha }{2}}}f(T), \qquad t>0. \end{aligned}$$

(2.28)

Therefore, it follows from (2.27) and (2.28) that

$$\begin{aligned} \begin{aligned} u(t+T)&=e^{-t(-\Delta )^{\frac{\alpha }{2}}}e^{-T(-\Delta )^{\frac{\alpha }{2}}}u_0+f(t+T)\\&=e^{-t(-\Delta )^{\frac{\alpha }{2}}}(u_0-f(T))+f(t)+e^{-t(-\Delta )^{\frac{\alpha }{2}}}f(T)\\&=u(t) \end{aligned} \end{aligned}$$

for all $t>0$. Hence, u(t) is T-time periodic. Next, we prove the uniqueness. Let $v_0$ be an arbitrary element of ${\dot{B}}_{p,q}^s({\mathbb {R}}^2)$ such that $v(t):=e^{-t(-\Delta )^{\frac{\alpha }{2}}}v_0+f(t)$ is a T-time periodic function. Then, since $u_0-v_0=u(NT)-v(NT)=e^{-NT(-\Delta )^{\frac{\alpha }{2}}}(u_0-v_0)$ holds for all $N\in {\mathbb {N}}$ by the periodicity, we obtain by (3) of Lemma 2.1 that

$$\begin{aligned} \Vert u_0-v_0\Vert _{{\dot{B}}_{p,q}^{s}} =\left\| e^{-NT(-\Delta )^{\frac{\alpha }{2}}}(u_0-v_0)\right\| _{{\dot{B}}_{p,q}^s} \rightarrow 0 \end{aligned}$$

as $N\rightarrow \infty$. Therefore, we have $u_0=v_0$ in ${\dot{B}}_{p,q}^s({\mathbb {R}}^2)$ and this completes the proof. $\square$

Finally, we recall a positivity lemma for the $L^p$-energy of the fractional dissipation:

Lemma 2.8

([4, 15]) Let $0<\alpha \leqslant 2$ and $2\leqslant p <\infty$. Then, there exists a positive constant $\lambda =\lambda (\alpha ,p)$ such that

$$\begin{aligned} \int _{{\mathbb {R}}^2} |\Delta _j f(x)|^{p-2}\Delta _jf(x)(-\Delta )^{\frac{\alpha }{2}}\Delta _jf(x) dx \geqslant \lambda 2^{\alpha j}\Vert \Delta _jf\Vert _{L^p}^p \end{aligned}$$

for all $j\in {\mathbb {Z}}$ and $f\in {\mathscr {S}}'_0({\mathbb {R}}^2)$ with $\Delta _jf \in L^p({\mathbb {R}}^2)$.

3 Proof of main results

In this section, we prove Theorems 1.1 and 1.3. Once Theorem 1.3 is proved, then we can get the time periodic solution on ${\mathbb {R}}$ by extending the solution on $[0,\infty )$ to the other half time line $(-\infty ,0)$ periodically. Therefore, we only prove Theorem 1.3.

Let T, $\alpha$, p, q and r satisfy the assumptions of Theorem 1.3 and let $\sigma$ satisfy $\alpha -2/p<\sigma <2/p$. We use the following notation for simplicity in this section:

$$\begin{aligned} \begin{aligned} s_\mathrm{c}&:=1+\frac{2}{p}-\alpha ,\\ X_T^{p,q}&:={\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,1}^0({\mathbb {R}}^2)) \cap {\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_\mathrm{c}}({\mathbb {R}}^2)). \end{aligned} \end{aligned}$$

We consider the successive approximation sequences $\{\theta _0^{(n)}\}_{n=0}^{\infty }\subset {\dot{B}}_{p,1}^0({\mathbb {R}}^2)\cap {\dot{B}}_{p,q}^{s_\mathrm{c}}({\mathbb {R}}^2)$ of the initial data and $\{\theta ^{(n)}\}_{n=0}^{\infty }\subset X_T^{p,q}$ of solutions to (1.5) defined inductively as follows:

First, let $\theta _0^{(0)}(x)=0$ and $\theta ^{(0)}(t,x)=0$. Next, if $\theta _0^{(n)}$ and $\theta ^{(n)}$ are determined, then we define $\theta _0^{(n+1)}$ and $\theta ^{(n+1)}$ by the following linear equation:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \theta ^{(n+1)} +(-\Delta )^{\frac{\alpha }{2}} \theta ^{(n+1)} +u^{(n)} \cdot \nabla \theta ^{(n+1)} =S_{n+4}F, &{}\qquad 0<t\leqslant T,x\in {\mathbb {R}}^2,\\ u^{(n)}={\mathcal {R}}^{\perp } \theta ^{(n)}, &{}\qquad 0\leqslant t\leqslant T, x\in {\mathbb {R}}^2,\\ \theta ^{(n+1)}(0,x)=S_{n+4}\theta _0^{(n+1)}, &{}\qquad x\in {\mathbb {R}}^2, \end{array}\right. } \end{aligned}$$

(3.1)

where $\theta _0^{(n+1)}$ is given by

$$\begin{aligned} \theta _0^{(n+1)} :=\sum _{k=0}^{\infty } e^{-Tk(-\Delta )^{\frac{\alpha }{2}}} \left( \theta ^{(n)}(T)-e^{-T(-\Delta )^{\frac{\alpha }{2}}}\theta ^{(n)}(0) \right) . \end{aligned}$$

(3.2)

For $n\in {\mathbb {N}}\cup \{0\}$, we put $\psi ^{(n)}(t):=\theta ^{(n)}(t)-e^{-t(-\Delta )^{\frac{\alpha }{2}}}\theta ^{(n)}(0)$ and

$$\begin{aligned} \begin{aligned} A_n&:=\max \left\{ \Vert \theta _0^{(n)}\Vert _{{\dot{B}}_{p,1}^0\cap {\dot{B}}_{p,q}^{s_\mathrm{c}}}, \Vert \theta ^{(n)}\Vert _{X_T^{p,q}} \right\} ,\\ B_n&:=\Vert \theta _0^{(n+1)}-\theta _0^{(n)}\Vert _{{\dot{B}}_{p,q}^{\sigma }} +\Vert \theta ^{(n+1)}-\theta ^{(n)}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{\sigma })}. \end{aligned} \end{aligned}$$

The well-definedness of the sequences is assured if the series in (3.2) converges in ${\dot{B}}_{p,1}^0({\mathbb {R}}^2)\cap {\dot{B}}_{p,q}^{s_\mathrm{c}}({\mathbb {R}}^2)$. In the following lemma, we check the convergence and derive some properties of the sequences.

Lemma 3.1

Let n be an positive integer. Assume that $\theta _0^{(n)}\in {\dot{B}}_{p,1}^0({\mathbb {R}}^2)\cap {\dot{B}}_{p,q}^{s_\mathrm{c}}({\mathbb {R}}^2)$ and $\theta ^{(n)}\in X_T^{p,q}$. Then, for every $F\in BC((0,\infty );{\dot{B}}_{r,\infty }^0({\mathbb {R}}^2))$, the series in (3.2) converges in ${\dot{B}}_{p,1}^0({\mathbb {R}}^2)\cap {\dot{B}}_{p,q}^{s_\mathrm{c}}({\mathbb {R}}^2)$ and it holds

$$\begin{aligned} (1-e^{-T(-\Delta )^{\frac{\alpha }{2}}})\theta _0^{(n+1)} =\theta ^{(n)}(T)-e^{-T(-\Delta )^{\frac{\alpha }{2}}}\theta ^{(n)}(0) \end{aligned}$$

(3.3)

in ${\dot{B}}_{p,1}^0({\mathbb {R}}^2)\cap {\dot{B}}_{p,q}^{s_\mathrm{c}}({\mathbb {R}}^2)$. Moreover, there exist positive constants $\delta _1=\delta _1(\alpha ,p,q,r,T)$ and $C_1=C_1(\alpha ,p,q,r,T)$ such that if F satisfies

$$\begin{aligned} \sup _{t>0}\Vert F(t)\Vert _{{\dot{B}}_{r,\infty }^0}\leqslant \delta _1, \end{aligned}$$

then it holds

$$\begin{aligned} \begin{aligned} \sup _{m\in {\mathbb {N}}\cup \{0\}}\Vert \theta _0^{(m)}\Vert _{{\dot{B}}_{p,1}^0\cap {\dot{B}}_{p,q}^{s_\mathrm{c}}}&\leqslant 2C_1\sup _{t>0}\Vert F(t)\Vert _{{\dot{B}}_{r,\infty }^0},\\ \sup _{m\in {\mathbb {N}}\cup \{0\}}\Vert \theta ^{(m)}\Vert _{X_T^{p,q}}&\leqslant 2C_1\sup _{t>0}\Vert F(t)\Vert _{{\dot{B}}_{r,\infty }^0}. \end{aligned} \end{aligned}$$

(3.4)

Proof

To prove the convergence of the series in (3.2) in ${\dot{B}}_{p,1}^0({\mathbb {R}}^2)\cap {\dot{B}}_{p,q}^{s_\mathrm{c}}({\mathbb {R}}^2)$, Lemma 2.6 yields that it suffices to check

$$\begin{aligned} \begin{aligned}&\theta ^{(n)}(T)-e^{-T(-\Delta )^\frac{\alpha }{2}}\theta ^{(n)}(0)=\psi ^{(n)}(T)\\&\quad \in \left( {\dot{B}}_{p,q}^{s_\mathrm{c}}({\mathbb {R}}^2)\cap {\dot{B}}_{p,q}^{s_\mathrm{c}-\alpha }({\mathbb {R}}^2) \right) \cap \left( {\dot{B}}_{p,1}^{0}({\mathbb {R}}^2)\cap {\dot{B}}_{p,1}^{-\alpha }({\mathbb {R}}^2) \right) . \end{aligned} \end{aligned}$$

Since $\psi ^{(n)}$ satisfies

$$\begin{aligned} \partial _t\psi ^{(n)}+(-\Delta )^{\frac{\alpha }{2}}\psi ^{(n)}+u^{(n-1)}\cdot \nabla \psi ^{(n)}\nonumber \\ +u^{(n-1)}\cdot \nabla e^{-t(-\Delta )^{\frac{\alpha }{2}}}\theta ^{(n)}(0)=S_{n+4}F, \end{aligned}$$

(3.5)

applying $\Delta _j$ to (3.5), we see that

$$\begin{aligned}&\partial _t\Delta _j\psi ^{(n)} +(-\Delta )^{\frac{\alpha }{2}}\Delta _j\psi ^{(n)}\nonumber \\&\qquad =S_{n+4}\Delta _jF +[u^{(n-1)},\Delta _j]\cdot \nabla \psi ^{(n)}\nonumber \\&\qquad \quad -u^{(n-1)}\cdot \nabla \Delta _j\psi ^{(n)} -\Delta _j(u^{(n-1)}\cdot \nabla e^{-t(-\Delta )^{\frac{\alpha }{2}}}\theta ^{(n)}(0)). \end{aligned}$$

(3.6)

Multiplying (3.6) by $p|\Delta _j \psi ^{(n)}|^{p-2}\Delta _j\psi ^{(n)}$ and integrating over ${\mathbb {R}}^2$, we have by the Hölder inequality that

$$\begin{aligned}&\frac{d}{dt}(\Vert \Delta _j\psi ^{(n)}(t)\Vert ^p_{L^p}) +p\int _{{\mathbb {R}}^2}|\Delta _j \psi ^{(n)}(t,x)|^{p-2}\Delta _j \psi ^{(n)}(t,x)(-\Delta )^{\frac{\alpha }{2}}\Delta _j\psi ^{(n)}(t,x) dx\nonumber \\&\qquad \leqslant Cp\Vert \Delta _j F(t)\Vert _{L^p}\Vert \Delta _j \psi ^{(n)}(t)\Vert _{L^p}^{p-1} +p\Vert [u^{(n-1)}(t),\Delta _j]\cdot \nabla \psi ^{(n)}(t)\Vert _{L^p}\Vert \Delta _j\psi (t)\Vert _{L^p}^{p-1}\nonumber \\&\qquad \quad -p\int _{{\mathbb {R}}^2} |\Delta _j \psi ^{(n)}(t,x)|^{p-2}\Delta _j \psi ^{(n)}(t,x) u^{(n-1)}(t,x)\cdot \nabla \Delta _j \psi ^{(n)}(t,x) dx\nonumber \\&\qquad \quad +p\Vert \Delta _j(u^{(n-1)}(t)\cdot \nabla e^{-t(-\Delta )^{\frac{\alpha }{2}}}\theta ^{(n)}(0))\Vert _{L^p}\Vert \Delta _j\psi ^{(n)}(t)\Vert _{L^p}^{p-1}. \end{aligned}$$

(3.7)

By Lemma 2.8, we obtain that

$$\begin{aligned} \int _{{\mathbb {R}}^2}|\Delta _j \psi ^{(n)}(t,x)|^{p-2}\Delta _j \psi ^{(n)}(t,x)(-\Delta )^{\frac{\alpha }{2}}\Delta _j\psi ^{(n)}(t,x) dx \geqslant \lambda 2^{\alpha j}\Vert \Delta _j \psi ^{(n)}(t)\Vert _{L^p}^p \end{aligned}$$

(3.8)

for some $\lambda =\lambda (\alpha ,p)>0$. On the other hand, it follows from the divergence free condition $\nabla \cdot u^{(n-1)}=0$ that

$$\begin{aligned} \int _{{\mathbb {R}}^2} |\Delta _j \psi ^{(n)}(t,x)|^{p-2}\Delta _j \psi ^{(n)}(t,x) u^{(n-1)}(t,x)\cdot \nabla \Delta _j \psi ^{(n)}(t,x) dx=0. \end{aligned}$$

(3.9)

Substituting (3.8) and (3.9) in (3.7), we have

$$\begin{aligned} \begin{aligned}&\frac{d}{dt}\Vert \Delta _j\psi ^{(n)}(t)\Vert _{L^p} +\lambda 2^{\alpha j}\Vert \Delta _j \psi ^{(n)}(t)\Vert _{L^p}\\&\qquad \leqslant C\Vert \Delta _j F(t)\Vert _{L^p} +\Vert [u^{(n-1)}(t),\Delta _j]\cdot \nabla \psi ^{(n)}(t)\Vert _{L^p}\\&\quad \qquad +\Vert \Delta _j(u^{(n-1)}(t)\cdot \nabla e^{-t(-\Delta )^{\frac{\alpha }{2}}}\theta ^{(n)}(0))\Vert _{L^p},\\ \end{aligned} \end{aligned}$$

which implies that

$$\begin{aligned} \Vert \Delta _j\psi ^{(n)}(T)\Vert _{L^p}&\leqslant C\int _0^T e^{-\lambda 2^{\alpha j}(T-\tau )}\Vert \Delta _j F(\tau )\Vert _{L^p} d\tau \nonumber \\&\quad +\int _0^T e^{-\lambda 2^{\alpha j}(T-\tau )}\Vert [u^{(n-1)}(\tau ),\Delta _j]\cdot \nabla \psi ^{(n)}(\tau )\Vert _{L^p} d\tau \nonumber \\&\quad +\int _0^T e^{-\lambda 2^{\alpha j}(T-\tau )}\Vert \Delta _j(u^{(n-1)}(\tau )\cdot \nabla e^{-\tau (-\Delta )^{\frac{\alpha }{2}}}\theta ^{(n)}(0))\Vert _{L^p} d\tau .\nonumber \\ \end{aligned}$$

(3.10)

Let $s\in \{s_\mathrm{c},s_\mathrm{c}-\alpha \}$. Multiplying (3.10) by $2^{sj}$, we have

$$\begin{aligned}&2^{sj}\Vert \Delta _j\psi ^{(n)}(T)\Vert _{L^p}\nonumber \\&\qquad \leqslant C\int _0^T 2^{(s+\frac{2}{r}-\frac{2}{p})j}e^{-\lambda 2^{\alpha j}(T-\tau )}\Vert \Delta _j F(\tau )\Vert _{L^r} d\tau \nonumber \\&\qquad \quad +\int _0^T 2^{(\alpha +s-s_\mathrm{c})j}e^{-\lambda 2^{\alpha j}(T-\tau )} d\tau \nonumber \\&\qquad \qquad \qquad \times 2^{(2s_\mathrm{c}-1-\frac{2}{p})j}\Vert [u^{(n-1)},\Delta _j]\cdot \nabla \psi ^{(n)}\Vert _{L^{\infty }(0,T;L^p)}\nonumber \\&\qquad \quad +\int _0^T 2^{(\alpha +s-s_\mathrm{c})j}e^{-\lambda 2^{\alpha j}(T-\tau )}\nonumber \\&\qquad \qquad \qquad \times 2^{(2s_\mathrm{c}-1-\frac{2}{p})j} \Vert \Delta _j(u^{(n-1)}(\tau )\cdot \nabla e^{-\tau (-\Delta )^{\frac{\alpha }{2}}}\theta ^{(n)}(0))\Vert _{L^p} d\tau .\nonumber \\ \end{aligned}$$

(3.11)

Taking $l^{q}({\mathbb {Z}})$-norm of (3.11), we see by (2.12) that

$$\begin{aligned} \begin{aligned}&\Vert \psi ^{(n)}(T)\Vert _{{\dot{B}}_{p,q}^s}\\&\leqslant C\sum _{j\in {\mathbb {Z}}}\int _0^T 2^{(s+\frac{2}{r}-\frac{2}{p})j}e^{-\lambda 2^{\alpha j}(T-\tau )}d\tau \sup _{t>0}\Vert F(t)\Vert _{{\dot{B}}_{r,\infty }^0}\\&\quad +CT^{\frac{s_\mathrm{c}-s}{\alpha }} \left\| \left\{ 2^{(2s_\mathrm{c}-1-\frac{2}{p})j}\Vert [u^{(n-1)},\Delta _j]\cdot \nabla \psi ^{(n)}\Vert _{L^{\infty }(0,T;L^p)} \right\} _{j\in {\mathbb {Z}}}\right\| _{l^q({\mathbb {Z}})}\\&\quad +\left\| \left\{ \int _0^T 2^{(\alpha +s-s_\mathrm{c})j}e^{-\lambda 2^{\alpha j}(T-\tau )}\right. \right. \\&\qquad \quad \left. \left. \times 2^{(2s_\mathrm{c}-1-\frac{2}{p})j} \Vert \Delta _j(u^{(n-1)}(\tau )\cdot \nabla e^{-\tau (-\Delta )^{\frac{\alpha }{2}}}\theta ^{(n)}(0))\Vert _{L^p} d\tau \right\} _{j\in {\mathbb {Z}}}\right\| _{l^q({\mathbb {Z}})}. \end{aligned} \end{aligned}$$

Hence, it follows from Lemma 2.3 and (2.5) that

$$\begin{aligned} \begin{aligned} \Vert \psi ^{(n)}(T)\Vert _{{\dot{B}}_{p,q}^s}&\leqslant CT^{1-\frac{1}{\alpha }(s+\frac{2}{r}-\frac{2}{p})} \sup _{t>0}\Vert F(t)\Vert _{{\dot{B}}_{r,\infty }^0}\\&\quad +CT^{\frac{s_\mathrm{c}-s}{\alpha }} \Vert u^{(n-1)}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_\mathrm{c}})} \Vert \psi ^{(n)}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_\mathrm{c}})}\\&\quad +CT^{\frac{s_\mathrm{c}-s}{\alpha }} (T^{-\frac{1-\alpha }{\alpha }}+1) \Vert u^{(n-1)}\Vert _{X_T^{p,q}} \Vert \theta ^{(n)}(0)\Vert _{{\dot{B}}_{p,q}^{s_\mathrm{c}}}. \end{aligned} \end{aligned}$$

Using

$$\begin{aligned} \begin{aligned} \Vert \psi ^{(n)}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_\mathrm{c}})}&\leqslant \Vert \theta ^{(n)}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_\mathrm{c}})} +\Vert e^{-t(-\Delta )^{\frac{\alpha }{2}}}\theta ^{(n)}(0)\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_\mathrm{c}})}\\&\leqslant \Vert \theta ^{(n)}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_\mathrm{c}})} +C\Vert \theta ^{(n)}(0)\Vert _{{\dot{B}}_{p,q}^{s_\mathrm{c}}}\\&\leqslant \Vert \theta ^{(n)}\Vert _{X_T^{p,q}} +C\Vert \theta _0^{(n)}\Vert _{{\dot{B}}_{p,q}^{s_\mathrm{c}}}\\&\leqslant CA_n \end{aligned} \end{aligned}$$

and the boundedness of the Riesz transform on the homogeneous space-time Besov spaces, we obtain

$$\begin{aligned} \Vert \psi ^{(n)}(T)\Vert _{{\dot{B}}_{p,q}^s}\leqslant & {} CT^{1-\frac{1}{\alpha }(s+\frac{2}{r}-\frac{2}{p})} \sup _{t>0}\Vert F(t)\Vert _{{\dot{B}}_{r,\infty }^0}\nonumber \\&\quad +CT^{\frac{s_\mathrm{c}-s}{\alpha }} \Vert \theta ^{(n-1)}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_\mathrm{c}})} A_n\nonumber \\&\quad +CT^{\frac{s_\mathrm{c}-s}{\alpha }} (T^{-\frac{1-\alpha }{\alpha }}+1) \Vert \theta ^{(n-1)}\Vert _{X_T^{p,q}} \Vert \theta _0^{(n)}\Vert _{{\dot{B}}_{p,q}^{s_\mathrm{c}}}. \end{aligned}$$

(3.12)

Hence, it follows from (3.12) and Lemma 2.6 that

$$\begin{aligned} \Vert \theta _0^{(n+1)}\Vert _{{\dot{B}}_{p,q}^{s_\mathrm{c}}}\leqslant & {} C(T^{-1}\Vert \psi ^{(n)}(T)\Vert _{{\dot{B}}_{p,q}^{s_\mathrm{c}-\alpha }} +\Vert \psi ^{(n)}(T)\Vert _{{\dot{B}}_{p,q}^{s_\mathrm{c}}})\nonumber \\\leqslant & {} CT^{\frac{1}{\alpha }(2\alpha -1-\frac{2}{r})}\sup _{t>0}\Vert F(t)\Vert _{{\dot{B}}_{r,\infty }^0} +C(1+T^{-\frac{1-\alpha }{\alpha }})A_{n-1}A_n.\quad \end{aligned}$$

(3.13)

Let $s':=s-s_\mathrm{c}\in \{-\alpha ,0\}$. Multiplying (3.10) by $2^{s'j}$ and

taking $l^1({\mathbb {Z}})$-norm, we see that

$$\begin{aligned} \begin{aligned}&\Vert \psi ^{(n)}(T)\Vert _{{\dot{B}}_{p,1}^{s'}}\\&\qquad \leqslant C\sum _{j\in {\mathbb {Z}}}\int _0^T 2^{(s'+\frac{2}{r}-\frac{2}{p})j}e^{-\lambda 2^{\alpha j}(T-\tau )}d\tau \sup _{t>0}\Vert F(t)\Vert _{{\dot{B}}_{r,\infty }^0}\\&\qquad \quad +\sum _{j\in {\mathbb {Z}}}\int _0^T 2^{(\alpha +s-2s_\mathrm{c})j}e^{-\lambda 2^{\alpha j}(T-\tau )} d\tau \\&\qquad \qquad \qquad \times \left\| \left\{ 2^{(2s_\mathrm{c}-1-\frac{2}{p})j}\Vert [u^{(n-1)},\Delta _j]\cdot \nabla \psi ^{(n)}\Vert _{L^{\infty }(0,T;L^p)} \right\} _{j\in {\mathbb {Z}}}\right\| _{l^q({\mathbb {Z}})}\\&\qquad \quad +\sum _{j\in {\mathbb {Z}}}\int _0^T 2^{(\alpha +s-2s_\mathrm{c})j}e^{-\lambda 2^{\alpha j}(T-\tau )}\\&\qquad \qquad \qquad \times \Vert u^{(n-1)}(\tau )\cdot \nabla e^{-\tau (-\Delta )^{\frac{\alpha }{2}}}\theta ^{(n)}(0)\Vert _ {{\dot{B}}_{p,q}^{2s_\mathrm{c}-1-\frac{2}{p}}} d\tau . \end{aligned} \end{aligned}$$

It follows from Lemma 2.3 and (2.6) that

$$\begin{aligned} \begin{aligned} \Vert \psi ^{(n)}(T)\Vert _{{\dot{B}}_{p,1}^{s'}}&\leqslant CT^{1-\frac{1}{\alpha }(s'+\frac{2}{r}-\frac{2}{p})} \sup _{t>0}\Vert F(t)\Vert _{{\dot{B}}_{r,\infty }^0}\\&\quad +CT^{\frac{2s_\mathrm{c}-s}{\alpha }} \Vert u^{(n-1)}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_\mathrm{c}})} \Vert \psi ^{(n)}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_\mathrm{c}})}\\&\quad +CT^{\frac{2s_\mathrm{c}-s}{\alpha }} (T^{-\frac{1-\alpha }{\alpha }}+1) \Vert u^{(n-1)}\Vert _{X_T^{p,q}} \Vert \theta ^{(n)}(0)\Vert _{{\dot{B}}_{p,q}^{s_\mathrm{c}}}. \end{aligned} \end{aligned}$$

Therefore, by the same argument as above, we have

$$\begin{aligned} \begin{aligned} \Vert \psi ^{(n)}(T)\Vert _{{\dot{B}}_{p,q}^{s'}}&\leqslant CT^{\frac{s_\mathrm{c}}{\alpha }}T^{\frac{s_\mathrm{c}-s}{\alpha }}T^{\frac{1}{\alpha }(2\alpha -1-\frac{2}{r})} \sup _{t>0}\Vert F(t)\Vert _{{\dot{B}}_{r,\infty }^0}\\&\quad +CT^{\frac{s_\mathrm{c}}{\alpha }}T^{\frac{s_\mathrm{c}-s}{\alpha }}(1+T^{-\frac{1-\alpha }{\alpha }})A_{n-1}A_n \end{aligned} \end{aligned}$$

and we see that the series in (3.2) converges in ${\dot{B}}_{p,1}^0({\mathbb {R}}^2)$ and

$$\begin{aligned} \Vert \theta _0^{(n+1)}\Vert _{{\dot{B}}_{p,1}^{0}}\leqslant & {} C(T^{-1}\Vert \psi ^{(n)}(T)\Vert _{{\dot{B}}_{p,1}^{-\alpha }} +\Vert \psi ^{(n)}(T)\Vert _{{\dot{B}}_{p,1}^{0}})\nonumber \\\leqslant & {} CT^{\frac{s_\mathrm{c}}{\alpha }}T^{\frac{1}{\alpha }(2\alpha -1-\frac{2}{r})}\sup _{t>0}\Vert F(t)\Vert _{{\dot{B}}_{r,\infty }^0} +CT^{\frac{s_\mathrm{c}}{\alpha }}(1+T^{-\frac{1-\alpha }{\alpha }}) A_{n-1}A_n.\nonumber \\ \end{aligned}$$

(3.14)

Combining (3.12) and (3.14), we find that Lemma 2.6 implies the series in (3.2) converges in ${\dot{B}}_{p,1}^0({\mathbb {R}}^2)\cap {\dot{B}}_{p,q}^{s_\mathrm{c}}({\mathbb {R}}^2)$ and (3.3) also holds.

Next, we prove (3.4). Since $\Delta _j \theta ^{(n+1)}$ satisfies

$$\begin{aligned} \begin{aligned}&\partial _t \Delta _j \theta ^{(n+1)} +(-\Delta )^{\frac{\alpha }{2}}\Delta _j\theta ^{(n+1)}\\&\qquad = S_{n+4}\Delta _jF +[u^{(n)},\Delta _j] \cdot \nabla \theta ^{(n+1)} -u^{(n)} \cdot \nabla \Delta _j \theta ^{(n+1)}, \end{aligned} \end{aligned}$$

the same argument as in the derivation of (3.10) yields that

$$\begin{aligned} \Vert \Delta _j \theta ^{(n+1)}(t)\Vert _{L^p}\leqslant & {} e^{-\lambda 2^{\alpha j}t}\Vert \Delta _j\theta ^{(n)}(0)\Vert _{L^p} +C\int _0^t e^{-\lambda 2^{\alpha j}(t-\tau )}\Vert \Delta _jF(\tau )\Vert _{L^p} d\tau \nonumber \\&\quad +\int _0^t e^{\lambda 2^{\alpha j}(t-\tau )}\Vert [u^{(n)}(\tau ),\Delta _j] \cdot \nabla \theta ^{(n+1)}(\tau )\Vert _{L^p} d\tau , \end{aligned}$$

(3.15)

which implies that

$$\begin{aligned}&2^{s_\mathrm{c}j}\Vert \Delta _j\theta ^{n+1}(t)\Vert _{L^p}\nonumber \\&\quad \leqslant C2^{s_\mathrm{c}j}\Vert \Delta _j \theta _0^{(n+1)}\Vert _{L^p} +C\int _0^t 2^{(1+\frac{2}{r}-\alpha )j}e^{-\lambda 2^{\alpha j}(t-\tau )}\Vert \Delta _jF(\tau )\Vert _{L^r} d\tau \nonumber \\&\qquad +\int _0^t 2^{\alpha j}e^{-\lambda 2^{\alpha j}(t-\tau )} d\tau 2^{(2s_\mathrm{c}-1-\frac{2}{p})j}\Vert [u^{(n)},\Delta _j] \cdot \nabla \theta ^{(n+1)}\Vert _{L^{\infty }(0,T;L^p)}.\quad \end{aligned}$$

(3.16)

Taking $L^{\infty }_t(0,T)$-norm of (3.16) and then taking $l^{q}({\mathbb {Z}})$-norm, we see by Lemma2.3 that

$$\begin{aligned}&\Vert \theta ^{(n+1)}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_\mathrm{c}})}\nonumber \\&\quad \leqslant C\Vert \theta _0^{(n+1)}\Vert _{{\dot{B}}_{p,q}^{s_\mathrm{c}}} +CT^{\frac{1}{\alpha }(2\alpha -1-\frac{2}{r})}\sup _{t>0}\Vert F(t)\Vert _{{\dot{B}}_{r,\infty }^0}\nonumber \\&\qquad +C\left\| \left\{ 2^{(2s_\mathrm{c}-1-\frac{2}{p})j}\Vert [u^{(n)},\Delta _j] \cdot \nabla \theta ^{(n+1)}\Vert _{L^{\infty }(0,T;L^p)} \right\} _{j\in {\mathbb {Z}}}\right\| _{l^{q}({\mathbb {Z}})}\nonumber \\&\quad \leqslant C\Vert \theta _0^{(n+1)}\Vert _{{\dot{B}}_{p,q}^{s_\mathrm{c}}} +CT^{\frac{1}{\alpha }(2\alpha -1-\frac{2}{r})}\sup _{t>0}\Vert F(t)\Vert _{{\dot{B}}_{r,\infty }^0}\nonumber \\&\qquad +C\Vert \theta ^{(n)}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_\mathrm{c}})} \Vert \theta ^{(n+1)}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_\mathrm{c}})}. \end{aligned}$$

(3.17)

On the other hand, by (3.15), we see that

$$\begin{aligned}&\Vert \Delta _j\theta ^{n+1}(t)\Vert _{L^p}\nonumber \\&\quad \leqslant C\Vert \Delta _j\theta _0^{(n+1)}\Vert _{L^p} +C\int _0^t 2^{(\frac{2}{r}-\frac{2}{p})j}e^{-\lambda 2^{\alpha j}(t-\tau )}d\tau \sup _{t>0}\Vert F(t)\Vert _{{\dot{B}}_{r,\infty }^0}\nonumber \\&\qquad +\int _0^t 2^{(\alpha -s_\mathrm{c})j}e^{-\lambda 2^{\alpha j}(t-\tau )} d\tau \nonumber \\&\qquad \quad \times \left\| \left\{ 2^{(2s_\mathrm{c}-1-\frac{2}{p})j}\Vert [u^{(n)},\Delta _j] \cdot \nabla \theta ^{(n+1)}\Vert _{L^{\infty }(0,T;L^p)} \right\} _{j\in {\mathbb {Z}}}\right\| _{l^q({\mathbb {Z}})}. \end{aligned}$$

(3.18)

Taking $L^{\infty }_t(0,T)$-norm and then $l^1({\mathbb {Z}})$-norm of (3.18), we have

$$\begin{aligned} \begin{aligned}&\Vert \theta ^{(n+1)}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,1}^0)}\\&\quad \leqslant C\Vert \theta _0^{(n+1)}\Vert _{{\dot{B}}_{p,1}^0} +C\sum _{j\in {\mathbb {Z}}}\sup _{0\leqslant t\leqslant T} \int _0^t 2^{(1+\frac{2}{r}-\alpha )j}e^{-\lambda 2^{\alpha j}(t-\tau )}d\tau \sup _{t>0}\Vert F(t)\Vert _{{\dot{B}}_{r,\infty }^0}\\&\qquad +\sum _{j\in {\mathbb {Z}}}\sup _{0\leqslant t\leqslant T} \int _0^t 2^{(\alpha -s_\mathrm{c})j}e^{-\lambda 2^{\alpha j}(t-\tau )} d\tau \\&\qquad \qquad \qquad \times \left\| \left\{ 2^{(2s_\mathrm{c}-1-\frac{2}{p})j}\Vert [u^{(n)},\Delta _j] \cdot \nabla \theta ^{(n+1)}\Vert _{L^{\infty }(0,T;L^p)} \right\} _{j\in {\mathbb {Z}}}\right\| _{l^q({\mathbb {Z}})}. \end{aligned} \end{aligned}$$

Using Lemma 2.3 and the second inequality of (2.16), we obtain that

$$\begin{aligned} \Vert \theta ^{(n+1)}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,1}^0)}\leqslant & {} C\Vert \theta _0^{(n+1)}\Vert _{{\dot{B}}_{p,1}^0} +CT^{\frac{s_\mathrm{c}}{\alpha }}T^{\frac{1}{\alpha }(2\alpha -1-\frac{2}{r})}\sup _{t>0}\Vert F(t)\Vert _{{\dot{B}}_{r,\infty }^0}\nonumber \\&\qquad +CT^{\frac{s_\mathrm{c}}{\alpha }} \Vert \theta ^{(n)}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_\mathrm{c}})} \Vert \theta ^{(n+1)}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_\mathrm{c}})}.\quad \end{aligned}$$

(3.19)

Hence, combining estimates (3.12), (3.14), (3.17) and (3.19), we obtain

$$\begin{aligned} A_{n+1} \leqslant C_1 \sup _{t>0} \Vert F(t)\Vert _{{\dot{B}}_{r,\infty }^0} +C_1 A_{n-1}A_n +C_1 A_nA_{n+1} \end{aligned}$$

(3.20)

for some $C_1=C_1(\alpha ,p,q,r,T)$.

On the other hand, since $\theta ^{(1)}$ satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \theta ^{(1)}+(-\Delta )^{\frac{\alpha }{2}}\theta ^{(1)}=S_4F, \qquad &{} t>0,x\in {\mathbb {R}}^2,\\ \theta ^{(1)}(0,x)=0, \qquad &{} x\in {\mathbb {R}}^2, \end{array}\right. } \end{aligned}$$

the simpler argument than above yields that

$$\begin{aligned} A_1\leqslant C_1\sup _{t>0}\Vert F(t)\Vert _{{\dot{B}}_{r,\infty }^0}. \end{aligned}$$

(3.21)

Hence, if F satisfies

$$\begin{aligned} \sup _{t>0}\Vert F(t)\Vert _{{\dot{B}}_{r,\infty }^0} \leqslant \delta _1:=\frac{1}{8C_1^2}, \end{aligned}$$

then by (3.20), (3.21) and the inductive argument, we obtain

$$\begin{aligned} A_m\leqslant 2C_1\sup _{t>0}\Vert F(t)\Vert _{{\dot{B}}_{r,\infty }^0} \end{aligned}$$

for all $m\in {\mathbb {N}}\cup \{0\}$. This completes the proof. $\square$

Next lemma ensures the convergence of the approximation sequences.

Lemma 3.2

There exists a positive constant $\delta _2=\delta _2(\alpha ,p,q,r,\sigma ,T)\leqslant \delta _1$ such that if $F\in BC((0,\infty );{\dot{B}}_{r,\infty }^0({\mathbb {R}}^2))$ satisfies

$$\begin{aligned} \sup _{t>0}\Vert F(t)\Vert _{{\dot{B}}_{r,\infty }^0}\leqslant \delta _2, \end{aligned}$$

then it holds

$$\begin{aligned} \sum _{n=0}^{\infty }\Vert \theta _0^{(n+1)}-\theta _0^{(n)}\Vert _{{\dot{B}}_{p,q}^{\sigma }} +\sum _{n=0}^{\infty }\Vert \theta ^{(n+1)}-\theta ^{(n)}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{\sigma })}<\infty . \end{aligned}$$

Proof

Due to

$$\begin{aligned} \theta _0^{(n+2)}-\theta _0^{(n+1)} =\sum _{k=0}^{\infty } e^{-Tk(-\Delta )^{\frac{\alpha }{2}}} (\psi ^{(n+1)}(T)-\psi ^{(n)}(T)) \end{aligned}$$

and Lemma 2.6, we consider the estimates of $\psi ^{(n+1)}(T)-\psi ^{(n)}(T)$ in ${\dot{B}}_{p,q}^{\sigma -\alpha }({\mathbb {R}}^2)\cap {\dot{B}}_{p,q}^{\sigma }({\mathbb {R}}^2)$. Since $\psi ^{(n+1)}-\psi ^{(n)}$ satisfies

$$\begin{aligned}&\partial _t (\psi ^{(n+1)}-\psi ^{(n)}) +(-\Delta )^{\frac{\alpha }{2}} (\psi ^{(n+1)}-\psi ^{(n)}) +u^{(n)} \cdot \nabla (\psi ^{(n+1)}-\psi ^{(n)})\nonumber \\&\quad +u^{(n)} \cdot \nabla e^{-t(-\Delta )^{\frac{\alpha }{2}}}(\theta ^{(n+1)}(0)-\theta ^{(n)}(0)) +(u^{(n)}-u^{(n-1)}) \cdot \nabla \theta ^{(n)} =\Delta _{n+1}F,\nonumber \\ \end{aligned}$$

(3.22)

we see that

$$\begin{aligned} \begin{aligned}&\partial _t \Delta _j(\psi ^{(n+1)}-\psi ^{(n)}) +(-\Delta )^{\frac{\alpha }{2}} \Delta _j(\psi ^{(n+1)}-\psi ^{(n)})\\&\quad =\Delta _{n+1}\Delta _jF +[u^{(n)},\Delta _j]\cdot \nabla (\psi ^{(n+1)}-\psi ^{(n)}) -u^{(n)} \cdot \nabla \Delta _j(\psi ^{(n+1)}-\psi ^{(n)})\\&\qquad -\Delta _j(u^{(n)} \cdot \nabla e^{-t(-\Delta )^{\frac{\alpha }{2}}}(\theta ^{(n+1)}(0)-\theta ^{(n)}(0))) -\Delta _j((u^{(n)}-u^{(n-1)}) \cdot \nabla \theta ^{(n)}). \end{aligned} \end{aligned}$$

Thus, the similar energy calculation as in the proof of Lemma 3.1 yields that

$$\begin{aligned} \begin{aligned}&\Vert \Delta _j(\psi ^{(n+1)}(T)-\psi ^{(n)}(T))\Vert _{L^p}\\&\quad \leqslant C\int _0^T e^{-\lambda 2^{\alpha j}(T-\tau )}2^{2(\frac{1}{r}-\frac{1}{p})j}\Vert \Delta _{n+1}\Delta _j F(\tau )\Vert _{L^r} d\tau \\&\qquad +\int _0^T e^{-\lambda 2^{\alpha j}(T-\tau )}\Vert [u^{(n)}(\tau ),\Delta _j]\cdot \nabla (\psi ^{(n+1)}(\tau )-\psi ^{(n)}(\tau ))\Vert _{L^p} d\tau \\&\qquad +\int _0^T e^{-\lambda 2^{\alpha j}(T-\tau )} \Vert \Delta _j(u^{(n)}(\tau ) \cdot \nabla e^{-\tau (-\Delta )^{\frac{\alpha }{2}}}(\theta ^{(n+1)}(0)-\theta ^{(n)}(0)))\Vert _{L^p} d\tau \\&\qquad +\int _0^T e^{-\lambda 2^{\alpha j}(T-\tau )}\Vert \Delta _j((u^{(n)}(\tau )-u^{(n-1)}(\tau )) \cdot \nabla \theta ^{(n)}(\tau ))\Vert _{L^p} d\tau . \end{aligned} \end{aligned}$$

Let $s\in \{\sigma ,\sigma -\alpha \}$. Multiplying this by $2^{sj}$ and

taking $l^q({\mathbb {Z}})$-norm of this, we obtain that

$$\begin{aligned} \begin{aligned}&\Vert \psi ^{(n+1)}(T)-\psi ^{(n)}(T)\Vert _{{\dot{B}}_{p,q}^s}\\&\quad \leqslant CT^{\frac{1}{\alpha }(2\alpha -1-\frac{2}{r})}2^{-(s_\mathrm{c}-\sigma )n}\sup _{t>0}\Vert F(t)\Vert _{{\dot{B}}_{r,\infty }^0}\\&\qquad +CT^{\frac{\sigma -s}{\alpha }} \left\| \left\{ 2^{(s_\mathrm{c}+(\sigma -1)-\frac{2}{p})j} \Vert [u^{(n)},\Delta _j]\cdot \nabla (\psi ^{(n+1)}-\psi ^{(n)})\Vert _{L^{\infty }(0,T;L^p)} \right\} _{j\in {\mathbb {Z}}}\right\| _{l^q({\mathbb {Z}})}\\&\qquad +\left\| \left\{ \int _0^T 2^{(\alpha +s-\sigma )j} e^{-\lambda 2^{\alpha j}(T-\tau )}2^{(s_\mathrm{c}+(\sigma -1)-\frac{2}{p})j} \right. \right. \\&\qquad \qquad \quad \left. \left. \times \Vert \Delta _j(u^{(n)} \cdot \nabla e^{-t(-\Delta )^{\frac{\alpha }{2}}}(\theta ^{(n+1)}(0)-\theta ^{(n)}(0)))\Vert _{L^p} d\tau \right\} _{j\in {\mathbb {Z}}}\right\| _{l^q({\mathbb {Z}})}\\&\qquad +CT^{\frac{\sigma -s}{\alpha }}\Vert (u^{(n)}-u^{(n-1)}) \cdot \nabla \theta ^{(n)}\Vert _ {{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{\sigma +(s_\mathrm{c}-1)-\frac{2}{p}})}. \end{aligned} \end{aligned}$$

Therefore, it follows from Lemmas 2.2, 2.3 and (2.5) that

$$\begin{aligned}&\Vert \psi ^{(n+1)}(T)-\psi ^{(n)}(T)\Vert _{{\dot{B}}_{p,q}^s}\nonumber \\&\quad \leqslant CT^{\frac{\sigma -s}{\alpha }}T^{\frac{1}{\alpha }(2\alpha -1-\frac{2}{r})}2^{-(s_\mathrm{c}-\sigma )n} \sup _{t>0}\Vert F(t)\Vert _{{\dot{B}}_{r,\infty }^0}\nonumber \\&\qquad +CT^{\frac{\sigma -s}{\alpha }} \Vert \theta ^{(n)}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_\mathrm{c}})} \Vert \psi ^{(n+1)}-\psi ^{(n)}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{\sigma })}\nonumber \\&\qquad +CT^{\frac{\sigma -s}{\alpha }} (1+T^{-\frac{1-\alpha }{\alpha }}) \Vert \theta ^{(n)}\Vert _{X_T^{p,q}} \Vert \theta ^{(n)}(0)-\theta ^{(n-1)}(0)\Vert _{{\dot{B}}_{p,q}^{\sigma }}\nonumber \\&\qquad +CT^{\frac{\sigma -s}{\alpha }} \Vert \theta ^{(n)}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_\mathrm{c}})} \Vert \theta ^{(n)}-\theta ^{(n-1)}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{\sigma })}. \end{aligned}$$

(3.23)

This gives

$$\begin{aligned}&\Vert \theta _0^{(n+1)}-\theta _0^{(n)}\Vert _{{\dot{B}}_{p,q}^{\sigma }}\nonumber \\&\quad \leqslant C(T^{-1}\Vert \psi ^{(n+1)}(T)-\psi ^{(n)}(T)\Vert _{{\dot{B}}_{p,q}^{\sigma -\alpha }}+\Vert \psi ^{(n+1)}(T)-\psi ^{(n)}(T)\Vert _{{\dot{B}}_{p,q}^{\sigma }})\nonumber \\&\quad \leqslant CT^{\frac{1}{\alpha }(2\alpha -1-\frac{2}{r})}2^{-(s_\mathrm{c}-\sigma )n} \sup _{t>0}\Vert F(t)\Vert _{{\dot{B}}_{r,\infty }^0}\nonumber \\&\qquad +CA_nB_n +C(1+T^{-\frac{1-\alpha }{\alpha }})A_nB_{n-1}. \end{aligned}$$

(3.24)

Since $\theta ^{(n+2)}-\theta ^{(n+1)}$ satisfy

$$\begin{aligned} \begin{aligned}&\partial _t (\theta ^{(n+2)}- \theta ^{(n+1)}) + (-\Delta )^{\frac{\alpha }{2}}(\theta ^{(n+2)}- \theta ^{(n+1)})\\&\qquad + u^{(n+1)} \cdot \nabla (\theta ^{(n+2)}- \theta ^{(n+1)}) + (u^{(n+1)}-u^{(n)}) \cdot \nabla \theta ^{(n+1)} = \Delta _{n+2}F, \end{aligned} \end{aligned}$$

we see that

$$\begin{aligned} \begin{aligned}&\partial _t \Delta _j(\theta ^{(n+2)}- \theta ^{(n+1)}) + (-\Delta )^{\frac{\alpha }{2}}\Delta _j(\theta ^{(n+2)}- \theta ^{(n+1)})\\&\quad = \Delta _j\Delta _{n+2}F + [u^{(n+1)}, \Delta _j ]\cdot \nabla (\theta ^{(n+2)}- \theta ^{(n+1)})\\&\qquad - u^{(n+1)} \cdot \nabla \Delta _j(\theta ^{(n+2)}-\theta ^{(n+1)}) - \Delta _j((u^{(n+1)}-u^{(n)}) \cdot \nabla \theta ^{(n+1)}). \end{aligned} \end{aligned}$$

By the similar energy calculation as in the proof of Lemma 3.1, we have

$$\begin{aligned}&\Vert \Delta _j(\theta ^{(n+2)}(t)- \theta ^{(n+1)}(t))\Vert _{L^p}\nonumber \\&\qquad \leqslant e^{-\lambda 2^{\alpha j}t}\Vert \Delta _j(\theta ^{(n+2)}(0)- \theta ^{(n+1)}(0))\Vert _{L^p}\nonumber \\&\qquad \quad +C\int _0^t 2^{(\frac{2}{r}-\frac{2}{p})j}e^{-\lambda 2^{\alpha j}(t-\tau )} \Vert \Delta _j\Delta _{n+2}F(\tau )\Vert _{L^r} d\tau \nonumber \\&\qquad \quad +\int _0^t e^{-\lambda 2^{\alpha j}(t-\tau )} \Vert [u^{(n+1)}(\tau ), \Delta _j ]\cdot \nabla (\theta ^{(n+2)}(\tau )-\theta ^{(n+1)}(\tau ))\Vert _{L^p} d\tau \nonumber \\&\qquad \quad +\int _0^t e^{-\lambda 2^{\alpha j}(t-\tau )} \Vert \Delta _j((u^{(n+1)}(\tau )-u^{(n)}(\tau )) \cdot \nabla \theta ^{(n+1)}(\tau ))\Vert _{L^p} d\tau . \end{aligned}$$

(3.25)

Multiplying (3.25) by $2^{\sigma j}$, we see that

$$\begin{aligned} \begin{aligned}&2^{\sigma j}\Vert \Delta _j(\theta ^{(n+2)}(t)- \theta ^{(n+1)}(t))\Vert _{L^p}\\&\leqslant 2^{\sigma j}\Vert \Delta _j(\theta ^{(n+2)}(0)- \theta ^{(n+1)}(0))\Vert _{L^p}\\&+C\int _0^t 2^{(1+\frac{2}{r}-\alpha )j}e^{-\lambda 2^{\alpha j}(t-\tau )}d\tau \sup _{t>0}\Vert \Delta _{n+2}F(t)\Vert _{{\dot{B}}_{r,\infty }^{(s_\mathrm{c}-\sigma )}}\\&+\int _0^t 2^{\alpha j}e^{-\lambda 2^{\alpha j}(t-\tau )} d\tau 2^{(s_\mathrm{c}+(\sigma -1)-\frac{2}{p})j} \Vert [u^{(n+1)}, \Delta _j ]\cdot \nabla (\theta ^{(n+2)}-\theta ^{(n+1)})\Vert _{L^{\infty }(0,T;L^p)} \\&+\int _0^t 2^{\alpha j}e^{-\lambda 2^{\alpha j}(t-\tau )} d\tau 2^{(\sigma +(s_\mathrm{c}-1)-\frac{2}{p})j} \Vert \Delta _j((u^{(n+1)}-u^{(n)}) \cdot \nabla \theta ^{(n+1)})\Vert _{L^{\infty }(0,T;L^p)}. \end{aligned} \end{aligned}$$

By taking $L^{\infty }_t(0,T)$ and then $l^q({\mathbb {Z}})$-norm, it follows from (2.16) and Lemmas 2.2 and 2.3 that

$$\begin{aligned}&\Vert \theta ^{(n+2)}- \theta ^{(n+1)}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{\sigma })}\nonumber \\&\qquad \quad \leqslant \Vert \theta ^{(n+2)}(0)- \theta ^{(n+1)}(0)\Vert _{{\dot{B}}_{p,q}^{\sigma }}\nonumber \\&\qquad \qquad +CT^{\frac{1}{\alpha }(2\alpha -1-\frac{2}{r})} 2^{-(s_\mathrm{c}-\sigma )n} \sup _{t>0}\Vert F(t)\Vert _{{\dot{B}}_{r,\infty }^0}\nonumber \\&\qquad \qquad +C\Vert \theta ^{(n+1)}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_\mathrm{c}})} \Vert \theta ^{(n+2)}- \theta ^{(n+1)}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{\sigma })} \nonumber \\&\qquad \qquad +C\Vert \theta ^{(n+1)}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_\mathrm{c}})} \Vert \theta ^{(n+1)}- \theta ^{(n)}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{\sigma })}. \end{aligned}$$

(3.26)

Since it holds

$$\begin{aligned} \begin{aligned} \Vert \theta ^{(n+2)}(0)- \theta ^{(n+1)}(0)\Vert _{{\dot{B}}_{p,q}^{\sigma }}&\leqslant \Vert S_{n+5}(\theta _0^{(n+2)}-\theta _0^{(n+1)})\Vert _{{\dot{B}}_{p,q}^{\sigma }} +\Vert \Delta _{n+2} \theta ^{(n+1)}\Vert _{{\dot{B}}_{p,q}^{\sigma }}\\&\leqslant C\Vert \theta _0^{(n+2)}-\theta _0^{(n+1)}\Vert _{{\dot{B}}_{p,q}^{\sigma }} +C 2^{-(s_\mathrm{c}-\sigma )n} \Vert \theta _0^{(n+1)}\Vert _{{\dot{B}}_{p,q}^{s_\mathrm{c}}}\\&\leqslant C\Vert \theta _0^{(n+2)}-\theta _0^{(n+1)}\Vert _{{\dot{B}}_{p,q}^{\sigma }} +C2^{-(s_\mathrm{c}-\sigma )n}A_{n+1}, \end{aligned} \end{aligned}$$

we have by (3.26) that

$$\begin{aligned}&\Vert \theta ^{(n+2)}-\theta ^{(n+1)}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{\sigma })}\nonumber \\&\qquad \leqslant C\Vert \theta _0^{(n+2)}-\theta _0^{(n+1)}\Vert _{{\dot{B}}_{p,q}^{\sigma }}\nonumber \\&\qquad \quad +C 2^{-(s_\mathrm{c}-\sigma )n} \left( T^{\frac{1}{\alpha }(2\alpha -1-\frac{2}{r})}\sup _{t>0}\Vert F(t)\Vert _{{\dot{B}}_{r,\infty }^0} +A_{n+1} \right) \nonumber \\&\qquad \quad +C \Vert \theta ^{(n+1)}\Vert _{X_T^{p,q}}\Vert \theta ^{(n+1)}-\theta ^{(n)}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{\sigma })}\nonumber \\&\qquad \quad +C \Vert \theta ^{(n+1)}\Vert _{X_T^{p,q}}\Vert \theta ^{(n+2)}-\theta ^{(n+1)}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{\sigma })}. \end{aligned}$$

(3.27)

Therefore, combining (3.24) and (3.27), we obtain

$$\begin{aligned} B_{n+1}\leqslant & {} C_2 2^{-(s_\mathrm{c}-\sigma )n} \left( \sup _{t>0}\Vert F(t)\Vert _{{\dot{B}}_{r,\infty }^0} +A_{n+1} \right) \nonumber \\&\quad +C_2 A_nB_{n-1} +C_2 (A_n+A_{n+1})B_n +C_2A_{n+1}B_{n+1} \end{aligned}$$

(3.28)

for some $C_2=C_2(\alpha ,p,q,r,\sigma ,T)>0$. Here, we assume that

$$\begin{aligned} \sup _{t>0}\Vert F(t)\Vert _{{\dot{B}}_{r,\infty }^0}\leqslant \delta _2 =:\min \left\{ \delta _1,\frac{1}{16C_1C_2} \right\} . \end{aligned}$$

Let $N\in {\mathbb {N}}$ satisfy $N\geqslant 2$. Then, summing (3.28) over $n=1,...,N-1$ and using Lemma 3.1, we have

$$\begin{aligned} \begin{aligned} \sum _{n=1}^{N-1}B_{n+1}&\leqslant C_T\delta _1 \sum _{n=1}^{N-1}2^{-(s_\mathrm{c}-\sigma )n}\\&\quad +2C_1C_2 \sum _{n=1}^{N-1}B_{n-1} +4C_1C_2 \sum _{n=1}^{N-1}B_n +2C_1C_2 \sum _{n=1}^{N-1}B_{n+1} \end{aligned} \end{aligned}$$

for some constant $C_T>0$ depending on T. This implies

$$\begin{aligned} \begin{aligned} \sum _{n=2}^{N}B_n&\leqslant C_T\delta _1 \sum _{n=1}^{N-1}2^{-(s_\mathrm{c}-\sigma )n} +\frac{1}{8}\sum _{n=0}^{N-2}B_n +\frac{2}{8} \sum _{n=1}^{N-1}B_n +\frac{1}{8}\sum _{n=2}^{N}B_n\\&\leqslant C_T\delta _1 \sum _{n=1}^{\infty }2^{-(s_\mathrm{c}-\sigma )n} +\frac{1}{2}\sum _{n=0}^{N}B_n.\\ \end{aligned} \end{aligned}$$

Hence, we have

$$\begin{aligned} \frac{1}{2}\sum _{n=0}^{\infty }B_n \leqslant C_T\delta _1\sum _{n=1}^{\infty }2^{-(s_\mathrm{c}-\sigma )n}+B_0+B_1 <\infty , \end{aligned}$$

which completes the proof. $\square$

Lemma 3.3

There exists a positive constant $C_3=C_3(\alpha ,p,q,r,T,\sigma )$ such that

$$\begin{aligned}&\Vert \theta -{\widetilde{\theta }}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{\sigma })}\nonumber \\&\quad \leqslant C_3 \left( \Vert \theta \Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_\mathrm{c}})} +\Vert {\widetilde{\theta }}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_\mathrm{c}})} \right) \Vert \theta -{\widetilde{\theta }}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{\sigma })} \end{aligned}$$

(3.29)

for all T-time periodic solutions $\theta \in BC([0,\infty );B_{p,q}^{s_\mathrm{c}}({\mathbb {R}}^2))\cap X_{T}^{p,q}$ and ${\widetilde{\theta }}\in BC([0,\infty );B_{p,q}^{s_\mathrm{c}}({\mathbb {R}}^2))\cap {\widetilde{L}}^{\infty }(0,\infty ;{\dot{B}}_{p,q}^{s_\mathrm{c}}({\mathbb {R}}^2))$ to (1.5) with the same T-time periodic external force F.

Proof

Since $\theta -{\widetilde{\theta }}$ satisfies

$$\begin{aligned} \partial _t(\theta -{\widetilde{\theta }}) +(-\Delta )^{\frac{\alpha }{2}}(\theta -{\widetilde{\theta }}) +u \cdot \nabla (\theta -{\widetilde{\theta }}) +(u-{\widetilde{u}}) \cdot \nabla {\widetilde{\theta }}=0, \end{aligned}$$

where $u={\mathcal {R}}^{\perp }\theta$, ${\widetilde{u}}={\mathcal {R}}^{\perp }{\widetilde{\theta }}$, we see that

$$\begin{aligned} \begin{aligned}&\partial _t\Delta _j(\theta -{\widetilde{\theta }}) +(-\Delta )^{\frac{\alpha }{2}}\Delta _j(\theta -{\widetilde{\theta }})\\&\qquad =[u,\Delta _j]\cdot \nabla (\theta -{\widetilde{\theta }}) -u\cdot \nabla \Delta _j(\theta -{\widetilde{\theta }}) -\Delta _j ((u-{\widetilde{u}}) \cdot \nabla {\widetilde{\theta }}). \end{aligned} \end{aligned}$$

Therefore, it follows from the similar energy calculation as in the derivation of (3.27) that

$$\begin{aligned}&\Vert \theta -{\widetilde{\theta }}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{\sigma })}\nonumber \\&\qquad \leqslant \Vert \theta (0)-{\widetilde{\theta }}(0)\Vert _{{\dot{B}}_{p,q}^{\sigma }}\nonumber \\&\qquad \quad +C\left( \Vert \theta \Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_\mathrm{c}})} +\Vert {\widetilde{\theta }}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_\mathrm{c}})} \right) \Vert \theta -{\widetilde{\theta }}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{\sigma })}. \end{aligned}$$

(3.30)

Next, we derive the estimate for $\theta (0)-{\widetilde{\theta }}(0)$. Since $\theta -{\widetilde{\theta }}$ is T-time periodic and the Duhamel principle gives

$$\begin{aligned} \theta (t)-{\widetilde{\theta }}(t) =e^{-t(-\Delta )^{\frac{\alpha }{2}}}(\theta (0)-{\widetilde{\theta }}(0))\\ -\int _0^t e^{-(t-\tau )(-\Delta )^{\frac{\alpha }{2}}}(u(\tau ) \cdot \nabla \theta (\tau )-{\widetilde{u}}(\tau ) \cdot \nabla {\widetilde{\theta }}(\tau )) d\tau , \end{aligned}$$

we have by Lemma 2.7 that

$$\begin{aligned} \Vert \theta (0)-{\widetilde{\theta }}(0)\Vert _{{\dot{B}}_{p,q}^{\sigma }} \leqslant C(T^{-1}\Vert \psi (T)-{\widetilde{\psi }}(T)\Vert _{{\dot{B}}_{p,q}^{\sigma -\alpha }} +\Vert \psi (T)-{\widetilde{\psi }}(T)\Vert _{{\dot{B}}_{p,q}^{\sigma }}), \end{aligned}$$

(3.31)

where $\psi (t):=\theta (t)-e^{-t(-\Delta )^{\frac{\alpha }{2}}}\theta (0)$ and ${\widetilde{\psi }}(t):={\widetilde{\theta }}(t)-e^{-t(-\Delta )^{\frac{\alpha }{2}}}{\widetilde{\theta }}(0)$. Since it holds

$$\begin{aligned}&\partial _t(\psi -{\widetilde{\psi }})+(-\Delta )^{\frac{\alpha }{2}}(\psi -{\widetilde{\psi }})\nonumber \\&\qquad =-u\cdot \nabla (\psi -{\widetilde{\psi }})-u \cdot \nabla e^{-t(-\Delta )^{\frac{\alpha }{2}}}(\theta (0)-{\widetilde{\theta }}(0)) -(u-{\widetilde{u}}) \cdot \nabla {\widetilde{\theta }}, \end{aligned}$$

(3.32)

applying $\Delta _j$ to (3.32), we have

$$\begin{aligned}&\partial _t\Delta _j(\psi -{\widetilde{\psi }})+(-\Delta )^{\frac{\alpha }{2}}\Delta _j(\psi -{\widetilde{\psi }})\ \\&\quad =[u,\Delta _j]\cdot \nabla (\psi -{\widetilde{\psi }})-u \cdot \nabla \Delta _j(\psi -{\widetilde{\psi }})\\&\quad -\Delta _j (u \cdot \nabla e^{-t(-\Delta )^{\frac{\alpha }{2}}}(\theta (0)-{\widetilde{\theta }}(0))) -\Delta _j ((u-{\widetilde{u}}) \cdot \nabla {\widetilde{\theta }}). \end{aligned}$$

Hence, by the similar energy calculation as in the derivation of (3.23), we obtain

$$\begin{aligned} \begin{aligned} \begin{aligned}&T^{-1}\Vert \psi (T)-{\widetilde{\psi }}(T)\Vert _{{\dot{B}}_{p,q}^{\sigma -\alpha }} +\Vert \psi (T)-{\widetilde{\psi }}(T)\Vert _{{\dot{B}}_{p,q}^{\sigma }}\\&\qquad \leqslant C(1+T^{-\frac{1-\alpha }{\alpha }})\left( \Vert \theta \Vert _{X_{T}^{p,q}} +\Vert {\widetilde{\theta }}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_\mathrm{c}})}\right) \Vert \theta -{\widetilde{\theta }}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{\sigma })}. \end{aligned} \end{aligned} \end{aligned}$$

(3.33)

Here, we have used

$$\begin{aligned} \Vert \theta (0)-{\widetilde{\theta }}(0)\Vert _{{\dot{B}}_{p,q}^{\sigma }} \leqslant \sup _{0\leqslant t\leqslant T}\Vert \theta (t)-{\widetilde{\theta }}(t)\Vert _{{\dot{B}}_{p,q}^{\sigma }} \leqslant \Vert \theta -{\widetilde{\theta }}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{\sigma })}. \end{aligned}$$

Combining (3.30), (3.31) and (3.33), we get (3.29). This completes the proof. $\square$

Now we are in a position to prove Theorem 1.3.

Proof of Theorem 1.3

Let $\alpha ,p,q,r$ and T satisfy the assumptions of Theorem 1.3 and let $\sigma :=\alpha /2$. Then, $\sigma$ satisfies $\alpha -2/p<\sigma <2/p$. We put

$$\begin{aligned} \delta :=\min \left\{ \delta _1,\delta _2,\frac{1}{8C_1C_3} \right\} \end{aligned}$$

and let $F\in BC((0,\infty );{\dot{B}}_{r,\infty }^0({\mathbb {R}}^2))$ satisfy

$$\begin{aligned} \sup _{t>0}\Vert F(t)\Vert _{{\dot{B}}_{r,\infty }^0}\leqslant \delta . \end{aligned}$$

It follows from (3.4) that

$$\begin{aligned} \sup _{n\in {\mathbb {N}}\cup \{0\}}\Vert \theta ^{(n)}\Vert _{X_T^{p,q}} \leqslant 2C_1\sup _{t>0}\Vert F(t)\Vert _{{\dot{B}}_{r,\infty }^0} \leqslant 2C_1\delta =:K. \end{aligned}$$

(3.34)

From Lemma 3.2, there exist limits $\theta _0\in {\dot{B}}_{p,q}^{\sigma }({\mathbb {R}}^2)$ and $\theta \in L^{\infty }(0,T;{\dot{B}}_{p,q}^{\sigma }({\mathbb {R}}^2))$ such that

$$\begin{aligned} \begin{aligned} \theta _0&=\sum _{n=0}^{\infty }(\theta ^{(n+1)}_0-\theta _0^{(n)})=\lim _{n\rightarrow \infty }\theta _0^{(n)} \qquad \mathrm{in }\ {\dot{B}}_{p,q}^{\sigma }({\mathbb {R}}^2),\\ \theta&=\sum _{n=0}^{\infty }(\theta ^{(n+1)}-\theta ^{(n)})=\lim _{n\rightarrow \infty }\theta ^{(n)} \qquad \mathrm{in }\ L^{\infty }(0,T;{\dot{B}}_{p,q}^{\sigma }({\mathbb {R}}^2)). \end{aligned} \end{aligned}$$

By Lemma 3.1 and (3.34), we see that $\theta _0\in {\dot{B}}_{p,q}^{s_\mathrm{c}}({\mathbb {R}}^2)$, $\theta \in X_T^{p,q}$ and

$$\begin{aligned} \Vert \theta \Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_\mathrm{c}})}\leqslant \Vert \theta \Vert _{X_T^{p,q}}\leqslant K. \end{aligned}$$

(3.35)

It is easy to check that $\theta$ is a solution to (1.5) on [0, T]. Next, we show the continuity in time of the solution $\theta$ by the idea in [7]. Let $(s,\rho )\in \{(s_\mathrm{c},q),(0,1)\}$. Since it holds $\partial _t\Delta _j\theta =\Delta _jF-(-\Delta )^{\frac{\alpha }{2}}\Delta _j\theta -\Delta _j(u\cdot \nabla \theta )$, we have

$$\begin{aligned} \begin{aligned}&\Vert \partial _t\Delta _j\theta \Vert _{{\dot{B}}_{p,\rho }^s}\\&\quad \leqslant C2^{sj}\Vert \partial _t\Delta _j\theta \Vert _{L^p}\\&\quad \leqslant C2^{sj+2(\frac{1}{r}-\frac{1}{p})j}\Vert \Delta _jF(t)\Vert _{L^r} +C2^{sj+\alpha j}\Vert \Delta _j\theta \Vert _{L^p} +C2^{sj+j}\Vert u(t)\Vert _{L^p}\Vert \theta (t)\Vert _{L^{\infty }}\\&\quad \leqslant C2^{sj+2(\frac{1}{r}-\frac{1}{p})j}\sup _{t>0}\Vert F(t)\Vert _{{\dot{B}}_{r,\infty }^0} +C2^{sj+\alpha j}\Vert \theta \Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,1}^0)} +C2^{sj+j}\Vert \theta \Vert _{X_T^{p,q}}^2, \end{aligned} \end{aligned}$$

which implies $\partial _t\Delta _j\theta \in L^{\infty }(0,T;{\dot{B}}_{p,\rho }^s({\mathbb {R}}^2))$. Therefore, we have

$$\begin{aligned} \Theta _m:=\sum _{|j|\leqslant m}\Delta _j \theta \in C([0,T];{\dot{B}}_{p,1}^0({\mathbb {R}}^2)\cap {\dot{B}}_{p,q}^{s_\mathrm{c}}({\mathbb {R}}^2)), \qquad m\in {\mathbb {N}}. \end{aligned}$$

It follows from $q<\infty$ and (3.35) that

$$\begin{aligned} \begin{aligned}&\Vert \Theta _m-\theta \Vert _{L^{\infty }(0,T;{\dot{B}}_{p,1}^0\cap {\dot{B}}_{p,q}^{s_\mathrm{c}})}\\&\quad \leqslant C\sum _{|j|\geqslant m}\Vert \Delta _j\theta \Vert _{L^{\infty }(0,T;L^p)} +C\left\| \left\{ 2^{s_\mathrm{c}j}\Vert \Delta _j\theta \Vert _{L^{\infty }(0,T;L^p)}\right\} _{\{j:|j|\geqslant m\}}\right\| _{l^q}\\&\quad \rightarrow 0, \qquad \mathrm{as}\ m\rightarrow \infty . \end{aligned} \end{aligned}$$

Hence, we see that $\theta \in C([0,T];{\dot{B}}_{p,1}^0({\mathbb {R}}^2)\cap {\dot{B}}_{p,q}^{s_\mathrm{c}}({\mathbb {R}}^2)) \subset C([0,T];B_{p,q}^{s_\mathrm{c}}({\mathbb {R}}^2))$. Since

$$\begin{aligned} \begin{aligned}&\Vert \theta ^{(n)}(0)-\theta _0\Vert _{{\dot{B}}_{p,q}^{\sigma }}\\&\quad \leqslant C\Vert S_{n+3}(\theta _0^{(n)}-\theta _0)\Vert _{{\dot{B}}_{p,q}^{\sigma }} +\Vert (1-S_{n+3})\theta _0\Vert _{{\dot{B}}_{p,q}^{\sigma }}\\&\quad \leqslant C\Vert \theta _0^{(n)}-\theta _0\Vert _{{\dot{B}}_{p,q}^{\sigma }} +\Vert (1-S_{n+3})\theta _0\Vert _{{\dot{B}}_{p,q}^{\sigma }} \rightarrow 0,\\&\Vert (1-e^{-T(-\Delta )^{\frac{\alpha }{2}}})\theta _0^{(n+1)} -(1-e^{-T(-\Delta )^{\frac{\alpha }{2}}})\theta _0\Vert _{{\dot{B}}_{p,q}^{\sigma }}\\&\quad \leqslant \Vert \theta _0^{(n+1)}-\theta _0\Vert _{{\dot{B}}_{p,q}^{\sigma }} +\Vert e^{-T(-\Delta )^{\frac{\alpha }{2}}}(\theta _0^{(n+1)}-\theta _0)\Vert _{{\dot{B}}_{p,q}^{\sigma }}\\&\quad \leqslant C\Vert \theta _0^{(n+1)}-\theta _0\Vert _{{\dot{B}}_{p,q}^{\sigma }}\rightarrow 0,\\&\Vert (\theta ^{(n)}(T)-e^{-(-\Delta )^{\frac{\alpha }{2}}}\theta ^{n}(0))-(\theta (T)-e^{-(-\Delta )^{\frac{\alpha }{2}}}\theta _0)\Vert _{{\dot{B}}_{p,q}^{\sigma }}\\&\quad \leqslant \Vert \theta ^{(n)}(T)-\theta (T)\Vert _{{\dot{B}}_{p,q}^{\sigma }}+\Vert e^{-T(-\Delta )^{\frac{\alpha }{2}}}(\theta ^{(n)}(0)-\theta _0)\Vert _{{\dot{B}}_{p,q}^{\sigma }}\\&\quad \leqslant \Vert \theta ^{(n)}-\theta \Vert _{L^{\infty }(0,T;{\dot{B}}_{p,q}^{\sigma })} +C\Vert \theta ^{(n)}(0)-\theta _0\Vert _{{\dot{B}}_{p,q}^{\sigma }} \rightarrow 0 \end{aligned} \end{aligned}$$

as $n\rightarrow \infty$, we obtain by letting $n\rightarrow \infty$ in (3.3) that

$$\begin{aligned} (1-e^{-T(-\Delta )^{\frac{\alpha }{2}}})\theta _0=\theta (T)-e^{-T(-\Delta )^{\frac{\alpha }{2}}}\theta _0,\qquad \theta (0)=\theta _0, \end{aligned}$$

which implies

$$\begin{aligned} \theta (T)=\theta (0)=\theta _0. \end{aligned}$$

(3.36)

Let us extend $\theta$ to the function on the interval $[0,\infty )$ periodically as

$$\begin{aligned} \theta (t)=\theta (t-NT),\qquad \mathrm{for}\ NT<t\leqslant (N+1)T,\ N\in {\mathbb {N}}. \end{aligned}$$

Then, $\theta \in BC([0,\infty );B_{p,q}^{s_\mathrm{c}}({\mathbb {R}}^2))$ and $\theta$ is a T-time periodic solution to (1.5) satisfying (1.4). Finally, we prove the uniqueness. Let ${\widetilde{\theta }}$ be arbitrary solution satisfying (1.7). Note that since $0<\sigma <s_c$, we see that $\theta ,{\widetilde{\theta }} \in L^{\infty }(0,T;L^p({\mathbb {R}}^2)) \cap {\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_\mathrm{c}}({\mathbb {R}}^2)) \subset {\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{\sigma }({\mathbb {R}}^2))$ holds by the similar calculation as (2.8). Then, it follows from (3.29) and (3.35) that

$$\begin{aligned} \begin{aligned} \Vert \theta -{\widetilde{\theta }}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{\sigma })}&\leqslant C_3 \left( \Vert \theta \Vert _{X_T^{p,q}} +\Vert {\widetilde{\theta }}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{s_\mathrm{c}})} \right) \Vert \theta -{\widetilde{\theta }}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{\sigma })}\\&\leqslant 2KC_3 \Vert \theta -{\widetilde{\theta }}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{\sigma })}\\&\leqslant \frac{1}{2} \Vert \theta -{\widetilde{\theta }}\Vert _{{\widetilde{L}}^{\infty }(0,T;{\dot{B}}_{p,q}^{\sigma })}. \end{aligned} \end{aligned}$$

Thus, we see that ${\widetilde{\theta }}=\theta$ on [0, T]. The periodicity of $\theta$ and ${\widetilde{\theta }}$ implies $\theta ={\widetilde{\theta }}$ on $[0,\infty )$. This completes the proof.