Global well-posedness, Gevrey class regularity and large time asymptotics for the dissipative quasi-geostrophic equation in Fourier–Besov spaces

Abstract

In this article, we consider the Cauchy problem of the dissipative quasi-geostrophic equation in critical Fourier–Besov spaces. Using the bilinear-type fixed point theory, we obtain the global well-posedness results and we prove the existence and uniqueness of analytic solutions with small initial data belonging to the critical Fourier–Besov spaces ${\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}$. In addition, we show the stability of global solutions.

1 Introduction

In this article, we consider the following Cauchy problem for the two-dimensional quasi-geostrophic equation with subcritical dissipation $\alpha >1 / 2$.

$$\begin{aligned} \left\{ \begin{array}{l} \partial _{t}v+k \Lambda ^{2 \alpha } v+u_v \cdot \nabla v=0, \quad x \in {\mathbb {R}}^{2},\;\; t>0, \\ v(0, x)=v_{0}(x), \end{array}\right. \end{aligned}$$

(1)

where $\alpha >1 / 2$ is a real number, $k>0$ is a dissipative coefficient. The scalar function v(x, t) represents the potential temperature, and $u_v=\left( -\mathfrak {R}_{2} v, \mathfrak {R}_{1}v\right)$ is the divergence free velocity $(\nabla \cdot u_v=0)$ which is determined by the Riesz transformation of v. $\Lambda$ is the operator defined by the fractional power of $-\Delta$:

$$\begin{aligned} \Lambda v=(-\Delta )^{1 / 2}v, \quad {\mathscr {F}}(\Lambda v)={\mathscr {F}}((-\Delta )^{1 / 2}v)=\vert \xi \vert {\mathscr {F}}(v), \end{aligned}$$

and more generally

$$\begin{aligned} {\mathscr {F}}(\Lambda ^{2\alpha } v)={\mathscr {F}}((-\Delta )^{\alpha }v)=\vert \xi \vert ^{2\alpha } {\mathscr {F}}(v), \end{aligned}$$

where ${\mathscr {F}}$ is the Fourier transform.

For the purpose of simplicity, we make the assumption that $k=1$ since we are interested in the dissipative case.

For fluid dynamics partial differential equations with singular data in various spaces, there is a large body of literature on global-in-time well-posedness, where the smallness conditions are considered in the norms of critical spaces (i.e., the norm is invariant under the scaling of the equation/system). For example, for Navier–Stokes equations, 2D quasi-geostrophic equations, and related models, we have well-posedness results in the critical case of the following spaces: Lebesgue space $L^{p}$ [33], Marcinkiewicz space $L^{p, \infty }$ [15, 46], Sobolev spaces $H^{s}$ [25], Lei-Lin spaces ${\mathscr {X}}^{s}$ [11, 13, 14], Besov spaces ${B}_{p, q}^{s}$ [47], Triebel-Lizorkin spaces ${F}_{p, q}^{s}$ [18], Morrey spaces ${M}^{\lambda }_{p}$ [34], Besov–Morrey spaces ${\mathscr {N}}_{p, \lambda , q}^{s}$ [38], Fourier–Besov spaces ${\mathscr {F}} {\mathscr {B}}_{p, q}^{s}$ [31], Fourier–Herz spaces ${\mathscr {F}} {\mathscr {B}}_{1, q}^{s}={\mathscr {B}}_{1, q}^{s}$ [17], Fourier–Besov–Morrey spaces ${\mathscr {F}} {\mathscr {N}}_{p, \lambda , q}^{s}$ [1,2,3,4,5,6, 23, 26], $B M O ^ { - 1 }$ [35], and pseudomeasure spaces ${\mathscr {P}}{\mathscr {M}}$ [40], among others. Moreover, in some of the above references, one can find results on regularity, decay and/or asymptotic behavior of solutions, such as the works [15, 16, 20, 26, 33, 34, 38, 46].

It is clear that the case $\alpha$ small is mathematically interesting to understand the effects of a milder dissipation in the model, in a similar spirit to studies that consider small viscosities. Moreover, although often considered a toy model, the 2D quasi-geostrophic equation with $\alpha =\frac{1}{2}$ presents some structural similarities with 3D Navier–Stokes as well as connections with geophysical models. In fact, the dissipative term being then of the form $\Lambda v$, the last one models the Ekman pumping observable in ocean flows for example. This phenomena appears when the winds blowing at the surface of the oceans compete with the Coriolis force (deflecting the water at the surface of the oceans, to the right in the northern hemisphere and to the left in the southern hemisphere) causing the water to rise to the surface. Mathematically, the power $\alpha =\frac{1}{2}$ corresponds to the index for which the nonlinear term and the dissipation are of the same order (in the sense that $u_v$ and $\Lambda v$ are two operators deriving once). The previous remarks motivate the following definition:

Definition 1

We define the following 3 cases:

• If $0 \le \alpha <\frac{1}{2}$, then Eq. (1) is called supercritical

• If $\alpha =\frac{1}{2}$, then Eq. (1) is called critical

• If $\frac{1}{2}<\alpha \le 1$, then Eq. (1) is called subcritical

Numerous works have been published in recent years on the study of the quasi-geostrophic equation, specifically the mathematical questions that naturally arise in the study of the Cauchy problem, namely questions of regularity, existence, blow-up, uniqueness, of solutions (strong, weak, mild). When we want to study questions about the existence (global or local) of solutions, it is important to know the scaling of the equation and to look for the critical spaces. For the quasi-geostrophic equation (1), we have the following scaling: if v solves (1) with initial data $v_{0}$, then $v_{\gamma }$ with $v_{\gamma }(x, t):=\gamma ^{2\alpha -1} v\left( \gamma x, \gamma ^{2\alpha } t\right)$ is also a solution to (1) with the initial data

$$\begin{aligned} \quad v_{0,\gamma }(x):=\gamma ^{2\alpha -1} v_{0}(\gamma x). \end{aligned}$$

(2)

Any Banach space $Y\subset {\mathscr {S}}^{\prime }\left( {\mathbb {R}}^{n}\right)$ whose norm is invariant under the scaling (2), i.e.,

$$\begin{aligned} \left\| v_{0,\gamma }(x)\right\| _{Y} \approx \left\| v_{0}(x)\right\| _{Y} \end{aligned}$$

is called critical for Eq. (1). We notice that the spaces $L^{\frac{2}{2 \alpha -1}}, \; {\dot{H}}^{2-2 \alpha }$, $\dot{{\mathscr {B}}}_{p, q}^{1-2\alpha +\frac{2}{p}}$ where $p, q \in [1, \infty ]$ and ${\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}$ with $p, q \in [1, \infty ]$ are critical for (1). In this respect, there are several papers on global and local existence for (1) in different critical spaces. For example:

-In the subcritical case, corresponding to the case $\frac{1}{2}<\alpha \le 1$, the global existence and uniqueness questions are well understood. This case corresponds to values of $\alpha$ for which the dissipation is an operator that derives strictly more than once, while the nonlinear term contains exactly one derivative. Constantin and Wu proved in [17] that for any regular initial data the solution remains regular for any time. We can recall for example the results in Lebesgue spaces $L^{\frac{2}{2 \alpha -1}}$ for $\alpha >\frac{1}{2}$ by Carrillo and Ferreira [16], Sobolev spaces ${\dot{H}}^{2-2 \alpha }$ for $\alpha >\frac{1}{2}$ by Ju [32]. In 2015, Benameur and Benhamed [13] proved the global existence of (QG) for small initial data and the local existence for large initial data in the Lei-Lin critical space ${\mathscr {X}}^{s}$ for $s=1-2 \alpha$, which are defined as follows [13]:

For $s\in {\mathbb {R}}$, ${\mathscr {X}}^{s}\left( {\mathbb {R}}^{n}\right) =\left\{ f \in {\mathscr {S}}^{\prime }\left( {\mathbb {R}}^{n}\right) ;\int _{{\mathbb {R}}^{n}}\vert \xi \vert ^{s}\vert {\widehat{f}}(\xi )\vert d\xi < \infty \right\} ,$ with the norm

$$\begin{aligned} \Vert f\Vert _{{\mathscr {X}}^{s}}=\int _{{\mathbb {R}}^{n}}\vert \xi \vert ^{s}\vert {\widehat{f}}(\xi )\vert d \xi . \end{aligned}$$

(3)

Recently, Benhamed and Abusalim [14] studied the asymptotic behavior of (1) in Lei-Lin space ${\mathscr {X}}^{1-2\alpha }$ with subcritical dissipation. Many other related results can be found in [12, 16, 21, 24].

-In the critical case $\alpha =\frac{1}{2}$, one of the first results (valid also in the supercritical case $0 \le \alpha <\frac{1}{2}$) was that of Resnick in 1995 [44] in which he proved the global existence of weak solutions with arbitrary large initial data $L^{\infty } \cap L^{2}$ and satisfying a "Leray" type energy inequality. In 2006, Marchand [42] extended Resnick’s weak solution existence result. He proved that for any initial data $v_{0} \in L^{p}$ with $p>\frac{4}{3}$ and for any $t \in (0, T)$, there exists a global weak solution verifying the following energy inequality:

$$\begin{aligned} \Vert v(x, t)\Vert _{p}^{p}+p k \int _{0}^{t} \int v \vert v \vert ^{p-2} \Lambda ^{2 \alpha } v d x d s \le \left\| v _{0}\right\| _{p}^{p}. \end{aligned}$$

If $v _{0} \in H^{-1 / 2}$ then there exists a solution $\left. v \in L^{1 / 2}\left( (0, T), H^{-1 / 2}\right) \cap L^{2}, H^{\alpha -1 / 2}\right)$ satisfying for all $t \in (0, T)$ the energy inequality

$$\begin{aligned} \Vert v (x, t)\Vert _{H^{-1 / 2}}+p k \int _{0}^{t} \int \left| \Lambda ^{\alpha -\frac{1}{2}} v \right| ^{2} d x d s \le \left\| v _{0}\right\| _{H^{-1 / 2}}^{2}. \end{aligned}$$

-In the supercritical case $(0 \le \alpha <\frac{1}{2})$. We have global existence results when the initial data are small or local existence when the initial data are large. A. Córdoba and D. Córdoba proved in [22] that there exists a global solution if the initial data are in the Sobolev space $H^{m}$ where $m<2$. Chae and Lee in [19] showed that if the initial data are small in the critical space $\dot{{\mathscr {B}}}_{2,1}^{2-\alpha }$ then there exists a unique global solution. This result was improved by Bae in [8] where he showed that it is sufficient that the initial data are small in $\dot{{\mathscr {B}}}_{p, q}^{1+\frac{2}{p}-2 \alpha }$ with $p<\infty$.

Writing the Cauchy problem (1) in integral form and solving the new problem using the fixed point method, we get the global solution. On the other hand, by setting: ${\tilde{V}}(t, x):=e^{ \sqrt{t} \vert D \vert ^{\alpha }} v(t, x)$ and considering the corresponding equations of ${\tilde{V}}(t, x)$, we also prove the Gevrey regularity of the solution in the same spaces, that is, ${\tilde{V}}(t, x)$ belong to the same spaces with v(t, x). Furthermore, we show that the solution v presents the asymptotic property $\Vert v(t)\Vert _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} }\longrightarrow 0$ as $t \longrightarrow \infty .$ For that, we use standard interpolation in the Fourier–Besov space ${\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}$, energy estimates in $L^{2}$, and Young’s inequality of convolutions, among others. The definition of the Fourier–Besov space is the Fourier transformation form of the Besov space. Exactly, analogous to that of a Besov space, the norm of the Littlewood–Paley blocks is taken in the frequency space instead of the physical space, that is, the homogeneous Fourier–Besov norm is defined as

$$\begin{aligned} \Vert v\Vert _{{{\mathscr {F}}B}^{s}_{p,q}}=\left\{ \sum _{j \in Z} 2^{j q s}\left\| \widehat{\Delta _{j}v} \right\| _{L^p}^{q}\right\} ^{\frac{1}{q}}. \end{aligned}$$

To solve the equation (1), we consider the following equivalent integral equation coming from Duhamel’s principle

$$\begin{aligned} v(t)={\mathscr {S}}_{\alpha }(t)v_0+ {\mathscr {B}}(v,v)(t), \end{aligned}$$

(4)

where ${\mathscr {S}}_{\alpha }:=e^{-t \Lambda ^{2\alpha }}$ denotes the fractional heat semigroup operator, which can be regarded as the convolution operator with the kernel $K_t(x)={\mathscr {F}}^{-1}(e^{-t \vert \xi \vert ^{2\alpha }}),$ and

$$\begin{aligned} {\mathscr {B}}(v, \psi )(t)=-\int _{0}^{t} {\mathscr {S}}_{\alpha }(t-\tau )\left( u_v\cdot \nabla \psi )\right) (\tau ) d \tau . \end{aligned}$$

(5)

First, we show the existence of global solutions for (1).

Theorem 1

(Well-posedness) Let $1\le p<\infty$, $1\le q\le \infty$ and $\frac{1}{2}<\alpha <1+\frac{1}{p'}$. There exists a constant $\gamma >0$ such that for any $v_0\in {\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}$ satisfying $\Vert v_0\Vert _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} }<\gamma$, the equation (1) admits a unique global solution

$$\begin{aligned} v\in {\mathscr {C}}\left( {\mathbb {R}}^{+}, {\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} \right) \cap {\mathscr {L}}^{1}\left( {\mathbb {R}}^{+}, {\mathscr {F}}B_{p,q}^{1+\frac{2}{p'}} \right) , \end{aligned}$$

such that

$$\begin{aligned} \Vert v\Vert _{X_{p,q}}\lesssim \Vert v_0\Vert _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}}, \end{aligned}$$

where $X_{p,q}={\mathscr {L}}^{\infty }\left( {\mathbb {R}}^{+}, {\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} \right) \cap {\mathscr {L}}^{1}\left( {\mathbb {R}}^{+}, {\mathscr {F}}B_{p,q}^{1+\frac{2}{p'}} \right) .$

Remark 1

Taking $n=2,\, a_{21}=1=-a_{12}, a_{11}=a_{22}=0, \left| P_{j}(\xi )\right| =\vert \xi \vert (j=1,2$,   $\beta =1$ ) in the expressions (1.2) to (1.5) in [26], we obtain the 2DQG (1). Then, using Theorem 3.1 in [26], we obtain a well-posedness result in Fourier–Besov–Morrey spaces $\mathscr {F}\mathscr {N}_{p, \lambda , \infty }^{1-2\alpha +\frac{\lambda }{p}+\frac{2}{p'}}$ for (1) in the subcritical case, which is related to Theorem 1 but is different. It is worth noticing that in [26] it is used the persistence norm $B C\left( {\mathbb {R}}^{+}, {\mathscr {F}} {\mathscr {N}}_{p, \lambda , \infty }^{1-2\alpha +\frac{\lambda }{p}+\frac{2}{p'}}\right)$ while Chemin–Lerner type norms are employed in our Theorem 1, namely the norms of ${\mathscr {L}}^{\infty }\left( {\mathbb {R}}^{+}, {\mathscr {F}} B_{p, q}^{1-2\alpha +\frac{2}{p'}}\right)$ and ${\mathscr {L}}^{1}\left( {\mathbb {R}}^{+}, {\mathscr {F}} B_{p, q}^{1+\frac{2}{p'}}\right)$.

Theorem 2

(Space analyticity) Under the assumptions of Theorem 1, the solution obtained in Theorem 1 is analytic in the sense that $e^{\sqrt{t} \vert D \vert ^{\alpha }}v\in {X}_{p,q}$ and

$$\begin{aligned} \Vert e^{\sqrt{t} \vert D \vert ^{\alpha }}v\Vert _{X_{p,q}}\lesssim \Vert v_0\Vert _{ {\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}}, \end{aligned}$$

where $e^{ \sqrt{t} \vert D \vert ^{\alpha }}v={\mathscr {F}}^{-1}(e^{\sqrt{t}\vert \xi \vert ^{\alpha }}{\hat{v}})$.

As an application of the analyticity of solutions obtained in the above theorem, we can get the time decay estimates of solutions.

Theorem 3

(Decay estimate) Under the assumptions of Theorem 2, for any $\sigma >0$, the global solution $v \in {X}_{p, q}$ and $e^{ \sqrt{t} \vert D \vert ^{\alpha }}v\in {X}_{p, q}$ achieved from Theorem 1 satisfies the following time decay estimate:

$$\begin{aligned} \left\| \Lambda ^{\sigma } v(t)\right\| _{ {\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}} \le C_{\alpha ,\sigma } t^{-\frac{\sigma }{2 \alpha }}\left\| v_{0}\right\| _{ {\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}}, \end{aligned}$$

where $C_{\alpha ,\sigma }$ is a constant depending on $\alpha$ and $\sigma$.

Next, we present an asymptotic stability result for solutions in the context of Fourier–Besov spaces.

Theorem 4

(Asymptotic stability) Let $\frac{2}{3}<\alpha < 1$, $1 \le p,q \le 2$, and $v\in {\mathscr {C}}\left( {\mathbb {R}}^{+}, {\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} \right)$ be a global solution of (1) given by Theorem 1. Then,

$$\begin{aligned} \lim _{t \longrightarrow \infty }\Vert v(t)\Vert _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} }=0. \end{aligned}$$

Remark 2

Theorems 1 and 4 extend the results of [13, 14] from Lei-Lin spaces to Fourier–Besov spaces. In fact, for $p=q=1$, ${\mathscr {F}}B_{1,1}^{s}={\mathscr {X}}^{s}$ (see Remark 3).

2 Preliminaries

The definitions and notations used in this paper are presented as follows:

• (1) The Lei-Lin space is defined by

  $$\begin{aligned} {\mathscr {X}}^{s}\left( {\mathbb {R}}^{n}\right) :=\left\{ f \in {\mathscr {S}}^{\prime }\left( {\mathbb {R}}^{n}\right) :\Vert f\Vert _{{\mathscr {X}}^{s}\left( {\mathbb {R}}^{n}\right) }=\int _{{\mathbb {R}}^{n}}\vert \xi \vert ^{s}\vert {\widehat{f}}(\xi )\vert \mathrm {d} \xi <\infty \right\} , \end{aligned}$$

  where ${\mathscr {S}}^{\prime }\left( {\mathbb {R}}^{n}\right)$ is the set of tempered distributions;

• (2) Let $1 \le p,q \le \infty$, the mixed Lebesgue-sequence space $l^{q}(L^p)$ consists of all sequences $\{f_i\}_{i\in {\mathbb {Z}}}$ of measurable functions in ${\mathbb {R}}^{n}$ such that $\Vert \{f_i\}_{i\in {\mathbb {Z}}}\Vert _{l^{q}(L^p)}<\infty .$ For $\{f_i\}_{i\in {\mathbb {Z}}}\in l^{q}(L^p)$ we define

  $$\begin{aligned} \Vert \{f_i\}_{i\in {\mathbb {Z}}}\Vert _{l^{q}(L^p)}:= \left( \sum _{j \in {\mathbb {Z}}}\left\| f_i\right\| _{L^p}^{q}\right) ^{\frac{1}{q}}; \end{aligned}$$

• (2) we will use $\Vert \cdot \Vert _{X}$ to denote $\Vert \cdot \Vert _{X\left( {\mathbb {R}}^{n}\right) }$;

• (3) For the convenience of description, we use ${\mathscr {F}}B_{p, q}^{s}$ to denote the homogeneous Fourier–Besov spaces, ${\dot{H}}^{s}$ to denote the usual homogeneous Sobolev space;

• (4) For f, we denote $u_f:=\left( -\mathfrak {R}_{2} f, \mathfrak {R}_{1}f\right)$, where $\mathfrak {R}_{j}=\partial _{x_{j}}(-\Delta )^{-1 / 2}$, $j=1,2$ are the Riesz transforms;

• (5) The convolution product of a suitable pair of functions f and g on ${\mathbb {R}}^{n}$ is given by

  $$\begin{aligned} (f * g)(x):=\int _{{\mathbb {R}}^{n}} f(y) g(x-y) d y; \end{aligned}$$

• (6) For any subset A of a set E, the symbol $\chi _{A}$ denote the characteristic function of A defined by $\chi _{A}(x)=1 \quad$ if $x \in A, \quad \chi _{A}(x)=0 \quad$ elsewhere;

• (7) Let X, Y be Banach spaces, we denote

  $$\begin{aligned} \Vert v\Vert _{X \cap Y}:=\Vert v\Vert _{X}+\Vert v\Vert _{Y} \quad \text {;} \quad \Vert (v, w)\Vert _{X}:=\Vert v\Vert _{X}+\Vert w\Vert _{X}; \end{aligned}$$

• (8) Throughout the paper, C stands for a generic positive constant, which may be different from line to line, and $A \lesssim B$ means that there is a constant $C>0$ such that

  $$\begin{aligned} A \le C B; \end{aligned}$$

• (9) $p^{\prime }$ is the conjugate of p satisfying $\frac{1}{p}+\frac{1}{p^{\prime }}=1$ for $1 \le p \le \infty ;$

The proofs of the results discussed in this work are based on a dyadic partition of unity in the Fourier variables, known as the homogeneous Littlewood–Paley decomposition. We present briefly this construction below. For more detail, we refer the reader to [10].

Let $f \in S^{\prime }\left( {\mathbb {R}}^{n}\right) .$ Define the Fourier transform as

$$\begin{aligned} {\hat{f}}(\xi )={\mathscr {F}} f(\xi )=(2 \pi )^{-\frac{n}{2}} \int _{{\mathbb {R}}^{n}} e^{-i x \cdot \xi } f(x) d x \end{aligned}$$

and its inverse Fourier transform as

$$\begin{aligned} \breve{f}(x)={\mathscr {F}}^{-1} f(x)=(2 \pi )^{-\frac{n}{2}} \int _{{\mathbb {R}}^{n}} e^{i x \cdot \xi } f(\xi ) d \xi . \end{aligned}$$

Let $\varphi \in S\left( {\mathbb {R}}^{d}\right)$ be such that $0 \le \varphi \le 1$ and ${\text {supp}}(\varphi ) \subset \left\{ \xi \in {\mathbb {R}}^{d}: \frac{3}{4} \le \vert \xi \vert \le \frac{8}{3}\right\}$ and

$$\begin{aligned} \sum _{j \in {\mathbb {Z}}} \varphi \left( 2^{-j} \xi \right) =1, \quad \text{ for } \text{ all } \xi \ne 0. \end{aligned}$$

We denote

$$\begin{aligned} \varphi _{j}(\xi )=\varphi \left( 2^{-j} \xi \right) , \quad {\dot{\Delta }}_{j} (\xi )=\sum _{k \le j-1} \varphi _{k}(\xi ) \end{aligned}$$

and

$$\begin{aligned} h(x)={\mathscr {F}}^{-1} \varphi (x), \quad g(x)={\mathscr {F}}^{-1} \delta (x). \end{aligned}$$

We now present some frequency localization operators:

$$\begin{aligned} {\dot{\Delta }}_{j} f={\mathscr {F}}^{-1}\varphi _{j}*f=2^{d j} \int _{{\mathbb {R}}^{d}} h\left( 2^{j} y\right) f(x-y) d y \end{aligned}$$

and

$$\begin{aligned} {\dot{S}}_{j} f=\sum _{k \le j-1} {\dot{\Delta }}_{k} f={\mathscr {F}}^{-1}{\dot{\Delta }}_{j} * f=2^{d j} \int _{{\mathbb {R}}^{d}} g\left( 2^{j} y\right) f(x-y) d y, \end{aligned}$$

where ${\dot{\Delta }}_{j}={\dot{S}}_{j}-{\dot{S}}_{j-1}$ is a frequency projection to the annulus $\{\vert \xi \vert \sim 2^{j}\}$ and ${\dot{S}}_{j}$ is a frequency to the ball $\{\vert \xi \vert \lesssim 2^{j}\}$.

From the definition of ${\dot{\Delta }}_{j}$ and ${\dot{S}}_{j}$, one easily derives that

$$\begin{aligned} \begin{aligned} {\dot{\Delta }}_{j} {\dot{\Delta }}_{k} f&=0, \quad \text{ if } |j-k| \ge 2 \\ {\dot{\Delta }}_{j}\left( {\dot{S}}_{k-1} f {\dot{\Delta }}_{k} f\right)&=0, \quad \text{ if } |j-k| \ge 5\\ \widehat{{\dot{\Delta }}_{j} f}&=\varphi _{j}{\widehat{f}}. \end{aligned} \end{aligned}$$

The following Bony paraproduct decomposition will be applied throughout the paper.

$$\begin{aligned} u v={\dot{T}}_{u} v+{\dot{T}}_{v} u+R(u, v) \end{aligned}$$

where

$$\begin{aligned} {\dot{T}}_{u} v=\sum _{j \in {\mathbb {Z}}} {\dot{S}}_{j-1} u {\dot{\Delta }}_{j} v, \quad {\dot{R}}(u, v)=\sum _{j \in {\mathbb {Z}}} {\dot{\Delta }}_{j} u {\tilde{\Delta }}_{j} v, \quad {\tilde{\Delta }}_{j} v=\sum _{\left| j^{\prime }-j\right| \le 1} {\dot{\Delta }}_{j^{\prime }} v. \end{aligned}$$

We give the definition of the homogeneous Fourier–Besov spaces ${\mathscr {F}}{B}_{p, q}^{s}$ [45].

Definition 2

(Homogeneous Fourier–Besov spaces) Let $1 \le p, q \le \infty$ and $s \in {\mathbb {R}}.$ The homogeneous Fourier–Besov space ${\mathscr {F}}{B}_{p, q}^{s}$ is defined as the set of all distributions $f \in {\mathscr {S}}^{\prime } \backslash {\mathscr {P}}, \; {\mathscr {P}}$ is the set of all polynomials, such that the norm $\Vert f\Vert _{{\mathscr {F}}{B}_{p, q}^{s} }$ is finite, where

$$\begin{aligned} \Vert f\Vert _{{\mathscr {F}}{B}_{p, q}^{s} } {\mathop {=}\limits ^{ \text{ def } }} \left\{ \begin{array}{ll} \left( \sum _{j \in {\mathbb {Z}}} 2^{j s q}\left\| \widehat{{\dot{\Delta }}_{j}f}\right\| _{L^{p}}^{q}\right) ^{\frac{1}{q}} \; \; for &{} q <\infty \\ \sup _{j \in {\mathbb {Z}}} 2^{j s}\left\| \widehat{{\dot{\Delta }}_{j}f}\right\| _{L^{p}} \;\; \;\; \; \; \;\; \; \;\; \; \;\; for &{} q=\infty , \end{array}\right. \end{aligned}$$

(6)

where $\widehat{{\dot{\Delta }}_{j}f}=\varphi _j {\hat{f}}.$

Remark 3

[36] Notice that in the case $p=q$ we have an equivalent norm on ${\mathscr {F}}{B}_{p, q}^{s}$, that is,

$$\begin{aligned} \Vert f\Vert _{{\mathscr {F}}{B}_{p, q}^{s}} \sim \left( \int _{{\mathbb {R}}^{n}}\vert \xi \vert ^{s p}\vert {\hat{f}}(\xi )\vert ^{p} d \xi \right) ^{1 / p}. \end{aligned}$$

Lemma 1

The derivation $\partial ^{\alpha }_{\xi }: {\mathscr {F}}B^{s+\vert \alpha \vert }_{p,q} \rightarrow {\mathscr {F}}B^{s}_{p,q}$ is a bounded operator.

Proof

We have

$$\begin{aligned} \left\| \partial ^{\alpha }_{\xi }v \right\| _{{\mathscr {F}}B^{s}_{p,q}}&=\left\| \{2^{js}\varphi _{j}\widehat{\partial ^{\alpha }_{\xi }v}\}_{j\in {\mathbb {Z}}}\right\| _{{l^{q}}(L^p )}\nonumber \\&=\left\| \{2^{js}\varphi _{j}\vert \xi \vert ^{\alpha }{\widehat{v}}\}_{j\in {\mathbb {Z}}}\right\| _{{l^{q}}(L^p )}\nonumber \\&\lesssim \left\| \{2^{js}2^{j\vert \alpha \vert }\widehat{{\dot{\Delta }}_{j}v}\}_{j\in {\mathbb {Z}}}\right\| _{{l^{q}}(L^p )}\nonumber \\&\lesssim \left\| v \right\| _{{\mathscr {F}}B^{s+\vert \alpha \vert }_{p,q}}, \end{aligned}$$

(7)

where in (7) we used the fact that $\vert \xi \vert \sim 2^j$ for all $j \in {\mathbb {Z}}.$ $\square$

Remark 4

As a consequence of Lemma 1, we have the following estimates:

$$\begin{aligned}&\Vert \text {div} (v) \Vert _{{\mathscr {F}}B^{s}_{p,q}} \lesssim \Vert v \Vert _{{\mathscr {F}}B^{s+1}_{p, q}} ,\\&\Vert \Delta v \Vert _{{\mathscr {F}}B^{s}_{p, q}} \lesssim \Vert v \Vert _{{\mathscr {F}}B^{s+2}_{p, q}} . \end{aligned}$$

Proposition 5

[23] (Sobolev-type embedding) For $p_{2} \le p_{1}$ and $s_{2} \le s_{1}$ satisfying $s_{2}+\frac{n}{p_{2}}=s_{1}+\frac{n}{p_{1}}$, we have the continuous inclusion

$$\begin{aligned} {\mathscr {F}} B_{p_{1}, r_{1}}^{s_{1}} \hookrightarrow {\mathscr {F}} B_{p_{2}, r_{2}}^{s_{2}} \end{aligned}$$

for all $1 \le r_{1} \le r_{2} \le \infty$.

The definition of mixed space-time spaces is given below.

Definition 3

[23] Let $s\in {\mathbb {R}},\;1\le p<\infty ,\; 1\le q,\rho \le \infty , \;$ and $I=[0,T),\;T\in (0,\infty ]$. The space-time norm is defined on u(t, x) by

$$\begin{aligned} \Vert u(t,x)\Vert _{{\mathscr {L}}^{\rho }(I,{\mathscr {F}}B_{p,q}^{s})}= \Big \{\sum _{j\in {\mathbb {Z}}}2^{jqs}\Vert \widehat{{\dot{\Delta }}_{j}u}\Vert _{L^{\rho }(I,L^p)} ^q \Big \}^{1/q}, \end{aligned}$$

and denote by ${\mathscr {L}}^{\rho }(I,{\mathscr {F}}B_{p,q}^{s})$ the set of distributions in $S'({\mathbb {R}}\times {\mathbb {R}}^{n})/{\mathscr {P}}$ with finite $\Vert .\Vert _{{\mathscr {L}}^{\rho }(I,{\mathscr {F}}B_{p,q}^{s})}$ norm.

According to Minkowski inequality, it is easy to verify that

$$\begin{aligned}&L^{\rho }\left( I ; {\mathscr {F}} B_{p, q}^{s}\right) \hookrightarrow {\mathscr {L}}^{\rho }\left( I, {\mathscr {F}}B_{p, q}^{s}\right) , \quad \text{ if } \rho \le q,\end{aligned}$$

(8)

$$\begin{aligned}&{\mathscr {L}}^{\rho }\left( I, {\mathscr {F}} B_{p, q}^{s}\right) \hookrightarrow L^{\rho }\left( I ; {\mathscr {F}}B_{p, q}^{s}\right) , \quad \text{ if } \rho \ge q, \end{aligned}$$

(9)

where $\displaystyle \Vert u(t, x)\Vert _{L^{\rho }\left( I; {\mathscr {F}}B_{p,q}^{s}\right) }:=\left( \int _{I}\Vert u(\tau , \cdot )\Vert _{{\mathscr {F}}B_{p, q}^{s}}^{\rho } d \tau \right) ^{1 / \rho }.$

At the end of this section, we will recall an existence and uniqueness result for an abstract operator equation in a Banach space that will be used to show Theorem 1 in the sequel. For the proof, we refer the reader to see [9, 41].

Proposition 6

Let X be a Banach space with norm $\Vert .\Vert _{X}$ and $B:X\times X\longmapsto X$ be a bounded bilinear operator satisfying

$$\begin{aligned} \Vert B(u,v)\Vert _{X}\le \eta \Vert u\Vert _{X}\Vert v\Vert _{X} \end{aligned}$$

for all $u,v\in X$ and a constant $\eta >0$. Then, if $0<\varepsilon <\frac{1}{4\eta }$ and if $y\in X$ such that $\Vert y\Vert _{X}\le \varepsilon$, the equation $x:=y+B(x,x)$ has a solution ${\overline{x}}$ in X such that $\Vert {\overline{x}}\Vert _{X}\le 2 \varepsilon$. This solution is the only one in the ball ${\overline{B}}(0,2\varepsilon )$. Moreover, the solution depends continuously on y in the sense: if $\Vert y'\Vert _{X}< \varepsilon ,\;x'=y'+B(x',x')$, and $\Vert x'\Vert _{X}\le 2\varepsilon$, then

$$\begin{aligned} \Vert {\overline{x}}-x'\Vert _{X}\le \frac{1}{1-4\varepsilon \eta }\Vert y-y'\Vert _{X}. \end{aligned}$$

3 Well-posedness

In this section, we present the proof Theorem 1. We have learned from Sect. 1 that the integral form of the equation (1) is as follows:

$$\begin{aligned} v(t)={\mathscr {S}}_{\alpha }(t)v_0+ {\mathscr {B}}(v,v)(t), \end{aligned}$$

(10)

where ${\mathscr {S}}_{\alpha }(t)v_0:={\mathscr {F}}^{-1}(e^{-t \vert \xi \vert ^{2\alpha }}\widehat{v_0})$ and

$$\begin{aligned} {\mathscr {B}}(v, \psi )(t)=-\int _{0}^{t} {\mathscr {S}}_{\alpha }(t-\tau )\left( u_v\cdot \nabla \psi )\right) (\tau ) d \tau . \end{aligned}$$

(11)

Lemma 2

(Linear estimate) Let $v_{0}\in {\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}$ for $1\le p<\infty$ and $1\le q\le \infty$. Then there exists a constant $C>0$ such that

$$\begin{aligned} \left\| {\mathscr {S}}_{\alpha }(t)v_0\right\| _{X_{p,q}}\le C\left\| v_{0}\right\| _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} }. \end{aligned}$$

(12)

Proof

Since ${\text {supp}} \varphi _{j} \subset \left\{ 2^{j-1} \le \vert \xi \vert \le 2^{j+1}\right\}$, we get

$$\begin{aligned} \left\| {\mathscr {F}}\left[ \Delta _{j} {\mathscr {S}}_{\alpha }(t) v_{0}\right] \right\| _{L^p}=\left\| \varphi _{j} e^{-t\vert \xi \vert ^{2\alpha }} \widehat{v_{0}}\right\| _{L^p} \le e^{-t 2^{2\alpha (j-1)}}\left\| \varphi _{j} \widehat{v_{0}}\right\| _{L^p}. \end{aligned}$$

Then, applying the Minkowski inequality, we obtain

$$\begin{aligned} \left\| {\mathscr {S}}_{\alpha }(t) v_{0} \right\| _{{\mathscr {L}}^{1}\left( {\mathbb {R}}^{+},{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}\right) }&\lesssim \left\| \left\{ 2^{j\left( 1+\frac{2}{p'}\right) }\int _{0}^{\infty } e^{-t 2^{2\alpha (j-1)}} d t\left\| \varphi _{j} \widehat{v_{0}}\right\| _{L^p}\right\} \right\| _{\ell ^{q}}\\&\lesssim \left\| \left\{ 2^{j\left( 1+\frac{2}{p'}\right) }\frac{1}{ 2^{2\alpha (j-1)}}\left\| \varphi _{j} \widehat{v_{0}}\right\| _{L^p}\right\} \right\| _{\ell ^{q}}\\&\lesssim \left\| v_0 \right\| _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}}. \end{aligned}$$

Similarly,

$$\begin{aligned} \left\| {\mathscr {S}}_{\alpha }(t) v_{0} \right\| _{{\mathscr {L}}^{\infty }\left( \mathbb { R}^{+},{\mathscr {F}}B_{p,q}^{1+\frac{2}{p'}}\right) }\lesssim \left\| v_0 \right\| _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}}. \end{aligned}$$

(13)

This finishes the proof of Lemma 2. $\square$

Lemma 3

(Bilinear estimate) Under the hypothesis of Theorem 1, there exists a constant $\eta > 0$ such that

$$\begin{aligned} \Vert {\mathscr {B}}(v, \psi )\Vert _{X_{p,q}} \le \eta \Vert v\Vert _{X_{p,q}}\Vert \psi \Vert _{X_{p,q}}, \end{aligned}$$

for all $v,\psi \in X_{p,q}.$

Proof

Recalling that $u_v=\left( -\mathfrak {R}_{2} v, \mathfrak {R}_{1}v\right) .$ Since $\widehat{\mathfrak {R}_{j}v(\xi )}=-i \frac{\xi _j}{\vert \xi \vert }{\widehat{v}}(\xi ), j=1,2,$ then $\nabla \cdot u_v=0$ and

$$\begin{aligned} \Vert \widehat{u_v}\Vert \lesssim \Vert {\widehat{v}}\Vert . \end{aligned}$$

(14)

Applying ${\dot{\Delta }}_{j}$ to ${\mathscr {B}}(v, \psi )$, then doing Fourier transform on it and taking the $L^{p}$-norm, using Minkowski’s inequality, we find that

$$\begin{aligned} \left\| \widehat{{\dot{\Delta }}_{j}{\mathscr {B}}(v, \psi )}\right\| _{L^{p}}&\lesssim \int _{0}^{t} \Vert {\mathscr {F}}\Big ({\dot{\Delta }}_{j} {\mathscr {S}}_{\alpha }(t-s) \nabla \cdot (u_v\psi )\Big ) \Vert _{L^{p}} d s \\&\lesssim \int _{0}^{t} e^{-(t-s) 2^{2 \alpha j}} 2^{j}\left\| \widehat{{\dot{\Delta }}_{j} u_v\psi }\right\| _{L^{p}} d s, \end{aligned}$$

where we have used the fact that $\vert \xi \vert \sim 2^{j}$.

According to Bony’s paraproduct decomposition and quasi-orthogonality property for Littlewood–Paley decomposition, for fixed j, we find that

$$\begin{aligned} {\dot{\Delta }}_{j}(u_v\psi )&=\sum _{ \vert k-j \vert \le 4}{\dot{\Delta }}_{j}({\dot{S}}_{k-1}u_v{\dot{\Delta }}_{k}\psi )+ \sum _{ \vert k-j \vert \le 4}{\dot{\Delta }}_{j}({\dot{S}}_{k-1}\psi {\dot{\Delta }}_{k}u_v)\\&\quad +\sum _{k\ge j-3}{\dot{\Delta }}_{j}({\dot{\Delta }}_{k}u_v \widetilde{{\dot{\Delta }}}_{k}\psi ). \end{aligned}$$

Then

$$\begin{aligned} \left\| \widehat{{\dot{\Delta }}_{j}{\mathscr {B}}(v, \psi )}\right\| _{L^{p}}&\lesssim \int _{0}^{t} e^{-(t-s) 2^{2 \alpha j}} 2^{j}\sum _{ \vert k-j \vert \le 4}\left\| {\mathscr {F}}( {\dot{\Delta }}_{j}({\dot{S}}_{k-1}u_v{\dot{\Delta }}_{k}\psi ))\right\| _{L^{p}} d s\\&+\int _{0}^{t} e^{-(t-s) 2^{2 \alpha j}} 2^{j}\sum _{ \vert k-j \vert \le 4}\left\| {\mathscr {F}}( {\dot{\Delta }}_{j}({\dot{S}}_{k-1}\psi {\dot{\Delta }}_{k}u_v))\right\| _{L^{p}} d s\\&+ \int _{0}^{t} e^{-(t-s) 2^{2 \alpha j}} 2^{j}\sum _{k\ge j-3}\left\| {\mathscr {F}}({\dot{\Delta }}_{j}({\dot{\Delta }}_{k}u_v\widetilde{{\dot{\Delta }}}_{k}\psi ))\right\| _{L^{p}} d s\\&:= I_1+I_2+I_3. \end{aligned}$$

Then, by the triangle inequalities in $L^p$ and in $\ell ^q$, we have

$$\begin{aligned} \left\| {\mathscr {B}}(v, \psi )\right\| _{{\mathscr {L}}^{\infty }\left( {\mathbb {R}}^{+},{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}\right) }&\lesssim \left( \sum _{j\in {\mathbb {Z}}}2^{j(1-2\alpha +\frac{2}{p'})q}\left\| I_1\right\| ^{q}_{L^{\infty }({\mathbb {R}}^{+})}\right) ^{\frac{1}{q}} \\&\quad + \left( \sum _{j\in {\mathbb {Z}}}2^{j(1-2\alpha +\frac{2}{p'})q}\left\| I_2\right\| ^{q}_{L^{\infty }({\mathbb {R}}^{+})}\right) ^{\frac{1}{q}}\\&\quad +\left( \sum _{j\in {\mathbb {Z}}}2^{j(1-2\alpha +\frac{2}{p'})q}\left\| I_3\right\| ^{q}_{L^{\infty }({\mathbb {R}}^{+})}\right) ^{\frac{1}{q}}. \end{aligned}$$

By applying Hölder’s inequality and Young’s inequality, we get

$$\begin{aligned} \sum _{ \vert k-j \vert \le 4}\left\| {\mathscr {F}}( {\dot{\Delta }}_{j}({\dot{S}}_{k-1}u_v{\dot{\Delta }}_{k}\psi ))\right\| _{L^{p}}&\lesssim \sum _{ \vert k-j \vert \le 4}\left\| \widehat{{\dot{S}}_{k-1}u_v}*\widehat{{\dot{\Delta }}_{k}\psi }\right\| _{L^{p}}\\&\lesssim \sum _{ \vert k-j \vert \le 4}\left\| \widehat{{\dot{S}}_{k-1}u_v}\right\| _{L^{1}}\left\| \widehat{{\dot{\Delta }}_{k}\psi }\right\| _{L^{p}}\\&\lesssim \sum _{ \vert k-j \vert \le 4} \sum _{l\le k-2}\Vert \widehat{{\dot{\Delta }}_{l}u_v}\Vert _{ L^1}\left\| \widehat{{\dot{\Delta }}_{k}\psi }\right\| _{L^{p}}\\&\lesssim \sum _{ \vert k-j \vert \le 4} \sum _{l\le k-2} 2^{l\frac{2}{p'}}\Vert \widehat{{\dot{\Delta }}_{l}v}\Vert _{ L^p}\left\| \widehat{{\dot{\Delta }}_{k}\psi }\right\| _{L^{p}}\\&\lesssim \sum _{ \vert k-j \vert \le 4} 2^{-k(1+\frac{2}{p'})}\sum _{l\le k-2} 2^{2\alpha l}2^{l(1-2\alpha +\frac{2}{p'})}\Vert \widehat{{\dot{\Delta }}_{l}v}\Vert _{ L^p}2^{k(1+\frac{2}{p'})}\left\| \widehat{{\dot{\Delta }}_{k}\psi }\right\| _{L^{p}}. \end{aligned}$$

Therefore,

$$\begin{aligned} I_1\lesssim \int _{0}^{t} e^{-(t-s) 2^{2 \alpha j}} 2^{j}\sum _{ \vert k-j \vert \le 4} 2^{-k(1+\frac{2}{p'})}\sum _{l\le k-2} 2^{2\alpha l}2^{l(1-2\alpha +\frac{2}{p'})}\Vert \widehat{{\dot{\Delta }}_{l}v}\Vert _{ L^p}2^{k(1+\frac{2}{p'})}\left\| \widehat{{\dot{\Delta }}_{k}\psi }\right\| _{L^{p}}d s. \end{aligned}$$

Taking the $L^{\infty }$-norm of $I_1$ in time, we can conclude that

$$\begin{aligned} \left\| I_1\right\| _{L^{\infty }({\mathbb {R}}^{+})}\lesssim 2^{-j(1-2\alpha +\frac{2}{p'})} \left\| v\right\| _{{\mathscr {L}}^{\infty }\left( \mathbb { R}^{+},{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}\right) } 2^{j(1+\frac{2}{p'})} \left\| \widehat{{\dot{\Delta }}_{k}\psi }\right\| _{{L}^{1}({\mathbb {R}}^{+}, L^p)}. \end{aligned}$$

(15)

Multiplying (15) by $2^{j(1-2\alpha +\frac{2}{p'})}$, and taking $\ell ^{q}$-norm with index j, using Minkowski’s inequality and Young’s inequality, we can see that

$$\begin{aligned} \left( \sum _{j\in {\mathbb {Z}}}2^{j(1-2\alpha +\frac{2}{p'})q}\left\| I_1\right\| ^{q}_{L^{\infty }({\mathbb {R}}^{+})}\right) ^{\frac{1}{q}} \lesssim \left\| v\right\| _{{\mathscr {L}}^{\infty }\left( \mathbb { R}^{+},{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}\right) } \left\| \psi \right\| _{{\mathscr {L}}^{1}\left( \mathbb { R}^{+},{\mathscr {F}}B_{p,q}^{1+\frac{2}{p'}}\right) }. \end{aligned}$$

Analogously, we have

$$\begin{aligned} \left( \sum _{j\in {\mathbb {Z}}}2^{j(1-2\alpha +\frac{2}{p'})q}\left\| I_2\right\| ^{q}_{L^{\infty }({\mathbb {R}}^{+})}\right) ^{\frac{1}{q}} \lesssim \left\| \psi \right\| _{{\mathscr {L}}^{\infty }\left( \mathbb { R}^{+},{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}\right) } \left\| v\right\| _{{\mathscr {L}}^{1}\left( \mathbb { R}^{+},{\mathscr {F}}B_{p,q}^{1+\frac{2}{p'}}\right) }. \end{aligned}$$

Next, we have

$$\begin{aligned} \sum _{k \ge j-3}\left\| {\mathscr {F}}( {\dot{\Delta }}_{j}({\dot{\Delta }}_{k}u_v\tilde{{\dot{\Delta }}}_{k}\psi ))\right\| _{L^{p}}&\lesssim \sum _{ k \ge j-3}\left\| \widehat{{\dot{\Delta }}_{k}u_v}*\widehat{\tilde{{\dot{\Delta }}}_{k}\psi }\right\| _{L^{p}}\\ {}&\lesssim \sum _{k \ge j-3} \Vert \widehat{{\dot{\Delta }}_{k}u_v}\Vert _{ L^p}\sum _{\vert l-k \vert \le 1}\left\| \widehat{{\dot{\Delta }}_{l}\psi }\right\| _{L^{1}}\\ {}&\lesssim \sum _{ k \ge j-3} \Vert \widehat{{\dot{\Delta }}_{k}v}\Vert _{ L^p}\sum _{\vert l-k \vert \le 1} 2^{l\frac{2}{p'}}\left\| \widehat{{\dot{\Delta }}_{l}\psi }\right\| _{L^{p}}. \end{aligned}$$

Therefore,

$$\begin{aligned} I_3\lesssim \int _{0}^{t} e^{-(t-s) 2^{2 \alpha j}} 2^{j}\sum _{ k \ge j-3} \Vert \widehat{{\dot{\Delta }}_{k}v}\Vert _{ L^p}\sum _{\vert l-k \vert \le 1} 2^{l\frac{2}{p'}}\left\| \widehat{{\dot{\Delta }}_{l}\psi }\right\| _{L^{p}}d s. \end{aligned}$$

Taking the $L^{\infty }$-norm of $I_3$ in time, we can conclude that

$$\begin{aligned} \left\| I_3\right\| _{L^{\infty }({\mathbb {R}}^{+})}\lesssim \Vert \psi \Vert _{{\mathscr {L}}^{\infty }(\mathbb { R}^{+},{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}})} \sum _{k\ge j-3} 2^{k(2\alpha -1)} \Vert \widehat{{\dot{\Delta }}_{k}v}\Vert _{{L}^{1}({\mathbb {R}}^{+}, L^p)}. \end{aligned}$$

(16)

Multiplying (16) by $2^{j(1-2\alpha +\frac{2}{p'})}$, and taking $\ell ^{q}$-norm with index j, using Young’s inequality, we can see that

$$\begin{aligned}&\left( \sum _{j\in {\mathbb {Z}}}2^{j(1-2\alpha +\frac{2}{p'})q}\left\| I_3\right\| ^{q}_{L^{\infty }({\mathbb {R}}^{+})}\right) ^{\frac{1}{q}} \\ {}&\lesssim \Vert \psi \Vert _{{\mathscr {L}}^{\infty }(\mathbb { R}^{+},{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}})}\left\{ \sum _{j\in {\mathbb {Z}}}2^{j(2-2 \alpha +\frac{2}{p'})q}\big (\sum _{k\ge j-3} 2^{k(2\alpha -1)}\Vert \widehat{{\dot{\Delta }}_{k}v}\Vert _{{L}^{1}(\mathbb { R}^{+},L^p)}\big )^q\right\} ^\frac{1}{q} \\&\lesssim \Vert \psi \Vert _{{\mathscr {L}}^{\infty }({\mathbb {R}}^{+},{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}})}\left\{ \sum _{j\in {\mathbb {Z}}}\big (\sum _{k\ge j-3}2^{(j-k)(2-2 \alpha +\frac{2}{p'})} 2^{k(1+\frac{2}{p'})}\Vert \widehat{{\dot{\Delta }}_{k}v}\Vert _{{L}^{1}({\mathbb {R}}^{+}, L^p)}\big )^q\right\} ^\frac{1}{q} \\ {}&\lesssim \Vert \psi \Vert _{{\mathscr {L}}^{\infty }({\mathbb {R}}^{+},{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}})} \Vert v\Vert _{{\mathfrak {L}}^{1}({\mathbb {R}}^{+},{\mathscr {F}}B_{p,q}^{1+\frac{2}{p'}})} \sum _{i\le 3}2^{i(2-2 \alpha +\frac{2}{p'})} \\ {}&\lesssim \Vert \psi \Vert _{{\mathscr {L}}^{\infty }({\mathbb {R}}^{+},{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}})} \Vert v\Vert _{{\mathscr {L}}^{1}({\mathbb {R}}^{+},{\mathscr {F}}B_{p,q}^{1+\frac{2}{p'}})}, \end{aligned}$$

where the condition $\alpha <1+\frac{1}{p'}$ ensures that the series $\sum _{i\le 3}2^{i(2-2 \alpha +\frac{2}{p'})}$ converges.

Hence,

$$\begin{aligned} \left\| {\mathscr {B}}(v, \psi )\right\| _{{\mathscr {L}}^{\infty }\left( \mathbb { R}^{+},{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}\right) }\lesssim \Vert v\Vert _{{\mathscr {L}}^{\infty }(\mathbb { R}^{+},{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}})} \Vert \psi \Vert _{{\mathscr {L}}^{1}(\mathbb { R}^{+},{\mathscr {F}}B_{p,q}^{1+\frac{2}{p'}})}. \end{aligned}$$

When taking the $L^1$-norm in time to $\left\| \widehat{ {\dot{\Delta }}_j {\mathscr {B}}(v, \psi )}\right\| _{L^p}$, we will obtain similar results that

$$\begin{aligned} \left\| {\mathscr {B}}(v, \psi )\right\| _{{\mathscr {L}}^{1}\left( {\mathbb {R}}^{+},{\mathscr {F}}B_{p,q}^{1+\frac{2}{p'}}\right) }\lesssim \Vert v\Vert _{{\mathscr {L}}^{\infty }({\mathbb {R}}^{+},{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}})} \Vert \psi \Vert _{{\mathscr {L}}^{1}({\mathbb {R}}^{+},{\mathscr {F}}B_{p,q}^{1+\frac{2}{p'}})}. \end{aligned}$$

This finishes the proof of Lemma 3.

From the above results and according to Proposition 6 (the fixed point theorem), we can prove Theorem 1. $\square$

4 Analyticity of global solutions

A number of researchers have focused on the analyticality of the solutions, especially with regard to the Navier–Stokes equations, for further information, see [7] and its references. In this section, we will show the Gevrey class regularity for (1) in the Fourier–Besov–Morrey space.

The next lemma is helpful in demonstrating spatial analyticality.

Lemma 4

Let $0<s \le t<\infty$ and $0\le \alpha \le 1$. Then the following inequality holds

$$\begin{aligned} t\vert \xi \vert ^{\alpha }-\frac{1}{2}(t^{2}-s^{2})\vert \xi \vert ^{2 \alpha }-s\vert \xi -y\vert ^{\alpha }-s\vert y\vert ^{\alpha }\le \frac{1}{2} \end{aligned}$$

(17)

for all $\xi ,\ y\in {\mathbb {R}}^{n}$.

4.1 Proof of Theorem 2

Let ${\tilde{V}} =e^{\sqrt{t} \vert D \vert ^{\alpha }}v$ and using the integral equation (10), we obtain

$$\begin{aligned} {\tilde{V}}&=e^{\sqrt{t} \vert D \vert ^{\alpha }}e^{-t (-\Delta )^{\alpha }}v_0+e^{\sqrt{t} \vert D \vert ^{\alpha }}{{\mathscr {B}}}(v,v).\\&=e^{\sqrt{t} \vert D \vert ^{\alpha }}e^{-t (-\Delta )^{\alpha }}v_0-e^{\sqrt{t} \vert D \vert ^{\alpha }}\int _{0}^{t} {\mathscr {H}}_{\alpha }(t-\tau )\left( u_v\cdot \nabla v)\right) (\tau ) \mathrm{d} \tau . \end{aligned}$$

Then

$$\begin{aligned} \widehat{{\tilde{V}} }=e^{\sqrt{t}\vert \xi \vert }e^{-{t}\vert \xi \vert ^{2}}\widehat{v_0}-e^{\sqrt{t}\vert \xi \vert }\int _{0}^{t} e^{\sqrt{\tau }\vert \xi \vert - (t-\tau )\vert \xi \vert ^{2}}\xi \cdot \int _{{\mathbb {R}}^{n}}e^{-\sqrt{\tau }\vert \xi -\eta \vert } \widehat{u_v}(\tau , \xi -\eta ) e^{-\sqrt{\tau }\vert \eta \vert } {\widehat{v}}(\tau , \eta )\mathrm{d}\eta \mathrm{d}\tau . \end{aligned}$$

Let $\rho =1$ (or $\infty$), multiplying with $2^{j (1-2\alpha +\frac{2}{p\prime }}\varphi _j$ and taking ${\mathscr {L}}^{\rho }(I,L^p)$-norm we can obtain

$$\begin{aligned}&\left\| 2^{j (1-2\alpha +\frac{2}{p\prime })}\varphi _j \widehat{{\tilde{V}} } \right\| _{{\mathscr {L}}^{\rho }(I,L^p)}\\ {}&\lesssim \left\| e^{\sqrt{t}\vert \xi \vert -{t}\vert \xi \vert ^{2}}2^{j (1-2\alpha +\frac{2}{p\prime })}\varphi _j \widehat{v_0 }\right\| _{{\mathscr {L}}^{\rho }(I,L^p)}+\Big \Vert 2^{j (2-2\alpha +\frac{2}{p\prime })}\varphi _j \int _{0}^{t} e^{- \frac{1}{2}(t-\tau )\vert \xi \vert ^{2}}\cdot \\ {}&\int _{{\mathbb {R}}^{n}}e^{\sqrt{\tau }\vert \xi \vert - \frac{1}{2}(t-\tau )\vert \xi \vert ^{2}-\sqrt{\tau }(\vert \xi -\eta \vert +\vert \eta \vert )} \widehat{u_v}(\tau , \xi -\eta ) {\widehat{v}}(\tau , \eta )\mathrm{d}\eta \mathrm{d}\tau \Big \Vert _{{\mathscr {L}}^{\rho }(I,L^p)}\\ {}&\lesssim \left\| e^{-\frac{t}{2}\vert \xi \vert ^{2}}\widehat{{\dot{\Delta }}_j v_0} \right\| _{{\mathscr {L}}^{\rho }(I,L^p)}+\left\| 2^{j (2-2\alpha +\frac{2}{p\prime })}\varphi _j\int _{0}^{t} e^{- \frac{1}{2}(t-\tau )\vert \xi \vert ^{2}}\int _{{\mathbb {R}}^{n}}\widehat{u_v}(\tau , \xi -\eta ) {\widehat{v}}(\tau , \eta )\mathrm{d}\eta \mathrm{d}\tau \right\| _{{\mathscr {L}}^{\rho }(I,L^p)}, \end{aligned}$$

where $e^{\sqrt{t}\vert \xi \vert -\frac{1}{2} t\vert \xi \vert ^{2}}=e^{-\frac{1}{2}\left( \sqrt{t}\vert \xi \vert -1\right) ^{2}+\frac{1}{2}} \le e^{\frac{1}{2}}$ and Lemma 4 are used. The remaining part of the proof is similar to the proof of Theorem 1, therefore the details can be omitted.

5 Time decay of mild solutions: proof of Theorem 3

Using the definition of the Fourier–Besov spaces, we have

$$\begin{aligned} \left\| \Lambda ^{\sigma } v(t)\right\| _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}}&=\left\| \Lambda ^{\sigma } e^{ -\sqrt{t} \vert D \vert ^{\alpha }}e^{ \sqrt{t} \vert D \vert ^{\alpha }} v(t)\right\| _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}} \\ {}&=\Big (\sum _{j \in {\mathbb {Z}}} 2^{j (1-2\alpha +\frac{2}{p\prime }) q}\left\| \varphi _{j} {\mathscr {F}}\Big (\Lambda ^{\sigma } e^{ -\sqrt{t} \vert D \vert ^{\alpha }}e^{ \sqrt{t} \vert D \vert ^{\alpha }} v(t)\Big )\right\| _{L^p }^{q}\Big )^{\frac{1}{q}} \\ {}&=\Big (\sum _{j \in {\mathbb {Z}}} 2^{j (1-2\alpha +\frac{2}{p\prime }) q}\left\| \vert \xi \vert ^{\sigma }e^{ -\sqrt{t}\vert \xi \vert ^{\alpha }}\varphi _{j} {\mathscr {F}}\Big (e^{ \sqrt{t} \vert D \vert ^{\alpha }} v(t)\Big )\right\| _{L^p }^{q}\Big )^{\frac{1}{q}}. \end{aligned}$$

Suppose the function $g(x)=x^{\sigma } e^{-\sqrt{t} x^{\alpha }}$, where $x \ge 0$, $\alpha$ is a constant greater than 0. From the derivation of the function g, one can infer that $g(x) \le g((\frac{x}{\alpha \sqrt{t}})^{\frac{1}{\alpha }}) \le$ $C_{\alpha ,\sigma } t^{-\frac{\sigma }{2 \alpha }}$. Therbey,

$$\begin{aligned} \left\| \Lambda ^{\sigma } v(t)\right\| _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}}&\le C_{\alpha ,\sigma } t^{-\frac{\sigma }{2 \alpha }} \left\| e^{ \sqrt{t} \vert D \vert ^{\alpha }} v(t)\right\| _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}} \\ {}&\le C_{\alpha ,\sigma } t^{-\frac{\sigma }{2 \alpha }} \left\| v_0\right\| _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}}. \end{aligned}$$

6 Asymptotic stability

We will establish some crucial lemmas in the proof of Theorem 4.

Lemma 5

Let $\frac{2}{3}<\alpha <1$ and $1 \le p, q \le 2$. Then we have

$$\begin{aligned} \Vert v\Vert _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}} \lesssim \Vert v\Vert _{L^{2}}^{\frac{3\alpha -2}{\alpha }}\Vert v\Vert _{{\dot{H}}^{\alpha }}^{\frac{ 2-2 \alpha }{\alpha }}. \end{aligned}$$

(18)

Proof

Using the definition of the Fourier–Besov spaces, and Hölder’s inequality, we have

$$\begin{aligned}&\Vert v\Vert _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}} \\&=\left\{ \sum _{j \in {\mathbb {Z}}} 2^{j\left( 1-2\alpha +\frac{2}{p'}\right) q}\left\| \widehat{{\dot{\Delta }}_ju}\right\| _{L^p}^{q}\right\} ^{1 / q}\\&\lesssim \left\{ \sum _{j \in {\mathbb {Z}}} 2^{j\left( 1-2\alpha +\frac{2}{p'}\right) q} 2^{j\left( \frac{2}{p}-1\right) q}\left\| \widehat{{\dot{\Delta }}_ju}\right\| _{L^{2}\left( B\left( x_{0}, r\right) \right) }^{q} \right\} ^{1 / q}\\&\lesssim \left\{ \sum _{j \le M} 2^{j\left( 2-2 \alpha \right) q}\left\| \widehat{{\dot{\Delta }}_ju}\right\| _{L^{2}\left( {\mathbb {R}}^{2}\right) }^{q}\right\} ^{1 / q}+\left\{ \sum _{j>M} 2^{j\left( 2-3\alpha \right) q} 2^{j \alpha q}\left\| \widehat{{\dot{\Delta }}_ju}\right\| _{L^{2}\left( {\mathbb {R}}^{2}\right) }^{q}\right\} ^{1 / q}\\&\lesssim 2^{\left( 2-2 \alpha \right) M}\left\{ \sum _{j \in {\mathbb {Z}}}\left\| \widehat{{\dot{\Delta }}_ju}\right\| _{L^{2}\left( {\mathbb {R}}^{2}\right) }^{2}\right\} ^{1 / 2}+2^{\left( 2-3\alpha \right) M}\left\{ \sum _{j \in {\mathbb {Z}}} 2^{2 \alpha j }\left\| \widehat{{\dot{\Delta }}_ju}\right\| _{L^{2}\left( {\mathbb {R}}^{2}\right) }^{2}\right\} ^{1 / 2}\\&\lesssim 2^{\left( 2-2 \alpha \right) M}\Vert v\Vert _{{\dot{FB}}_{2,2}^{0}} +2^{\left( 2-3\alpha \right) M}\Vert v\Vert _{{\dot{FB}}_{2,2}^{\alpha }}. \end{aligned}$$

From Remark 3, one can see that ${\dot{FB}}_{2,2}^{\alpha }={\dot{H}}^{\alpha }$ and ${\dot{FB}}_{2,2}^{0}=L^{2}$. Then, by Taking M such that $2^{M}=\left( \frac{\Vert v\Vert _{{\dot{H}}^{\alpha }}}{\Vert v\Vert _{L^{2}}}\right) ^{1/\alpha }$, we obtain

$$\begin{aligned} \Vert v\Vert _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}}&\lesssim \left( \frac{\Vert v\Vert _{{\dot{H}}^{\alpha }}}{\Vert v\Vert _{L^{2}}}\right) ^{\frac{2-\alpha }{\alpha }} \Vert v\Vert _{L^{2}}+\left( \frac{\Vert v\Vert _{{\dot{H}}^{\alpha }}}{\Vert v\Vert _{L^{2}}}\right) ^{\frac{2-3\alpha }{\alpha }}\Vert v\Vert _{{\dot{H}}^{\alpha }}\\&\lesssim \Vert v\Vert _{{\dot{H}}^{\alpha }}^{\frac{2-\alpha }{\alpha }} \Vert v\Vert _{L^{2}}^{\frac{3\alpha -2}{\alpha }}. \end{aligned}$$

$\square$

Lemma 6

Let $\frac{1}{2}<\alpha <1$, $1 \le p<\infty$ and $1\le q\le 2$. Then we have

$$\begin{aligned} \Vert u\theta \Vert _{{\dot{H}}^{1-\alpha }} \lesssim \Vert u\Vert _{L^{2}}\Vert \theta \Vert _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}}+\Vert u\Vert _{{\dot{H}}^{\alpha }}\Vert \theta \Vert _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}}. \end{aligned}$$

(19)

Proof

To prove this lemma, we will follow the method described in the proof of Lemma 3, for fixed j, we have

$$\begin{aligned} {\dot{\Delta }}_{j}(u\theta )&=\sum _{\vert k-j \vert \le 4}{\dot{\Delta }}_{j}({\dot{S}}_{k-1}u{\dot{\Delta }}_{k}\theta )+ \sum _{\vert k-j\vert \le 4}{\dot{\Delta }}_{j}({\dot{S}}_{k-1}\theta {\dot{\Delta }}_{k}u)\\&\quad +\sum _{k\ge j-3}{\dot{\Delta }}_{j}({\dot{\Delta }}_{k}u \widetilde{{\dot{\Delta }}}_{k}\theta )\\&:=II^1_{j}+II^2_{j}+II^3_{j}\,. \end{aligned}$$

One can write

$$\begin{aligned} \Vert u\theta \Vert _{{\dot{H}}^{1-\alpha }}&=\Vert u\theta \Vert _{F {\dot{B}}_{2,2}^{1-\alpha }}\\&\le \left\{ \sum _{j \in {\mathbb {Z}}} 2^{2 j(1-\alpha )}\left\| \widehat{II^1_{j}}\right\| _{L^{2}}^{2}\right\} ^{1 / 2}+\left\{ \sum _{j \in {\mathbb {Z}}} 2^{2 j(1-\alpha )}\left\| \widehat{II^2_{j}}\right\| _{L^{2}}^{2}\right\} ^{1 / 2} \\&\quad +\left\{ \sum _{j \in {\mathbb {Z}}} 2^{2 j(1-\alpha )}\left\| \widehat{II^3_{j}}\right\| _{L^{2}}^{2}\right\} ^{1 / 2}\\&:=\mathrm {R}_1+\mathrm {R}_2+\mathrm {R}_2. \end{aligned}$$

Then

$$\begin{aligned} \left\| {\widehat{II}}^{1}_{j}\right\| _{L^{2}}&\le \sum _{ \vert k-j \vert \le 4}\left\| {\dot{S}}_{k-1}u {\widehat{\Delta }}_{k} \theta \right\| _{L^{2}} \\&\le \sum _{ \vert k-j \vert \le 4}\left\| {\widehat{\theta }}_{k}\right\| _{L^{2}} \sum _{l \le k-2}\left\| {\widehat{u}}_{l}\right\| _{L^{1}} \\&\lesssim \sum _{ \vert k-j \vert \le 4}\left\| {\widehat{\theta }}_{k}\right\| _{L^{2}} \sum _{l \le k-2} 2^{l\left( \frac{2}{p'}\right) }\left\| {\widehat{u}}_{l}\right\| _{L^p}\\&\lesssim \sum _{ \vert k-j \vert \le 4}\left\| {\widehat{\theta }}_{k}\right\| _{L^{2}} \sum _{l \le k-2} 2^{l\left( \frac{2}{p'}\right) } 2^{-l(2 \alpha -1)} 2^{l(2 \alpha -1)}\left\| {\widehat{u}}_{l}\right\| _{L^p} \\&\lesssim \sum _{ \vert k-j \vert \le 4}\left\| {\widehat{\theta }}_{k}\right\| _{L^{2}}\left( \sum _{l \le k-2} 2^{l(2 \alpha -1) q^{\prime }}\right) ^{\frac{1}{q^{\prime }}}\Vert u\Vert _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}}\\&\lesssim \sum _{ \vert k-j \vert \le 4} 2^{k(2 \alpha -1)}\left\| \widehat{\theta }_{k}\right\| _{L^{2}}\Vert u\Vert _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}}. \end{aligned}$$

Multiplying by $2^{2j(1-\alpha )}$, and taking $l^2$-norm of both sides in the above estimate, we obtain

$$\begin{aligned} \mathrm {R}_1\lesssim \Vert u\Vert _{{\dot{H}}^{\alpha }}\Vert \theta \Vert _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}}. \end{aligned}$$

(20)

Similarly,

$$\begin{aligned} \left\| {\widehat{II}}^{2}_{j}\right\| _{L^{2}}&\le \sum _{ \vert k-j \vert \le 4}\left\| {\dot{S}}_{k-1}\theta {\widehat{\Delta }}_{k} u\right\| _{L^{2}} \\&\le \sum _{ \vert k-j \vert \le 4}\left\| {\widehat{u}}_{k}\right\| _{L^{1}} \sum _{l \le k-2}\left\| {\widehat{\theta }}_{l}\right\| _{L^{2}} \\&\lesssim \left\| \theta \right\| _{L^{2}} \sum _{ \vert k-j \vert \le 4} 2^{k\left( \frac{2}{p'}\right) }\left\| {\widehat{u}}_{k}\right\| _{L^p}. \end{aligned}$$

Therefore, by the Young inequality for series, we get

$$\begin{aligned} \mathrm {R}_2&\lesssim \left\| \theta \right\| _{L^{2}}\left\{ \sum _{j\in {\mathbb {Z}}}2^{2j(1-\alpha )}\big ( \sum _{ \vert k-j \vert \le 4} 2^{\frac{2}{p'}k}\left\| {\widehat{u}}_{k}\right\| _{L^p }\big )^2\right\} ^\frac{1}{2} \\ {}&\lesssim \left\| \theta \right\| _{L^{2}} \left\{ \sum _{j\in {\mathbb {Z}}}\big (\sum _{ \vert k-j \vert \le 4} 2^{(j-k)(1-\alpha )} 2^{k(1-2\alpha +\frac{2}{p'})}\left\| {\widehat{u}}_{k}\right\| _{L^p }\big )^2\right\} ^\frac{1}{2} \\ {}&\lesssim \left\| \theta \right\| _{L^{2}} \Vert u\Vert _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} } \sum _{i\le 3}2^{i(1-\alpha )} \\ {}&\lesssim \left\| \theta \right\| _{L^{2}} \Vert u\Vert _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}}, \end{aligned}$$

where we have used Proposition 5 and the fact that $\alpha <1.$

Now we deal with the third term $\mathrm {R}_3$, Again employing Young’s inequality and Hölder’s inequality and the fact that $\alpha <1$, we obtain

$$\begin{aligned} \Vert \widehat{II^{3}_{j}}\Vert _{L^2}&\le \sum _{k\ge j-3} \Vert \widehat{({\dot{\Delta }}_k u \widetilde{{\dot{\Delta }}_k}\theta )}\Vert _{L^2} \\ {}&\le \sum _{k\ge j-3} \Vert \widehat{({\dot{\Delta }}_k u }*\widehat{\widetilde{{\dot{\Delta }}}_{k}\theta }\Vert _{L^2} \\ {}&\le \sum _{k\ge j-3} \Vert \widehat{{\dot{\Delta }}_k u} \Vert _{L^2}\sum _{ \vert l-k\vert \le 1}\Vert \widehat{{\dot{\Delta }}_l \theta } \Vert _{L^1} \\ {}&\lesssim \sum _{k\ge j-3} \Vert \widehat{{\dot{\Delta }}_k u} \Vert _{L^2}\sum _{ \vert l-k\vert \le 1} 2^{\frac{2}{p'}l}\Vert \widehat{{\dot{\Delta }}_l \theta } \Vert _{L^p } \\ {}&\lesssim \sum _{k\ge j-3} \Vert \widehat{{\dot{\Delta }}_k u} \Vert _{L^2}\Big (\sum _{ \vert l-k\vert \le 1} 2^{l(2\alpha -1)q'}\Big )^\frac{1}{q'} \Vert \theta \Vert _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}} \\ {}&\lesssim \Vert \theta \Vert _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}} \sum _{k\ge j-3} 2^{k(2\alpha -1)} \Vert \widehat{{\dot{\Delta }}_k u} \Vert _{L^2}. \end{aligned}$$

Thus,

$$\begin{aligned} \mathrm {R}_3&\lesssim \Vert \theta \Vert _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}}\left\{ \sum _{j\in {\mathbb {Z}}}2^{2j(1-\alpha )}\big (\sum _{k\ge j-3} 2^{k(2\alpha -1)}\Vert \widehat{{\dot{\Delta }}_k u} \Vert _{L^2}\big )^2\right\} ^\frac{1}{2} \\ {}&\lesssim \Vert \theta \Vert _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}}\left\{ \sum _{j\in {\mathbb {Z}}}\big (\sum _{k\ge j-3}2^{(j-k)(1-\alpha )} 2^{k\alpha }\Vert \widehat{{\dot{\Delta }}_k u} \Vert _{L^2}\big )^2\right\} ^\frac{1}{2} \\ {}&\lesssim \Vert \theta \Vert _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}} \Vert u \Vert _{{\mathscr {F}}B^{\alpha }_{2,2}} \sum _{i\le 3}2^{i(1-\alpha )} \\ {}&\lesssim \Vert u \Vert _{{{\dot{H}}}^\alpha } \Vert \theta \Vert _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}}. \end{aligned}$$

Combining $\mathrm {R}_1$, $\mathrm {R}_2$ and $\mathrm {R}_3$, we conclude the desired result. $\square$

6.1 Proof of Theorem 4

To show the asymptotic stability for the global solution, we follow ideas from [28], in which the authors used standard interpolation in the Fourier space and energy estimates in $L^{2}$. (see also [12, 14, 45]).

Let $\varepsilon >0$, such that $\varepsilon \le \sigma$. For $m \in {\mathbb {N}}$, setting

$$\begin{aligned} \mathrm {S}_{m}=\left\{ \xi \in {\mathbb {R}}^{2};\vert \xi \vert \le m \text{ and } \left| {\widehat{v}}_{0}(\xi )\right| \le m \right\} . \end{aligned}$$

Clearly ${\mathscr {F}}^{-1}\left( \chi _{\mathrm {S}_{m}} {\hat{v}}_{0}\right) \rightarrow v_0$ in ${\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}}.$ Then, there is $m_0\in {\mathbb {N}}$ such that

$$\begin{aligned} \left\| v_{0}-{\mathscr {F}}^{-1}\left( \chi _{\mathrm {S}_{m}} {\hat{v}}_{0}\right) \right\| _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} } \le \frac{\varepsilon }{2}, \quad \forall m\ge m_0. \end{aligned}$$

Let m be a fixed integer such that $m\ge m_0.$

Set $v_{0, m}={\mathscr {F}}^{-1}\left( \chi _{\mathrm {S}_{m}} {\hat{v}}_{0}\right) , w_{0,m} =v_{0}-{\mathscr {F}}^{-1}\left( \chi _{\mathrm {S}_{m}} {\hat{v}}_{0}\right) .$ Therefore, we have proved that

$$\begin{aligned}\left\| w_{0,m} \right\| _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} } \le \frac{\varepsilon }{2},\\ v_{0, m} \in {\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} \cap \mathrm {L}^{2}. \end{aligned}$$

Next, we consider the system

$$\begin{aligned} \left\{ \begin{array}{l} \partial _{t}w_m + \Lambda ^{2 \alpha } w_m +u_{w_m }\cdot \nabla w_m =0, \quad x \in {\mathbb {R}}^{2}, t>0, \\ w_m (0, x)=w_{0,m} (x). \end{array}\right. \end{aligned}$$

(21)

For all $m\ge m_0$, we have $\left\| w_{0,m} \right\| _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} } \le \frac{\varepsilon }{2}$. So, we can conclude from Theorem 1 that there exists a unique global solution $w_m \in {\mathscr {C}}\left( {\mathbb {R}}^{+}, {\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} \right) \cap {\mathscr {L}}^{1}\left( {\mathbb {R}}^{+}, {\mathscr {F}}B_{p,q}^{1+\frac{2}{p'}} \right)$.

In addition, $\forall t\ge 0$ we have

$$\begin{aligned} \Vert w_m (t)\Vert _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} }+\Vert w_m \Vert _{ {\mathscr {L}}^{1}\left( {\mathbb {R}}^{+}, {\mathscr {F}}B_{p,q}^{1+\frac{2}{p'}} \right) }\lesssim \Vert w_{0,m} \Vert _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} }. \end{aligned}$$

Put

$$\begin{aligned} v=v-w_m +w_m:=v_m+w_m. \end{aligned}$$

with v is the solution of the Eq. (1). Then, $v_m$ is a solution of the system:

$$\begin{aligned} \left\{ \begin{array}{l} \partial _{t}v_m+ \Lambda ^{2 \alpha } v_m+u_{v_m} \cdot \nabla v_m+u_{v_m} \cdot \nabla w_m +u_{w_m } \cdot \nabla v_m=0, \quad x \in {\mathbb {R}}^{2}, t>0, \\ v_m(0, x)=v_{0,m}(x)\in {\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} \cap \mathrm {L}^{2}. \end{array}\right. \end{aligned}$$

Furthermore, $v_m\in {\mathscr {C}}\left( {\mathbb {R}}^{+}, {\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} \right) \cap {\mathscr {L}}^{1}\left( {\mathbb {R}}^{+}, {\mathscr {F}}B_{p,q}^{1+\frac{2}{p'}} \right) .$ By taking the inner products in $L^{2}({\mathbb {R}}^{2})$ with $v_m$ and integrating by parts, we can get

$$\begin{aligned} \frac{1}{2} \frac{d}{d t}\left\| v_{m}\right\| _{L^{2}}^{2}+\left\| \Lambda ^{\alpha } v_{m}\right\| _{L^{2}}^{2} \le \left| <u_{v_{m}} \cdot \nabla w_m, v_{m} >_{L^2}\right| . \end{aligned}$$

Since

$$\begin{aligned} \left\| \Lambda ^{\alpha } v_{m}\right\| _{L^{2}}=\left\| v_m \right\| _{{\dot{H}}^{\alpha }}. \end{aligned}$$

Thus, Cauchy–Schwarz inequality, the estimate (14), Lemma 6, and Young’s inequality give

$$\begin{aligned}&\frac{1}{2} \frac{d}{d t}\left\| v_{m}\right\| _{L^{2}}^{2}+\left\| v_m \right\| ^{2}_{{\dot{H}}^{\alpha }}\nonumber \\&\le \left| < \text {div} (u_{v_{m}}w_m ), v_{m} >_{L^2}\right| \nonumber \\&\lesssim \left\| \Lambda ^{1-\alpha }(u_{v_m}w_m ) \right\| _{L^2} \left\| \Lambda ^{\alpha }v_m \right\| _{L^2} \nonumber \\&\lesssim \left\| u_{v_m}w_m \right\| _{{\dot{H}}^{1-\alpha }} \left\| v_m \right\| _{{\dot{H}}^{\alpha }}\nonumber \\&\lesssim \left\| v_m w_m \right\| _{{\dot{H}}^{1-\alpha }} \left\| v_m \right\| _{{\dot{H}}^{\alpha }}\nonumber \\&\lesssim \left\| v_m \right\| _{L^2}\left\| w_m \right\| _{{\mathscr {F}}B_{p,q}^{1-\alpha +\frac{2}{p'}} }\left\| v_m \right\| _{{\dot{H}}^{\alpha }}+ \left\| w_m \right\| _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} }\left\| v_m \right\| ^{2}_{{\dot{H}}^{\alpha }}\nonumber \\&\lesssim \left\| v_m \right\| ^{2}_{L^2}\left\| w_m \right\| ^{2}_{{\mathscr {F}}B_{p,q}^{1-\alpha +\frac{2}{p'}} }+\left\| v_m \right\| ^{2}_{{\dot{H}}^{\alpha }} + \left\| w_m \right\| _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} }\left\| v_m \right\| ^{2}_{{\dot{H}}^{\alpha }}\\&\lesssim \left\| v_m \right\| ^{2}_{L^2}\left\| w_m \right\| ^{2}_{{\mathscr {F}}B_{p,q}^{1-\alpha +\frac{2}{p'}} }+\left\| v_m \right\| ^{2}_{{\dot{H}}^{\alpha }},\nonumber \end{aligned}$$

(22)

where in (22) we used that $\left\| w_m \right\| _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} }\le \frac{\varepsilon }{2}.$

Consequently,

$$\begin{aligned} \frac{d}{d t}\left\| v_{m}\right\| _{L^{2}}^{2}+\left\| v_{m}\right\| ^{2}_{{\dot{H}}^{\alpha }} \lesssim \left\| v_m \right\| ^{2}_{L^2}\left\| w_m \right\| ^{2}_{{\mathscr {F}}B_{p,q}^{1-\alpha +\frac{2}{p'}} }. \end{aligned}$$

Integrating with respect to time, we obtain

$$\begin{aligned} \left\| v_{m}\right\| _{L^{2}}^{2}+ \int _{0}^{t}\left\| v_{m}\right\| _{{\dot{H}}^{\alpha }}^{2} \lesssim \left\| v_{0, m}\right\| _{L^{2}}^{2}+\int _{0}^{t}\left\| v_m \right\| ^{2}_{L^2}\left\| w_m \right\| ^{2}_{{\mathscr {F}}B_{p,q}^{1-\alpha +\frac{2}{p'}} }. \end{aligned}$$

(23)

By Gronwall’s lemma, we get

$$\begin{aligned} \left\| v_{m}\right\| _{L^{2}}^{2}&\lesssim \left\| v_{0, m}\right\| _{L^{2}}^{2}\exp {\int _{0}^{t}\left\| w_m \right\| ^{2}_{{\mathscr {F}}B_{p,q}^{1-\alpha +\frac{2}{p'}} }}\nonumber \\&\lesssim \left\| v_{0, m}\right\| _{L^{2}}^{2}, \end{aligned}$$

(24)

where the following fact is used in the last inequality: $\displaystyle \int _{0}^{t}\left\| w_m \right\| ^{2}_{{\mathscr {F}}B_{p,q}^{1-\alpha +\frac{2}{p'}} }\le C.$

Indeed:

Since $q\le 2$, then by Minkowski’s inequality (9) we have

$$\begin{aligned} {\mathscr {L}}^{2}\left( I, {\mathscr {F}}B_{p,q}^{1-\alpha +\frac{2}{p'}} \right) \hookrightarrow L^{2}\left( I ; {\mathscr {F}}B_{p,q}^{1-\alpha +\frac{2}{p'}} \right) . \end{aligned}$$

(25)

Thus, using (25) and Hölder’s inequality for series, we get

$$\begin{aligned}&\int _{0}^{t}\left\| w_m \right\| ^{2}_{{\mathscr {F}}B_{p,q}^{1-\alpha +\frac{2}{p'}} }\\&\lesssim \left\| 2^{j\left( 1-2\alpha +\frac{2}{p'}\right) }\left\| \widehat{{\dot{\Delta }}_j w_m} \right\| _{L^{2}([0,t),L^p )}\right\| _{l^q}^{2} \\&\lesssim \left\| 2^{\frac{j}{2}\left( 1-2\alpha +\frac{2}{p'}\right) }2^{\frac{j}{2}\left( 1+\frac{2}{p'}\right) }\left\| \widehat{{\dot{\Delta }}_j w_m} \right\| ^{\frac{1}{2}}_{L^{\infty }([0,t),L^p )}\left\| \widehat{{\dot{\Delta }}_j w_m} \right\| ^{\frac{1}{2}}_{L^{1}([0,t),L^p )}\right\| _{l^q}^{2}\\&\lesssim \left\| 2^{j\left( 1-2\alpha +\frac{2}{p'}\right) }\left\| \widehat{{\dot{\Delta }}_j w_m} \right\| _{L^{\infty }([0,t),L^p )}\right\| _{l^q} \left\| 2^{j\left( 1+\frac{2}{p'}\right) }\left\| \widehat{{\dot{\Delta }}_j w_m} \right\| _{L^{1}([0,t),L^p )}\right\| _{l^q}\\&\lesssim \left\| w_m \right\| _{{\mathscr {L}}^{\infty }\left( [0,t), {\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} \right) } \left\| w_m \right\| _{{\mathscr {L}}^{1}\left( [0,t), {\mathscr {F}}B_{p,q}^{1+\frac{2}{p'}} \right) }\\&\lesssim \left( \left\| w_m \right\| _{{\mathscr {L}}^{\infty }\left( [0,t), {\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} \right) }+ \left\| w_m \right\| _{{\mathscr {L}}^{1}\left( [0,t), {\mathscr {F}}B_{p,q}^{1+\frac{2}{p'}} \right) }\right) ^{2}\\&\lesssim \left\| w_{0,m} \right\| ^{2}_{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} }\\&\le C. \end{aligned}$$

Combining (23) and (24), we obtain

$$\begin{aligned} \left\| v_{m}\right\| _{L^{2}}^{2}+ \int _{0}^{t}\left\| v_{m}\right\| _{{\dot{H}}^{\alpha }}^{2}&\le \left\| v_{0, m}\right\| _{L^{2}}^{2}+\left\| v_{0, m}\right\| _{L^{2}}^{2}\int _{0}^{t}\left\| w_m \right\| ^{2}_{{\mathscr {F}}B_{p,q}^{1-\alpha +\frac{2}{p'}} }\\&\le C. \end{aligned}$$

Now, applying Lemma 5 we deduce

$$\begin{aligned} \Vert v_m\Vert ^{\frac{\alpha }{1-\alpha }}_{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} } \lesssim \Vert v_m\Vert _{L^{2}}^{\frac{3\alpha -2}{1-\alpha }}\Vert v_m\Vert _{{\dot{H}}^{\alpha }}^{2}. \end{aligned}$$

(26)

Therefore,

$$\begin{aligned} \Vert v_m\Vert ^{\frac{\alpha }{1-\alpha }}_{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} } \lesssim \Vert v_m\Vert _{{\dot{H}}^{\alpha }}^{2}, \end{aligned}$$

(27)

where we have used (24).

Finally, by integrating in time between 0 and $\infty$, we obtain

$$\begin{aligned} \int ^{\infty }_{0}\Vert v_m\Vert ^{\frac{\alpha }{1-\alpha }}_{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} } \le C \int ^{\infty }_{0}\Vert v_m\Vert _{{\dot{H}}^{\alpha }}^{2}. \end{aligned}$$

(28)

Let us consider the following subset of $[0,\infty [$:

$$\begin{aligned} {\mathscr {A}}_{\varepsilon }=\left\{ t\ge 0; \;\;\Vert v_m(t)\Vert _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} \left( {\mathbb {R}}^2\right) }\ge \frac{\varepsilon }{2}\right\} . \end{aligned}$$

Then, for all $\frac{1}{2}<\alpha <1,$

$$\begin{aligned} \int _{0}^{\infty }\Vert v_m(t)\Vert ^{\frac{\alpha }{1-\alpha }}_{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} \left( {\mathbb {R}}^2\right) }dt\ge \int _{{\mathscr {A}}_{\varepsilon }}\Vert v_m(t)\Vert ^{\frac{\alpha }{1-\alpha }}_{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} \left( {\mathbb {R}}^2\right) }dt\ge \mu \left( {{\mathscr {A}}_{\varepsilon }}\right) \left( {\frac{\varepsilon }{2}}\right) ^{\frac{\alpha }{1-\alpha }},\end{aligned}$$

(29)

where $\mu \left( {{\mathscr {A}}_{\varepsilon }}\right)$ is the Lebesgue measure of ${\mathscr {A}}_{\varepsilon }$.

Using the estimates (28) and (29), we get $\mu \left( {{\mathscr {A}}_{\varepsilon }}\right) <\infty$ and $\displaystyle \mu \left( {{\mathscr {A}}_{\varepsilon }}\right) \lesssim \left( {\frac{2}{\varepsilon }}\right) ^{\frac{\alpha }{1-\alpha }}\int ^{\infty }_{0}\Vert v_m\Vert _{{\dot{H}}^{\alpha }}^{2}.$ For $\eta >0,$ there exists $t_0\in [0,\mu \left( {{\mathscr {A}}_{\varepsilon }}\right) +\eta ]$ such that $t_0 \notin {\mathscr {A}}_{\varepsilon }.$

Then,

$$\begin{aligned} \Vert v_m(t_0)\Vert _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} \left( {\mathbb {R}}^2\right) }\le \frac{\varepsilon }{2}. \end{aligned}$$

Therefore, we have

$$\begin{aligned} \Vert v(t_0)\Vert _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} \left( {\mathbb {R}}^2\right) }&\le \Vert v_m(t_0)\Vert _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} \left( {\mathbb {R}}^2\right) }+\Vert w_m (t_0)\Vert _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} \left( {\mathbb {R}}^2\right) }\\&\le \frac{\varepsilon }{2}+\Vert w_{0,m} \Vert _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} \left( {\mathbb {R}}^2\right) }\\&\le \frac{\varepsilon }{2}+\frac{\varepsilon }{2}. \end{aligned}$$

Consequently,

$$\begin{aligned} \Vert v(t_0)\Vert _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} \left( {\mathbb {R}}^2\right) }\le \varepsilon . \end{aligned}$$

(30)

Now, we consider the quasi-geostrophic equation starting at $t=t_0$.

$$\begin{aligned} \left\{ \begin{array}{l} \partial _{t}\theta +u_{\theta } \cdot \nabla \theta + \Lambda ^{2 \alpha } \theta =0, \quad x \in {\mathbb {R}}^{2},\; t>0, \\ \theta (0, x)=\theta _0=v(t_0). \end{array}\right. \end{aligned}$$

(31)

Using inequality (30) and Theorem 1, we infer that there exists a unique solution $\theta \in {\mathscr {C}}\left( {\mathbb {R}}^{+}, {\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} \right) \cap {\mathscr {L}}^{1}\left( {\mathbb {R}}^{+}, {\mathscr {F}}B_{p,q}^{1+\frac{2}{p'}} \right)$ of the problem (1).

The existence and uniqueness of a solution to the quasi-geostrophic equation gives $\forall t \ge 0 \quad \theta (t)=v\left( t_{0}+t\right)$. Then,

$$\begin{aligned} \Vert v(t_0+t)\Vert _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} \left( {\mathbb {R}}^2\right) }&=\Vert \theta (t)\Vert _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} \left( {\mathbb {R}}^2\right) }\\&\le \Vert \theta _0\Vert _{{\mathscr {F}}B_{p,q}^{1-2\alpha +\frac{2}{p'}} \left( {\mathbb {R}}^2\right) }\\&\le \varepsilon . \end{aligned}$$

This completes the proof of Theorem 4.

Remark 5

If $k\ne 1$, the results of Theorem 1 and Theorem 4 remain true, but with more complicated calculations.