1 Introduction

Rotating fluid systems appear in many applications of geophysical fluid mechanics, in particular in the models describing large-scale ocean and atmosphere flows, see the monographs [10, 20, 37] for instance. The Coriolis force, arising from the rotation of the Earth, plays a significant role in such systems. In 1868, Kelvin first observed that a sphere moving along the axis of uniformly rotating water takes with it a column of liquid as if this were a rigid mass, and pioneered the research on the motion of rotating fluid, see [17]. Since then a lot of researchers, particularly Taylor [41] and Proudmann [39], have made great contributions to this subject. Mathematical studies on the rotating fluid are initiated by Poincaré [38], and have drawn increasing attention during recent years, see [14, 9, 14, 19, 22, 26, 27, 30, 31] and references therein.

In this paper, we study unique existence and analyticity of global mild solutions to the initial value problem of the rotating Navier–Stokes equations in \(\mathbb {R}^3\), describing the rotating flow of an incompressible viscous fluid:

$$\begin{aligned} \left\{ \begin{array}{ll} \partial _t u-\Delta u+\Omega e_3\times u+(u\cdot \nabla )u+\nabla p=0, \ \ \ \ \ \ \ \ \ \ &{}\text {in}~~\mathbb {R}^3\times (0,\infty ), \\ \text {div} u=0,&{}\text {in}~~\mathbb {R}^3\times (0,\infty ), \\ u|_{t=0}=u_0,&{}\text {in}~~\mathbb {R}^3, \end{array} \right. \end{aligned}$$
(1.1)

and that of mild solutions to the time-periodic problem of the three-dimensional rotating Navier–Stokes equations with external force in the whole time line:

$$\begin{aligned} \left\{ \begin{array}{ll} \partial _t u-\Delta u+\Omega e_3\times u+(u\cdot \nabla )u+\nabla p=f,\ \ \ \ \ \ \ \ \ \ &{}\text {in}~~\mathbb {R}^3\times \mathbb {R}, \\ \text {div} u=0,&{}\text {in}~~\mathbb {R}^3\times \mathbb {R}, \\ u(t+\omega ,\cdot )=u(t,\cdot ),&{} \text {for}~~t\in \mathbb {R}, \end{array} \right. \end{aligned}$$
(1.2)

where the unknown functions \(u=(u_1, u_2, u_3)\) and p denote velocity field and pressure, respectively, \(u_0\) is the given initial velocity, f is the given time-periodic external force with period \(\omega \). Note that \(\Omega \in \mathbb {R}\) is the Coriolis parameter, which is twice angular velocity of the rotation around the vertical unit vector \(e_3=(0,0,1)\), and \(\Omega e_3\times u\) represents the so-called Coriolis force. Here the kinematic viscosity coefficient has been normalized into one by rescaling. We refer to [10, 20, 37] for derivation of this model and more detailed discussions on its physical background.

If \(\Omega =0\), (1.1) and (1.2) become the problems related to the classical Navier–Stokes equations, which have been widely studied during the past seventy years. It has been proved that problem (1.1) with \(\Omega =0\) is globally well-posed for small initial data in a family of function spaces including particularly the following ones:

$$\begin{aligned} \dot{H}^{\frac{1}{2}}(\mathbb {R}^3)\hookrightarrow L^3(\mathbb {R}^3)\hookrightarrow \dot{B}^{-1+\frac{3}{p}}_{p,\infty }(\mathbb {R}^3)\;(3<p<\infty )\;\hookrightarrow \text {BMO}^{-1}(\mathbb {R}^3), \end{aligned}$$

see Fujita and Kato [18], Kato [28], Cannone [8], Koch and Tataru [29]. For problem (1.2) with \(\Omega =0\), based on the formalism of mild solutions developed by Fujita and Kato [18] for the study of Cauchy problem, Kozono and Nakao [32] studied the existence and uniqueness of time-periodic solutions for \(f\in BC(\mathbb {R},L^r(\mathbb {R}^3)\cap \dot{H}^{-1}_p(\mathbb {R}^3))\) under some smallness conditions. Yamazaki generalized the result of [32] to Morrey space in [44] and proved that the Lorentz space \(L^\infty _t L^{3,\infty }\) is also well fit to the search of time-periodic solutions in [45].

If \(\Omega \in \mathbb {R}\setminus \{0\}\), it is a remarkable fact that (1.1) and (1.2) admit global mild solutions and time-periodic mild solutions for arbitrarily large initial velocity \(u_0\) and arbitrarily large external force f, respectively, provided that the speed of rotation is fast enough. It was proved by Chemin et al. in [9, 10] that for any given \(u_0\in L^2(\mathbb {R}^2) + H^{\frac{1}{2}}(\mathbb {R}^3)\) with \(\text {div}u_0=0\), there exists a corresponding constant \(\Omega _0>0\) such that for any \(|\Omega |\ge \Omega _0\), problem (1.1) admits a unique global mild solution. Iwabuchi and Takada [26] studied unique existence of global mild solutions to problem (1.1) for \(u_0\in \dot{H}^s(\mathbb {R}^3)\) with \(\frac{1}{2}<s<\frac{3}{4}\) under the size condition

$$\begin{aligned} \Vert u_0\Vert _{\dot{H}^{s}}\le C|\Omega |^{\frac{1}{2}(s-\frac{1}{2})}, \end{aligned}$$
(1.3)

which gives us an explicit characterization of the relationship between the size of initial velocity and the speed of rotation. Furthermore, by establishing an improved dispersive estimate for the Coriolis linear group \(\{e^{\pm i\Omega t\frac{D_3}{|D|}}\}_{t\in \mathbb {R}}\), Koh, Lee and Takada [30] enlarged the range of regular index s into \(\frac{1}{2}<s<\frac{9}{10}\). We also mention that Babin, Mahalov and Nicolaenko [2, 3] proved global existence and regularity of solutions to problem (1.1) for periodic initial data under large \(|\Omega |\). To the global well-posedness of problem (1.1) for the uniformly small initial data \(u_0\) with respect to \(|\Omega |\), we refer interested readers to [14, 19, 22, 27, 31].

For problem (1.2), Konieczny and Yoneda [31] verified unique existence of stationary mild solutions for any given f belonging to \(\dot{FB}^{-\frac{3}{p}}_{p,p}\) with \(3<p<\infty \) under large \(|\Omega |\). Iwabuchi and Takada [25] obtained existence of unique time-periodic solutions to problem (1.2) for \(f\in BC(\mathbb {R},\dot{B}^{s_1}_{p_1,2}(\mathbb {R}^3) \cap \dot{B}^{s_2}_{p_2,2}(\mathbb {R}^3))\) satisfying some size conditions with prescribed \(s_1, s_2, p_1\) and \(p_2\). In [30], Koh, Lee and Takada proved unique existence of time-periodic solutions to problem (1.2) for \(f\in BC(\mathbb {R},L^r(\mathbb {R}^3))\) with \(1<r<\frac{15}{11}\) under the size condition

$$\begin{aligned} \sup _{t\in \mathbb {R}}\Vert f\Vert _{L^r}\le C|\Omega |^{\frac{3}{2}(1-\frac{1}{r})}. \end{aligned}$$
(1.4)

The goals of the present paper are to establish the global existence and the space analyticity of mild solutions to the three-dimensional rotating Navier–Stokes equations in some more general function spaces, and to provide explicit estimates for the analyticity radius as a function of time. In fluid dynamics, the space analyticity radius has an important physical interpretation: at this length scale the viscous effects and the (nonlinear) inertial effects are roughly comparable. Below this length scale, the viscous effects dominate the inertial effects and the Fourier spectrum decays exponentially, see [13, 16, 23]. This fact can be used to show that the finite-dimensional Galerkin approximations converge exponentially in these cases, see [12]. Other applications of analyticity radius occur in establishing sharp temporal decay rates of solutions in higher Sobolev norms, see [36], and in establishing geometric regularity criteria for the Navier–Stokes equations, see [21].

For the issue of existence, by striking new balances between the smoothing effects of heat flow and the dispersive effects of Coriolis force, we obtain the existence and uniqueness of global mild solutions and time-periodic mild solutions to problems (1.1) and (1.2), respectively. These results improve the related ones of Iwabuchi and Takada [26] and Koh, Lee and Takada [30], and the result for problem (1.2) can also be regarded as an enhancement and complement of that of Iwabuchi and Takada [25].

For the issue of analyticity, we follow the Gevrey class approach pioneered by Foias and Temam [15] for estimating space analyticity radius of the Navier–Stokes equations and subsequently developed by [5, 24, 36]. More precisely, let \(\mathcal {X}\) be a Besov space, and we will show that a solution \(u(t,\cdot )\in \mathcal {X}\) satisfies

$$\begin{aligned} \sup _{0<t\le T}\Vert e^{\sqrt{t}\Lambda _1}u\Vert _{\mathcal {X}}<\infty \ \ \ \text {for arbitrary}\ \ T>0, \end{aligned}$$
(1.5)

where \(\Lambda _1\) is the Fourier multiplier whose symbol is given by \(|\xi |_1:=\sum _{i=1}^3|\xi _i|\). This approach enables one to avoid cumbersome recursive estimation of higher-order derivatives. For the studies on the space analyticity for some other models, we refer to [34, 35, 46, 47].

Let X be a Banach space. For a operator \(T: X\rightarrow X\), we denote

$$\begin{aligned} TX:=\{Tf:\ f\in X\}\ \ \ \ \text {and} \ \ \ \Vert f\Vert _{TX}:=\Vert Tf\Vert _{X}. \end{aligned}$$

The first two results in this paper are the following ones for problem (1.1) concerning \(u_0\in \dot{B}^{s}_{p,r}(\mathbb {R}^3)\) with \(1< p<2\) and \(p=2\), respectively.

Theorem 1.1

Let \(\theta \in \{0,1\}\). Let \(p\in (\frac{3}{2},2)\) and \(r\in [1, \infty )\). Let \(s\in \mathbb {R}\) satisfy

$$\begin{aligned} -1+\frac{3}{p}<s<3-\frac{3}{p}, \end{aligned}$$

and let \(\delta \in [2,\infty )\) satisfy

$$\begin{aligned} 0<\frac{1}{\delta }< \frac{2}{p}-1\ \ \ \ \text {and} \ \ \ \ \frac{1}{2}\left( s+1-\frac{3}{p}\right) +1-\frac{2}{p}<\frac{1}{\delta } <\frac{1}{4}\left( s+1-\frac{3}{p}\right) . \end{aligned}$$

Then, there exists a positive constant C such that for \(\Omega \in \mathbb {R}\setminus \{0\}\) and \(u_0\in \dot{B}^{s}_{p,r}(\mathbb {R}^3)\) satisfying \(\text {div} u_0 = 0\) and

$$\begin{aligned} \Vert u_0\Vert _{\dot{B}^{s}_{p,r}}\le C|\Omega |^{\frac{1}{2}\left( s+1-\frac{3}{p}\right) }, \end{aligned}$$

(1.1) possesses a unique mild solution

$$\begin{aligned} u\in C\big ([0,\infty ); e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s}_{p,r}(\mathbb {R}^3)\big )\bigcap {\tilde{L}}^\delta \left( 0,\infty ;e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s+\frac{3}{p'}-\frac{3}{p}}_{p',r}(\mathbb {R}^3)\right) , \end{aligned}$$

where \(\frac{1}{p'}+\frac{1}{p}=1\).

Theorem 1.2

Let \(\theta \in \{0,1\}\). Let \(s\in (\frac{1}{2},1)\) and \(r\in [1, \infty )\). Let \(q\in (2,3]\) satisfy

$$\begin{aligned} \frac{s}{6}+\frac{1}{4}<\frac{1}{q}< -\frac{s}{6}+\frac{7}{12}, \end{aligned}$$

and let \(\delta \in [2,\infty )\) satisfy

$$\begin{aligned} \frac{1}{2}\left( s-\frac{1}{2}\right) +\frac{2}{q}-1<\frac{1}{\delta } <\min \left\{ \frac{1}{4}\left( s-\frac{1}{2}\right) ,\ \frac{1}{2}-\frac{1}{q}\right\} . \end{aligned}$$

Then, there exists a positive constant C such that for \(\Omega \in \mathbb {R}\setminus \{0\}\) and \(u_0\in \dot{B}^{s}_{2,r}(\mathbb {R}^3)\) satisfying \(\text {div} u_0 = 0\) and

$$\begin{aligned} \Vert u_0\Vert _{\dot{B}^{s}_{2,r}}\le C|\Omega |^{\frac{1}{2}\left( s-\frac{1}{2}\right) }, \end{aligned}$$
(1.6)

(1.1) possesses a unique mild solution

$$\begin{aligned} u\in C\big ([0,\infty ); e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s}_{2,r}(\mathbb {R}^3)\big )\bigcap {\tilde{L}}^\delta \big (0,\infty ;e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s+\frac{3}{q}-\frac{3}{2}}_{q,r}(\mathbb {R}^3)\big ). \end{aligned}$$

Remark 1.3

  1. (i)

    In the case of \(\theta =0\), Theorems 1.1 and 1.2 show that for any given \(u_0\in \dot{B}^{s}_{p,r}(\mathbb {R}^3)\) with \(p\in (\frac{3}{2},2]\) and s, r prescribed above, there exists a positive parameter \(\Omega _0\) defined by

    $$\begin{aligned} { \Omega _0:=\left( C^{-1}\Vert u_0\Vert _{\dot{B}^{s}_{p,r}}\right) ^{\frac{2}{s+1-\frac{3}{p}}}}. \end{aligned}$$

    If \(|\Omega |\ge \Omega _0\), then problem (1.1) admits a unique global mild solution. In the case of \(\theta =1\), Theorems 1.1 and 1.2 indicate that the obtained solution satisfies the Gevrey estimate (1.5) with \(\mathcal {X}=\dot{B}^{s}_{p,r}(\mathbb {R}^3)\), which implies that this solution is analytic in the spatial variables.

  2. (ii)

    The global existence of mild solutions has been obtained to problem (1.1) for \(u_0\in \dot{H}^{s}(\mathbb {R}^3)\) under the size condition (1.3) with \(\frac{1}{2}<s<\frac{3}{4}\) in [26] and \(\frac{1}{2}<s<\frac{9}{10}\) in [30]. Due to \(\dot{H}^{s}=\dot{B}^s_{2,2}\) and \(\dot{H}^s\hookrightarrow \dot{B}^s_{2,r}\) for \(r>2\), Theorem 1.2 improves [26] and [30] in the sense that it enlarges the range of regular index s into \(\frac{1}{2}<s<1\) when \(u_0\in \dot{H}^{s}(\mathbb {R}^3)\), and the initial velocity \(u_0\) is permitted in a larger function space \(\dot{B}^s_{2,r}(\mathbb {R}^3)\), \(r>2\).

The third result is concerned with the time-periodic problem (1.2) and reads as follows:

Theorem 1.4

Let \(\theta \in \{0,1\}\). Let \(p\in (\frac{3}{2}, 2)\)\(r\in [1, \infty ]\), and let \(s\in \mathbb {R}\) satisfy

$$\begin{aligned} -3+\frac{3}{p}<s<\min \left\{ \frac{11}{p}-7, -1+\frac{1}{p}\right\} \end{aligned}$$
(1.7)

and let \(\beta \in \mathbb {R}\) satisfy

$$\begin{aligned} s+5-\frac{7}{p}<\beta<s+3-\frac{3}{p},~~~~0<\beta <\min \left\{ \frac{4}{p}-2, 4-\frac{6}{p}\right\} . \end{aligned}$$

Then, there exist positive constants C and M such that for \(\Omega \in \mathbb {R}\setminus \{0\}\) and external force \(f\in BC(\mathbb {R};e^{\theta \sqrt{t}\Lambda _1}\dot{B}_{p,r}^{s}(\mathbb {R}^3))\) satisfying \(f(t)=f(t + \omega )\) for all \(t\in \mathbb {R}\) with \(\omega > 0\) and

$$\begin{aligned} \sup _{t\in \mathbb {R}}\Vert f(t)\Vert _{e^{\theta \sqrt{t}\Lambda _1}\dot{B}_{p,r}^{s}}\le C|\Omega |^{\frac{1}{2}\left( s+3-\frac{3}{p}\right) }, \end{aligned}$$
(1.8)

(1.2) possesses a unique mild solution \(u\in Z_M\) satisfying \(u(t) = u(t + \omega )\) for all \(t\in \mathbb {R}\). Here

$$\begin{aligned} Z_M:=\Big \{u\in BC\left( \mathbb {R};e^{\theta \sqrt{t}\Lambda _1}\dot{B}_{p',r}^{-1+\frac{3}{p'}+\beta }(\mathbb {R}^3)\right) :~~\sup _{t\in \mathbb {R}} \Vert e^{\theta \sqrt{t}\Lambda _1}u(t)\Vert _{\dot{B}_{p',r}^{-1+\frac{3}{p'}+\beta }}\le |\Omega |^{\frac{\beta }{2}}M\Big \}, \end{aligned}$$

where \(\frac{1}{p'}+\frac{1}{p}=1\).

Remark 1.5

  1. (i)

    In the case of \(\theta =0\), Theorem 1.4 implies that for any given external force \(f\in BC(\mathbb {R};\dot{B}_{p,r}^{s}(\mathbb {R}^3))\) with \(p\in (\frac{3}{2},2)\) and s, r prescribed above, there exists a positive parameter \(\Omega _0\) defined by

    $$\begin{aligned} \Omega _0:=\left( C^{-1}\sup _{t\in \mathbb {R}}\Vert f(t)\Vert _{\dot{B}_{p,r}^{s}}\right) ^{\frac{2}{s+3-\frac{3}{p}}}. \end{aligned}$$

    If \(|\Omega |\ge \Omega _0\), then problem (1.2) admits a unique time-periodic mild solution. In the case of \(\theta =1\), the size condition (1.8), which means that f is analytic in the spatial variables, still ensures the unique existence of mild solutions to problem (1.2) due to \(\Vert f(t)\Vert _{\dot{B}_{p,r}^{s}}\le C\Vert e^{\sqrt{t}\Lambda _1}f(t)\Vert _{\dot{B}_{p,r}^{s}}\). Theorem 1.4 therefore indicates that if the external force f is analytic in the spatial variables, so is the obtained mild solution.

  2. (ii)

    In [30], the unique existence of mild solutions to problem (1.2) was proved for \(f\in BC(\mathbb {R};L^\rho (\mathbb {R}^3))\) with \(1<\rho <\frac{15}{11}\) under the size condition (1.4). However, by Lemma 2.4, we note that

    $$\begin{aligned} L^\rho (\mathbb {R}^3)\hookrightarrow \dot{B}_{\rho ,2}^{0}(\mathbb {R}^3)\hookrightarrow \dot{B}_{p,2}^{-3\left( \frac{1}{\rho }-\frac{1}{p}\right) }(\mathbb {R}^3),~~~~\text {for}\ \ \rho \in \left( 1,\frac{15}{11}\right) \ \ \text {and}\ \ p\in \left[ \frac{3}{2}, 2\right) . \end{aligned}$$

    It is not difficult to check that, for each \(\rho \in (1, \frac{15}{11})\), there always exists p in \((\frac{3}{2}, 2)\) such that \(-3(\frac{1}{\rho }-\frac{1}{p})\) satisfies (1.7), namely

    $$\begin{aligned} -3+\frac{3}{p}<-3\left( \frac{1}{\rho }-\frac{1}{p}\right) <\min \left\{ \frac{11}{p}-7, -1+\frac{1}{p}\right\} . \end{aligned}$$
    (1.9)

    Particularly, it is obvious that (1.9) holds for all \(\rho \in (1, \frac{15}{11})\) by taking \(p=\frac{5}{3}\). So, Theorem 1.4 improves the corresponding result in [30] by providing some much more general external force spaces.

  3. (iii)

    In [25], the unique existence of mild solutions to problem (1.2) was studied for external force \(f\in BC(\mathbb {R};\dot{B}_{p_1,2}^{s_1}(\mathbb {R}^3))\cap BC(\mathbb {R};\dot{B}_{p_2,2}^{s_2}(\mathbb {R}^3))\) with

    $$\begin{aligned}&\frac{3}{2}<p_1<2,\ \ -5+\frac{6}{p_1}<s_1<-6 +\frac{8}{p_1}\\&\quad \text {and} \ \ \frac{12}{7}\le p_2<2, \ \ -4+\frac{6}{p_2}<s_2<-5+\frac{8}{p_2} \end{aligned}$$

    under the size condition

    $$\begin{aligned} \sup _{t\in \mathbb {R}}\Vert f(t)\Vert _{\dot{B}_{p_1,2}^{s_1}}+\sup _{t\in \mathbb {R}}\Vert f(t)\Vert _{\dot{B}_{p_2,2}^{s_2}}<C\min \bigg \{|\Omega |^{\frac{1}{2}(s_1+5-\frac{6}{p_1})}, |\Omega |^{\frac{1}{2}(s_2+4-\frac{6}{p_2})}\bigg \}. \end{aligned}$$

    For \(\frac{3}{2}<p<2\), it is obvious that

    $$\begin{aligned} {-5+\frac{6}{p}< -3+\frac{3}{p}<-6+\frac{8}{p}<\min \left\{ \frac{11}{p}-7, -1+\frac{1}{p}\right\} .} \end{aligned}$$

    Thus, Theorem 1.4 strengthens [25] by demanding that f belongs only to \(BC(\mathbb {R};\dot{B}_{p,r}^{s}(\mathbb {R}^3))\) with \(p\in (\frac{3}{2},2)\), \(r\in [1,\infty ]\) and s satisfying (1.7). Theorem 1.4 can also be regarded as a complement to [25] because it provides a new range \([-6+\frac{8}{p},\min \{\frac{11}{p}-7, -1+\frac{1}{p}\})\) for regular index s.

The rest of this paper is organized as follows. In Sect. 2, we collect some basic facts on Littlewood–Paley theory and introduce the Stokes–Coriolis semigroup \(\{T_\Omega (t)\}_{t\ge 0}\). In Sect. 3, by using the smoothing effects of heat flow and the dispersive effects of Coriolis force, we derive some new linear estimates for semigroup \(\{T_\Omega (t)\}_{t\ge 0}\). In Sect. 4, by using Bony’s decomposition, we establish a useful product law and further get the bilinear estimates. Finally, we present the proofs of the main results.

Throughout this paper, for any scaler function space X, we shall use the same notation X to denote its 3-vector counterpart to simplify the notation. Both \(\mathcal {F}g\) and \({\hat{g}}\) stand for Fourier transform of g with respect to space variables, while \(\mathcal {F}^{-1}\) stands for the inverse Fourier transform. We shall use C to denote universal constant whose value may change from line to line. In particular, if we write \(C= C(., \ldots , .)\), then we mean that this constant depends only on the quantities appearing in the parentheses. The linear space of all multipliers on \(L^p\) is denoted by \(M_p\).

2 Preliminaries

Let X be a Banach space endowed with the norm \(\Vert \cdot \Vert _X\), we denote by \(BC(\mathbb {R}; X)\) the set of all functions \(u\in C(\mathbb {R}; X)\) such that \(\sup _{t\in \mathbb {R}}\Vert u\Vert _X<\infty \).

Let \(\mathscr {S}(\mathbb {R}^3)\) be the Schwartz space of smooth functions over \(\mathbb {R}^3\), and let \(\mathscr {S}'(\mathbb {R}^3)\) be the space of tempered distributions. First, we recall the homogeneous Littlewood–Paley decomposition.

Let \(\varphi , \psi \in \mathscr {S}(\mathbb {R}^3)\) be two radial functions such that their Fourier transforms \({\hat{\varphi }}\) and \({\hat{\psi }}\) satisfy the following properties:

$$\begin{aligned} \text {supp}~{\hat{\varphi }}\subset \mathcal {B}:= & {} \left\{ \xi \in \mathbb {R}^3:|\xi |\le \frac{4}{3}\right\} ,\\ \text {supp}~{\hat{\psi }}\subset \mathcal {C}:= & {} \left\{ \xi \in \mathbb {R}^3:\frac{3}{4}\le |\xi |\le \frac{8}{3}\right\} \end{aligned}$$

and

$$\begin{aligned} \sum _{j\in \mathbb {Z}} {\hat{\psi }}(2^{-j}\xi )=1 ~\quad \text {for all } \xi \in \mathbb {R}^3\setminus \{0\}. \end{aligned}$$

Let \(\varphi _j(x):=2^{3j}\varphi (2^jx)\) and \(\psi _j(x):=2^{3j}\psi (2^jx)\) for \(j\in \mathbb {Z}\). We define by \(\Delta _j\) and \(S_j\) the following operators in \(\mathscr {S}'(\mathbb {R}^3)\):

$$\begin{aligned} \Delta _j f:=\psi _j *f~~\text {and}~~S_j f:=\varphi _j*f \quad \text {for}~j\in \mathbb {Z}\ \ \text {and}\ \ f\in \mathscr {S}'(\mathbb {R}^3). \end{aligned}$$

Define \(\mathscr {S}'_h(\mathbb {R}^3):=\mathscr {S}'(\mathbb {R}^3)/\mathcal {P}[\mathbb {R}^3]\), where \(\mathcal {P}[\mathbb {R}^3]\) denotes the linear space of polynomials on \(\mathbb {R}^3\). It is known that there hold the following decompositions:

$$\begin{aligned} f=\sum _{j\in \mathbb {Z}}\Delta _j f \ \ \ \text{ and }\ \ \ \ S_j f=\sum _{j'\le j-1}\Delta _{j'}f \ \ \ \ \text{ in }\ \ \ \mathscr {S}'_h(\mathbb {R}^3), \end{aligned}$$

and the operators \(\Delta _{j}\) and \(S_{j}\) map \(L^{p}\) into \(L^{p}\) with norms independent of j and p, see [6, 43]. Moreover, with our choice of \(\varphi \) and \(\psi \), it is easy to verify that

$$\begin{aligned} \Delta _j\Delta _k f=0\ \ \ \ \ \hbox {if}\ |j-k|\ge 2 \ \ \hbox {and}\quad \Delta _j(S_{k-1}f\Delta _kf)=0\ \ \ \ \hbox {if}\ |j-k|\ge 5. \end{aligned}$$

In the sequel, we will use Bony’s decomposition from [7]:

$$\begin{aligned} uv=T_uv+T_vu+R(u,v) \end{aligned}$$

with

$$\begin{aligned} T_uv=\sum _{j\in \mathbb {Z}}S_{j-1}u\Delta _jv \ \ \ \ \text {and} \ \ \ \ R(u,v)=\sum _{j\in \mathbb {Z}}\Delta _{j}u{\tilde{\Delta }}_jv, \end{aligned}$$
(2.1)

where \({\tilde{\Delta }}_jv:=(\Delta _{j-1}+\Delta _{j}+\Delta _{j+1})v\).

Now, we introduce the definitions of homogeneous Besov space \(\dot{B}^{s}_{p,r}(\mathbb {R}^3)\) and Chemin-Lerner type space \({\tilde{L}}^\delta (0,\infty ; \dot{B}^{s}_{p,r}(\mathbb {R}^3))\).

Definition 2.1

[43] Let \(s\in \mathbb {R}\) and \(1\le p,r\le \infty \), and let \(u\in \mathscr {S}'_h(\mathbb {R}^3)\), we set (with the usual convection if \(r=\infty \))

$$\begin{aligned} \Vert u\Vert _{\dot{B}^{s}_{p,r}}:=\left( \sum _{j\in \mathbb {Z}}2^{jsr}\Vert \Delta _j u\Vert ^r_{L^p}\right) ^\frac{1}{r}. \end{aligned}$$
  • For \(s<\frac{3}{p}\) \((\text {or}~s=\frac{3}{p}, \text {if}~r=1)\), we define \(\dot{B}^{s}_{p,r}(\mathbb {R}^3):=\{u\in \mathscr {S}'_h(\mathbb {R}^3)|\Vert u\Vert _{\dot{B}^{s}_{p,r}}<\infty \};\)

  • If \(k\in \mathbb {N}\), \(\frac{3}{p}+k\le s<\frac{3}{p}+k+1\) \((\text {or}~s=\frac{3}{p}+k+1, \text {if}~r=1)\) then \(\dot{B}^{s}_{p,r}(\mathbb {R}^3)\) is defined as the subset of distributions \(u\in \mathscr {S}'_h(\mathbb {R}^3)\) such that \(\partial ^\delta u\in \dot{B}^{s-k}_{p,r}(\mathbb {R}^3)\) whenever \(|\delta |=k\).

Definition 2.2

([6, 11]) For \(s\in \mathbb {R}\) and \(1\le r,\delta \le \infty \), we set (with the usual convection if \(r=\infty \))

$$\begin{aligned} \Vert u\Vert _{{\tilde{L}}^\delta (0,\infty ;\dot{B}^{s}_{p,r})} :=\left( \sum _{j\in \mathbb {Z}}2^{jsr}\Vert \Delta _j u\Vert ^r_{L^\delta (0,\infty ; L^p)}\right) ^{\frac{1}{r}}. \end{aligned}$$

We then define the space \({\tilde{L}}^\delta (0,\infty ; \dot{B}^{s}_{p,r}(\mathbb {R}^3))\) as the set of temperate distributions u over \((0,\infty )\times \mathbb {R}^3\) such that \(\lim \nolimits _{j\rightarrow -\infty } S_ju=0\) in \(\mathscr {S}'(0,\infty \times \mathbb {R}^3)\) and \(\Vert u\Vert _{{\tilde{L}}^\delta (0,\infty ;\dot{B}^{s}_{p,r})}<\infty \).

For the convenience of the reader, we recall some basic facts on Littlewood–Paley theory and embedding theorems on Besov spaces, one may refer to [6, 43] for more details.

Lemma 2.3

(Bernstein) Let \(\mathcal {B}\) be a ball and \(\mathcal {C}\) a ring centred at origin of \(\mathbb {R}^3\). A constant C exists such that for any positive real number \(\lambda \), any non-negative integer k, and any couple of real numbers (ab) with \(b \ge a \ge 1\), there hold

  • \(\text {Supp}\hat{u} \subset \lambda \mathcal {B} \Longrightarrow \sup _{|\alpha |=k}\Vert \partial ^\alpha u\Vert _{L^b}\le C^{k+1}\lambda ^{k+3(\frac{1}{a}-\frac{1}{b})}\Vert u\Vert _{L^a}\);

  • \(\text {Supp}\hat{u} \subset \lambda \mathcal {C} \Longrightarrow C^{-(k+1)}\lambda ^{k}\Vert u\Vert _{L^a}\le \sup _{|\alpha |=k}\Vert \partial ^\alpha u\Vert _{L^a}\le C^{k+1}\lambda ^{k}\Vert u\Vert _{L^a}\).

Lemma 2.4

(Embedding theorem)

  1. (i)

    Let \(p\in (1,2]\), then \(L^{p}(\mathbb {R}^3)\) is continuously embedded in \(\dot{B}^{0}_{p,2}(\mathbb {R}^3)\), and \(\dot{B}^{0}_{p',2}(\mathbb {R}^3)\) is continuously embedded in \(L^{p'}(\mathbb {R}^3)\), where \(\frac{1}{p}+\frac{1}{p'}=1\).

  2. (ii)

    Let \(1\le p_1\le p_2\le \infty \) and \(1\le r_1\le r_2\le \infty \). Then, for any \(s\in \mathbb {R}\), \(\dot{B}^{s}_{p_1,r_1}(\mathbb {R}^3)\) is continuously embedded in \(\dot{B}^{s-(\frac{3}{p_1}-\frac{3}{p_2})}_{p_2,r_2}(\mathbb {R}^3)\).

Let \(R_j\) (\(j=1,2,3\)) be the Riesz transforms on \(\mathbb {R}^3\) and \(\delta _{ij}\) be the Kronecker’s delta notation. To study the problem (1.1), we consider the following equivalent integral equation

$$\begin{aligned} u(t)=T_\Omega (t)u_0-\int ^t_0T_\Omega (t-\tau )\mathbb {P}\nabla \cdot (u(\tau )\otimes u(\tau ))\hbox {d}\tau , \end{aligned}$$
(2.2)

where \(\mathbb {P}=(\delta _{ij}+R_iR_j)_{1\le i,j\le 3}\) denotes the Helmholtz projection onto the divergence-free vector fields, \(\{T_\Omega (t)\}_{t\ge 0}\) denotes the Stokes–Coriolis semigroup, which is given explicitly by

$$\begin{aligned} T_\Omega (t)f:=\mathcal {F}^{-1}\bigg [\cos \bigg (\Omega \frac{\xi _3}{|\xi |}t\bigg )e^{-t|\xi |^2}I\hat{f}(\xi ) +\sin \bigg (\Omega \frac{\xi _3}{|\xi |}t\bigg )e^{-t|\xi |^2}R(\xi )\hat{f}(\xi )\bigg ] \end{aligned}$$
(2.3)

for \(t\ge 0\) and divergence-free vector field f. Here, I is the identity matrix in \(\mathbb {R}^3\), \(R(\xi )\) is the skew-symmetric matrix defined by

$$\begin{aligned} R(\xi ):=\frac{1}{|\xi |}\left( \begin{array}{ccc} 0~ &{} \xi _3~ &{} -\xi _2 \\ -\xi _3~ &{} 0~ &{} \xi _1 \\ \xi _2~ &{} -\xi _1~ &{} 0 \\ \end{array} \right) ~~~~~~\text {for}~~\xi \in \mathbb {R}^3\setminus \{0\}. \end{aligned}$$

We refer to Hieber and Shibata [22] for the derivation of \(\{T_\Omega (t)\}_{t\ge 0}\).

Next, we define the operators \(\mathcal {G}_\pm (t)\) by

$$\begin{aligned} \mathcal {G}_\pm (t) f(x):=e^{\pm it\frac{D_3}{|D|}}f(x) :=\int _{\mathbb {R}^3}e^{ix\cdot \xi \pm it\frac{\xi _3}{|\xi |}}\hat{f}(\xi )d\xi , ~~~~~~\text {for}~~x\in \mathbb {R}^3~~\text {and}~~t\in \mathbb {R}. \end{aligned}$$
(2.4)

Then, we can rewrite the operator \(T_\Omega (t)\) as

$$\begin{aligned} T_{\Omega }(t)f=\frac{1}{2}\mathcal {G}_+(\Omega t)[e^{t\Delta }(I+\mathcal {R})f]+\frac{1}{2}\mathcal {G}_-(\Omega t)[e^{ t\Delta }(I-\mathcal {R})f] \end{aligned}$$
(2.5)

for all \(t\ge 0\), where \(\mathcal {R}\) denotes the matrix of singular integral operators defined by

$$\begin{aligned} \mathcal {R}:=\left( \begin{array}{ccc} 0 ~~&{} R_3~~ &{} -R_2 \\ -R_3~~ &{} 0~~ &{} R_1 \\ R_2~~ &{} -R_1~~ &{} 0 \\ \end{array} \right) . \end{aligned}$$
(2.6)

The operators \(\mathcal {G}_\pm (t)\) represent the oscillation parts of \(T_{\Omega }(t)\). Note that Riesz transforms \(\{R_j\}_{1\le j\le 3}\) are bounded in \(L^p(\mathbb {R}^3)(1\le p\le \infty )\) when localized in dyadic annulus in the Fourier space, and further in Besov space \(\dot{B}^s_{p,r}(\mathbb {R}^3)\) for all \(s\in \mathbb {R}\) and \(1\le p,r\le \infty \), and so are operators \(\mathbb {P}\) and \(\mathcal {R}\).

To solve the problem (1.2), we follow the approach of Kozono and Nakao [32] and transform the original equations into the following equivalent integral equation by the integration on the infinite time interval \((-\infty , t)\):

$$\begin{aligned} u(t)=\int ^t_{-\infty }T_\Omega (t-\tau )\mathbb {P}f(\tau )\hbox {d}\tau - \int ^t_{-\infty }T_\Omega (t-\tau )\mathbb {P}\nabla \cdot (u(\tau )\otimes u(\tau ))\hbox {d}\tau , \end{aligned}$$
(2.7)

where \(T_\Omega (\cdot )\) is defined as above. We say that u is a mild solution to problem (1.1) or (1.2) if u satisfies the integral equation (2.2) or (2.7) in an appropriate function space.

As a direct application of Mihlin’s theorem, we may verify that semigroup \(\{T_\Omega (t)\}_{t\ge 0}\) is a \(C_0\)-semigroup on \(L^p_\sigma (\mathbb {R}^3):=\{u\in L^p(\mathbb {R}^3) | \text {div} u=0\}\). Moreover, Mihlin’s theorem implies the following \(L^p-L^p\) type estimate for operators \(\mathcal {G}_\pm (t)\).

Lemma 2.5

For \(1< p<\infty \), there exists a positive constant \(C=C(p)\) such that

$$\begin{aligned} \Vert \mathcal {G}_\pm (t) f\Vert _{L^{p}}\le C(1+|t|)^2\Vert f\Vert _{L^p} \end{aligned}$$

for all \(t\in \mathbb {R}\) and \(f\in L^p(\mathbb {R}^3)\). Moreover,

$$\begin{aligned} \Vert \mathcal {G}_\pm (t) f\Vert _{\dot{B}^s_{p,r}}\le C(1+|t|)^2\Vert f\Vert _{\dot{B}^s_{p,r}} \end{aligned}$$

for all \(t\in \mathbb {R}\) and \(f\in \dot{B}^s_{p,r}(\mathbb {R}^3)\) with \(s\in \mathbb {R}\), \(p\in (1,\infty )\) and \(r\in [1,\infty ]\).

The dispersive effect of Coriolis force and the smoothing effect of heat flow are the keys to obtaining our main results. The following dispersive estimate for operators \(\mathcal {G}_\pm (t)\) is due to Koh et al. [30].

Lemma 2.6

[30] Let \(1\le p\le 2\). Then, there exists a positive constant \(C=C(p)\) such that

$$\begin{aligned} \Vert \Delta _j \mathcal {G}_\pm (t) f\Vert _{L^{p'}}\le C(1+|t|)^{-(1-\frac{2}{p'})}2^{3 j(1-\frac{2}{p'})}\Vert \Delta _j f\Vert _{L^p} \end{aligned}$$

for all \(t\in \mathbb {R}, j\in \mathbb {Z}\) and \(f\in \mathscr {S}'(\mathbb {R}^3)\), where \(\frac{1}{p}+\frac{1}{p'}=1\). Moreover,

$$\begin{aligned} \Vert \mathcal {G}_\pm (t) f\Vert _{\dot{B}^{s}_{p',r}}\le C(1+|t|)^{-(1-\frac{2}{p'})}\Vert f\Vert _{\dot{B}^{s+3(1-\frac{2}{p'})}_{p,r}} \end{aligned}$$

for all \(t\in \mathbb {R}\) and \(f\in \dot{B}^{s+3(1-\frac{2}{p'})}_{p,r}(\mathbb {R}^3)\) with \(s\in \mathbb {R}\) and \(1\le r\le \infty \), where \(\frac{1}{p'}+\frac{1}{p}=1\).

The following \(L^p-L^p\) estimate for operator \(e^{t\Delta }\) was studied by Kozono et al. [33].

Lemma 2.7

[33] Let \( -\infty< s_0\le s_1< \infty \). Then there exists a positive constant \(C = C(s_0, s_1,p)\) such that

$$\begin{aligned} \Vert \Delta _j e^{t\Delta }f\Vert _{L^p}\le C 2^{-(s_1- s_0)j}t^{-\frac{1}{2}(s_1-s_0)}\Vert \Delta _jf\Vert _{L^p}, \end{aligned}$$

for all \(t>0\), \(j\in \mathbb {Z}\), \(1\le p\le \infty \) and \(f\in \mathscr {S}'(\mathbb {R}^3)\). Moreover,

$$\begin{aligned} \Vert e^{t\Delta }f\Vert _{\dot{B}^{s_1}_{p,r}}\le C t^{-\frac{1}{2}(s_1-s_0)}\Vert f\Vert _{\dot{B}^{s_0}_{p,r}} \end{aligned}$$

for all \(t>0\) and \(f\in \dot{B}^{s_0}_{p,r}(\mathbb {R}^3)\) with \(1\le p, r\le \infty \).

The following two Lemmas are useful to obtain the Gevrey estimates.

Lemma 2.8

[5] The operator \(E = e^{\frac{1}{2}a\Delta +\sqrt{a}\Lambda _1}\) is a Fourier multiplier which maps boundedly \(L^{ p}\rightarrow L^{p}, 1< p < \infty \), and its operator norm is uniformly bounded with respect to \(a\ge 0\).

Lemma 2.9

[5] Consider the operator \(E := e^{-[\sqrt{t-s}+\sqrt{s}-\sqrt{t}]\Lambda _1}\) for \(0 \le s \le t\). Then, E is either the identity operator or an \(L^{1}\) kernel whose \(L^{1}\) norm is bounded independent of s and t.

3 Linear estimates

In this section, we mainly obtain the linear estimates for the Stokes–Coriolis semigroup \(\{T_{\Omega }(t)\}_{t\ge 0}\).

Lemma 3.1

Let \(\theta \in \{0,1\}\). Let \(s\in \mathbb {R}\), \(p\in (1,2)\) and \(r\in [1,\infty ]\), and let \(\delta \in [1,\infty ]\) satisfy

$$\begin{aligned} 0<\frac{1}{\delta }<\frac{2}{p}-1. \end{aligned}$$

Then, there exists a positive constant \(C=C(p,\delta )\) such that

$$\begin{aligned} \Vert T_\Omega (t)f\Vert _{{\tilde{L}}^\delta (0,\infty ;e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s+\frac{3}{p'}-\frac{3}{p}}_{p',r})}\le C|\Omega |^{-\frac{1}{\delta }}\Vert f\Vert _{\dot{B}^{s}_{p,r}} \end{aligned}$$

for \(\Omega \in \mathbb {R}\backslash \{0\}\) and \(f\in \dot{B}^{s}_{p,r}(\mathbb {R}^3)\), where \(\frac{1}{p}+\frac{1}{p'}=1\).

Proof

By Definition 2.2, we have

$$\begin{aligned} \Vert T_\Omega (t)f\Vert _{{\tilde{L}}^\delta \left( 0, \infty ; e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s+\frac{3}{p'}-\frac{3}{p}}_{p',r}\right) }= \bigg (\sum _{j\in \mathbb {Z}}2^{(s+\frac{3}{p'}-\frac{3}{p})jr} \Vert \Delta _je^{\theta \sqrt{t}\Lambda _1} T_\Omega (t)f\Vert ^r_{L^\delta (0,\infty ;L^{p'})}\bigg )^{\frac{1}{r}}. \end{aligned}$$

Since \(\mathcal {R}\) is bounded in \(L^p(\mathbb {R}^3)(1\le p\le \infty )\) when localized in dyadic annulus in the Fourier space, by (2.5) it suffices to show that

$$\begin{aligned} \Vert \Delta _je^{\theta \sqrt{t}\Lambda _1}\mathcal {G}_\pm (\Omega t)e^{t\Delta }f\Vert _{L^\delta (0,\infty ;L^{p'})}\le C|\Omega |^{-\frac{1}{\delta }}2^{j(\frac{3}{p}-\frac{3}{p'})}\Vert \Delta _jf\Vert _{L^{p}} \end{aligned}$$

for all \(t>0\) and \(\Omega \in \mathbb {R}\backslash \{0\}\).

In fact, it follows from Lemma 2.8 that

$$\begin{aligned} \Vert \Delta _j e^{\theta \sqrt{t}\Lambda _1}\mathcal {G}_\pm (\Omega t) e^{t\Delta }f\Vert _{L^{p'}}= & {} \Vert e^{\theta \sqrt{t}\Lambda _1+\frac{t}{2}\Delta } \mathcal {G}_\pm (\Omega t)e^{\frac{t}{2}\Delta }\Delta _jf\Vert _{L^{p'}}\\\le & {} C \Vert \mathcal {G}_\pm (\Omega t)e^{\frac{t}{2}\Delta }\Delta _jf\Vert _{L^{p'}}. \end{aligned}$$

By using Lemmas 2.6 and 2.7, one has

$$\begin{aligned} \Vert \mathcal {G}_\pm (\Omega t)e^{\frac{t}{2}\Delta }\Delta _jf\Vert _{L^{p'}}\le C(1+|\Omega |t)^{-(1-\frac{2}{p'})}2^{j(\frac{3}{p}-\frac{3}{p'})}\Vert \Delta _jf\Vert _{L^{p}}. \end{aligned}$$

Due to \(0<\frac{1}{\delta }<\frac{2}{p}-1\), there exists a positive constant \(C=C(p, \delta )\) such that

$$\begin{aligned} \Big (\int ^\infty _0(1+|\Omega |t)^{-(1-\frac{2}{p'})\delta }\hbox {d}t\Big )^\frac{1}{\delta }\le C|\Omega |^{-\frac{1}{\delta }} \ \ \ \ \text {for} \ \Omega \in \mathbb {R}\backslash \{0\}, \end{aligned}$$

the end of the proof is then straightforward. \(\square \)

Lemma 3.2

Let \(\theta \in \{0,1\}\). Let \(p\in (1,2)\) and \(r\in [1, \infty ]\). Let \(s\in \mathbb {R}\) satisfy

$$\begin{aligned} -1+\frac{3}{p}<s<-2+\frac{7}{p}, \end{aligned}$$

and let \(\delta \in [2, \infty )\) satisfy

$$\begin{aligned} \frac{1}{2}\left( s+1-\frac{3}{p}\right) +1-\frac{2}{p}<\frac{1}{\delta } <\frac{1}{2}\left( s+1-\frac{3}{p}\right) . \end{aligned}$$

Then, there exists a positive constant \(C = C(s, p, \delta )\) such that

$$\begin{aligned}&\left\| \int _0^t T_\Omega (t-\tau )\mathbb {P}\nabla f(\tau )\mathrm{{d}}\tau \right\| _{{\tilde{L}}^\delta \left( 0, \infty ;e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s+\frac{3}{p'}-\frac{3}{p}}_{p',r}\right) }\\&\quad \le C|\Omega |^{-[\frac{1}{2}(s+1-\frac{3}{p})-\frac{1}{\delta }]} \Vert f\Vert _{{\tilde{L}}^\frac{\delta }{2}\left( 0, \infty ;e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{2s-\frac{3}{p}}_{p,r}\right) } \end{aligned}$$

for \(\Omega \in \mathbb {R}\backslash \{0\}\) and \(f\in {\tilde{L}}^\frac{\delta }{2}(0, \infty ;e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{2s-\frac{3}{p}}_{p,r}(\mathbb {R}^3))\), where \(\frac{1}{p'}+\frac{1}{p}=1\).

Proof

By Definition 2.2, one sees

$$\begin{aligned}&\left\| \int _0^t T_\Omega (t-\tau )\mathbb {P}\nabla f(\tau )\hbox {d}\tau \right\| _{{\tilde{L}}^\delta \left( 0, \infty ; e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s+\frac{3}{p'}-\frac{3}{p}}_{p',r}\right) }\\&\quad =\left\| \Big \{2^{j(s+\frac{3}{p'}-\frac{3}{p})}\right\| \Delta _je^{\theta \sqrt{t}\Lambda _1} \int _0^t T_\Omega (t-\tau ) \mathbb {P}\nabla f(\tau )\hbox {d}\tau \Big \Vert _{L^\delta (0,\infty ; L^q)}\Big \}_{j\in \mathbb {Z}}\Big \Vert _{\ell ^r(\mathbb {Z})}. \end{aligned}$$

Since \(\mathcal {R}\) and \(\mathbb {P}\) are bounded in \(L^p(\mathbb {R}^3)(1\le p\le \infty )\) when localized in dyadic annulus in the Fourier space, by (2.5) it suffices to show that

$$\begin{aligned}&\left\| \int _0^t \right\| e^{\theta \sqrt{t}\Lambda _1}\mathcal {G}_\pm (\Omega (t-\tau ))e^{(t-\tau )\Delta }\nabla \Delta _j f(\tau )\big \Vert _{L^{p'}}\hbox {d}\tau \Big \Vert _{L^\delta (0,\infty )}\\&\le C|\Omega |^{-[\frac{1}{2}(s+1-\frac{3}{p})-\frac{1}{\delta }]} 2^{j(s-\frac{3}{p'})} \big \Vert \Delta _j f\big \Vert _{L^\frac{\delta }{2}(0,\infty ;e^{\theta \sqrt{t}\Lambda _1}L^{p})}. \end{aligned}$$

In fact, by applying Lemmas 2.8 and 2.9, one has

$$\begin{aligned}&\left\| \int _0^t \right\| e^{\theta \sqrt{t}\Lambda _1} \mathcal {G}_\pm (\Omega (t-\tau ))e^{(t-\tau )\Delta }\nabla \Delta _j f(\tau )\left\| _{L^{p'}}\hbox {d}\tau \right\| _{L^\delta (0,\infty )}\\&\quad =\left\| \int _0^t \right\| e^{\theta (\sqrt{t}-\sqrt{\tau }-\sqrt{t-\tau })\Lambda _1} e^{\theta \sqrt{t-\tau }\Lambda _1 +\frac{t-\tau }{2}\Delta }\nonumber \\&\quad \quad \mathcal {G}_ \pm (\Omega (t-\tau )) e^{\frac{t-\tau }{2}\Delta }\nabla e^{\theta \sqrt{\tau }\Lambda _1}\Delta _j f(\tau )\left\| _{L^{p'}}\hbox {d}\tau \right\| _{L^\delta (0,\infty )}\nonumber \\&\quad \le C\left\| \int _0^t \right\| \mathcal {G}_\pm (\Omega (t-\tau ))e^{\frac{t-\tau }{2}\Delta }\nabla e^{\theta \sqrt{\tau }\Lambda _1}\Delta _j f(\tau )\left\| _{L^{p'}}\hbox {d}\tau \right\| _{L^\delta (0,\infty )}. \end{aligned}$$

Moreover, by using Lemmas 2.3, 2.6 and 2.7, as well as Young’s inequality, one deduces

$$\begin{aligned}&\Big \Vert \int _0^t \big \Vert \mathcal {G}_\pm (\Omega (t-\tau ))e^{\frac{t-\tau }{2}\Delta }\nabla e^{\theta \sqrt{\tau }\Lambda _1}\Delta _j f(\tau )\big \Vert _{L^{p'}}\hbox {d}\tau \Big \Vert _{L^\delta (0,\infty )}\\&\quad \le C2^{j(s-\frac{3}{p'})}\Big \Vert \int _0^t K_\Omega (t-\tau ) \big \Vert e^{\theta \sqrt{\tau }\Lambda _1}\Delta _jf(\tau )\big \Vert _{L^{p}} \hbox {d}\tau \Big \Vert _{L^\delta (0,\infty )}\nonumber \\&\quad \le C2^{j(s-\frac{3}{p'})}\Vert K_\Omega \Vert _{L^{\delta '}(0,\infty )} \big \Vert \Delta _jf\big \Vert _{L^\frac{\delta }{2}(0,\infty ;e^{\theta \sqrt{t}\Lambda _1}L^{p})}, \end{aligned}$$

where \(K_\Omega (t):=(1+|\Omega |t)^{-(1-\frac{2}{p'})} t^{-\frac{1}{2}(1+\frac{3}{p}-s)}\) and \(\frac{1}{\delta }+\frac{1}{\delta '}=1\). Since \(\delta \in [2, \infty )\) satisfies

$$\begin{aligned} \frac{1}{2}\left( s+1-\frac{3}{p}\right) +1-\frac{2}{p}<\frac{1}{\delta } <\frac{1}{2}\left( s+1-\frac{3}{p}\right) , \end{aligned}$$

there exists a positive constant \(C=C(s,p,\delta )\) such that

$$\begin{aligned} \Vert K_\Omega \Vert _{L^{\delta '}(0, \infty )}= & {} |\Omega |^{-\left[ \frac{1}{2}\left( s+1-\frac{3}{p}\right) -\frac{1}{\delta }\right] } \left( \int ^\infty _0(1+\tau )^{-\left( 1-\frac{2}{p'}\right) \delta '} \tau ^{-\frac{1}{2}\left( 1+\frac{3}{p}-s\right) \delta '}\hbox {d}\tau \right) ^\frac{1}{\delta '}\\\le & {} C|\Omega |^{-[\frac{1}{2}(s+1-\frac{3}{p})-\frac{1}{\delta }]}, \end{aligned}$$

for \(\Omega \in \mathbb {R}\setminus \{0\}\). This completes the proof. \(\square \)

Lemmas 3.1 and 3.2 are aimed to establish the linear estimates involving parameter \(|\Omega |\), which is crucial to obtain the desired size condition in our main theorem for \(u_0\in \dot{B}^{s}_{p,r}(\mathbb {R}^3)\) with \(p\in (1,2)\). However, when \(u_0\in \dot{B}^{s}_{2,r}(\mathbb {R}^3)\), by the similar argument as in Lemma 3.1, we are unable to get the parameter \(|\Omega |\) in the estimation from a direct application of Lemma 2.6 with \(p=2\). The following two lemmas are concerned with the endpoint case \(p=2\), in which we may use the \(TT^\star \) argument.

Lemma 3.3

Let \(\theta \in \{0,1\}\). Let \(s\in \mathbb {R}\), \(q\in (2, \infty )\), \(r\in [1, \infty ]\), and let \(\delta \in [2,\infty )\) satisfy \(0<\frac{1}{\delta }<\frac{1}{2}-\frac{1}{q}.\) Then, there exists a positive constant \(C = C(s, q, \delta )\) such that

$$\begin{aligned} \Vert T_\Omega (t)f\Vert _{{\tilde{L}}^\delta \left( 0, \infty ; e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s+\frac{3}{q}-\frac{3}{2}}_{q,r}\right) }\le C|\Omega |^{-\frac{1}{\delta }}\Vert f\Vert _{\dot{B}^{s}_{2,r}} \end{aligned}$$

for \(\Omega \in \mathbb {R}\backslash \{0\}\) and \(f\in \dot{B}^{s}_{2,r}(\mathbb {R}^3)\).

Proof

By Definition 2.2, we have

$$\begin{aligned} \Vert T_\Omega (t)f\Vert _{{\tilde{L}}^\delta \left( 0, \infty ; e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s+\frac{3}{q}-\frac{3}{2}}_{q,r}\right) }= \left( \sum _{j\in \mathbb {Z}}2^{(s+\frac{3}{q}-\frac{3}{2})jr} \Vert \Delta _je^{\theta \sqrt{t}\Lambda _1} T_\Omega (t)f\Vert ^r_{L^\delta (0,\infty ;L^{q})}\right) ^{\frac{1}{r}}. \end{aligned}$$

Thanks to the boundedness property of operator \(\mathcal {R}\), it suffices to prove that

$$\begin{aligned} \Vert \Delta _je^{\theta \sqrt{t}\Lambda _1}\mathcal {G}_\pm (\Omega t)e^{t\Delta }f\Vert _{L^\delta (0,\infty ;L^{q})}\le C2^{(\frac{3}{2}-\frac{3}{q})j} |\Omega |^{-\frac{1}{\delta }}\Vert \Delta _jf\Vert _{L^2},~~~~j\in \mathbb {Z}. \end{aligned}$$

In fact, it follows from Lemma 2.8 that

$$\begin{aligned} \Vert \Delta _j e^{\theta \sqrt{t}\Lambda _1}\mathcal {G}_\pm (\Omega t) e^{t\Delta }f\Vert _{L^{q}}= & {} \Vert e^{\theta \sqrt{t}\Lambda _1+\frac{t}{2}\Delta } \mathcal {G}_\pm (\Omega t)e^{\frac{t}{2}\Delta }\Delta _jf\Vert _{L^{q}}\nonumber \\\le & {} C \Vert \mathcal {G}_\pm (\Omega t)e^{\frac{t}{2}\Delta }\Delta _jf\Vert _{L^{q}}. \end{aligned}$$
(3.1)

We claim for \(q\in (2,\infty )\) and \(0<\frac{1}{\delta }<\frac{1}{2}-\frac{1}{q}\) that

$$\begin{aligned} \Vert \mathcal {G}_\pm (\Omega t)e^{\frac{t}{2}\Delta }\Delta _jf\Vert _{L^{\delta }(0,\infty ;L^{q})} \le C2^{(\frac{3}{2}-\frac{3}{q})j} |\Omega |^{-\frac{1}{\delta }}\Vert \Delta _jf\Vert _{L^2},~~~~j\in \mathbb {Z}. \end{aligned}$$
(3.2)

The proof of (3.2) is based on the usual \(TT^\star \) argument, which goes back to Tomas [42] (see also Strichartz [40]). Indeed, by duality it suffices to prove that

$$\begin{aligned} \bigg |\int _0^\infty \int _{\mathbb {R}^3}\mathcal {G}_\pm (\Omega t)e^{\frac{t}{2}\Delta }\Delta _jf(x)\overline{\varphi (t,x)}\hbox {d}x\hbox {d}t\bigg |\le C2^{j(\frac{3}{2}-\frac{3}{q})} |\Omega |^{-\frac{1}{\delta }}\Vert \Delta _jf\Vert _{L^2} \Vert \varphi \Vert _{L^{\delta '}(0,\infty ;L^{q'})} \end{aligned}$$

for \(\varphi \in C^\infty _0((0,\infty )\times \mathbb {R}^3)\), where \(\frac{1}{q'}+\frac{1}{q}=1\) and \(\frac{1}{\delta }+\frac{1}{\delta '}=1\).

Here, we introduce a new Littlewood–Paley operator \({\tilde{\Delta }}_j\) defined by

$$\begin{aligned} {\tilde{\Delta }}_jf:=(\psi _{j-1}+\psi _j+\psi _{j+1}) *f,~~~~j\in \mathbb {Z},~\forall f\in \mathscr {S}'(\mathbb {R}^3). \end{aligned}$$

It is easy to check that \({\tilde{\Delta }}_j\Delta _j f=\Delta _j f\) for all \(j\in \mathbb {Z}\) and \(f\in \mathscr {S}'(\mathbb {R}^3)\), and \({\tilde{\Delta }}_j\) is also bounded in \(L^p(\mathbb {R}^3)\).

By Parseval formula and Hölder’s inequality, we deduce

$$\begin{aligned} \begin{array}{ll} &{}\displaystyle \Big | \int _0^\infty \int _{\mathbb {R}^3}\mathcal {G}_\pm (\Omega t)e^{\frac{t}{2}\Delta }\Delta _jf(x)\overline{\varphi (t,x)}\hbox {d}x\hbox {d}t\Big |\\ &{}\quad \le \ \displaystyle \Big |\int _0^\infty \int _{\mathbb {R}^3}\Delta _jf(x) \overline{\mathcal {G}_\mp (\Omega t)e^{\frac{t}{2}\Delta }{\tilde{\Delta }}_j\varphi (t,x)}\hbox {d}x\hbox {d}t\Big |\\ &{}\quad \le \ \displaystyle \Vert \Delta _jf\Vert _{L^2}\Big \Vert \int _0^\infty \mathcal {G}_\mp (\Omega t)e^{\frac{t}{2}\Delta }{\tilde{\Delta }}_j\varphi (t)\hbox {d}t\Big \Vert _{L^2}. \end{array} \end{aligned}$$
(3.3)

Moreover, it follows from Parseval formula and Hölder’s inequality that

$$\begin{aligned}&\Big \Vert \int _0^\infty \mathcal {G}_\mp (\Omega t)e^{\frac{t}{2}\Delta }{\tilde{\Delta }}_j\varphi (t)\hbox {d}t\Big \Vert ^2_{L^2}\nonumber \\&\quad = \int _{\mathbb {R}^3}\int _0^\infty \int _0^\infty \mathcal {G}_\mp (\Omega t)e^{\frac{t}{2}\Delta }{\tilde{\Delta }}_j\varphi (t,x)\overline{\mathcal {G}_\mp (\Omega \tau )e^{\frac{\tau }{2}\Delta }{\tilde{\Delta }}_j\varphi (\tau ,x)}\hbox {d}t\hbox {d}\tau \hbox {d}x\nonumber \\&\quad \le \int _0^\infty \int _0^\infty \Vert \varphi (t)\Vert _{L^{q'}}\Big \Vert \mathcal {G}_\pm (\Omega (t-\tau ))e^{\frac{1}{2}(t+\tau )\Delta }{\tilde{\Delta }}_j\varphi (\tau )\Big \Vert _{L^{q}}\hbox {d}t\hbox {d}\tau . \end{aligned}$$
(3.4)

Applying Lemmas 2.6 and 2.7 yields

$$\begin{aligned} \Big \Vert \mathcal {G}_\pm (\Omega (t-\tau ))e^{\frac{1}{2}(t+\tau )\Delta }{\tilde{\Delta }}_j\varphi (\tau )\Big \Vert _{L^{q}} \le C (1+|\Omega ||t-\tau |\big )^{-(1-\frac{2}{q})}2^{3j(1-\frac{2}{q})} \Vert {\tilde{\Delta }}_j\varphi (\tau )\Vert _{L^{q'}}.\nonumber \\ \end{aligned}$$
(3.5)

Substituting (3.5) into (3.4), together with Hölder’s inequality and Young’s inequality, we obtain

$$\begin{aligned}&\Big \Vert \int _0^\infty \mathcal {G}_\mp (\Omega t)e^{\frac{t}{2}\Delta }{\tilde{\Delta }}_j\varphi (t)\hbox {d}t\Big \Vert ^2_{L^2}\nonumber \\&\quad \le C 2^{3j(1-\frac{2}{q})} \Vert \varphi \Vert _{L^{\delta '}(0,\infty ;L^{q'})}\Big \Vert \int _0^\infty K_\Omega (t-\tau )\Vert \varphi (\tau )\Vert _{L^{q'}}\hbox {d}\tau \Big \Vert _{L^\delta (0,\infty )}\nonumber \\&\quad \le C 2^{3j(1-\frac{2}{q})}\Vert \varphi \Vert ^2_{L^{\delta '}(0,\infty ;L^{q'})} \Vert K_\Omega \Vert _{L^{\frac{\delta }{2}}(0,\infty )}, \end{aligned}$$
(3.6)

where \(K_{\Omega }(t):=(1+|\Omega ||t|)^{-(1-\frac{2}{q})}\). Due to \(0<\frac{1}{\delta }<\frac{1}{2}-\frac{1}{q}\), there exists a positive constant \(C=C(\delta ,q)\) such that

$$\begin{aligned} \Vert K_\Omega \Vert _{L^{\frac{\delta }{2}}(0,\infty )}= |\Omega |^{-\frac{2}{\delta }} \bigg (\int _0^\infty (1+\tau )^{-\frac{\delta }{2}(1-\frac{2}{q})} \hbox {d}\tau \bigg )^{\frac{2}{\delta }}\le C|\Omega |^{-\frac{2}{\delta }}, \end{aligned}$$

which implies

$$\begin{aligned} \Big \Vert \int _0^\infty \mathcal {G}_\mp (\Omega t)e^{\frac{t}{2}\Delta }{\tilde{\Delta }}_j\varphi (t)\hbox {d}t\Big \Vert ^2_{L^2} \le C 2^{3j(1-\frac{2}{q})} |\Omega |^{-\frac{2}{\delta }} \Vert \varphi \Vert ^2_{L^{\delta '}\left( 0,\infty ;L^{q'}\right) }. \end{aligned}$$
(3.7)

Substituting (3.7) into (3.3) yields the desired estimate. This completes the proof. \(\square \)

Lemma 3.4

Let \(\theta \in \{0,1\}\). Let \(s\in (\frac{1}{2}, \frac{5}{2}]\), \(r\in [1, \infty ]\), \(q\in (2, \infty )\) satisfy \(\frac{1}{q}<-\frac{s}{4}+\frac{7}{8}\), and let \(\delta \in [2, \infty )\) satisfy

$$\begin{aligned} \frac{1}{2}\left( s-\frac{1}{2}\right) +\frac{2}{q}-1<\frac{1}{\delta } <\frac{1}{2}\left( s-\frac{1}{2}\right) . \end{aligned}$$

Then, there exists a positive constant \(C = C(s, q, \delta )\) such that

$$\begin{aligned}&\Big \Vert \int _0^t T_\Omega (t-\tau )\mathbb {P}\nabla f(\tau )\mathrm{{d}}\tau \Big \Vert _{{\tilde{L}}^\delta \left( 0, \infty ;e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s+\frac{3}{q}-\frac{3}{2}}_{q,r}\right) }\\&\quad \le C|\Omega |^{-[\frac{1}{2}\left( s-\frac{1}{2}\right) -\frac{1}{\delta }]}\Vert f\Vert _{{\tilde{L}}^\frac{\delta }{2}\left( 0, \infty ;e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{2s+\frac{3}{q'}-3}_{q',r}\right) } \end{aligned}$$

for \(\Omega \in \mathbb {R}\backslash \{0\}\) and \(f\in {\tilde{L}}^\frac{\delta }{2}(0, \infty ;e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{2s+\frac{3}{q'}-3}_{q',r}(\mathbb {R}^3))\), where \(\frac{1}{q'}+\frac{1}{q}=1\).

Proof

By Definition 2.2, one sees

$$\begin{aligned}&\left\| \int _0^t T_\Omega (t-\tau )\mathbb {P}\nabla f(\tau )\hbox {d}\tau \right\| _{{\tilde{L}}^\delta (0, \infty ; e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s+\frac{3}{q}-\frac{3}{2}}_{q,r})}\\&\quad =\Big \Vert \Big \{2^{j(s+\frac{3}{q}-\frac{3}{2})}\Big \Vert \Delta _je^{\theta \sqrt{t}\Lambda _1} \int _0^t T_\Omega (t-\tau ) \mathbb {P}\nabla f(\tau )\hbox {d}\tau \Big \Vert _{L^\delta (0,\infty ; L^q)}\Big \}_{j\in \mathbb {Z}}\Big \Vert _{\ell ^r(\mathbb {Z})}. \end{aligned}$$

Thanks to the boundedness properties of operators \(\mathcal {R}\) and \(\mathbb {P}\), it suffices to show that

$$\begin{aligned}&\Big \Vert \int _0^t \big \Vert e^{\theta \sqrt{t}\Lambda _1}\mathcal {G}_\pm (\Omega (t-\tau ))e^{(t-\tau )\Delta }\nabla \Delta _j f(\tau )\big \Vert _{L^q}\hbox {d}\tau \Big \Vert _{L^\delta (0,\infty )}\\&\quad \le C|\Omega |^{-[\frac{1}{2}(s-\frac{1}{2})-\frac{1}{\delta }]} 2^{j(s+\frac{3}{q'}-\frac{3}{q}-\frac{3}{2})} \big \Vert \Delta _j f\big \Vert _{L^\frac{\delta }{2}(0,\infty ;e^{\theta \sqrt{t}\Lambda _1}L^{q'})}. \end{aligned}$$

In fact, by Lemmas 2.8 and 2.9, one has

$$\begin{aligned}&\Big \Vert \int _0^t \big \Vert e^{\theta \sqrt{t}\Lambda _1} \mathcal {G}_\pm (\Omega (t-\tau ))e^{(t-\tau )\Delta }\nabla \Delta _j f(\tau )\big \Vert _{L^q}\hbox {d}\tau \Big \Vert _{L^\delta (0,\infty )}\\&\quad =\Big \Vert \int _0^t \big \Vert e^{\theta (\sqrt{t}-\sqrt{\tau }-\sqrt{t-\tau })\Lambda _1} e^{\theta \sqrt{t-\tau }\Lambda _1+\frac{t-\tau }{2}\Delta }\\&\qquad \mathcal {G}_\pm (\Omega (t-\tau )) e^{\frac{t-\tau }{2}\Delta }\nabla e^{\theta \sqrt{\tau }\Lambda _1}\Delta _j f(\tau )\big \Vert _{L^q}\hbox {d}\tau \Big \Vert _{L^\delta (0,\infty )}\\&\quad \le C\Big \Vert \int _0^t \big \Vert \mathcal {G}_\pm (\Omega (t-\tau ))e^{\frac{t-\tau }{2}\Delta }\nabla e^{\theta \sqrt{\tau }\Lambda _1}\Delta _j f(\tau )\big \Vert _{L^q}\hbox {d}\tau \Big \Vert _{L^\delta (0,\infty )}. \end{aligned}$$

By using Lemmas 2.3, 2.6 and 2.7, as well as Young’s inequality, one then obtains

$$\begin{aligned}&\left\| \int _0^t \big \Vert e^{\theta \sqrt{t}\Lambda _1} \mathcal {G}_\pm (\Omega (t-\tau ))e^{(t-\tau )\Delta }\nabla \Delta _j f(\tau )\big \Vert _{L^q}\hbox {d}\tau \right\| _{L^\delta (0,\infty )}\nonumber \\&\quad \le C2^{j(s+\frac{3}{q'}-\frac{3}{q}-\frac{3}{2})}\Big \Vert \int _0^t K_\Omega (t-\tau ) \big \Vert e^{\theta \sqrt{\tau }\Lambda _1}\Delta _jf(\tau )\big \Vert _{L^{q'}} \hbox {d}\tau \Big \Vert _{L^\delta (0,\infty )}\nonumber \\&\quad \le C2^{j(s+\frac{3}{q'}-\frac{3}{q}-\frac{3}{2})}\Vert K_\Omega \Vert _{L^{\delta '}(0,\infty )} \big \Vert \Delta _jf\big \Vert _{L^\frac{\delta }{2}(0,\infty ;e^{\theta \sqrt{t}\Lambda _1}L^{q'})}, \end{aligned}$$
(3.8)

where \(K_\Omega (t):=(1+|\Omega |t)^{-(1-\frac{2}{q})}t^{-\frac{1}{2}(\frac{5}{2}-s)}\) and \(\frac{1}{\delta }+\frac{1}{\delta '}=1\). Due to

$$\begin{aligned} \frac{1}{2}\left( s-\frac{1}{2}\right) -\left( 1-\frac{2}{q}\right)<\frac{1}{\delta } <\frac{1}{2}\left( s-\frac{1}{2}\right) , \end{aligned}$$

there exists a positive constant \(C=C(s,q,\delta )\) such that

$$\begin{aligned} \Vert K_\Omega \Vert _{L^{\delta '}(0, \infty )}= & {} |\Omega |^{-\left[ \frac{1}{2}\left( s-\frac{1}{2}\right) -\frac{1}{\delta }\right] } \bigg (\int _0^\infty (1+\tau )^{-\delta '(1-\frac{2}{q})}\tau ^{-\frac{\delta '}{2}(\frac{5}{2}-s)} \hbox {d}\tau \bigg )^{\frac{1}{\delta '}}\nonumber \\\le & {} C |\Omega |^{-\left[ \frac{1}{2}\left( s-\frac{1}{2}\right) -\frac{1}{\delta }\right] } \end{aligned}$$
(3.9)

for all \(\Omega \in \mathbb {R}{\setminus }\{0\}\). Plugging (3.9) into (3.8), one concludes the proof. \(\square \)

The following lemmas are used for the time-periodic case.

Lemma 3.5

Let \(\theta \in \{0,1\}\). Let \(s\in \mathbb {R}\), \(p\in (1,2)\)\(r\in [1, \infty ]\), and \(s+5-\frac{7}{p}<\beta <s+3-\frac{3}{p}\). Then there exists a positive constant \(C = C(s, p, \beta )\) such that

$$\begin{aligned} \sup _{t\in \mathbb {R}}\left\| \int _{-\infty }^t T_\Omega (t-\tau )\mathbb {P} f(\tau ) \mathrm{d}\tau \right\| _{e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{-1+\frac{3}{p'}+\beta }_{p',r}}\le C|\Omega |^{-\frac{1}{2}(s+3-\frac{3}{p}-\beta )}\sup _{t\in \mathbb {R}}\Vert f\Vert _{e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s}_{p,r}} \end{aligned}$$

for \(\Omega \in \mathbb {R}\backslash \{0\}\) and \(f\in BC(\mathbb {R}; e^{\theta \sqrt{t}\Lambda _1} \dot{B}^{s}_{p,r}(\mathbb {R}^3))\), where \(\frac{1}{p}+\frac{1}{p'}=1\).

Proof

By applying Lemmas 2.6 \(\sim \) 2.9 and Young’s inequality, together with the boundedness properties of operators \(\mathcal {R}\) and \(\mathbb {P}\), we see

$$\begin{aligned}&\sup _{t\in \mathbb {R}}\Big \Vert \int _{-\infty }^t T_\Omega (t-\tau )\mathbb {P} f(\tau )\hbox {d}\tau \Big \Vert _{e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{-1+\frac{3}{p'}+\beta }_{p',r}}\nonumber \\&\quad \le C\sup _{t\in \mathbb {R}}\int _{-\infty }^t \Big \Vert \Big \{2^{j(-1+\frac{3}{p'}+\beta )} \big \Vert e^{\theta (\sqrt{t}-\sqrt{\tau }-\sqrt{t-\tau })\Lambda _1} e^{\theta \sqrt{t-\tau }\Lambda _1 +\frac{t-\tau }{2}\Delta }\nonumber \\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathcal {G}_\pm (\Omega (t-\tau ))e^{\frac{t-\tau }{2}\Delta }e^{\theta \sqrt{\tau }\Lambda _1} \Delta _jf(\tau )\big \Vert _{L^{p'}}\Big \}_{j\in \mathbb {Z}}\Big \Vert _{\ell ^r}\hbox {d}\tau \nonumber \\&\quad \le C\sup _{t\in \mathbb {R}}\int _{-\infty }^tK_\Omega (t-\tau ) \Vert e^{\theta \sqrt{\tau }\Lambda _1}f(\tau )\Vert _{\dot{B}^{s}_{p,r}}\hbox {d}\tau \nonumber \\&\quad \le C\Vert K_\Omega \Vert _{L^1(0,\infty )}\sup _{t\in \mathbb {R}}\Vert f\Vert _{e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s}_{p,r}}, \end{aligned}$$
(3.10)

where \(K_\Omega (t):=(1+|\Omega |t)^{-(1-\frac{2}{p'})}t^{-\frac{1}{2}(\beta -s-1+\frac{3}{p})}\). Due to \(s+5-\frac{7}{p}<\beta <s+3-\frac{3}{p}\), there exists a positive constant \(C=C(s,p,\beta )\) such that

$$\begin{aligned} \Vert K_\Omega \Vert _{L^1(0,\infty )}= & {} |\Omega |^{-\frac{1}{2}\left( s+3-\frac{3}{p}-\beta \right) } \int ^\infty _0(1+\tau )^{-(1-\frac{2}{p'})} \tau ^{-\frac{1}{2}\left( \beta -s-1+\frac{3}{p}\right) }\hbox {d}\tau \nonumber \\\le & {} C|\Omega |^{-\frac{1}{2}\left( s+3-\frac{3}{p}-\beta \right) } \end{aligned}$$
(3.11)

for all \(\Omega \in \mathbb {R}\setminus \{0\}\). Plugging (3.11) into (3.10) finishes the proof. \(\square \)

Lemma 3.6

Let \(\theta \in \{0,1\}\). Let \(p\in (1,2)\)\(r\in [1, \infty ]\) and \(\beta \in (0,\frac{4}{p}-2)\). Then, there exists a positive constant \(C = C(p, \beta )\) such that

$$\begin{aligned} \sup _{t\in \mathbb {R}}\left\| \int _{-\infty }^t T_\Omega (t-\tau )\mathbb {P}\nabla f(\tau ) \mathrm{d}\tau \right\| _{e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{-1+\frac{3}{p'}+\beta }_{p',r}}\le C|\Omega |^{-\frac{\beta }{2}}\sup _{t\in \mathbb {R}}\Vert f(t)\Vert _{e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{-2+\frac{3}{p}+2\beta }_{p,r}} \end{aligned}$$

for \(\Omega \in \mathbb {R}\backslash \{0\}\) and \(f\in BC(\mathbb {R}; e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{-2+\frac{3}{p}+2\beta }_{p,r}(\mathbb {R}^3))\), where \(\frac{1}{p}+\frac{1}{p'}=1\).

Proof

By using Lemmas 2.3, 2.6 \(\sim \) 2.9 and Young’s inequality, together with the boundedness properties of operators \(\mathcal {R}\) and \(\mathbb {P}\), we obtain

$$\begin{aligned}&\sup _{t\in \mathbb {R}}\left\| \int _{-\infty }^t T_\Omega (t-\tau )\mathbb {P}\nabla f(\tau )\hbox {d}\tau \right\| _{e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{-1+\frac{3}{p'}+\beta }_{p',r}}\nonumber \\&\quad \le C\sup _{t\in \mathbb {R}}\int _{-\infty }^t \Big \Vert \Big \{2^{j(\frac{3}{p'}+\beta )} \Vert e^{\theta (\sqrt{t}-\sqrt{\tau }-\sqrt{t-\tau })\Lambda _1} e^{\theta \sqrt{t-\tau }\Lambda _1 +\frac{t-\tau }{2}\Delta }\nonumber \\&\quad \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathcal {G}_\pm (\Omega (t-\tau )) e^{\frac{t-\tau }{2}\Delta }e^{\theta \sqrt{\tau }\Lambda _1} \Delta _jf(\tau )\Vert _{L^{p'}}\Big \}_{j\in \mathbb {Z}}\Big \Vert _{\ell ^r}\hbox {d}\tau \nonumber \\&\quad \le C\sup _{t\in \mathbb {R}}\int _{-\infty }^tK_\Omega (t-\tau )\Vert e^{\theta \sqrt{\tau }\Lambda _1}f(\tau )\Vert _{\dot{B}^{s}_{p,r}}\hbox {d}\tau \nonumber \\&\quad \le \ C\Vert K_\Omega \Vert _{L^1(0,\infty )}\sup _{t\in \mathbb {R}}\Vert f\Vert _{e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s}_{p,r}}, \end{aligned}$$
(3.12)

where \(K_\Omega (t):=(1+|\Omega |t)^{-(1-\frac{2}{p'})}t^{-\frac{1}{2}(2-\beta )}\). Due to \(0<\beta <\frac{4}{p}-2\), there exists a positive constant \(C=C(p,\beta )\) such that

$$\begin{aligned} \Vert K_\Omega \Vert _{L^1(0,\infty )}= |\Omega |^{-\frac{\beta }{2}} \int ^\infty _0(1+\tau )^{-(1-\frac{2}{p'})}\tau ^{-\frac{1}{2}(2-\beta )}\hbox {d}\tau \le C|\Omega |^{-\frac{\beta }{2}} \end{aligned}$$
(3.13)

for all \(\Omega \in \mathbb {R}\setminus \{0\}\). Substituting (3.13) into (3.12) finishes the proof. \(\square \)

4 Product law and bilinear estimates

To get the estimates on the bilinear terms in (2.2) and (2.7), we establish the following product law:

Lemma 4.1

Let \(\theta \in \{0,1\}\). Let \(p_0\in (1,\infty )\) and \((p_1, p_2, r, \lambda _1, \lambda _2)\in [1,\infty ]^5\) such that \(\frac{1}{p_0}\le \frac{1}{p_1}+\frac{1}{p_2}\)\(p_1\le \lambda _2\)\(p_2\le \lambda _1\), \(\frac{1}{p_0}\le \frac{1}{p_1}+\frac{1}{\lambda _1}\le 1\) and \(\frac{1}{p_0}\le \frac{1}{p_2}+\frac{1}{\lambda _2}\le 1\). If \(s_1+s_2+3\inf \{0, 1-\frac{1}{p_1}-\frac{1}{p_2}\}>0\), \(s_1+\frac{3}{\lambda _2}<\frac{3}{p_1}\) and \(s_2+\frac{3}{\lambda _1}<\frac{3}{p_2}\), then

$$\begin{aligned} \Vert uv\Vert _{e^{\theta \sqrt{t}\Lambda _1} \dot{B}^{s_1+s_2-3(\frac{1}{p_1}+\frac{1}{p_2}-\frac{1}{p_0})}_{p_0,r}}\le C \Vert u\Vert _{e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s_1}_{p_1,r}} \Vert v\Vert _{e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s_2}_{p_2,\infty }} \end{aligned}$$
(4.1)

for all \(u\in e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s_1}_{p_1,r}(\mathbb {R}^3)\) and \(v\in e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s_2}_{p_2,\infty }(\mathbb {R}^3)\).

Proof

Denote \((U, V):=(e^{\theta \sqrt{t}\Lambda _1}u, e^{\theta \sqrt{t}\Lambda _1}v)\). Thanks to Bony’s decomposition [7], we compute

$$\begin{aligned} \begin{aligned} e^{\theta \sqrt{t}\Lambda _1}\Delta _{j}(u v) =&\,e^{\theta \sqrt{t}\Lambda _1}\Delta _{j}( T_{u}v+T_{v}u+R(u,v))\\ =&\,e^{\theta \sqrt{t}\Lambda _1}\Delta _{j}\Big ( T_{e^{-\theta \sqrt{t}\Lambda _1}U} e^{-\theta \sqrt{t}\Lambda _1}V +T_{e^{-\theta \sqrt{t}\Lambda _1}V}e^{-\theta \sqrt{t}\Lambda _1}U\\&+R(e^{-\theta \sqrt{t}\Lambda _1}U, e^{-\theta \sqrt{t}\Lambda _1}V)\Big ). \end{aligned} \end{aligned}$$
(4.2)

To estimate the right-hand side of (4.2), we first introduce the bilinear operator \(B^\theta _{t}(\mathfrak {f}, \mathfrak {g})\) of the form for \(\theta \in \{0,1\}\):

$$\begin{aligned} \begin{aligned} B_{t}^{\theta }(\mathfrak {f}, \mathfrak {g}) :=&\ e^{\theta \sqrt{t}\Lambda _1}(e^{-\theta \sqrt{t}\Lambda _1}\mathfrak {f} e^{-\theta \sqrt{t}\Lambda _1}\mathfrak {g}) \\ =&\int _{\mathbb {R}^{3}}\int _{\mathbb {R}^{3}}e^{ix\xi }e^{\theta \sqrt{t}(| \xi |_{1}-|\xi -\eta |_{1}-|\eta |_{1})}{\widehat{\mathfrak {f}}}(\xi -\eta ) {\widehat{\mathfrak {g}}}(\eta )d\eta d\xi . \end{aligned} \end{aligned}$$
(4.3)

Claim that

$$\begin{aligned} \Vert B_{t}^{\theta }(\mathfrak {f}, \mathfrak {g})\Vert _{L^{p_0}}\le C\Vert \mathfrak {f}\Vert _{L^{p_{1}}}\Vert \mathfrak {g}\Vert _{L^{p_{2}}}, \ \ \ \text {for}\ \ 1< p_0 < \infty \ \ \ \text {and} \ \ \ \frac{1}{p_0}=\frac{1}{p_{1}}+\frac{1}{p_{2}}. \end{aligned}$$
(4.4)

Indeed, when \(\theta =0\), it is obvious to get

$$\begin{aligned} \Vert B_{t}^{0}(\mathfrak {f}, \mathfrak {g})\Vert _{L^{p_0}}\le C\Vert \mathfrak {f}\Vert _{L^{p_{1}}}\Vert \mathfrak {g}\Vert _{L^{p_{2}}}, \ \ \ \text {for}\ \ 1\le p_0 \le \infty \ \ \ \text {and} \ \ \ \frac{1}{p_0}=\frac{1}{p_{1}}+\frac{1}{p_{2}}. \end{aligned}$$

Next, we consider the case of \(\theta =1\). Denote for \(\rho =(\rho _{1},\rho _{2},\rho _{3}), \mu = (\mu _{1},\mu _{2},\mu _{3}), \nu = (\nu _{1},\nu _{2},\nu _{3})\) with \(\rho _{i},\mu _{i}, \nu _{i}\in \{1,-1\}\) that

$$\begin{aligned} \begin{aligned}&D_{\rho }:=\{\eta \in \mathbb {R}^3: \rho _{i}\eta _{i}\ge 0, i=1, 2, 3\},\\&D_{\mu }:= \{\xi -\eta \in \mathbb {R}^3: \mu _{i}(\xi _{i}-\eta _{i}) \ge 0, i=1,2,3\},\\&D_{\nu }:=\{\xi \in \mathbb {R}^3: \nu _{i}\xi _{i}\ge 0, i = 1,2,3\}, \end{aligned} \end{aligned}$$

and we also denote by \(\chi _{D}\) the characteristic function on D. Then, we can rewrite \(B_{t}^{1}(\mathfrak {f}, \mathfrak {g})\) as

$$\begin{aligned}&B_{t}^{1}(\mathfrak {f}, \mathfrak {g})=\sum _{\rho _{i}, \mu _{j}, \nu _{k}\in \{-1,1\}}\int _{\mathbb {R}^{n}}\int _{\mathbb {R}^{n}}e^{ix\xi } \chi _{D_{\nu }}(\xi )e^{\sqrt{t}(| \xi |_{1}-|\xi -\eta |_{1} -|\eta |_{1})}\\&\quad \times \chi _{D_{\mu }}(\xi -\eta ) {\widehat{\mathfrak {f}}}(\xi -\eta )\chi _{D_{\rho }}(\eta ){\widehat{\mathfrak {g}}}(\eta )d\eta d\xi . \end{aligned}$$

Notice that for \(\eta \in D_{\rho }\), \(\xi -\eta \in D_{\mu }\) and \(\xi \in D_{\nu }\), \(e^{\sqrt{t}(|\xi _{i}|-|\xi _{i}-\eta _{i}|-|\eta _{j}|)}\) must belong to the following set

$$\begin{aligned} \mathfrak {M}:=\{1, \ e^{-2\sqrt{t}|\xi _{i}|}, \ e^{-2\sqrt{t}|\xi _{i}-\eta _{i}|}, \ e^{-2\sqrt{t}|\eta _{i}|},\ \ \ i=1,2,3\}. \end{aligned}$$

For \(1<p_0<\infty \), \(\chi _{D}\in M_{p_0}\) and \(m\in M_{p_0}\) for any \(m \in \mathfrak {M}\). Hence, it follows from the algebra property of \(M_{p_0}\) that

$$\begin{aligned} \Vert B_{t}^{1}(\mathfrak {f}, \mathfrak {g})\Vert _{L^{p_0}}\le C\Vert \mathfrak {f}\Vert _{L^{p_{1}}}\Vert \mathfrak {g}\Vert _{L^{p_{2}}},\ \ \ \ \ \text {for}\ \ 1<p_0<\infty \ \ \text {and}\ \ \frac{1}{p_0}=\frac{1}{p_{1}}+\frac{1}{p_{2}}. \end{aligned}$$

Now, we estimate the terms of paraproduct in (4.2). By Lemma 2.3 and (4.4), we see

$$\begin{aligned}&\Vert e^{\theta \sqrt{t}\Lambda _1}\Delta _j T_uv \Vert _{L^{p_0}} \le \sum _{|k-j|\le 4}\Big \Vert \Delta _{j}e^{\theta \sqrt{t}\Lambda _1}(S_{k-1}e^{-\theta \sqrt{t}\Lambda _1}U \Delta _{k} e^{-\theta \sqrt{t}\Lambda _1}V)\Big \Vert _{L^{p_0}}\nonumber \\&\quad \le C\sum _{|k-j|\le 4}2^{j\left( \frac{3}{p_2}+\frac{3}{\lambda _2}-\frac{3}{p_0}\right) } \Big \Vert e^{\theta \sqrt{t}\Lambda _1}(e^{-\theta \sqrt{t}\Lambda _1}S_{k-1}U e^{-\theta \sqrt{t}\Lambda _1}\Delta _{k}V)\Big \Vert _{L^{\frac{p_{2}\lambda _2}{\lambda _2+p_{2}}}}\nonumber \\&\quad \le C 2^{j\left( \frac{3}{p_2}+\frac{3}{\lambda _2}-\frac{3}{p_0}\right) }\sum _{|k-j| \le 4}\Vert S_{k-1}U\Vert _{L^{\lambda _2}}\Vert \Delta _kV\Vert _{L^{p_2}}. \end{aligned}$$

Since \(s_1+\frac{3}{\lambda _2}\le \frac{3}{p_1}\), by the definition of \(S_{k-1}\) and Lemma 2.3, we obtain

$$\begin{aligned} \Vert S_{k-1}U\Vert _{L^{\lambda _2}}\le & {} C\sum _{k'\le k-2} 2^{k'\left( \frac{3}{p_1}-\frac{3}{\lambda _2}-s_1\right) } 2^{s_1 k'}\Vert \Delta _{k'}U\Vert _{L^{p_1}}\\\le & {} C \ 2^{k\left( \frac{3}{p_1}-\frac{3}{\lambda _2}-s_1\right) }\sum _{k'\le k-2} 2^{(k'-k)\left( \frac{3}{p_1}-\frac{3}{\lambda _2}-s_1\right) } a_{k'}\Vert U\Vert _{\dot{B}^{s_1}_{p_1, r}}\\\le & {} C \ 2^{k\left( \frac{3}{p_1}-\frac{3}{\lambda _2}-s_1\right) } \tilde{a}_{k}\Vert U\Vert _{\dot{B}^{s_1}_{p_1, r}}, \end{aligned}$$

where \(\Vert \{a_{k'}\}_{k'\in \mathbb {Z}}\Vert _{\ell ^r}\le 1\) and \(\{\tilde{a}_{k}\}_{k\in \mathbb {Z}}:=\{\sum _{k'\le k-2} 2^{(k'-k)(\frac{3}{p_1}-\frac{3}{\lambda _2}-s_1)} a_{k'}\}_{k\in \mathbb {Z}}\in \ell ^r\). We therefore get

$$\begin{aligned}&2^{j\left( s_1+s_2+\frac{3}{p_0}-\frac{3}{p_1}-\frac{3}{p_2}\right) } \Vert e^{\theta \sqrt{t}\Lambda _1}\Delta _j T_uv\Vert _{L^{p_0}} \\&\quad \le C\sum _{|k-j|\le 4}2^{(j-k)\left( s_1+s_2+\frac{3}{\lambda _2}-\frac{3}{p_1}\right) } \tilde{a}_{k}2^{ks_2}\Vert \Delta _kV\Vert _{L^{p_2}}\Vert U\Vert _{\dot{B}^{s_1}_{p_1, r}}, \end{aligned}$$

which, together with Young’s inequality, implies that

$$\begin{aligned} \Vert e^{\theta \sqrt{t}\Lambda _1}T_uv\Vert _{\dot{B}_{p_0,r}^{s_1+s_2+\frac{3}{p_0}-\frac{3}{p_1}-\frac{3}{p_2}}}\le & {} C \left( \sum _{-4\le l\le 4}2^{l(s_1+s_2+\frac{3}{\lambda _2}-\frac{3}{p_1})}\right) \Vert \{\tilde{a}_{j}\} _{j\in \mathbb {Z}}\Vert _{l^r} \Vert U\Vert _{\dot{B}^{s_1}_{p_1, r}}\Vert V\Vert _{\dot{B}^{s_2}_{p_2, \infty }} \nonumber \\\le & {} C\Vert U\Vert _{\dot{B}^{s_1}_{p_1, r}}\Vert V\Vert _{\dot{B}^{s_2}_{p_2, \infty }}. \end{aligned}$$
(4.5)

On the other hand, we obtain by Lemma 2.3 and (4.4) that

$$\begin{aligned}&\Vert e^{\theta \sqrt{t}\Lambda _1}\Delta _jT_vu \Vert _{L^{p_0}} \le \sum _{|k-j|\le 4}\Vert \Delta _{j}e^{\theta \sqrt{t}\Lambda _1}(S_{k-1}e^{-\theta \sqrt{t}\Lambda _1}V \Delta _{k} e^{-\theta \sqrt{t}\Lambda _1}U)\Vert _{L^{p_0}}\nonumber \\&\quad \le C\sum _{|k-j|\le 4}2^{j\left( \frac{3}{p_1}+\frac{3}{\lambda _1}-\frac{3}{p_0}\right) } \Vert e^{\theta \sqrt{t}\Lambda _1}(S_{k-1}e^{-\theta \sqrt{t}\Lambda _1}V \Delta _{k} e^{-\theta \sqrt{t}\Lambda _1}U)\Vert _{L^{\frac{p_{1}\lambda _1}{\lambda _1+p_{1}}}}\nonumber \\&\quad \le C2^{j\left( \frac{3}{p_1}+\frac{3}{\lambda _1}-\frac{3}{p_0}\right) }\sum _{|k-j| \le 4}\Vert S_{k-1}V\Vert _{L^{\lambda _1}}\Vert \Delta _kU\Vert _{L^{p_1}}. \end{aligned}$$

Since \(s_2+\frac{3}{\lambda _1}\le \frac{3}{p_2}\), by the definition of \(S_{k-1}\) and Lemma 2.3, we obtain

$$\begin{aligned} \Vert S_{k-1}V\Vert _{L^{\lambda _1}}\le & {} C\sum _{k'\le k-2} 2^{k'\left( \frac{3}{p_2}-\frac{3}{\lambda _1}-s_2\right) } 2^{s_2 k'}\Vert \Delta _{k'}V\Vert _{L^{p_2}}\\\le & {} C 2^{k\left( \frac{3}{p_2}-\frac{3}{\lambda _1}-s_2\right) } \Vert V\Vert _{\dot{B}^{s_2}_{p_2, \infty }}, \end{aligned}$$

hence,

$$\begin{aligned}&2^{j\left( s_1+s_2+\frac{3}{p_0}-\frac{3}{p_1}-\frac{3}{p_2}\right) } \Vert e^{\theta \sqrt{t}\Lambda _1}\Delta _j T_vu\Vert _{L^{p_0}}\\&\quad \le C\sum _{|k-j|\le 4}2^{(j-k)\left( s_1+s_2+\frac{3}{\lambda _1}-\frac{3}{p_2}\right) } 2^{ks_1}\Vert \Delta _kU\Vert _{L^{p_1}}\Vert V\Vert _{\dot{B}^{s_2}_{p_2, \infty }}, \end{aligned}$$

which, together with Young’s inequality, implies that

$$\begin{aligned} \Vert e^{\theta \sqrt{t}\Lambda _1}T_vu\Vert _{\dot{B}_{p_0,r}^{s_1+s_2+\frac{3}{p_0}-\frac{3}{p_1}-\frac{3}{p_2}}}\le & {} C \left( \sum _{-4\le l\le 4}2^{l(s_1+s_2+\frac{3}{\lambda _1}-\frac{3}{p_1})}\right) \Vert U\Vert _{\dot{B}^{s_1}_{p_1, r}}\Vert V\Vert _{\dot{B}^{s_2}_{p_2, \infty }}\nonumber \\\le & {} C\Vert U\Vert _{\dot{B}^{s_1}_{p_1, r}}\Vert V\Vert _{\dot{B}^{s_2}_{p_2, \infty }}. \end{aligned}$$
(4.6)

Finally, we consider the remainder term in (4.2). We divide our estimation into two cases: \(\frac{1}{p_1}+\frac{1}{p_2}\le 1\) and \(\frac{1}{p_1}+\frac{1}{p_2}>1\).

If \(\frac{1}{p_1}+\frac{1}{p_2}\le 1\), by Lemma 2.3 and (4.4), we have

$$\begin{aligned} \Vert e^{\theta \sqrt{t}\Lambda _1}\Delta _j R(u,v)\Vert _{L^{p_0}}\le & {} \sum _{k\ge j-2}\left\| \Delta _{j}e^{\theta \sqrt{t}\Lambda _1} \left( \Delta _{k}e^{-\theta \sqrt{t}\Lambda _1}U{\widetilde{\Delta }}_{k} e^{-\theta \sqrt{t}\Lambda _1}V\right) \right\| _{L^{p_0}}\\\le & {} 2^{j(\frac{3}{p_1}+\frac{3}{p_2}-\frac{3}{p_0})}\sum _{k\ge j-2}\Vert \Delta _{k}U\Vert _{L^{p_1}}\Vert {\tilde{\Delta }}_kV\Vert _{L^{p_2}}, \end{aligned}$$

hence,

$$\begin{aligned}&2^{j\left( s_1+s_2+\frac{3}{p_0}-\frac{3}{p_1}-\frac{3}{p_2}\right) }\Vert \Delta _j e^{\theta \sqrt{t}\Lambda _1}R(u,v)\Vert _{L^{p_0}}\\&\quad \le \sum _{k\ge j-2}2^{(j-k)(s_1+s_2)}2^{ks_1}\Vert \Delta _{k}U\Vert _{L^{p_1}}2^{ks_2} \Vert {\tilde{\Delta }}_kV\Vert _{L^{p_2}}, \end{aligned}$$

which, together with Young’s inequality and \(s_1+s_2>0\), implies that

$$\begin{aligned} \Vert e^{\theta \sqrt{t}\Lambda _1}R(u,v)\Vert _{\dot{B}_{p,r}^{s_1+s_2+\frac{3}{p_0}-\frac{3}{p_1}-\frac{3}{p_2}}}\le & {} C \Vert \{2^{ks_1}\Vert \Delta _{k}U\Vert _{L^{p_1}}\}_{k\in \mathbb {Z}}\Vert _{\ell ^r} \Big (\sup _{k\in \mathbb {Z}}2^{ks_2}\Vert \Delta _{k}U\Vert _{L^{p_2}}\Big ) \nonumber \\\le & {} C\Vert U\Vert _{\dot{B}_{p_1,r}^{s_1}}\Vert V\Vert _{\dot{B}_{p_2,\infty }^{s_2}}. \end{aligned}$$
(4.7)

If \(\frac{1}{p_1}+\frac{1}{p_2}>1\), without loss of generality, we assume \(p_2\le p'_1\) with \(\frac{1}{p_1}+\frac{1}{p'_1}=1\). By Lemma 2.3 and (4.4), we get

$$\begin{aligned} \Vert e^{\theta \sqrt{t}\Lambda _1}\Delta _j R(u,v)\Vert _{L^{p_0}}\le & {} C2^{3j(1-\frac{1}{p_0})} \sum _{k\ge j-2}\Vert \Delta _{k}U\Vert _{L^{p_1}}\Vert {\tilde{\Delta }}_kV\Vert _{L^{p'_1}} \\\le & {} C2^{3j(1-\frac{1}{p_0})} \sum _{k\ge j-2}\Vert \Delta _{k}U\Vert _{L^{p_1}} 2^{k\left( \frac{3}{p_2}-\frac{3}{p'_1}\right) }\Vert {\tilde{\Delta }}_kV\Vert _{L^{p_2}}, \end{aligned}$$

hence,

$$\begin{aligned}&\ 2^{j(s_1+s_2+\frac{3}{p_0}-\frac{3}{p_1}-\frac{3}{p_2})}\Vert \Delta _j e^{\theta \sqrt{t}\Lambda _1}R(u,v)\Vert _{L^{p_0}}\\&\quad \le \ \sum _{k\ge j-2}2^{(j-k)(s_1+s_2+3-\frac{3}{p_1}-\frac{3}{p_2})}2^{ks_1} \Vert \Delta _{k}U\Vert _{L^{p_1}}2^{ks_2}\Vert {\tilde{\Delta }}_kV\Vert _{L^{p_2}}, \end{aligned}$$

which, together with Young’s inequality and \(s_1+s_2+3-\frac{3}{p_1}-\frac{3}{p_2}>0\), implies that

$$\begin{aligned} \Vert e^{\theta \sqrt{t}\Lambda _1}R(u,v)\Vert _{\dot{B}_{p,r}^{s_1+s_2+\frac{3}{p_0}-\frac{3}{p_1}-\frac{3}{p_2}}}\le & {} C \Vert \{2^{ks_1}\Vert \Delta _{k}U\Vert _{L^{p_1}}\}_{k\in \mathbb {Z}}\Vert _{\ell ^r} \Big (\sup _{k\in \mathbb {Z}}2^{ks_2}\Vert \Delta _{k}V\Vert _{L^{p_2}}\Big ) \nonumber \\\le & {} C\Vert U\Vert _{\dot{B}_{p_1,r}^{s_1}}\Vert V\Vert _{\dot{B}_{p_2,\infty }^{s_2}}. \end{aligned}$$
(4.8)

Combining (4.5) \(\sim \) (4.8) implies (4.1). This completes our proof. \(\square \)

Remark 4.2

It is easy to generalize Lemma 4.1 to the space \({\tilde{L}}^\delta (0,\infty ; e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s}_{p,r}(\mathbb {R}^3))\). The general principle is that the indices spr behave just as in the stationary case, and the index \(\delta \) behaves according to Hölder’s inequality.

Before showing the proofs of our main results, we need the following bilinear estimates. Define

$$\begin{aligned} N_1(u,v):=\int ^t_0T_\Omega (t-\tau )\mathbb {P}\nabla \cdot [u(\tau )\otimes v(\tau )]\hbox {d}\tau ,~~~~~t\ge 0, \end{aligned}$$
(4.9)

and

$$\begin{aligned} N_2(u,v):=\int ^t_{-\infty }T_\Omega (t-\tau )\mathbb {P}\nabla \cdot [u(\tau )\otimes v(\tau )]\hbox {d}\tau ,~~~~~t\in \mathbb {R}. \end{aligned}$$
(4.10)

The first two lemmas for problem (1.1) correspond to the cases of \(u_0\in \dot{B}^s_{p,r}(\mathbb {R}^3)\) with \(1\le p<2\) and \(u_0\in \dot{B}^s_{2,r}(\mathbb {R}^3)\), respectively, and the last one concerns the time-periodic case.

Lemma 4.3

Let \(\theta \in \{0, 1\}\). Let \(p\in (\frac{3}{2}, 2)\) and \(r\in [1, \infty ]\). Let \(s\in \mathbb {R}\) satisfy \(-1+\frac{3}{p}<s<-2+\frac{7}{p}\), and let \(\delta \in [2, \infty )\) satisfy

$$\begin{aligned} \frac{1}{2}(s+1-\frac{3}{p})+1-\frac{2}{p}<\frac{1}{\delta } <\frac{1}{2}(s+1-\frac{3}{p}). \end{aligned}$$

Then, there exists a positive constant \(C=C(s,p,\delta )\) such that

$$\begin{aligned}&\Vert N_1(u,v)\Vert _{{\tilde{L}}^\delta \left( 0,\infty ; e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s+\frac{3}{p'}-\frac{3}{p}}_{p',r}\right) }\\&\quad \le C|\Omega |^{-\left[ \frac{1}{2}\left( s+1-\frac{3}{p}\right) -\frac{1}{\delta }\right] } \Vert u\Vert _{{\tilde{L}}^\delta \left( 0,\infty ; e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s+\frac{3}{p'}-\frac{3}{p}}_{p',r}\right) } \Vert v\Vert _{{\tilde{L}}^\delta \left( 0,\infty ; e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s+\frac{3}{p'}-\frac{3}{p}}_{p',r}\right) } \end{aligned}$$

for all \(\Omega \in \mathbb {R}\setminus \{0\}\), where \(\frac{1}{p}+\frac{1}{p'}=1\).

Proof

Note that the indices s, \(\delta \), p and r satisfy the assumptions in Lemma 3.2. Thus, by Lemma 3.2, we deduce

$$\begin{aligned} \Vert N_1(u,v)\Vert _{{\tilde{L}}^\delta \left( 0,\infty ; e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s+\frac{3}{p'}-\frac{3}{p}}_{p',r}\right) }\le C|\Omega |^{-\left[ \frac{1}{2}\left( s+1-\frac{3}{p}\right) -\frac{1}{\delta }\right] }\Vert u\otimes v\Vert _{{\tilde{L}}^\frac{\delta }{2}\left( 0, \infty ; e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{2s-\frac{3}{p}}_{p,r}\right) }.\nonumber \\ \end{aligned}$$
(4.11)

By taking \(s_1=s_2=s+\frac{3}{p'}-\frac{3}{p}\), \(p_0=p\), \(p_1=p_2=p'\) and \(\lambda _1=\lambda _2=\frac{p'}{p'-2}\) in Lemma 4.1, combining with Remark 4.2, we arrive at

$$\begin{aligned} \Vert u\otimes v\Vert _{{\tilde{L}}^\frac{\delta }{2}\left( 0,\infty ; e^{\theta \sqrt{t}\Lambda _1}\dot{B}_{p,r}^{2s-\frac{3}{p}}\right) } \le C\Vert u\Vert _{{\tilde{L}}^\delta \left( 0,\infty ; e^{\theta \sqrt{t}\Lambda _1}\dot{B}_{p',r}^{s+\frac{3}{p'}-\frac{3}{p}}\right) } \Vert v\Vert _{{\tilde{L}}^\delta \left( 0,\infty ; e^{\theta \sqrt{t}\Lambda _1}\dot{B}_{p',r}^{s+\frac{3}{p'}-\frac{3}{p}}\right) }.\nonumber \\ \end{aligned}$$
(4.12)

Substituting (4.12) into (4.11) finishes the proof. \(\square \)

Lemma 4.4

Let \(\theta \in \{0, 1\}\). Let \(s\in (\frac{1}{2}, \frac{3}{2})\) and \(r\in [1, \infty ]\). Let \(q\in (2, 3]\) satisfy \(\frac{s}{6}+\frac{1}{4}<\frac{1}{q}<-\frac{s}{4}+\frac{7}{8}\), and let \(\delta \in [2, \infty )\) satisfy

$$\begin{aligned} \frac{1}{2}\left( s-\frac{1}{2}\right) +\frac{2}{q}-1<\frac{1}{\delta } <\frac{1}{2}\left( s-\frac{1}{2}\right) . \end{aligned}$$

Then, there exists a positive constant \(C=C(s, q, \delta )\) such that

$$\begin{aligned}&\Vert N_1(u,v)\Vert _{{\tilde{L}}^\delta \left( 0,\infty ; e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s+\frac{3}{q}-\frac{3}{2}}_{q,r}\right) }\le C|\Omega |^{-\left[ \frac{1}{2}\left( s-\frac{1}{2}\right) -\frac{1}{\delta }\right] }\\&\quad \times \Vert u\Vert _{{\tilde{L}}^\delta \left( 0,\infty ; e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s+\frac{3}{q}-\frac{3}{2}}_{q,r}\right) } \Vert v\Vert _{{\tilde{L}}^\delta \left( 0,\infty ; e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s+\frac{3}{q}-\frac{3}{2}}_{q,r}\right) } \end{aligned}$$

for all \(\Omega \in \mathbb {R}\setminus \{0\}\).

Proof

Note that the indices s, \(\delta \), q and r satisfy the assumptions in Lemma 3.4. Thus, by Lemma 3.4, we have

$$\begin{aligned}&\Vert N_1(u,v)\Vert _{{\tilde{L}}^\delta \left( 0,\infty ; e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s+\frac{3}{q}-\frac{3}{2}}_{q,r}\right) }\le C|\Omega |^{-\left[ \frac{1}{2}\left( s-\frac{1}{2}\right) -\frac{1}{\delta }\right] }\nonumber \\&\quad \times \Vert u\otimes v\Vert _{{\tilde{L}}^\frac{\delta }{2}\left( 0, \infty ; e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{2s+\frac{3}{q'}-3}_{q',r}\right) }. \end{aligned}$$
(4.13)

By taking \(s_1=s_2=s+\frac{3}{q}-\frac{3}{2}\), \(p_0=q'\), \(p_1=p_2=q\) and \(\lambda _1=\lambda _2=\frac{q}{q-2}\) in Lemma 4.1, combining with Remark 4.2, we see

$$\begin{aligned}&\Vert u\otimes v\Vert _{{\tilde{L}}^\frac{\delta }{2}\left( 0,\infty ; e^{\theta \sqrt{t}\Lambda _1}\dot{B}_{q',r}^{2s+\frac{3}{q'}-3}\right) } \le C\Vert u\Vert _{{\tilde{L}}^\delta \left( 0,\infty ; e^{\theta \sqrt{t}\Lambda _1}\dot{B}_{q,r}^{s+\frac{3}{q}-\frac{3}{2}}\right) }\nonumber \\&\quad \times \Vert v\Vert _{{\tilde{L}}^\delta \left( 0,\infty ; e^{\theta \sqrt{t}\Lambda _1}\dot{B}_{q,r}^{s+\frac{3}{q}-\frac{3}{2}}\right) }. \end{aligned}$$
(4.14)

Substituting (4.13) into (4.14) finishes the proof. \(\square \)

Lemma 4.5

Let \(\theta \in \{0, 1\}\). Let \(p\in (\frac{3}{2}, 2)\), \(r\in [1, \infty ]\) and \(\beta \in (\frac{3}{p}-2, 4-\frac{6}{p})\cap (0, \frac{4}{p}-2)\). Then, there exists a positive constant \(C=C(\beta , p)\) such that

$$\begin{aligned}&\Vert N_2(u,v)\Vert _{BC\left( \mathbb {R}; e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{-1+\frac{3}{p'}+\beta }_{p',r}\right) }\le C|\Omega |^{-\frac{\beta }{2}} \Vert u\Vert _{BC\left( \mathbb {R}; e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{-1+\frac{3}{p'}+\beta }_{p',r}\right) }\\&\quad \times \Vert v\Vert _{BC\left( \mathbb {R}; e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{-1+\frac{3}{p'}+\beta }_{p',r}\right) } \end{aligned}$$

for all \(\Omega \in \mathbb {R}\setminus \{0\}\).

Proof

Note that the indices \(\beta \), p and r satisfy the assumptions in Lemma 3.6. Thus, by Lemma 3.6, we obtain

$$\begin{aligned} \Vert N_2(u,v)\Vert _{BC\left( \mathbb {R}; e^{\theta \sqrt{t}\Lambda _1}\dot{B}_{p',r}^{-1+\frac{3}{p'}+\beta }\right) } \le C|\Omega |^{-\frac{\beta }{2}} \Vert u\otimes v\Vert _{BC\left( \mathbb {R}; e^{\theta \sqrt{t}\Lambda _1}\dot{B}_{p,r}^{-2+\frac{3}{p}+2\beta }\right) }. \end{aligned}$$
(4.15)

By taking \(s_1=s_2=-1+\frac{3}{p'}+\beta \), \(p_0=p\), \(p_1=p_2=p'\) and \(\lambda _1=\lambda _2=\frac{p'}{p'-2}\) in Lemma 4.1, combining with Remark 4.2, we have

$$\begin{aligned}&\Vert u\otimes v\Vert _{BC\left( \mathbb {R}; e^{\theta \sqrt{t}\Lambda _1}\dot{B}_{p,r}^{-2+\frac{3}{p}+2\beta }\right) } \le C \Vert u\Vert _{BC\left( \mathbb {R}; e^{\theta \sqrt{t}\Lambda _1}\dot{B}_{p',r}^{-1+\frac{3}{p'}+\beta }\right) }\nonumber \\&\quad \times \Vert v\Vert _{BC\left( \mathbb {R}; e^{\theta \sqrt{t}\Lambda _1}\dot{B}_{p',r}^{-1+\frac{3}{p'}+\beta }\right) }. \end{aligned}$$
(4.16)

Substituting (4.16) into (4.15) finishes the proof. \(\square \)

5 Proof of main results

Proof of Theorem 1.1

It is not difficult to examine that the indices spr and \(\delta \) given in Theorem 1.1 satisfy the assumptions of Lemmas 3.1 and 4.3.

Let \(\Omega \in \mathbb {R}{\setminus }\{0\}\) and \(u_0\in \dot{B}_{p,r}^{s}(\mathbb {R}^3)\) satisfy \(\text {div} u_0=0\). It follows from Lemma 3.1 that there exists a positive constant \(C_0\) such that

$$\begin{aligned} \Vert T_\Omega (t)u_0\Vert _{{\tilde{L}}^\delta \left( 0, \infty ; e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s+\frac{3}{p'}-\frac{3}{p}}_{p',r}\right) }\le C_0|\Omega |^{-\frac{1}{\delta }}\Vert u_0\Vert _{\dot{B}^{s}_{p,r}}. \end{aligned}$$
(5.1)

Define the mapping \(\mathscr {B}\) and the solution space Y by

$$\begin{aligned} \mathscr {B}(u)(t):=T_\Omega (t)u_0+N_1(u,u)(t) \end{aligned}$$
(5.2)

and

$$\begin{aligned} Y:=\left\{ u\in {\tilde{L}}^\delta \left( 0,\infty ; e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s+\frac{3}{p'}-\frac{3}{p}}_{p',r}(\mathbb {R}^3)\right) :\Vert u\Vert _Y\le 2C_0|\Omega |^{-\frac{1}{\delta }}\Vert u_0\Vert _{\dot{B}^{s}_{p,r}}\right\} , \end{aligned}$$
(5.3)

where \(\Vert \cdot \Vert _Y=\Vert \cdot \Vert _{{\tilde{L}}^\delta (0,\infty ; e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s+\frac{3}{p'}-\frac{3}{p}}_{p',r})}\) and \(N_1(u,u)\) is defined in (4.9). From (5.1) and Lemma 4.3, there exists a positive constant \(C_1\) such that

$$\begin{aligned} \Vert \mathscr {B}(u)\Vert _Y~~\le & {} C_0|\Omega |^{-\frac{1}{\delta }}\Vert u_0\Vert _{\dot{B}^{s}_{p,r}}+C_1|\Omega |^{-\left[ \frac{1}{2}\left( s+1-\frac{3}{p}\right) -\frac{1}{\delta }\right] } \Vert u\Vert ^2_{Y}\nonumber \\\le & {} C_0|\Omega |^{-\frac{1}{\delta }}\Vert u_0\Vert _{\dot{B}^{s}_{p,r}} +4C_0^2C_1|\Omega |^{-\frac{2}{\delta }} |\Omega |^{-\left[ \frac{1}{2}\left( s+1-\frac{3}{p}\right) -\frac{1}{\delta }\right] }\Vert u_0\Vert ^2_{\dot{B}^{s}_{p,r}}\nonumber \\\le & {} C_0|\Omega |^{-\frac{1}{\delta }}\Vert u_0\Vert _{\dot{B}^{s}_{p,r}}\bigg \{1+4C_0C_1|\Omega |^{-\frac{1}{2}\left( s+1-\frac{1}{p}\right) } \Vert u_0\Vert _{\dot{B}^{s}_{p,r}}\bigg \} \end{aligned}$$
(5.4)

for all \(u\in Y\). Moreover, by using Lemma 4.3, there exists a positive constant \(C_2\) such that

$$\begin{aligned}&\Vert \mathscr {B}(u)-\mathscr {B}(v)\Vert _Y\nonumber \\&\quad =\Big \Vert \int ^t_0 T_\Omega (t-\tau )\mathbb {P}\nabla \cdot \big [u(\tau )\otimes (u(\tau )-v(\tau ))+(u(\tau )-v(\tau )) \otimes v(\tau )\big ]\hbox {d}\tau \Big \Vert _Y\nonumber \\&\quad \le C_2|\Omega |^{-[\frac{1}{2}(s+1-\frac{3}{p})-\frac{1}{\delta }]}(\Vert u\Vert _Y+\Vert v\Vert _Y)\Vert u-v\Vert _Y\nonumber \\&\quad \le 4C_0C_2|\Omega |^{-\frac{1}{2}(s+1-\frac{3}{p})}\Vert u_0\Vert _{\dot{B}^{s}_{p,r}}\Vert u-v\Vert _Y \end{aligned}$$
(5.5)

for \(u, v \in Y\).

Now, let us assume that initial velocity \(u_0\in \dot{B}^{s}_{p,r}(\mathbb {R}^3)\) satisfies

$$\begin{aligned} \Vert u_0\Vert _{\dot{B}^{s}_{p,r}}\le \min \bigg \{\frac{1}{4C_0C_1}, \frac{1}{8C_0C_2}\bigg \}|\Omega |^{\frac{1}{2}\left( s+1-\frac{3}{p}\right) }, \end{aligned}$$

(5.4) and (5.5) immediately imply that

$$\begin{aligned} \Vert \mathscr {B}(u)\Vert _Y\le 2C_0|\Omega |^{-\frac{1}{\delta }}\Vert u_0\Vert _{\dot{B}^{s}_{p,r}}\ \ \ \ \text {and}\ \ \ \ \Vert \mathscr {B}(u)-\mathscr {B}(v)\Vert _Y<\frac{1}{2}\Vert u-v\Vert _Y \end{aligned}$$

for \(u,v\in Y\). Therefore, by the contraction mapping principle, there exists a unique solution \(u\in Y\) satisfying (2.2) for all \(t > 0\).

It remains to show that the solution \(u \in Y\) also belongs to \(C([0,\infty ); e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s}_{p,r}(\mathbb {R}^3))\). By using Lemmas 2.3, 2.5, 2.7 \(\sim \) 2.9, 4.3 and Young’s inequality, together with the boundedness properties of \(\mathcal {R}\) and \(\mathbb {P}\), we obtain

$$\begin{aligned}&\Vert u(t)\Vert _{e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s}_{p,r}}~~\nonumber \\&\quad \le C\bigg \Vert \Big \{2^{sjr} \Big \Vert e^{\theta \sqrt{t}\Lambda _1+\frac{t}{2}\Delta } \mathcal {G}_\pm (\Omega t)e^{\frac{t}{2}\Delta }\Delta _ju_0\Big \Vert _{L^{p}} \Big \}_{j\in \mathbb {Z}}\bigg \Vert _{\ell ^r}\nonumber \\&\qquad +C\bigg \Vert \Big \{2^{js}\int _0^t \Big \Vert \Delta _je^{\theta \sqrt{t}\Lambda _1} e^{(t-\tau )\Delta }\mathcal {G}_\pm (\Omega t)\nabla \cdot (u(\tau )\otimes u(\tau ))\Big \Vert _{L^p}\hbox {d}\tau \Big \}_{j\in \mathbb {Z}}\bigg \Vert _{\ell ^r}\nonumber \\&\quad \le C(1+|\Omega |t)^2\Vert u_0\Vert _{\dot{B}^{s}_{p,r}}\nonumber \\&\qquad +C\bigg \Vert \Big \{2^{j(2s-\frac{3}{p})}\int _0^t K_\Omega (t-\tau ) \Big \Vert \Delta _je^{\theta \sqrt{\tau }\Lambda _1} (u(\tau )\otimes u(\tau ))\Big \Vert _{L^{p}}\hbox {d}\tau \Big \}_{j\in \mathbb {Z}}\bigg \Vert _{\ell ^r}\nonumber \\&\quad \le C(1+|\Omega |t)^2\Vert u_0\Vert _{\dot{B}^{s}_{p,r}}+C\Vert K_\Omega \Vert _{L^{(\frac{\delta }{2})'}(0,t)} \Vert u\otimes u\Vert _{{\tilde{L}}^{\frac{\delta }{2}}\left( 0,\infty ; e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{2s-\frac{3}{p}}_{p,r}\right) },\nonumber \\&\quad \le C(1+|\Omega |t)^2\Vert u_0\Vert _{\dot{B}^{s}_{p,r}}+C\Vert K_\Omega \Vert _{L^{(\frac{\delta }{2})'}(0,t)}\Vert u\Vert ^2_{Y}, \end{aligned}$$
(5.6)

where \(K_\Omega (t):=(1+|\Omega |t)^2 t^{-\frac{1}{2}(1+\frac{3}{p}-s)}\) and \(\frac{1}{(\frac{\delta }{2})'}+\frac{1}{(\frac{\delta }{2})}=1\). Due to \(\frac{1}{\delta }<\frac{1}{4}(s+1-\frac{3}{p})\), it is easy to check that there exists a continuous function \(C(t)>0\) defined on \([0,\infty )\) such that

$$\begin{aligned} \int _0^t (1+|\Omega |\tau )^{2(\frac{\delta }{2})'} \tau ^{-\frac{1}{2}(1+\frac{3}{p}-s)(\frac{\delta }{2})'}\hbox {d}\tau \le C(t). \end{aligned}$$
(5.7)

Substituting (5.7) into (5.6) implies that \(u(t)\in e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s}_{p,r}(\mathbb {R}^3)\) for \(t\ge 0\). Similarly, we see that \(u\in C([0,\infty ); e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s}_{p,r}(\mathbb {R}^3))\). This completes the proof of Theorem 1.1. \(\square \)

Proof of Theorem 1.2

The proof of Theorem 1.2 is quite similar to that of Theorem 1.1. By replacing the solution space Y with

$$\begin{aligned} Y:=\bigg \{u\in {\tilde{L}}^\delta (0,\infty ; e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s+\frac{3}{q}-\frac{3}{2}}_{q,r}(\mathbb {R}^3)):\Vert u\Vert _Y\le 2C_0|\Omega |^{-\frac{1}{\delta }}\Vert u_0\Vert _{\dot{B}^{s}_{2,r}}\bigg \}, \end{aligned}$$

we shall verify that there exists a unique solution \(u\in Y\) satisfying (2.2) for all \(t>0\).

It remains to show that the solution \(u \in Y\) also belongs to \(C([0,\infty ); e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s}_{2,r}(\mathbb {R}^3))\). By applying Lemmas 2.3, 2.6 \(\sim \) 2.8, 4.4 and Young’s inequality, together with the boundedness properties of \(\mathcal {R}\) and \(\mathbb {P}\), we obtain

$$\begin{aligned}&\Vert u(t)\Vert _{e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s}_{2,r}}~~\nonumber \\&\quad \le \Vert T_\Omega (t) u_0\Vert _{e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s}_{2,r}} +\bigg \Vert \Big \{2^{js}\int _0^t \Big \Vert \Delta _je^{\theta \sqrt{t}\Lambda _1} T_\Omega (t-\tau ) \mathbb {P}\nabla \cdot (u(\tau )\otimes u(\tau ))\Big \Vert _{L^2}\hbox {d}\tau \Big \}_{j\in \mathbb {Z}}\bigg \Vert _{\ell ^r}\nonumber \\&\quad \le C\Vert u_0\Vert _{\dot{B}^{s}_{2,r}}+C\bigg \Vert \Big \{2^{j\left( 2s+\frac{3}{q'}-3\right) }\int _0^t (t-\tau )^{-\frac{1}{2}\left( \frac{5}{2}-s\right) } \Big \Vert \Delta _j e^{\theta \sqrt{\tau }\Lambda _1}(u(\tau )\otimes u(\tau ))\Big \Vert _{L^{q'}}\hbox {d}\tau \Big \}_{j\in \mathbb {Z}}\bigg \Vert _{\ell ^r}\nonumber \\&\quad \le C\Vert u_0\Vert _{\dot{B}^{s}_{2,r}}+C\Big (\int _0^t \tau ^{-\frac{1}{2}(\frac{5}{2}-s) (\frac{\delta }{2})'}\hbox {d}\tau \Big )^{\frac{1}{(\frac{\delta }{2})'}} \Vert u\otimes u\Vert _{{\tilde{L}}^{\frac{\delta }{2}}\left( 0,\infty ; e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{2s+\frac{3}{q'}-3}_{q',r}\right) }\nonumber \\&\quad \le C\Vert u_0\Vert _{\dot{B}^{s}_{2,r}}+C\bigg (\int _0^t \tau ^{-\frac{1}{2}(\frac{5}{2}-s)(\frac{\delta }{2})'} \hbox {d}\tau \bigg )^{\frac{1}{(\frac{\delta }{2})'}}\Vert u\Vert ^2_{Y}, \end{aligned}$$

Thanks to \(\frac{1}{\delta }<\frac{1}{4}(s-\frac{1}{2})\), it is easy to check that

$$\begin{aligned} \bigg (\int _0^t \tau ^{-\frac{1}{2}\left( \frac{5}{2}-s\right) (\frac{\delta }{2})'}\hbox {d}\tau \bigg )^{\frac{1}{\left( \frac{\delta }{2}\right) '}} \le Ct^{\frac{1}{2}\left( s-\frac{1}{2}\right) -\frac{2}{\delta }}, \end{aligned}$$

which implies that u(t) belongs to \(e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s}_{2,r}(\mathbb {R}^3)\) for all \(t\ge 0\). Similarly, we see that \(u\in C([0,\infty ); e^{\theta \sqrt{t}\Lambda _1}\dot{B}^{s}_{2,r}(\mathbb {R}^3))\). This completes the proof of Theorem 1.2. \(\square \)

Proof of Theorem 1.4

It is not difficult to check that the indices spr and \(\beta \) given in the Theorem 1.4 satisfy the assumptions of Lemmas 3.5 and 4.5. We consider the Banach space Z, which is defined by

$$\begin{aligned} Z:=\bigg \{u\in BC\big (\mathbb {R}; e^{\theta \sqrt{t}\Lambda _1}\dot{B}_{p',r}^{-1+\frac{3}{p'}+\beta }(\mathbb {R}^3)\big ):~\Vert u\Vert _Z:=|\Omega |^{-\frac{\beta }{2}} \sup _{t\in \mathbb {R}}\Vert u\Vert _{e^{\theta \sqrt{t}\Lambda _1}\dot{B}_{p',r}^{-1+\frac{3}{p'}+\beta }}< \infty \bigg \}. \end{aligned}$$

We construct a time-periodic mild solution to problem (1.2) by successive approximation as follows:

$$\begin{aligned} u_0(t):= & {} \int _{-\infty }^tT_\Omega (t-\tau )\mathbb {P} f(\tau )\hbox {d}\tau , \end{aligned}$$
(5.8)
$$\begin{aligned} u_{n+1}(t):= & {} u_0(t)-N_2(u_n,u_n)(t),~~~~~n\in \mathbb {N}\cup \{0\}, \end{aligned}$$
(5.9)

where \(N_2(u,v)\) is given by (4.10). By Lemma 3.5, we get that \(u_0\in Z\) and there exists a positive constant \(C_1\) such that

$$\begin{aligned} \Vert u_0\Vert _Z\le & {} C_1|\Omega |^{-\frac{\beta }{2}}|\Omega |^{\frac{\beta }{2} -\frac{1}{2}(s+3-\frac{3}{p})}\sup _{t\in \mathbb {R}} \Vert f(t)\Vert _{e^{\theta \sqrt{t}\Lambda _1}\dot{B}_{p,r}^{s}}\nonumber \\= & {} C_1|\Omega |^{-\frac{1}{2}(s+3-\frac{3}{p})}\sup _{t\in \mathbb {R}} \Vert f(t)\Vert _{e^{\theta \sqrt{t}\Lambda _1}\dot{B}_{p,r}^{s}} \end{aligned}$$
(5.10)

Moreover, since f is time-periodic with the period \(\omega \), \(u_0\) is also time-periodic with the same period \(\omega \). By an inductive argument and Lemma 4.5, it is easy to examine that \(u_n\) belongs to Z, and is periodic with the same period \(\omega \) for all \(n\in \mathbb {N}\cup \{0\}\). Furthermore, by Lemma 4.5 there exists a positive constant \(C_2\) such that

$$\begin{aligned} \Vert u_{n+1}\Vert _Z\le & {} \Vert u_0\Vert _Z+\Vert N_2(u_n,u_n)\Vert _Z\nonumber \\\le & {} \Vert u_0\Vert _Z+C_2|\Omega |^{-\frac{\beta }{2}}|\Omega |^{-\frac{\beta }{2}} \Big (\sup _{t\in \mathbb {R}}\Vert u_n(t)\Vert _{e^{\theta \sqrt{t}\Lambda _1}\dot{B}_{p,r}^{-1+\frac{3}{p}+\beta }}\Big )^2\nonumber \\\le & {} \Vert u_0\Vert _Z+C_2\Vert u_n\Vert _Z^2. \end{aligned}$$
(5.11)

Hence, if the external force f satisfies

$$\begin{aligned} \sup _{t\in \mathbb {R}}\Vert f(t)\Vert _{e^{\theta \sqrt{t}\Lambda _1}\dot{B}_{p,r}^{s}}<\frac{1}{4C_1C_2}|\Omega |^{\frac{1}{2}(s+3-\frac{3}{p})}. \end{aligned}$$

It follows form (5.10) that \(\Vert u_0\Vert _Z<\frac{1}{4C_2}\), and from (5.11) and an inductive argument that

$$\begin{aligned} \Vert u_n\Vert _Z\le \frac{1-\sqrt{1-4C_2\Vert u_0\Vert _Z}}{2C_2}:=M \end{aligned}$$
(5.12)

for all \(n\in \mathbb {N}\cup \{0\}\). Setting \(v_n:=u_n-u_{n-1}\) for \(n\in \mathbb {N}\cup \{0\}\) with \(u_{-1}:=0\), it follows from Lemma 4.5 and (5.12) that

$$\begin{aligned} \Vert v_{n+1}\Vert _Z\le & {} \Vert N_2(v_n,u_n)\Vert _Z+\Vert N_2(u_{n-1},v_n)\Vert _Z\nonumber \\\le & {} C_2(\Vert v_n\Vert _Z\Vert u_n\Vert _Z+\Vert u_{n-1}\Vert _Z\Vert v_n\Vert _Z)\nonumber \\\le & {} 2C_2M\Vert v_n\Vert _Z\nonumber \\&~~~~~~~~\vdots \nonumber \\\le & {} (2C_2M)^{n+1}\Vert u_0\Vert _Z. \end{aligned}$$
(5.13)

Owing to \(u_n(t)=\sum _{j=0}^n v_j(t)\) and \(0< 2C_2M < 1\), we obtain by (5.13) that \(u_n\) converges:

$$\begin{aligned} u_n\rightarrow u,~~~~\text {in}~~Z\ \ \text {as}\ \ n\rightarrow \infty . \end{aligned}$$
(5.14)

Furthermore, it is easy to see that the limit \(u\in Z\) is also time-periodic with the same period \(\omega \) as f. By Lemma 4.5 and (5.12), we also obtain

$$\begin{aligned} \Vert N_2(u_n,u_n)-N_2(u,u)\Vert _Z\le & {} \Vert N_2(u_n-u,u_n)\Vert _Z+\Vert N_2(u,u_n-u)\Vert _Z\nonumber \\\le & {} C_2(\Vert u_n-u\Vert _Z\Vert u_n\Vert _Z+\Vert u\Vert _Z\Vert u_n-u\Vert _Z)\nonumber \\\le & {} 2C_2M\Vert u_n-u\Vert _Z\rightarrow 0 \end{aligned}$$
(5.15)

as \(n\rightarrow \infty \). Now, taking the limit \(n\rightarrow \infty \) in (5.9), by (5.14) and (5.15), it is easy to see that the limit \(u\in Z\) is a desired time-periodic mild solution to problem (1.2) with \(\Vert u\Vert _Z\le M\).

It remains to show the uniqueness of solutions. Let \(v\in Z\) be another time-periodic mild solution to problem (1.2) satisfying \(\Vert v\Vert _Z\le M\) with M defined in (5.12). Then, by Lemma 4.5 and (5.12), we see

$$\begin{aligned} \Vert u-v\Vert _Z\le & {} \Vert N_2(u-v,u)\Vert _Z+\Vert N_2(v,u-v)\Vert _Z\\\le & {} C_2(\Vert u-v\Vert _Z\Vert u\Vert _Z+\Vert v\Vert _Z\Vert u-v\Vert _Z)\\\le & {} 2C_2M\Vert u-v\Vert _Z, \end{aligned}$$

which yields that \((1-2C_2M)\Vert u-v\Vert _Z\le 0\). Therefore, \(u = v\) in Z since \(0< 2C_2M < 1\). This completes the proof of Theorem 1.4. \(\square \)