Well-Posedness for Stochastic Fractional Navier–Stokes Equation in the Critical Fourier–Besov Space

Abstract

The well-posedness of stochastic Navier–Stokes equations with various noises is a hot topic in the area of stochastic partial differential equations. Recently, the consideration of stochastic Navier–Stokes equations involving fractional Laplacian has received more and more attention. Due to the scaling-invariant property of the fractional stochastic equations concerned, it is natural and also very important to study the well-posedness of stochastic fractional Navier–Stokes equations in the associated critical Fourier–Besov spaces. In this paper, we are concerned with the three-dimensional stochastic fractional Navier–Stokes equation driven by multiplicative noise. We aim to establish the well-posedness of solutions of the concerned equation. To this end, by utilising the Fourier localisation technique, we first establish the local existence and uniqueness of the solutions in the critical Fourier–Besov space $\dot{\mathcal {B}}^{4-2\alpha -\frac{3}{p}}_{p,r}$. Then, under the condition that the initial date is sufficiently small, we show the global existence of the solutions in the probabilistic sense.

1 Introduction

In this paper, we are concerned with the following three-dimensional stochastic incompressible fractional Navier–Stokes equation

$$\begin{aligned} \left\{ \begin{array}{ll} du+[(-\Delta )^{\alpha }u+u\cdot \nabla u+\nabla \pi ]dt =\sum _{k\ge 1}g_k(t,u)dB_k,\quad t>0\\ \mathord {\mathrm{div}}u=0,\\ u(0)=u_0,\\ \end{array}\right. \end{aligned}$$

(1.1)

for unknown random field $u=(u_1,u_2,u_3)\in \mathbb {R}^3$ representing the velocity of a fluid, where $\pi$ stands for the pressure and the fractional Laplace operator $(-\Delta )^{\alpha },~{\alpha \in (0,1]}$ is the Fourier multiplier with symbol $|\xi |^{2\alpha }$, $g_k, k\ge 1$ are jointly measurable coefficients, $\{B_k,k\ge 1\}$ is a sequence of one-dimensional independent Brownian motions defined in a given completed filtered probability space $(\Omega ,\mathcal {F},\mathcal {F}_t,\mathbb {P})$ (see e.g. [9]). When $\alpha =1$, the equation (1.1) becomes the well-known stochastic Navier–Stokes equation (SNS in short). The SNS has been intensively studied due to its feature simulation for fluid flow dynamics. Especially, Holz and Ziane [8] obtained the local well-posedness of the strong solution for the multiplicative SNS in bounded domains when the initial data are in $H^1$. Sritharan and Sundar [13] established Wentzell-Freidlin-type large deviation principle for the two-dimensional SNS with multiplicative Gaussian noise. Caraballo, Langa and Taniguchi [2] proved that the weak solutions for the two-dimensional SNS converge exponentially in the mean square and almost surely exponentially to the stationary solutions. Xu and Zhang [18] discussed the small time asymptotics of two-dimensional SNS in the state space C([0, T], H). Recently, the study of well-posedness for the SNS in Besov spaces has attracted the interest of many scholars. In particular. Du and Zhang [6] obtained local and global existence of strong solutions for the SNS in the critical Besov space $\dot{B}^{\frac{d}{p}-1}_{p,r}$. Chang and Yang [3] studied the initial-boundary value problem of the SNS in the half space.

If $g:=\{g_k,k\ge 1\}\equiv 0$, the system (1.1) reduces to incompressible fractional Navier–Stokes equations (FNS in short). It is scaling invariant under certain changes of spatial and temporal variables. To be more precise, one has the following

$$\begin{aligned} u_\lambda (t,x)=\lambda ^{2\alpha -1}u(\lambda ^{2\alpha }t,\lambda x),~ \pi _{\lambda }(t,x)=\lambda ^{4\alpha -2}\pi (\lambda x, \lambda ^{2\alpha }t). \end{aligned}$$

This scaling invariant property naturally leads to the definition of the critical space for the equation (1.1). Recall that a functional space X endowed with norm $\Vert \cdot \Vert _X$ is critical for the equation (1.1) if it satisfies $\Vert \varphi _\lambda \Vert _X=\Vert \varphi \Vert _X$, where $\varphi _\lambda (x)=\lambda ^{2\alpha -1}\varphi (x)$. To date, the well-posedness for FNS has been studied by many scholars in different critical spaces. For example, Wu [17] studied the well-posedness for FNS in the Besov space $\dot{B}^{1-2\alpha +\frac{3}{p}}_{p,r}$, Wang and Wu [15] proved the global well-posedness of mild solution and Gevrey class regularity for FNS in the Lei Lin space $\chi ^{1-2\alpha }$, Ru and Abidin [12] discussed the global well-posedness for FNS in the variable exponent Fourier–Besov spaces $\dot{\mathcal {B}}^{4-2\alpha -\frac{3}{p(\cdot )}}_{p(\cdot ),r}$, just mention a few. More discussions on the global well-posedness can be found in [4, 10] (and references therein).

Motivated by the above investigations, in this paper, we want to study the well-posedness of the equation (1.1) in the critical Fourier–Besov space $\dot{\mathcal {B}}^{4-2\alpha -\frac{3}{p}}_{p,r}$, aiming to extend the well-posedness results of [12, 16]. To this end, we first derive the local well-posedness for the equation (1.1) in the space $\dot{\mathcal {B}}^{4-2\alpha -\frac{3}{p}}_{p,r}$. Then for sufficiently small initial data, we show that the solution is global in the probabilistic sense.

The rest of the paper is organised as follows. In Section 2, we briefly recall some harmonic analysis tools including the Littlewood-Paley theory and the definition of the Fourier–Besov space. Section 3 is devoted to establishing our main well-posedness results.

2 Preliminaries

In this section, we recall the homogeneous Littlewood-Paley decomposition and the definition of Fourier–Besov spaces. For more details, the reader is referred to [1].

Let $\mathcal {S}(\mathbb {R}^d),~d\ge 1$, be the Schwarz space of all smooth functions that are rapidly decreasing infinite functions along with all partial derivatives and $\mathcal {S}'(\mathbb {R}^d)$ be the space of tempered distributions. Let the dual pairing between $\mathcal {S}(\mathbb {R}^d)$ and $\mathcal {S}'(\mathbb {R}^d)$ be denoted by $\langle \cdot ,\cdot \rangle$. For any $f\in \mathcal {S}(\mathbb {R}^d)$, the Fourier transform and the inverse Fourier transform of f are defined, respectively, by

$$\begin{aligned} \mathcal {F}(f)(\xi ):= & {} \hat{f}(\xi ):=\frac{1}{(2\pi )^{\frac{d}{2}}}\int _{\mathbb {R}^d}e^{ix\cdot \xi }f(x)dx,\\ \mathcal {F}^{-1}(f)(\xi ):= & {} \check{f}(x):=\frac{1}{(2\pi )^{\frac{d}{2}}}\int _{\mathbb {R}^d}e^{-ix\cdot \xi }f(\xi )d\xi . \end{aligned}$$

Furthermore, for any $\phi \in \mathcal {S}'(\mathbb {R}^d)$, we define its Fourier transform and inverse Fourier transform as

$$\begin{aligned} \langle \hat{\phi }, f\rangle :=\langle \phi , \hat{f}\rangle ,~ \langle \check{\varphi }, f\rangle :=\langle \phi , \check{f}\rangle ,~\forall ~f\in \mathcal {S}(\mathbb {R}^d). \end{aligned}$$

Let $\mathcal {C}:=\bigl \{\xi \in \mathbb {R}^d: \frac{3}{4}\le |\xi |\le \frac{8}{3}\bigr \}$ and $\mathcal {D}(\mathcal {C})$ be the space of all test functions on $\mathcal {C}$, that is, the totality of all smooth functions on $\mathcal {C}$ that have compact support. Then there exists a non-negative radial function $\varphi \in \mathcal {D}(\mathcal {C})$, such that

$$\begin{aligned} \sum _{j\in \mathbb {Z}}\varphi _j(\xi )=1,~\forall ~\xi \in \mathbb {R}^d\backslash \{0\}, \end{aligned}$$

where $\varphi _j(\xi )=\varphi (2^{-j}\xi )$. The homogeneous dyadic blocks is defined in the following manner

$$\begin{aligned} \dot{\Delta }_ju:=\check{\varphi }_j*u,~j\in \mathbb {Z}. \end{aligned}$$

We further set the quotient space $\mathcal {S}'_h(\mathbb {R}^d):=\mathcal {S}'(\mathbb {R}^d)/ \mathcal {P}$, where $\mathcal {P}$ is the space of all polynomials. For any $u\in \mathcal {S}'_h(\mathbb {R}^d)$, we define

$$\begin{aligned} \dot{S}_ju:=\sum _{j'\le j-1}\dot{\Delta }_{j'}u. \end{aligned}$$

We have

Definition 1

Let $1\le p,r\le \infty , s\in \mathbb {R}$, the homogeneous Fourier–Besov space is defined as

$$\begin{aligned} \dot{\mathcal {B}}^{s}_{p,r}=\Bigl \{u\in \mathcal {S}'_h(\mathbb {R}^d): \Vert u\Vert _{{\dot{\mathcal {B}}}^{s}_{p,r}} :=\bigl \{2^{js}\Vert \widehat{\dot{\Delta }_ju}\Vert _{L^p_\xi }\bigr \}_{l_j^r}<\infty \Bigr \}. \end{aligned}$$

Taking the time variable and random variables into account, we also need the following definition of Chemin–Lerner-type spaces, see, [1, 16].

Definition 2

Let $1\le p,q,r,\sigma \le \infty$, $s\in \mathbb {R},~T>0$, we define

$$\begin{aligned} \mathcal {L}^q_T\dot{\mathcal {B}}^s_{p,r}:= & {} \Bigl \{u:~\forall ~ a.e.~ t\in [0,T], ~u(t,\cdot )\in \mathcal {S}'_h(\mathbb {R}^d),\\&\quad \Vert u\Vert _{ \mathcal {L}^q_T\dot{\mathcal {B}}^s_{p,r} }:= \bigl \{2^{js}\Vert \widehat{\dot{\Delta }_ju}\Vert _{L^q_TL^p_\xi }\bigr \}_{l_j^r}<\infty \Bigr \};\\ \mathcal {L}^q_T\mathcal {L}^\sigma _\omega \dot{\mathcal {B}}^s_{p,r}:= & {} \Bigl \{u:~\forall ~ a.e.~(\omega ,t) \in \Omega \times [0,T],~u(\omega ,t,\cdot )\in \mathcal {S}'_h(\mathbb {R}^d),\\&\quad \Vert u\Vert _{ \mathcal {L}^q_T\mathcal {L}^\sigma _\omega \dot{\mathcal {B}}^s_{p,r} }:= \bigl \{2^{js}\Vert \widehat{\dot{\Delta }_ju}\Vert _{L^q_TL^\sigma _\omega L^p_\xi }\bigr \}_{l_j^r}<\infty \Bigr \};\\ \mathcal {L}^q_T\mathcal {L}^\sigma _\omega \dot{\mathbf {B}}^s_{p,r}:= & {} \Bigl \{f=(f_k,k\ge 1):~\forall ~ k\ge 1,~f_k\in \mathcal {L}^q_T\mathcal {L}^\sigma _\omega \dot{\mathcal {B}}^s_{p,r},\\&\quad \Vert f\Vert _{\mathcal {L}^q_T\mathcal {L}^\sigma _\omega \dot{\mathbf {B}}^s_{p,r}}:= \bigl \{2^{js}\Vert \widehat{\dot{\Delta }_jf_k}\Vert _{L^q_TL^\sigma _\omega l^2_k L^p_\xi }\bigr \}_{l_j^r}<\infty \Bigr \}. \end{aligned}$$

We list here some of the properties that will be used in the sequel. Let $u,v \in \mathcal {S}'_h(\mathbb {R}^d)$, then

• $|k-j|\ge 2\Rightarrow \dot{\Delta }_k\dot{\Delta }_ju=0$;

• $|k-j|\ge 5\Rightarrow \dot{\Delta }_j(\dot{S}_{k-1}u\dot{\Delta }_kv)=0$.

• For $1\!\le \! p,r\!\le \! \infty ,~s\!<\!0$, it holds that $u\!\in \! \dot{\mathcal {B}}^{s}_{p,r}$ if and only if $\{2^{js}\Vert \widehat{\dot{S}_ju}\Vert _{L^\xi _p}\}_{l^r_j}\!<\!+\!\infty$.

Finally, we introduce the homogeneous Bony decomposition. For more details, we refer readers to [1, 11] and the references therein. Let $u,v \in \mathcal {S}'_h(\mathbb {R}^d)$, then the homogeneous Bony decomposition of uv is defined by

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

where $\dot{T}_uv$ and $\dot{T}_vu$ denote the homogenous paraproduct of v by u and u by v, respectively

$$\begin{aligned}&\dot{T}_uv:=\sum _{j\in \mathbb {Z}}\dot{S}_{j-1}u\dot{\Delta }_jv,\\&\dot{T}_vu:=\sum _{j\in \mathbb {Z}}\dot{S}_{j-1}v\dot{\Delta }_ju, \end{aligned}$$

and $\dot{R}(u,v)$ denotes the homogeneous remainder of u and v:

$$\begin{aligned} \dot{R}(u,v):= & {} \sum _{|j-k|\le 1}\dot{\Delta }_ju\dot{\Delta }_kv\\= & {} \sum _{j\in \mathbb {Z}}\dot{\Delta }_ju\widetilde{\dot{\Delta }}_jv \end{aligned}$$

with $\widetilde{\dot{\Delta }}_k:={\dot{\Delta }}_{k-1}+{\dot{\Delta }}_{k}+{\dot{\Delta }}_{k+1}$.

3 Main Results

Acting the Leray-Hopf operator $\mathbf {P}:=I+\nabla (-\Delta )^{-1}div$ (see [1, 11]) on equation (1.1), we then have

$$\begin{aligned} \left\{ \begin{array}{ll} du+[(-\Delta )^\alpha u+\mathbf {P}div (u\otimes u)]dt =\sum _{k\ge 1}\mathbf {P}g_k(t,u)dB_k,\\ u(0)=u_0,\\ \end{array}\right. \end{aligned}$$

(3.1)

where ${\alpha \in (0,1]}$. Following [6], we introduce the definition of local and global strong solutions for the equation (1.1).

Definition 3

Let $2\le p,r\le \infty$ and the initial data $u_0\in \dot{\mathcal {B}}^{4-2\alpha -\frac{3}{p}}_{p,r}$ be $\mathcal {F}_0$-measurable.

(1) $(u,\tau _R)$ is called a local strong solution for the equation (1.1), if

(i) u is a progressively measurable process and for any $0<T<\infty$,

$$\begin{aligned}&u\in L^r_\omega \mathcal {L}^{\infty }_T\dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}}\cap L^r_\omega \mathcal {L}^2_T\dot{\mathcal {B}}_{p,r}^{4-\alpha -\frac{3}{p}}, \\&\tau _R(\omega )=\inf \Bigl \{t\ge 0; \Vert u\Vert _{\mathcal {L}^{2}_{T} \dot{\mathcal {B}}_{p,q}^{4-\alpha -\frac{3}{p}}}\ge R\Bigr \}, \end{aligned}$$

where R is a positive constant.

(ii) For almost all $\omega \in \Omega$, $u(t,x)\in C([0,\tau _R(\omega ));\dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}})$, and the following equality

$$\begin{aligned} u(t\wedge \tau _R)=u_0-\int ^{t\wedge \tau _R}_0[(-\Delta )^\alpha u+\mathbf {P}div (u\otimes u)]ds +\sum _{k\ge 1}\int ^{t\wedge \tau _R}_0\mathbf {P}g_k(s,u)dB_k \end{aligned}$$

holds $\mathbb {P}$-a.s. in $\mathcal {S}'(\mathbb {R}^d)$.

(2) We say that the local strong solution is unique, if $(\tilde{u}, \tilde{\tau }_R)$ is another strong solution, then

$$\begin{aligned} \mathbb {P}(\{\omega \in \Omega :~u=\tilde{u},~\forall ~0\le t\le \tau _R\wedge \tilde{\tau }_R\})=1. \end{aligned}$$

Definition 4

Let $2\le p,r\le \infty$ and $u_0\in \dot{\mathcal {B}}^{4-2\alpha -\frac{3}{p}}_{p,r}$ be $\mathcal {F}_0$-measurable. We say that u is a global strong solution for the equation (1.1), if the following two conditions are fulfilled

(1) u is a progressively measurable process and for any $0<T<\infty$, we have

$$\begin{aligned} u\in L^r_\omega \mathcal {L}^{\infty }_T\dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}}\cap L^r_\omega \mathcal {L}^2_T\dot{\mathcal {B}}_{p,r}^{4-\alpha -\frac{3}{p}}. \end{aligned}$$

(2) For almost all $\omega \in \Omega$, $u(t,x)\in C([0,\infty );\dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}})$, and for all $0<t<\infty$, the following equality

$$\begin{aligned} u(t)=u_0-\int ^{t}_0[(-\Delta )^\alpha u+\mathbf {P}div (u\otimes u)]ds +\sum _{k\ge 1}\int ^{t}_0\mathbf {P}g_k(s,u)dB_k \end{aligned}$$

holds $\mathbb {P}$-a.s. in $\mathcal {S}'(\mathbb {R}^d)$.

We are now in the position to state our two main results of this paper.

Theorem 1

Let $2\le p,r\le \infty$ and $u_0\in \dot{\mathcal {B}}^{4-2\alpha -\frac{3}{p}}_{p,r}$ be $\mathcal {F}_0$-measurable. Assume that

$$\begin{aligned}&g(t,u): [0,T]\times (\mathcal {L}^2_T \mathcal {L}^r_\omega \dot{\mathcal {B}}^{4-2\alpha -\frac{3}{p}}_{p,r}\cap L^r_\omega \mathcal {L}^2_T\dot{\mathcal {B}}^{4-\alpha -\frac{3}{p}}_{p,r})\\&\quad \rightarrow \mathcal {L}^2_T\mathcal {L}^r_\omega \dot{\mathbf {B}}^{4-2\alpha -\frac{3}{p}}_{p,r} \end{aligned}$$

fulfils

$$\begin{aligned} \Vert \mathbf {P}g(t,u)\Vert _{\mathcal {L}^2_T\mathcal {L}^r_\omega \dot{\mathbf {B}}^{4-2\alpha -\frac{3}{p}}_{p,r}}\le & {} L_1\Vert u\Vert _{\mathcal {L}^2_T\mathcal {L}^r_\omega \dot{\mathcal {B}}^{4-2\alpha -\frac{3}{p}}_{p,r}}\nonumber \\&+ L_2\Vert u\Vert _{L^r_\omega \mathcal {L}^2_T\dot{\mathcal {B}}^{4-\alpha -\frac{3}{p}}_{p,r}}, \end{aligned}$$

(3.2)

$$\begin{aligned} \Vert \mathbf {P}g(t,u)-\mathbf {P}g(t,v)\Vert _{\mathcal {L}^2_T\mathcal {L}^r_\omega \dot{\mathbf {B}}^{4-2\alpha -\frac{3}{p}}_{p,r}}\le & {} L_1\Vert u-v\Vert _{\mathcal {L}^2_T\mathcal {L}^r_\omega \dot{\mathcal {B}}^{4-2\alpha -\frac{3}{p}}_{p,r}}\nonumber \\&+ L_2\Vert u-v\Vert _{L^r_\omega \mathcal {L}^2_T\dot{\mathcal {B}}^{4-\alpha -\frac{3}{p}}_{p,r}}, \end{aligned}$$

(3.3)

where $L_1,L_2>0$ and $L_2$ is small enough. Then, there is a constant $R>0$ such that there exists a unique local solution $(u,\tau _R)$ to Eq. (1.1) with

$$\begin{aligned} \mathbb {P}(\tau _R>0)=1. \end{aligned}$$

Theorem 2

With the same preamble as in Theorem 1. If further for sufficiently small $L_3>0$,

$$\begin{aligned} \Vert \mathbf {P}g(t,u)\Vert _{\mathcal {L}^2_T\mathcal {L}^r_\omega \dot{\mathbf {B}}^{4-2\alpha -\frac{3}{p}}_{p,r}}\le & {} L_3\Vert u\Vert _{{L}^r_\omega \mathcal {L}^\infty _T\dot{\mathcal {B}}^{4-2\alpha -\frac{3}{p}}_{p,r}}\nonumber \\&+ L_2\Vert u\Vert _{L^r_\omega \mathcal {L}^2_T\dot{\mathcal {B}}^{4-\alpha -\frac{3}{p}}_{p,r}}, \end{aligned}$$

(3.4)

and for any $\varepsilon >0$, there exists a constant $\gamma =\gamma (\varepsilon )>0$ such that

$$\begin{aligned} \Vert u_0\Vert _{L_\omega ^r\dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}}} <\gamma . \end{aligned}$$

(3.5)

Then,

$$\begin{aligned} \mathbb {P}(\tau _R=\infty )\ge 1-\varepsilon . \end{aligned}$$

Example 1

Here, we give an example to show that the noise coefficient in the above theorems is non-empty. Motivated by [6], let $M>0$ be arbitrary, we take $g_k(t,u)=\frac{1}{\sqrt{2}k}e^{-M(1+t)}u,~k\ge 1$. Then

$$\begin{aligned} \Vert \mathbf {P}g(t,u)-\mathbf {P}g(t,v)\Vert _{\mathcal {L}^2_T\mathcal {L}^r_\omega \dot{\mathbf {B}}^{4-2\alpha -\frac{3}{p}}_{p,r}} \le e^{-M}\Vert u-v\Vert _{\mathcal {L}^2_T\mathcal {L}^r_\omega \dot{\mathbf {B}}^{4-2\alpha -\frac{3}{p}}_{p,r}}, \end{aligned}$$

Thus, conditions (3.2) and (3.3) are satisfied. Moreover, It is easy to verify that condition (3.4) also holds.

In order to prove the main results, we need the following lemmas.

Lemma 1

([14]) Assume $0<T\le \infty ,~\frac{1}{2}<\alpha \le 1, 1\le p,q,\rho \le \infty$, $s\in \mathbb {R}$. Let u be a solution of

$$\begin{aligned} \left\{ \begin{array}{ll} \partial _tu+(-\Delta )^{\alpha }u=f(t,x),~~(t,x)\in [0,T)\times \mathbb {R}^3,\\ u(0)=u_0,\\ \end{array}\right. \end{aligned}$$

where $f\in \mathcal {L}^\rho _{T}\dot{\mathcal {B}}^{s+\frac{2\alpha }{\rho }-2\alpha }_{p,q}$. Then, $u\in \mathcal {L}^\infty _{T}\dot{\mathcal {B}}^s_{p,q}\cap \mathcal {L}^\rho _{T} \dot{\mathcal {B}}^{s+\frac{2\alpha }{\rho }-2\alpha }_{p,q}$. Moreover, for any $\rho \le \rho _1\le \infty$, the following inequality

$$\begin{aligned} \Vert u\Vert _{\mathcal {L}^{\rho _1}_{T}\dot{\mathcal {B}}^{s+\frac{2\alpha }{\rho _1}}_{p,q}} \le C \Bigl (\Vert u_0\Vert _{\dot{\mathcal {B}}^s_{p,q}}+\Vert f\Vert _{\mathcal {L}^\rho _{T} \dot{\mathcal {B}}^{s+\frac{2\alpha }{\rho }-2\alpha }_{p,q}}\Bigr ) \end{aligned}$$

holds for some positive constant C.

Lemma 2

Let $0<T\le \infty$, $\frac{1}{2}<\alpha \le 1$, $2\le p<\infty$, $2\le q\le r<\infty$ and $s\in \mathbb {R}$. Then, the following stochastic fractional heat equation

$$\begin{aligned} \left\{ \begin{array}{ll} du+(-\Delta )^{\alpha }udt=\sum _{k\ge 1}g_kdW_k(t),\\ u(0)=0,\\ \end{array}\right. \end{aligned}$$

(3.6)

has a unique solution $u\in L^r(\Omega ;\mathcal {L}^{q}_T\dot{\mathcal {B}}^{s}_{p,r} \cap C_T\dot{\mathcal {B}}^{s-\frac{2\alpha }{q}}_{p,r})$, where $g=\{ g_k,k\ge 1\}\in \mathcal {L}^q_T\mathcal {L}^r_\omega {\dot{\mathbf {B}}}^{s-\alpha }_{p,r}$ is progressively measurable. Moreover, there exists a constant $C>0$, such that for any $q\le q_1\le \infty$,

$$\begin{aligned} \Vert u\Vert _{L^r_\omega \mathcal {L}^{q_1}_T\dot{\mathcal {B}}^{s+\frac{2\alpha }{q_1}-\frac{2\alpha }{q}}_{p,r}} \le C\Vert g\Vert _{\mathcal {L}^q_T\mathcal {L}^r_\omega {\dot{\mathbf {B}}}^{s-\alpha }_{p,r}}. \end{aligned}$$

(3.7)

Proof

Taking the Fourier transform on both sides of (3.6), we conclude that the unique solution u satisfies

$$\begin{aligned} \hat{u}(t,\xi )=\sum _{k\ge 1}\int ^t_0e^{-(t-t')|\xi |^{2\alpha }}\hat{g}_k(t',\xi )dW_k(t'). \end{aligned}$$

Therefore, by using Minkowski’s inequality, Young’s inequality and [6, Lemma 2.5], we get

$$\begin{aligned} \begin{aligned}&\Vert \varphi _j(\xi ) \hat{u}(t,\xi )\Vert _{L^r_\omega L_T^qL^p_\xi } \le \Vert \varphi _j(\xi ) \hat{u}(t,\xi )\Vert _{L_T^qL^r_\omega L^p_\xi }\\&\quad =\Bigl \Vert \sum _{k\ge 1}\int ^t_0e^{-(t-t')|\xi |^{2\alpha }}\varphi _j(\xi )\hat{g}_k(t',\xi )dW_k(t') \Bigr \Vert _{L_T^qL^r_\omega L^p_\xi }\\&\quad \le C\Bigl \Vert e^{-(t-t')|\xi |^{2\alpha }}\varphi _j(\xi )\hat{g}_k(t',\xi )\Bigr \Vert _{L^q_T L^r_\omega L^2_tl_k^2L^p_\xi }\\&\quad \le C\Bigl \Vert e^{-c(t-t')2^{2\alpha j}}\varphi _j(\xi )\hat{g}_k(t',\xi )\Bigr \Vert _{L^q_T L^2_tL^r_\omega l_k^2L^p_\xi }\\&\quad \le C\Bigl \Vert \int ^t_0e^{-2c(t-t')2^{2\alpha j}}\Vert \varphi _j(\xi )\hat{g}_k(t',\xi )\Vert ^2_{L^r_\omega l^2_kL^p_\xi }dt'\Bigr \Vert ^{\frac{1}{2}}_{L^{q/2}_T}\\&\quad \le C\bigl \Vert e^{-2c2^{2\alpha j}t}\bigr \Vert ^{1/2}_{L^1_T}\bigl \Vert \varphi _j(\xi )\hat{g}_k(t',\xi )\bigr \Vert _{L^q_TL^r_\omega l^2_kL^p_\xi }\\&\quad \le C2^{-\alpha j}\bigl \Vert \varphi _j(\xi )\hat{g}_k(t',\xi )\bigr \Vert _{L^q_TL^r_\omega l^2_kL^p_\xi }. \end{aligned} \end{aligned}$$

Consequently,

$$\begin{aligned} \begin{aligned} \Vert u\Vert _{L^r_\omega \mathcal {L}^q_T\dot{\mathcal {B}}^{s+\frac{2\alpha }{q}}_{p,r}}&= \Bigl \Vert 2^{j(s+\frac{2\alpha }{q})}\Vert \varphi _j(\xi )\hat{u}(t,\xi )\Vert _{L^r_\omega L^q_TL^p_\xi }\Bigr \Vert _{l^r_j}\\&\le C\Bigl \Vert 2^{j(s+\frac{2\alpha }{q}-\alpha )}\bigl \Vert \varphi _j(\xi )\hat{g}_k(t',\xi ) \bigr \Vert _{L^q_TL^r_\omega l^2_kL^p_\xi }\Bigr \Vert _{l^r_j}\\&=C\Vert g\Vert _{\mathcal {L}^q_T\mathcal {L}^r_\omega {\dot{\mathbf {B}}}^{s+\frac{2\alpha }{q}-\alpha }_{p,r}}. \end{aligned} \end{aligned}$$

(3.8)

Let $p'$ be the conjugate number of p, then for any $\beta \in (0,\frac{1}{2})$, we have

$$\begin{aligned} \begin{aligned} \Vert \varphi _j\hat{u}\Vert _{{L}^r_\omega {L}^\infty _T{L}^p_\xi }&\le C\Bigl \Vert \int ^t_0e^{-(t-t')|\xi |^{2\alpha }}(t-s)^{\beta -1}\times \\&\quad \Bigl (\sum _{k\ge 1}\int ^s_0e^{-(s-t')|\xi |^{2\alpha }}(s-t')^{-\beta }\varphi _j\hat{g}_k dW_k\Bigr )ds\Bigr \Vert _{{L}^r_\omega {L}^\infty _T{L}^p_\xi }\\&\le C\Bigl \Vert \int ^t_0e^{-c(t-t')2^{2\alpha j}}(t-s)^{\beta -1}\times \\&\quad \Bigl \Vert \sum _{k\ge 1}\int ^s_0e^{-(s-t')|\xi |^{2\alpha }}(s-t')^{-\beta }\varphi _j\hat{g}_kdW_k \Bigr \Vert _{{L}^p_\xi }ds\Bigr \Vert _{{L}^r_\omega {L}^\infty _T}\\&\le C\bigl \Vert e^{-ct2^{2\alpha j}}t^{\beta -1}\bigr \Vert _{L^{q'}_T}\Bigl \Vert \sum _{k\ge 1}\int ^s_0 e^{-(s-t')|\xi |^{2\alpha }}(s-t')^{-\beta }\varphi _j\hat{g}_kdW_k\Bigr \Vert _{{L}^r_\omega {L}^q_T{L}^p_\xi }, \end{aligned} \end{aligned}$$

where we have used the factorisation formula (see e.g. [5])

$$\begin{aligned} \begin{aligned}&\sum _{k\ge 1}\int ^t_0e^{-(t-t')|\xi |^{2\alpha }}\varphi _j(\xi )\hat{g}_k(t',\xi )dW_k(t')\\&\quad =\frac{\sin \beta \pi }{\pi }\int ^t_0e^{-(t-t')|\xi |^{2\alpha }}(t-s)^{\beta -1}\\&\qquad \Bigl (\sum _{k\ge 1}\int ^s_0e^{-(s-t')|\xi |^{2\alpha }}(s-t')^{-\beta }\varphi _j\hat{g}_kdW_k\Bigr )ds. \end{aligned} \end{aligned}$$

(3.9)

Basic calculus then implies that

$$\begin{aligned} \bigl \Vert e^{-ct2^{2\alpha j}}t^{\beta -1}\bigr \Vert _{L^{q'}_T}\le C2^{2j\alpha (\frac{1}{q}-\beta )}. \end{aligned}$$

Substituting this estimate into (3.8), we obtain

$$\begin{aligned} \Vert \varphi _j\hat{u}\Vert _{{L}^r_\omega {L}^\infty _T{L}^p_\xi }\le & {} C2^{2j\alpha (\frac{1}{q}-\beta )}\\&\Bigl \Vert \sum _{k\ge 1}\int ^s_0e^{-(s-t') |\xi |^{2\alpha }}(s-t')^{-\beta }\varphi _j\hat{g}_kdW_k\Bigr \Vert _{L^q_TL^r_\omega L^p_\xi }\\\le & {} C2^{2j\alpha (\frac{1}{q}-\beta )}\Bigl \Vert e^{-(s-t')|\xi |^{2\alpha }}(s-t')^{-\beta }\varphi _j\hat{g}_k \Bigr \Vert _{L^q_TL^r_\omega L^2_sl_k^2 L^p_\xi }\\\le & {} C2^{2j\alpha (\frac{1}{q}-\beta )}\Bigl \Vert e^{-(s-t')|\xi |^{2\alpha }}(s-t')^{-\beta }\varphi _j\hat{g}_k \Bigr \Vert _{L^q_TL^2_sL^r_\omega l_k^2 L^p_\xi }\\\le & {} C2^{2j\alpha (\frac{1}{q}-\beta )}\Bigl \Vert \int ^s_0e^{-2c2^{2j\alpha }(s-t')}(s-t')^{-2\beta } \Vert \varphi _j\hat{g}_k\Vert _{L^r_\omega l_k^2 L^p_\xi }dt'\Bigr \Vert ^{\frac{1}{2}}_{L^{\frac{q}{2}}_T}\\\le & {} C2^{2j\alpha (\frac{1}{q}-\beta )}\Vert e^{-c2^{2j\alpha }t}t^{-2\beta }\Vert ^{\frac{1}{2}}_{L^1_T} \Vert \varphi _j\hat{g}_k\Vert _{L^q_TL^r_\omega l^2_kL^p_\xi }\\\le & {} C2^{j\alpha (\frac{2}{q}-1)}\Vert \varphi _j\hat{g}_k\Vert _{L^q_TL^r_\omega l^2_kL^p_\xi }. \end{aligned}$$

Therefore,

$$\begin{aligned} \begin{aligned} \Vert u\Vert _{L^r_\omega \mathcal {L}^\infty _T\dot{\mathcal {B}}^s_{p,r}}&= \Bigl \Vert 2^{js}\Vert \varphi _j\hat{u}\Vert _{{L}^r_\omega {L}^\infty _T{L}^p_\xi }\Bigr \Vert _{l_r}\\&\le C\Bigl \Vert 2^{j(s+\frac{2\alpha }{q}-\alpha )}\Vert \varphi _j\hat{g}_k\Vert _{L^q_T L^r_\omega l^2_kL^p_\xi }\Bigr \Vert _{l^r_j}\\&\le C\Vert g\Vert _{\mathcal {L}^q_T\mathcal {L}^r_\omega {\dot{\mathbf {B}}}^{s+\frac{2\alpha }{q}-\alpha }_{p,r}}. \end{aligned} \end{aligned}$$

(3.10)

Using the interpolation inequality for (3.8) and (3.10), we obtain (3.7), Finally, $u\in L^r(\Omega ;C_T\dot{\mathcal {B}}^{s-\frac{2\alpha }{q}}_{p,r})$ holds by utilising factorisation formula (3.9) again (see e.g. [5]). We thus complete the proof. $\square$

Lemma 3

Let $1< p, \rho ,q\le \infty$, $\frac{1}{2}<\alpha \le 1$ with $\frac{2}{\rho }+1-2\alpha <0$ and $5-4\alpha -\frac{3}{p}+\frac{4\alpha }{\rho }>0$. Then, there exists a constant $C>0$ such that

$$\begin{aligned} \Vert uv\Vert _{\mathcal {L}^{\frac{\rho }{2}}_{T}\dot{\mathcal {B}}_{p,q}^{5-4\alpha -\frac{3}{p}+\frac{4\alpha }{\rho }}}\le & {} C\Vert u\Vert _{\mathcal {L}^{\rho }_{T}\dot{\mathcal {B}}_{p,q}^{4-2\alpha -\frac{3}{p}+\frac{2\alpha }{\rho }}}\\&\Vert v\Vert _{\mathcal {L}^{\rho }_{T}\dot{\mathcal {B}}_{p,q}^{4-2\alpha -\frac{3}{p}+\frac{2\alpha }{\rho }}}. \end{aligned}$$

Proof

According to Bony’s decomposition, we have

$$\begin{aligned} \begin{aligned} \dot{\Delta }_j(uv)&=\dot{\Delta }_j\dot{T}_uv+\dot{\Delta }_j\dot{T}_vu+\dot{\Delta }_jR(u,v)\\&=\sum _{|k-j|\le 4}\dot{\Delta }_j(\dot{S}_{k-1}u\dot{\Delta }_kv)+\sum _{|k-j|\le 4} \dot{\Delta }_j(\dot{S}_{k-1}v\dot{\Delta }_ku)\\&\quad +\sum _{k-j\ge 3}\dot{\Delta }_j\Bigl (\sum _{k}\widetilde{\dot{\Delta }}_ku\dot{\Delta }_kv\Bigr )\\&=:I _1+I _2+I _3. \end{aligned} \end{aligned}$$

For the term $I _1$, by Young’s inequality and $l^q\hookrightarrow l^\infty$, we get

$$\begin{aligned} \Vert \widehat{I }_1\Vert _{L^{\frac{\rho }{2}}_TL^p_\xi }\le & {} \sum _{|k-j|\le 4}\Vert \varphi _j (\widehat{\dot{S}_{k-1}u\dot{\Delta }_kv})\Vert _{L^\frac{\rho }{2}_TL^p_\xi }\\\le & {} \sum _{|k-j|\le 4}\Vert \widehat{\dot{S}_{k-1}u}*\widehat{\dot{\Delta }_kv}\Vert _{L^\frac{\rho }{2}_TL^p_\xi }\\\le & {} \sum _{|k-j|\le 4}\Vert \varphi _k\hat{v}\Vert _{L^\rho _TL^p_\xi } \Vert \widehat{\dot{S}_{k-1}u}\Vert _{L^\rho _TL^1_\xi }\\\le & {} \sum _{|k-j|\le 4}2^{-k(1-2\alpha +\frac{2\alpha }{\rho })}\Bigl \Vert 2^{k(1-2\alpha +\frac{2\alpha }{\rho })} \Vert \widehat{\dot{S}_{k-1}u}\Vert _{L^\rho _TL^1_\xi }\Bigr \Vert _{l^\infty }\Vert \varphi _k\hat{v}\Vert _{L^\rho _TL^p_\xi }\\\le & {} \sum _{|k-j|\le 4}2^{-k(1-2\alpha +\frac{2\alpha }{\rho })}\Vert \varphi _k\hat{v}\Vert _{L^\rho _TL^p_\xi } \Vert u\Vert _{\mathcal {L}^\rho _T\dot{\mathcal {B}}^{1-2\alpha +\frac{2\alpha }{\rho }}_{1,q}}. \end{aligned}$$

Therefore,

$$\begin{aligned} \begin{aligned} \Bigl \Vert&2^{j(5-4\alpha -\frac{3}{p}+\frac{4\alpha }{\rho })} \Vert I _1\Vert _{\mathcal {L}_T^{\frac{\rho }{2}}L^p_\xi }\Bigr \Vert _{l^q_j}\\&\le \Bigl \Vert \sum _{|k-j|\le 4}2^{(j-k)(5-4\alpha -\frac{3}{p}+\frac{4\alpha }{\rho })} 2^{k(4-2\alpha -\frac{3}{p}+\frac{2\alpha }{\rho })}\Vert \varphi _k\hat{v}\Vert _{L^\rho _TL^p_\xi }\Bigr \Vert _{l^q_j} \Vert u\Vert _{\mathcal {L}^\rho _T\dot{\mathcal {B}}^{1-2\alpha +\frac{2}{\rho }}_{1,q}}\\&\le \sum _{|j|\le 4}2^{j(5-4\alpha -\frac{3}{p}+\frac{4\alpha }{\rho })} \Bigl \Vert 2^{j(4-2\alpha -\frac{3}{p}+\frac{2\alpha }{\rho })}\Vert \varphi _j\hat{v}\Vert _{L^\rho _TL^p_\xi }\Bigr \Vert _{l^q_j} \Vert u\Vert _{\mathcal {L}^\rho _T\dot{\mathcal {B}}^{1-2\alpha +\frac{2\alpha }{\rho }}_{1,q}}\\&\le C\Vert v\Vert _{\mathcal {L}^\rho _T\dot{\mathcal {B}}^{4-2\alpha -\frac{3}{p}+\frac{2\alpha }{\rho }}_{p,q}} \Vert u\Vert _{\mathcal {L}^\rho _T\dot{\mathcal {B}}^{1-2\alpha +\frac{2\alpha }{\rho }}_{1,q}}\\&\le C\Vert v\Vert _{\mathcal {L}^\rho _T\dot{\mathcal {B}}^{4-2\alpha -\frac{3}{p}+\frac{2\alpha }{\rho }}_{p,q}} \Vert u\Vert _{\mathcal {L}^\rho _T\dot{\mathcal {B}}^{4-2\alpha -\frac{3}{p}+\frac{2\alpha }{\rho }}_{p,q}}, \end{aligned} \end{aligned}$$

where we have used the fact that $\dot{\mathcal {B}}^{s_1}_{p_1,q}\hookrightarrow \dot{\mathcal {B}}^{s_2}_{p_2,q},~p_2\le p_1, s_1+\frac{d}{p_1}=s_2+\frac{d}{p_2}$. Similarly,

$$\begin{aligned} \begin{aligned} \Bigl \Vert 2^{j(5-4\alpha -\frac{3}{p}+\frac{4\alpha }{\rho })} \Vert I _2\Vert _{\mathcal {L}_T^{\frac{\rho }{2}}L^p_\xi }\Bigr \Vert _{l^q_j} \le C\Vert v\Vert _{\mathcal {L}^\rho _T\dot{\mathcal {B}}^{4-2\alpha -\frac{3}{p}+\frac{2\alpha }{\rho }}_{p,q}} \Vert u\Vert _{\mathcal {L}^\rho _T\dot{\mathcal {B}}^{4-2\alpha -\frac{3}{p}+\frac{2\alpha }{\rho }}_{p,q}}. \end{aligned} \end{aligned}$$

For the term $I _3$, utilising Bernstein inequality, we get

$$\begin{aligned}&2^{j(5-4\alpha -\frac{3}{p}+\frac{4\alpha }{\rho })}\Vert \widehat{I }_3\Vert _{L^{\frac{\rho }{2}}_TL^p_\xi }\\&\quad \le 2^{j(5-4\alpha -\frac{3}{p}+\frac{4\alpha }{\rho })}\sum _{k-j\ge 3}\Bigl \Vert \varphi _j \widehat{\tilde{\dot{\Delta }}_ku\dot{\Delta }_kv}\Bigr \Vert _{L^{\frac{\rho }{2}}_TL^p_\xi }\\&\quad \le 2^{j(5-4\alpha -\frac{3}{p}+\frac{4\alpha }{\rho })}\sum _{k-j\ge 3}\Bigl \Vert \widehat{\tilde{\dot{\Delta }}_ku} *\widehat{\dot{\Delta }_kv}\Bigr \Vert _{L^{\frac{\rho }{2}}_TL^p_\xi }\\&\quad \le 2^{j(5-4\alpha -\frac{3}{p}+\frac{4\alpha }{\rho })} \sum _{k-j\ge 3}\Bigl \Vert \widehat{\tilde{\dot{\Delta }}_ku} \Bigr \Vert _{L^{\rho }_TL^p_\xi }\Bigl \Vert \widehat{\dot{\Delta }_kv}\Bigr \Vert _{L^{\rho }_TL^1_\xi }\\&\quad \le 2^{j(5-4\alpha -\frac{3}{p}+\frac{4\alpha }{\rho })}\sum _{k-j\ge 3}2^{k(3-\frac{3}{p})} \Bigl \Vert \widehat{\tilde{\dot{\Delta }}_ku}\Bigr \Vert _{L^{\rho }_TL^p_\xi } \Bigl \Vert \widehat{\dot{\Delta }_kv}\Bigr \Vert _{L^{\rho }_TL^p_\xi }\\&\quad \le \Vert v\Vert _{\mathcal {L}^\rho _{T}\dot{\mathcal {B}}^{4-2\alpha -\frac{3}{p}+\frac{2\alpha }{\rho }}_{p,\infty }} \sum _{k-j\ge 3}2^{(j-k)(5-4\alpha -\frac{3}{p}+\frac{4\alpha }{\rho })}2^{k(4-2\alpha -\frac{3}{p}+\frac{2\alpha }{\rho })} \Bigl \Vert \widehat{\tilde{\dot{\Delta }}_ku}\Bigr \Vert _{L^{\rho }_TL^p_\xi }. \end{aligned}$$

Thus,

$$\begin{aligned} \begin{aligned}&\Bigl \Vert 2^{j(5-4\alpha -\frac{3}{p}+\frac{4\alpha }{\rho })}\Vert \widehat{I }_3\Vert _{L^{\frac{\rho }{2}}_TL^p_\xi }\Bigr \Vert _{l^q_j}\\&\quad \le \Vert v\Vert _{\mathcal {L}^\rho _{T}\dot{\mathcal {B}}^{4-2\alpha -\frac{3}{p}+ \frac{2\alpha }{\rho }}_{p,\infty }}\sum _{l\le -3}2^{l(5-4\alpha -\frac{3}{p}+\frac{4\alpha }{\rho })} \Vert u\Vert _{\mathcal {L}^\rho _T\dot{\mathcal {B}}^{4-2\alpha -\frac{3}{p}+\frac{2\alpha }{\rho }}_{p,q}}\\&\quad \le C\Vert v\Vert _{\mathcal {L}^\rho _{T}\dot{\mathcal {B}}^{4-2\alpha -\frac{3}{p}+\frac{2\alpha }{\rho }}_{p,\infty }} \Vert u\Vert _{\mathcal {L}^\rho _T\dot{\mathcal {B}}^{4-2\alpha -\frac{3}{p}+\frac{2\alpha }{\rho }}_{p,q}}\\&\quad \le C\Vert v\Vert _{\mathcal {L}^\rho _{T}\dot{\mathcal {B}}^{4-2\alpha -\frac{3}{p}+\frac{2\alpha }{\rho }}_{p,q}} \Vert u\Vert _{\mathcal {L}^\rho _T\dot{\mathcal {B}}^{4-2\alpha -\frac{3}{p}+\frac{2\alpha }{\rho }}_{p,q}}. \end{aligned} \end{aligned}$$

The proof is completed. $\square$

Let $0<R\le 1$, which will be determined later. We introduce a continuous decreasing function $\vartheta :~[0, \infty )\rightarrow [0,1]$ defined via

$$\begin{aligned} \vartheta (x):=\{0\vee (2-R^{-1}x)\}\wedge 1. \end{aligned}$$

We consider the following modified system of the equation (1.1)

$$\begin{aligned} \left\{ \begin{array}{ll} du+[(-\Delta )^\alpha u+\mathbf {P}div (\chi _uu\otimes u)]dt =\sum _{k\ge 1}\mathbf {P}g_k(t,u)dB_k,\\ u(0)=u_0,\\ \end{array}\right. \end{aligned}$$

(3.11)

where $\chi _u(t)=\vartheta \bigl (\Vert u\Vert _{\mathcal {L}^2_T\dot{\mathcal {B}}_{p,r}^{4-\alpha -\frac{3}{p}}}\bigr )$.

Proposition 1

Under the assumptions of Theorem 1. Equation (3.11) has a global strong solution.

Proof

Equation (3.11) can be rewritten as

$$\begin{aligned} u(t)=e^{-t(-\Delta )^\alpha }u_0+S(\chi _uu\otimes u)+K(t,u), \end{aligned}$$

where S and K are, respectively, the solutions to the fractional heat equation

$$\begin{aligned} \left\{ \begin{array}{ll} \partial _t S(A)+(-\Delta )^\alpha S(A)=-\mathbf {P}div A,\\ S(A)|_{t=0}=0, \end{array}\right. \end{aligned}$$

and stochastic fractional heat equation

$$\begin{aligned} \left\{ \begin{array}{ll} dK(t,u)+(-\Delta )^{\alpha }K(t,u)dt=\sum _{k\ge 1}g_kdW_k(t),\\ K(t,u)|_{t=0}=0.\\ \end{array}\right. \end{aligned}$$

Define

$$\begin{aligned} \Psi (u):=e^{-t(-\Delta )^\alpha }u_0+S(\chi _uu\otimes u)+K(t,u). \end{aligned}$$

Let

$$\begin{aligned} X_T:=\{w: w~\text{ is } \text{ progressively } \text{ measurable }, \Vert w\Vert _{X_T}<+\infty \} \end{aligned}$$

with

$$\begin{aligned} \Vert w\Vert _{X_T}:=\Vert u\Vert _{L^r_\omega \mathcal {L}^{\infty }_T\dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}}} +\Vert u\Vert _{L^r_\omega \mathcal {L}^2_T\dot{\mathcal {B}}_{p,r}^{4-\alpha -\frac{3}{p}}}. \end{aligned}$$

For any $p_1\ge 2$, we obtain

$$\begin{aligned} \begin{aligned}&\Vert e^{-t(-\Delta )^\alpha }u_0\Vert _{L^r_\omega \mathcal {L}^{q_1}_{T} \dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}+\frac{2\alpha }{q_1}}}\\&\quad =\bigl \Vert 2^{j(4-2\alpha -\frac{3}{p}+\frac{2\alpha }{q_1})}\Vert \varphi _j e^{-t|\xi |^{2\alpha }}\hat{u}_0\Vert _{L^{q_1}_TL^p_\xi }\bigr \Vert _{L^r_\omega l_j^r}\\&\quad \le \bigl \Vert 2^{j(4-2\alpha -\frac{3}{p}+\frac{2\alpha }{q_1})}\Vert \varphi _j e^{-ct2^{2j\alpha }}\hat{u}_0\Vert _{L^{q_1}_TL^p_\xi }\bigr \Vert _{L^r_\omega l_j^r}\\&\quad \le C\bigl \Vert 2^{j(4-2\alpha -\frac{3}{p})}\Vert \varphi _j\hat{u}_0\Vert _{L^p_\xi }\bigr \Vert _{L^r_\omega l_j^r}\\&\quad =C\Vert u_0\Vert _{L_\omega ^r\dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}}}. \end{aligned} \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert e^{-t(-\Delta )^\alpha }u_0\Vert _{X_T}\le C\Vert u_0\Vert _{L_\omega ^r\dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}}}. \end{aligned}$$

(3.12)

By Lemmas 1 and 3, we have

$$\begin{aligned} \begin{aligned} \Vert S(\chi _uu\otimes u)\Vert _{X_T}&\le C\Vert \chi _uu\otimes u\Vert _{L^r_\omega \mathcal {L}^{1}_{T} \dot{\mathcal {B}}_{p,q}^{5-2\alpha -\frac{3}{p}}}\\&\le C\Bigl \Vert \Vert \chi _uu\Vert _{\mathcal {L}^{2}_{T}\dot{\mathcal {B}}_{p,q}^{4-\alpha -\frac{3}{p}}} \Vert u\Vert _{\mathcal {L}^{2}_{T}\dot{\mathcal {B}}_{p,q}^{4-\alpha -\frac{3}{p}}}\Bigr \Vert _{L^r_\omega }\\&\le CR\Vert u\Vert _{L^r_\omega \mathcal {L}^{2}_{T}\dot{\mathcal {B}}_{p,q}^{4-\alpha -\frac{3}{p}}}. \end{aligned} \end{aligned}$$

(3.13)

For the stochastic term, by Lemma 2, we get

$$\begin{aligned} \begin{aligned} \Vert K(t,u)\Vert _{X_T}&\le C\Vert \mathbf {P}f(t,u)\Vert _{\mathcal {L}^2_T\mathcal {L}^r_\omega \dot{\mathbf {B}}^{4-2\alpha -\frac{3}{p}}_{p,r}}\\&\le C\Bigl (L_1T^{\frac{1}{2}}\Vert u\Vert _{L^r_\omega \mathcal {L}^\infty _T \dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}}}+L_2\Vert u\Vert _{L^r_\omega \mathcal {L}^2_T \dot{\mathcal {B}}_{p,r}^{4-\alpha -\frac{3}{p}}}\Bigr ). \end{aligned} \end{aligned}$$

(3.14)

Combining (3.12), (3.13) and (3.14), we obtain

$$\begin{aligned} \begin{aligned} \Vert \Psi (u)\Vert _{X_T}&\le C_1\Bigl [\Vert u_0\Vert _{L_\omega ^r\dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}}}+ R\Vert u\Vert _{L^r_\omega \mathcal {L}^{2}_{T}\dot{\mathcal {B}}_{p,q}^{4-\alpha -\frac{3}{p}}}\\&\qquad +L_1T^{\frac{1}{2}}\Vert u\Vert _{L^r_\omega \mathcal {L}^\infty _T\dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}}} +L_2\Vert u\Vert _{L^r_\omega \mathcal {L}^2_T\dot{\mathcal {B}}_{p,r}^{4-\alpha -\frac{3}{p}}}\Bigr ]\\&\le C_1\Bigl [\Vert u_0\Vert _{L_\omega ^r\dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}}} +(R+L_1T^{\frac{1}{2}}+L_2)\Vert u\Vert _{S_T}\Bigr ]. \end{aligned} \end{aligned}$$

(3.15)

Next, we estimate the term $\Psi (u)-\Psi (v)$ and we have

$$\begin{aligned} \begin{aligned} \Psi (u)-\Psi (v)&=S(\chi _uu\otimes u-\chi _v v\otimes v)+[K(t,u)-K(t,v)]\\&=:II _1+II _2. \end{aligned} \end{aligned}$$

In order to estimate $II _1$. Firstly, similar to [3], one can get that

$$\begin{aligned} |\chi _u-\chi _v|_{L^\infty _T}\le R^{-1}\Vert u-v\Vert _{\mathcal {L}^2_T\dot{\mathcal {B}}_{p,r}^{4-\alpha -\frac{3}{p}}}. \end{aligned}$$

(3.16)

We now divide $II _1$ into the three following cases.

(1) If $\chi _u>0,~\chi _v>0$, then

$$\begin{aligned} \begin{aligned} |\chi _uu\otimes u-\chi _vv\otimes v|\le&|(\chi _u-\chi _v)u\otimes u|+|\chi _v(u-v)\otimes u|\\&+|\chi _vv\otimes (u-v)|. \end{aligned} \end{aligned}$$

Furthermore, by Lemma 3 and (3.16), we then have

$$\begin{aligned} \begin{aligned} \Vert (\chi _u-\chi _v)u\otimes u\Vert _{\mathcal {L}^1_T\mathcal {B}^{5-2\alpha -\frac{3}{p}}_{p,r}}&\le |\chi _u-\chi _v|_{L^\infty _T}\Vert u\otimes u\Vert _{\mathcal {L}^1_T\dot{\mathcal {B}}^{5-2\alpha -\frac{3}{p}}_{p,r}}\\&\le |\chi _u-\chi _v|_{L^\infty _T}\Vert u\Vert ^2_{\mathcal {L}^2_T\dot{\mathcal {B}}^{4-\alpha -\frac{3}{p}}_{p,r}}\\&\le 4R\Vert u-v\Vert _{\mathcal {L}^2_T\dot{\mathcal {B}}_{p,r}^{4-\alpha -\frac{3}{p}}}, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned}&\Vert \chi _u(u-v)\otimes u\Vert _{\mathcal {L}^1_T\mathcal {B}^{5-2\alpha -\frac{3}{p}}_{p,r}} +\Vert \chi _vv\otimes (u-v)\Vert _{\mathcal {L}^1_T\mathcal {B}^{5-2\alpha -\frac{3}{p}}_{p,r}}\\&\quad \le \Vert u-v\Vert _{\mathcal {L}^2_T\dot{\mathcal {B}}_{p,r}^{4-\alpha -\frac{3}{p}}} \Bigl (\Vert u\Vert _{\mathcal {L}^2_T\dot{\mathcal {B}}_{p,r}^{4-\alpha -\frac{3}{p}}}+ \Vert v\Vert _{\mathcal {L}^2_T\dot{\mathcal {B}}_{p,r}^{4-\alpha -\frac{3}{p}}}\Bigr )\\&\quad \le 4R\Vert u-v\Vert _{\mathcal {L}^2_T\dot{\mathcal {B}}_{p,r}^{4-\alpha -\frac{3}{p}}}. \end{aligned} \end{aligned}$$

(2) If $\chi _u>0, \chi _v=0$, then

$$\begin{aligned} \begin{aligned} |\chi _uu\otimes u-\chi _vv\otimes v|\le&|(\chi _u-\chi _v)u\otimes u|, \end{aligned} \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert \chi _uu\otimes u-\chi _vv\otimes v\Vert _{\mathcal {L}^1_T\mathcal {B}^{5-2\alpha -\frac{3}{p}}_{p,r}} \le 4R\Vert u-v\Vert _{\mathcal {L}^2_T\dot{\mathcal {B}}_{p,r}^{4-\alpha -\frac{3}{p}}}. \end{aligned}$$

(3) If $\chi _u=0, \chi _v>0$, then by the similar argument, we have

$$\begin{aligned} \Vert \chi _uu\otimes u-\chi _vv\otimes v\Vert _{\mathcal {L}^1_T\mathcal {B}^{5-2\alpha -\frac{3}{p}}_{p,r}} \le 4R\Vert u-v\Vert _{\mathcal {L}^2_T\dot{\mathcal {B}}_{p,r}^{4-\alpha -\frac{3}{p}}}. \end{aligned}$$

Combing all the above estimates, we obtain

$$\begin{aligned} \begin{aligned} \Vert II _1\Vert _{X_T}&\le C\Vert \chi _uu\otimes u-\chi _vv\otimes v\Vert _{L^r_\omega \mathcal {L}^1_T\mathcal {B}^{5-2\alpha -\frac{3}{p}}_{p,r}}\\&\le CR\Vert u-v\Vert _{L^r_\omega \mathcal {L}^2_T\dot{\mathcal {B}}_{p,r}^{4-\alpha -\frac{3}{p}}}. \end{aligned} \end{aligned}$$

To estimate the term $II _2$, we observe that Lemma 3 implies

$$\begin{aligned} \begin{aligned} \Vert II _2\Vert _{X_T}&\le C\Vert \mathbf {P}f(t,u)-\mathbf {P}f(t,v)\Vert _{\mathcal {L}^2_T \mathcal {L}^r_\omega \dot{\mathbf {B}}_{p,r}^{4-2\alpha -\frac{3}{p}}}\\&\le C\Bigl (L_1\Vert u-v\Vert _{\mathcal {L}^2_T\mathcal {L}^r_\omega \dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}}}+L_2\Vert u-v\Vert _{L^r_\omega \mathcal {L}^2_T \dot{\mathcal {B}}_{p,r}^{4-\alpha -\frac{3}{p}}}\Bigr )\\&\le C\Bigl (L_1T^{\frac{1}{2}}\Vert u-v\Vert _{L^r_\omega \mathcal {L}^\infty _T \dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}}}+L_2\Vert u-v\Vert _{L^r_\omega \mathcal {L}^2_T \dot{\mathcal {B}}_{p,r}^{4-\alpha -\frac{3}{p}}}\Bigr ). \end{aligned} \end{aligned}$$

Consequently,

$$\begin{aligned} \Vert \Psi (u)-\Psi (v)\Vert _{X_T}\le C_2(R+L_1T^{\frac{1}{2}}+L_2)\Vert u-v\Vert _{X_T}. \end{aligned}$$

(3.17)

Define

$$\begin{aligned} M:= & {} 2C_1\Vert u_0\Vert _{L_\omega ^r\dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}}}, \end{aligned}$$

(3.18)

$$\begin{aligned} R:= & {} \min \{1, (8\min \{C_1,C_2\})^{-1}\}, \end{aligned}$$

(3.19)

$$\begin{aligned} X_{T,M}:= & {} \{w: w\in X_T, \Vert w\Vert _{X_T}\le M\}, \end{aligned}$$

(3.20)

$$\begin{aligned} \tilde{T}:= & {} \min \Bigl \{1, \Bigl (\frac{1}{4L_1\max \{C_1,C_2\}}\Bigr )^2\Bigr \}. \end{aligned}$$

(3.21)

If $L_2$ is sufficiently small such that

$$\begin{aligned} \max \{C_1,C_2\}L_2<\frac{1}{8}, \end{aligned}$$

(3.22)

then the mapping $\Psi$ becomes a contracting mapping on $X_{\tilde{T},M}$. Therefore, the equation (3.11) has a unique strong solution u on the interval $[0,\tilde{T}]$ by Banach’s fixed point theorem (see [1, Lemma5.5]). Repeating this procedure, we obtain a global strong solution u for the equation (3.11). $\square$

With all these in hand, we proceed to show our two main theorems.

Proof of Theorem 1

Let u be the solution for the equation (3.11). Define the stopping time

$$\begin{aligned} \tau _R(\omega ):=\inf \Bigl \{t\ge 0; \Vert u\Vert _{\mathcal {L}^{2}_{T} \dot{\mathcal {B}}_{p,q}^{4-\alpha -\frac{3}{p}}}\ge R\Bigr \} \end{aligned}$$

with the convention $\inf \emptyset =\infty$. Then for almost all $\omega \in \Omega$, it holds that

$$\begin{aligned} u\in C([0,\tau _R(\omega )); \dot{\mathcal {B}}_{p,q}^{4-\alpha -\frac{3}{p}}). \end{aligned}$$

If $\tau _R(\omega )<+\infty$, we set $u(t):=0,~\forall ~t\ge \tau _R(\omega )$. Then, $(u,\tau _R)$ is the unique local strong solution to (3.1). According to (3.15), we can show that

$$\begin{aligned} \begin{aligned} \Vert u\Vert _{L^r_\omega \mathcal {L}^\infty _{\frac{1}{k}\wedge \tau _R}\dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}}}&\le C_1\Bigl [\Vert u_0\Vert _{L_\omega ^r\dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}}}+ R\Vert u\Vert _{L^r_\omega \mathcal {L}^{2}_{\frac{1}{k}\wedge \tau _R}\dot{\mathcal {B}}_{p,q}^{4-\alpha -\frac{3}{p}}}\\&\quad +\frac{L_1}{k^{\frac{1}{2}}}\Vert u\Vert _{L^r_\omega \mathcal {L}^\infty _{\frac{1}{k}\wedge \tau _R} \dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}}}+L_2\Vert u\Vert _{L^r_\omega \mathcal {L}^2_{\frac{1}{k}\wedge \tau _R}\dot{\mathcal {B}}_{p,r}^{4-\alpha -\frac{3}{p}}}\Bigr ]\\&\le C_1\Vert u_0\Vert _{L_\omega ^r\dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}}} +\frac{1}{2}\Vert u\Vert _{L^r_\omega \mathcal {L}^\infty _{\frac{1}{k}\wedge \tau _R} \dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}}}\\&\quad +\frac{1}{4}\Vert u\Vert _{L^r_\omega \mathcal {L}^2_{\frac{1}{k}\wedge \tau _R} \dot{\mathcal {B}}_{p,r}^{4-\alpha -\frac{3}{p}}} \end{aligned} \end{aligned}$$

holds for all $k>(2C_1L_1)^{2}$. Consequently,

$$\begin{aligned} \begin{aligned} \Vert u\Vert _{L^r_\omega \mathcal {L}^\infty _{\frac{1}{k}\wedge \tau _R}\dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}}}&\le 2C_1\Vert u_0\Vert _{L_\omega ^r\dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}}} +\frac{1}{2}\Vert u\Vert _{L^r_\omega \mathcal {L}^2_{\frac{1}{k}\wedge \tau _R} \dot{\mathcal {B}}_{p,r}^{4-\alpha -\frac{3}{p}}}. \end{aligned} \end{aligned}$$

Similarly, we get

$$\begin{aligned} \begin{aligned} \Vert u\Vert _{L^r_\omega \mathcal {L}^2_{\frac{1}{k}\wedge \tau _R}\dot{\mathcal {B}}_{p,r}^{4-\alpha -\frac{3}{p}}}&\le \Vert e^{-t(-\Delta )^\alpha }u_0\Vert _{L^r_\omega \mathcal {L}^{2}_{\frac{1}{k}}\dot{\mathcal {B}}_{p,r}^{4 -\alpha -\frac{3}{p}}}+\frac{C_1L_1}{k^{\frac{1}{2}}}\Vert u\Vert _{L^r_\omega \mathcal {L}^\infty _{\frac{1}{k}\wedge \tau _R} \dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}}}\\&\quad +\frac{1}{4}\Vert u\Vert _{L^r_\omega \mathcal {L}^2_{\frac{1}{k}\wedge \tau _R} \dot{\mathcal {B}}_{p,r}^{4-\alpha -\frac{3}{p}}}\\&\le \Vert e^{-t(-\Delta )^\alpha }u_0\Vert _{L^r_\omega \mathcal {L}^{2}_{\frac{1}{k}} \dot{\mathcal {B}}_{p,r}^{4-\alpha -\frac{3}{p}}}+\frac{2C_1^2L_1}{k^{\frac{1}{2}}} \Vert u_0\Vert _{L_\omega ^r\dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}}}\\&\quad +\frac{1}{2}\Vert u\Vert _{L^r_\omega \mathcal {L}^2_{\frac{1}{k}\wedge \tau _R} \dot{\mathcal {B}}_{p,r}^{4-\alpha -\frac{3}{p}}}. \end{aligned} \end{aligned}$$

Therefore,

$$\begin{aligned} \begin{aligned} \Vert u\Vert _{L^r_\omega \mathcal {L}^2_{\frac{1}{k}\wedge \tau _R}\dot{\mathcal {B}}_{p,r}^{4-\alpha -\frac{3}{p}}} \le 2 \Vert e^{-t(-\Delta )^\alpha }u_0\Vert _{L^r_\omega \mathcal {L}^{2}_{\frac{1}{k}} \dot{\mathcal {B}}_{p,r}^{4-\alpha -\frac{3}{p}}}+\frac{4C_1^2L_1}{k^{\frac{1}{2}}} \Vert u_0\Vert _{L_\omega ^r\dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}}}. \end{aligned} \end{aligned}$$

It follows from Chebyshev’s inequality that

$$\begin{aligned} \begin{aligned} \mathbb {P}\Bigl (\tau _R\le \frac{1}{k}\Bigr )&\le \mathbb {P}\Bigl (\Vert u\Vert _{\mathcal {L}^{2}_{\tau _R \wedge \frac{1}{k}}\dot{\mathcal {B}}_{p,q}^{4-\alpha -\frac{3}{p}}}\ge R\Bigr )\\&\le \frac{1}{R}\Vert u\Vert _{L^1_\omega \mathcal {L}^{2}_{\tau _R\wedge \frac{1}{k}} \dot{\mathcal {B}}_{p,q}^{4-\alpha -\frac{3}{p}}}\\&\le \frac{1}{R}\Vert u\Vert _{L^r_\omega \mathcal {L}^{2}_{\tau _R\wedge \frac{1}{k}} \dot{\mathcal {B}}_{p,q}^{4-\alpha -\frac{3}{p}}}\\&\le \frac{1}{R}\Bigl (2 \Vert e^{-t(-\Delta )^\alpha }u_0\Vert _{L^r_\omega \mathcal {L}^{2}_{\frac{1}{k}} \dot{\mathcal {B}}_{p,r}^{4-\alpha -\frac{3}{p}}}+\frac{4C_1^2L_1}{k^{\frac{1}{2}}} \Vert u_0\Vert _{L_\omega ^r\dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}}}\Bigr ). \end{aligned} \end{aligned}$$

For any $0<\varepsilon <1$, there exists $\varrho _\varepsilon >0$, such that

$$\begin{aligned} \left[ \mathbb {E}\sum _{j\ge \varrho _\varepsilon }2^{jr(4-2\alpha -\frac{3}{p})} \Vert \varphi _j\widehat{u}_0\Vert ^r_{L^p_\xi }\right] ^{\frac{1}{r}}\le \frac{R\varepsilon }{8}. \end{aligned}$$

Thus,

$$\begin{aligned} \begin{aligned} \Vert e^{-t(-\Delta )^\alpha }u_0\Vert _{L^r_\omega \mathcal {L}^{2}_{\frac{1}{k}} \dot{\mathcal {B}}_{p,r}^{4-\alpha -\frac{3}{p}}}&=\left( \mathbb {E}\sum _{j\in \mathbb {Z}}2^{jr(4-\alpha -\frac{3}{p})} \Vert e^{-t|\xi |^{2\alpha }}\varphi _j\widehat{u}_0\Vert ^r_{L^2_{\frac{1}{k}}L^p_\xi }\right) ^{\frac{1}{r}}\\&\le \frac{R\varepsilon }{8}+\left( \mathbb {E}\sum _{j\le J_\varepsilon }2^{jr(4-\alpha -\frac{3}{p})} \Vert e^{-t|\xi |^{2\alpha }}\varphi _j\widehat{u}_0\Vert ^r_{L^2_{\frac{1}{k}}L^p_\xi }\right) ^{\frac{1}{r}}\\&\le \frac{R\varepsilon }{8}+C_3k^{-\frac{1}{2}} \Vert u_0\Vert _{L_\omega ^r\dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}}}. \end{aligned} \end{aligned}$$

If we choose

$$\begin{aligned} k\ge \max \Bigl \{(2C_1L_1)^{2}, 16R^{-2}\varepsilon ^{-2}(2C_3+4C_1L_1)^2 \Vert u_0\Vert ^2_{L_\omega ^r\dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}}}\Bigr \}, \end{aligned}$$

then we have

$$\begin{aligned} \mathbb {P}\Bigl (\tau _R\le \frac{1}{k}\Bigr )\le \varepsilon . \end{aligned}$$

Thereby,

$$\begin{aligned} \begin{aligned} \mathbb {P}\bigl (\tau _R> 0\bigr )&=1-\mathbb {P}\bigl (\tau _R= 0\bigr )\\&\ge 1-\lim _{k\rightarrow \infty }\mathbb {P}\Bigl (\tau _R\le \frac{1}{k}\Bigr )\\&\ge 1-\varepsilon . \end{aligned} \end{aligned}$$

Since $\varepsilon$ is arbitrary small, we get $\mathbb {P}\bigl (\tau _R> 0\bigr )=1$. The proof is thus completed. $\square$

Proof of Theorem 2

Similar to (3.15), for any $0<t<\infty$, we have

$$\begin{aligned} \begin{aligned} \Vert u\Vert _{X_{t\wedge \tau _R}}&\le C_1\Bigl [\Vert u_0\Vert _{L_\omega ^r\dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}}} +L_3\Vert u\Vert _{L^r_\omega \mathcal {L}^\infty _{t\wedge \tau _R}\dot{\mathcal {B}}^{4-2\alpha -\frac{3}{p}}_{p,r}}\\&\quad +(R+L_2)\Vert u\Vert _{L^r_\omega \mathcal {L}^2_T\dot{\mathcal {B}}^{4-\alpha -\frac{3}{p}}_{p,r}}\Bigr ]. \end{aligned} \end{aligned}$$

(3.23)

Choose $L_3>0$ such that $C_1L_3<1$. Utilising (3.19) and (3.22), we obtain

$$\begin{aligned} \begin{aligned} \Vert u\Vert _{L^r_\omega \mathcal {L}^2_{t\wedge \tau _R}\dot{\mathcal {B}}^{4-\alpha -\frac{3}{p}}_{p,r}} \le 2C_1\Vert u_0\Vert _{L_\omega ^r\dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}}}. \end{aligned} \end{aligned}$$

By Fatou’s lemma (see [7, Theorem 1.5.4]), we have

$$\begin{aligned} \begin{aligned} R\mathbb {P}(\tau _R<\infty )&=\mathbb {E}\Bigl (1_{(\tau _R<\infty )}\lim _{t\rightarrow \infty } \Vert u\Vert _{\mathcal {L}^2_{t\wedge \tau _R}\dot{\mathcal {B}}^{4-\alpha -\frac{3}{p}}_{p,r}}\Bigr )\\&\le \liminf _{t\rightarrow \infty }\mathbb {E}\Bigl (1_{(\tau _R<\infty )} \Vert u\Vert _{\mathcal {L}^2_{t\wedge \tau _R}\dot{\mathcal {B}}^{4-\alpha -\frac{3}{p}}_{p,r}}\Bigr )\\&\le 2C_1\Vert u_0\Vert _{L_\omega ^r\dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}}}. \end{aligned} \end{aligned}$$

Hence,

$$\begin{aligned} \mathbb {P}(\tau _R<\infty )\le \frac{2C_1}{R}\Vert u_0\Vert _{L_\omega ^r\dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}}}. \end{aligned}$$

Thus,

$$\begin{aligned} \mathbb {P}(\tau _R=\infty )\ge 1-\frac{2C_1}{R}\Vert u_0\Vert _{L_\omega ^r\dot{\mathcal {B}}_{p,r}^{4-2\alpha -\frac{3}{p}}}. \end{aligned}$$

It follows that for any $\varepsilon >0$, choosing $\gamma =\frac{\varepsilon R}{2C_1}$, we obtain

$$\begin{aligned} \mathbb {P}(\tau _R=\infty )\ge 1-\varepsilon . \end{aligned}$$

We thus complete the proof of Theorem 2. $\square$