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

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 B˙p,r4-2α-3p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{\mathcal {B}}^{4-2\alpha -\frac{3}{p}}_{p,r}$$\end{document}. Then, under the condition that the initial date is sufficiently small, we show the global existence of the solutions in the probabilistic sense.


Introduction
In this paper, we are concerned with the following three-dimensional stochastic incompressible fractional Navier-Stokes equation for unknown random field u = (u 1 , u 2 , u 3 ) ∈ R 3 representing the velocity of a fluid, where π stands for the pressure and the fractional Laplace operator (− ) α , α ∈ (0, 1] is the Fourier multiplier with symbol |ξ | 2α , g k , k ≥ 1 are jointly measurable coefficients, {B k , k ≥ 1} is a sequence of one-dimensional independent Brownian motions defined in a given completed filtered probability space ( , F, F t , P) (see e.g. [9]). When α = 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 spaceḂ d p −1 p,r . Chang and Yang [3] studied the initial-boundary value problem of the SNS in the half space.
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 · X is critical for the equation 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 spaceḂ 1−2α+ 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 χ 1−2α , Ru and Abidin [12] discussed the global well-posedness for FNS in the variable exponent Fourier-Besov spacesḂ 4−2α− 3 p(·) p(·),r , just mention a few. More discussions on the global well-posedness can be found in [4,10]  , 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 spaceḂ . 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.

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 S(R d ), d ≥ 1, be the Schwarz space of all smooth functions that are rapidly decreasing infinite functions along with all partial derivatives and S (R d ) be the space of tempered distributions. Let the dual pairing between S(R d ) and S (R d ) be denoted by ·, · . For any f ∈ S(R d ), the Fourier transform and the inverse Fourier transform of f are defined, respectively, by Furthermore, for any φ ∈ S (R d ), we define its Fourier transform and inverse Fourier transform as Let C := ξ ∈ R d : 3 4 ≤ |ξ | ≤ 8 3 and D(C) be the space of all test functions on C, that is, the totality of all smooth functions on C that have compact support. Then there exists a non-negative radial function ϕ ∈ D(C), such that where ϕ j (ξ ) = ϕ(2 − j ξ). The homogeneous dyadic blocks is defined in the following manner˙ j u :=φ j * u, j ∈ Z.
We further set the quotient space S h (R d ) := S (R d )/P, where P is the space of all polynomials. For any u ∈ S h (R d ), we definė We have Definition 1 Let 1 ≤ p, r ≤ ∞, s ∈ R, the homogeneous Fourier-Besov space is defined asḂ Taking the time variable and random variables into account, we also need the following definition of Chemin-Lerner-type spaces, see, [1,16].
We list here some of the properties that will be used in the sequel.
Finally, we introduce the homogeneous Bony decomposition. For more details, we refer readers to [1,11] and the references therein. Let u, v ∈ S h (R d ), then the homogeneous Bony decomposition of uv is defined by whereṪ u v andṪ v u denote the homogenous paraproduct of v by u and u by v, respec-tivelyṪ andṘ(u, v) denotes the homogeneous remainder of u and v:
Definition 3 Let 2 ≤ p, r ≤ ∞ and the initial data u 0 ∈Ḃ is called a local strong solution for the equation (1.1), if (i) u is a progressively measurable process and for any 0 < T < ∞, , and the following equality (2) We say that the local strong solution is unique, if (ũ,τ R ) is another strong solution, then be 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 < ∞, we have , and for all 0 < t < ∞, the following equality We are now in the position to state our two main results of this paper.
Theorem 2 With the same preamble as in Theorem 1. If further for sufficiently small L 3 > 0, (3.4) and for any ε > 0, there exists a constant γ = γ (ε) > 0 such that Then, 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 Then 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.
ρ −2α p,q holds for some positive constant C.
Then, the following stochastic fractional heat equation has a unique solution u ∈ L r ( ; L q TḂ s p,r ∩ C TḂ s−α p,r is progressively measurable. Moreover, there exists a constant C > 0, such that for any q ≤ q 1 ≤ ∞, Proof Taking the Fourier transform on both sides of (3.6), we conclude that the unique solution u satisfiesû Therefore, by using Minkowski's inequality, Young's inequality and [6, Lemma 2.5], we get Consequently, Let p be the conjugate number of p, then for any β ∈ (0, 1 2 ), we have where we have used the factorisation formula (see e.g. [5]) (3.9) Basic calculus then implies that Substituting this estimate into (3.8), we obtain Therefore, (3.10) Using the interpolation inequality for (3.8) and (3.10), we obtain (3.7), Finally, u ∈ L r ( ; C TḂ s− 2α q p,r ) holds by utilising factorisation formula (3.9) again (see e.g. [5]). We thus complete the proof.
Proof According to Bony's decomposition, we havė For the term I 1 , by Young's inequality and l q → l ∞ , we get where we have used the fact thatḂ s 1 For the term I 3 , utilising Bernstein inequality, we get The proof is completed.
Let 0 < R ≤ 1, which will be determined later. We introduce a continuous decreas- We consider the following modified system of the equation (1.1)

Proposition 1 Under the assumptions of Theorem 1. Equation (3.11) has a global strong solution.
Proof Equation (3.11) can be rewritten as where S and K are, respectively, the solutions to the fractional heat equation and stochastic fractional heat equation K (t, u).
For any p 1 ≥ 2, we obtain Therefore, (3.12) By Lemmas 1 and 3, we have For the stochastic term, by Lemma 2, we get (3.14) Combining (3.12), (3.13) and (3.14), we obtain Next, we estimate the term (u) − (v) and we have In order to estimate II 1 . Firstly, similar to [3], one can get that (3.16) We now divide II 1 into the three following cases.
With all these in hand, we proceed to show our two main theorems.
We thus complete the proof of Theorem 2.
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