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.
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1 Introduction
In this paper, we are concerned with the following three-dimensional stochastic incompressible fractional Navier–Stokes equation
for unknown random field \(u=(u_1,u_2,u_3)\in \mathbb {R}^3\) 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
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
Furthermore, for any \(\phi \in \mathcal {S}'(\mathbb {R}^d)\), we define its Fourier transform and inverse Fourier transform as
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
where \(\varphi _j(\xi )=\varphi (2^{-j}\xi )\). The homogeneous dyadic blocks is defined in the following manner
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
We have
Definition 1
Let \(1\le p,r\le \infty , s\in \mathbb {R}\), the homogeneous Fourier–Besov space is defined as
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
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
where \(\dot{T}_uv\) and \(\dot{T}_vu\) denote the homogenous paraproduct of v by u and u by v, respectively
and \(\dot{R}(u,v)\) denotes the homogeneous remainder of u and v:
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
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 \(0<T<\infty \),
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
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
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
(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
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
fulfils
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
Theorem 2
With the same preamble as in Theorem 1. If further for sufficiently small \(L_3>0\),
and for any \(\varepsilon >0\), there exists a constant \(\gamma =\gamma (\varepsilon )>0\) 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 \(g_k(t,u)=\frac{1}{\sqrt{2}k}e^{-M(1+t)}u,~k\ge 1\). 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.
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
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
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
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 \(q\le q_1\le \infty \),
Proof
Taking the Fourier transform on both sides of (3.6), we conclude that the unique solution u satisfies
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 \(\beta \in (0,\frac{1}{2})\), we have
where we have used the factorisation formula (see e.g. [5])
Basic calculus then implies that
Substituting this estimate into (3.8), we obtain
Therefore,
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
Proof
According to Bony’s decomposition, we have
For the term \(I _1\), by Young’s inequality and \(l^q\hookrightarrow l^\infty \), we get
Therefore,
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,
For the term \(I _3\), utilising Bernstein inequality, we get
Thus,
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
We consider the following modified system of the equation (1.1)
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
where S and K are, respectively, the solutions to the fractional heat equation
and stochastic fractional heat equation
Define
Let
with
For any \(p_1\ge 2\), we obtain
Therefore,
For the stochastic term, by Lemma 2, we get
Combining (3.12), (3.13) and (3.14), we obtain
Next, we estimate the term \(\Psi (u)-\Psi (v)\) and we have
In order to estimate \(II _1\). Firstly, similar to [3], one can get that
We now divide \(II _1\) into the three following cases.
(1) If \(\chi _u>0,~\chi _v>0\), then
Furthermore, by Lemma 3 and (3.16), we then have
and
(2) If \(\chi _u>0, \chi _v=0\), then
Therefore,
(3) If \(\chi _u=0, \chi _v>0\), then by the similar argument, we have
Combing all the above estimates, we obtain
To estimate the term \(II _2\), we observe that Lemma 3 implies
Consequently,
Define
If \(L_2\) is sufficiently small such that
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
with the convention \(\inf \emptyset =\infty \). Then for almost all \(\omega \in \Omega \), it holds that
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
holds for all \(k>(2C_1L_1)^{2}\). Consequently,
Similarly, we get
Therefore,
It follows from Chebyshev’s inequality that
For any \(0<\varepsilon <1\), there exists \(\varrho _\varepsilon >0\), such that
Thus,
If we choose
then we have
Thereby,
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
Choose \(L_3>0\) such that \(C_1L_3<1\). Utilising (3.19) and (3.22), we obtain
By Fatou’s lemma (see [7, Theorem 1.5.4]), we have
Hence,
Thus,
It follows that for any \(\varepsilon >0\), choosing \(\gamma =\frac{\varepsilon R}{2C_1}\), we obtain
We thus complete the proof of Theorem 2. \(\square \)
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Acknowledgements
The authors would like to thank the referee for their insightful comments which have led us to improve the presentation of the paper. This research was supported by the Natural Science Foundation of China (No. 11901005 and No. 12071003) and the Natural Science Foundation of Anhui Province (No. 2008085QA20), respectively.
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Xiuwei Yin and Guangjun Shen were supported by the Natural Science Foundation of China (Nos. 11901005 and 12071003) and the Natural Science Foundation of Anhui Province (No. 2008085QA20).
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XY was involved in the conceptualisation and proving and writing for the original draft. J-LW contributed by proving, reviewing, editing, and coordinating. GS contributed to methodology and proving for the original draft.
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Yin, X., Wu, JL. & Shen, G. Well-Posedness for Stochastic Fractional Navier–Stokes Equation in the Critical Fourier–Besov Space. J Theor Probab 35, 2940–2959 (2022). https://doi.org/10.1007/s10959-021-01152-y
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DOI: https://doi.org/10.1007/s10959-021-01152-y