1 Introduction
The
$3$
-dimensional (
$3$
D) generalized incompressible micropolar system can be written as
$$ \begin{align} \left\{ \begin{array}{@{}lcl} \partial_t u + u \cdot \nabla u +(\chi+\nu)(-\Delta)^{\gamma} u + \nabla \Pi - 2\chi\nabla \times w =0,\\ \partial_t w + u \cdot \nabla w + \mu(-\Delta)^{\gamma}w+4\chi w-\kappa\nabla\nabla\cdot w -2\chi\nabla \times u =0,\\ \nabla \cdot u=0,\\ u(x, 0) = u_{0}(x),\ w(x, 0)=w_{0}(x). \end{array} \right. \end{align} $$
where
$x\in \mathbb R^3$
with
$t>0$
,
$u=u(x,t)$
,
$\Pi =\Pi (x,t)$
and
$w=w(x,t)$
denote the velocity of the fluid, the scalar pressure and the micro-rotational velocity, respectively. The nonnegative parameters
$\chi $
and
$\nu $
are the kinematic viscosity,
$\mu $
and
$\kappa $
are the vortex viscosity. The specific values of these parameters are not important, we assume that
$\chi =\nu =\frac {1}{2}$
and
$\mu =\kappa =1$
for simplicity.
When
$\gamma =1$
, the system (1.1) is the standard micropolar fluid system, which was first proposed by Eringen [Reference Eringen6], it describes many physical phenomena which are difficult to be treated by the classical Navier–Stokes system for the numerous incompressible fluids. The system (1.1) was presented as an essential modification to the conventional Navier–Stokes equations for the purpose to better describe the motion of numerous real fluids (such as blood) by considering the fact that such fluids usually consist of rigid, randomly oriented (or spherical) particles suspended in a viscous medium, with the ignored deformation of fluid particles. Actually, there are several experiments indicating that solutions of the micropolar fluid system do better mimic behavior of many fluids than solutions of the Navier–Stokes equations, see [Reference Migun and Prohorenko14–Reference Popel, Regirer and Usick16]. For more physical background of the micropolar fluid system, refer to [Reference Lukaszewicz12, Reference Lukaszewicz13].
The micropolar fluid system has recently attracted considerable attention and many interesting results have been obtained in the literature. For example, Galdi and Rionero [Reference Galdi and Rionero9] first considered some existence and uniqueness theorems of solution of the incompressible micropolar system. Baraka and Toumlilin [Reference El Baraka and Toumlilin3] established that the global well-posedness and decay results for 3D generalized magneto-hydrodynamic equations in critical Fourier–Besov–Morrey spaces. In the critical Besov spaces
$\dot { B}_{p,\infty }^{\frac 3p-1}(\mathbb R^3)$
for
$1\leq p<6$
, Chen and Miao [Reference Chen and Miao2] established global well-posedness of the micropolar fluid system (1.1) with small initial data.
Recently, Baraka and Toumlilin [Reference El Baraka and Toumlilin5] showed that the uniform well-posedness and stability for fractional Navier–Stokes equations with coriolis force in critical Fourier–Besov–Morrey Spaces. The local well-posedness results ([Reference El Baraka and Toumlilin5]) were established in
$\mathcal L^{4}([0,T);\mathcal {F\dot {N}}^{1-\frac 32\gamma + \frac {3}{p'}+\frac {\lambda }{p}}_{p,\lambda ,q})$
with
$\max \{1,\frac {3-\lambda }{2}\}\leq p<\infty $
and
$\frac 23<\gamma \leq \frac 23+\frac {1}{p'}+\frac {\lambda }{3p}$
. We wonder that if we lower the requirement for
$\frac 12<\gamma \leq 1$
, does the local well-posedness still hold? the answer is positive. In this article, by constructing the work space
$\mathcal L^{2r}([0,T);\mathcal {F\dot {N}}^{4+ \frac {\lambda -3}{p}-\frac {2\gamma }{(2r)'}}_{p,\lambda ,q})$
with
$1<\frac {2\gamma }{(2r)'}<\frac 52+\frac {\lambda -3}{2p}$
, we can get the local well-posedness results with
$\frac 12<\gamma \leq 1$
. We also refer readers to see previous works [Reference El Baraka and Toumlilin4] and [Reference Xiao, Chen, Fan and Zhou19] for Navier–Stokes equations in Fourier–Besov–Morrey spaces.
In this article, we prove the well-posedness for the 3D micropolar fluid system in Fourier–Besov–Morrey spaces. To this end, we sketch the main difficulty and the strategy to overcome it.
Applying the Leray projection
$\mathbb P$
to both sides of the first equation of (1.1) to eliminate the pressure
$\Pi $
, one has
$$ \begin{align} \left\{ \begin{array}{@{}lcl} \partial_t u +(-\Delta)^{\gamma} u +\mathbb P( u \cdot \nabla u)-\nabla \times w =0,\\ \partial_t w+(-\Delta)^{\gamma}w+u \cdot \nabla w+2 w-\nabla\nabla\cdot w -\nabla \times u =0 ,\\ \nabla \cdot u=0,\\ u(x, 0) = u_{0}(x),\ w(x, 0)=w_{0}(x). \end{array} \right. \end{align} $$
where
$\mathbb P=(\delta _{ij}+R_iR_j)_{1\leq i,j\leq 3}$
denotes the Helmholtz projection onto the divergence-free vector fields. Inspired by [Reference Ferreira and Villamizar-Roa8], we will study the following linear system of (1.2):
$$ \begin{align} \left\{ \begin{array}{@{}lcl} \partial_t u + (-\Delta)^{\gamma} u -\nabla \times w =0,\\ \partial_t w + (-\Delta)^{\gamma}w+2 w-\nabla\nabla\cdot w -\nabla \times u =0,\\ \nabla \cdot u=0,\\ u(x, 0) = u_{0}(x),\ w(x, 0)=w_{0}(x). \end{array} \right. \end{align} $$
Taking the Fourier transform of (1.3)
$_1$
and (1.3)
$_2$
, one obtains
$$ \begin{align} \hat {u} _t+|\xi|^{2\gamma}\hat u +B(\xi)\hat w=0, \end{align} $$
and
$$ \begin{align} \hat w_t+(|\xi|^{2\gamma}+2)\hat w+B(\xi)\hat u+C(\xi)\hat w=0, \end{align} $$
where
$$ \begin{align*}B(\xi)=i\left[\begin{aligned} ~~0~~~~~~\xi_3~~~~-\xi_2~~\\ -\xi_3~~~~~~0~~~~~~~\xi_1~~\\ ~~\xi_2~~~~~-\xi_1~~~~~~0~~\\ \end{aligned}\right] \mathrm {and}\ C(\xi)=\left[\begin{aligned} ~~\xi^2_1~~~~~~\xi_1\xi_2~~~~\xi_1\xi_3~~\\ ~~\xi_1\xi_2~~~~~\xi^2_2~~~~~~~\xi_2\xi_3~~\\ ~~\xi_1\xi_3~~~~~\xi_2\xi_3~~~~~~\xi^2_3~~\\ \end{aligned}\right]. \end{align*} $$
Now, defining
$G:=(u,w)^T$
, rewrite the system (1.4) and (1.5) as
$$ \begin{align} \widehat G_t+A(\xi)\widehat G=0, \end{align} $$
where
$$ \begin{align} A(\xi)=\left[\begin{aligned} ~~~|\xi|^{2\gamma}I~~~~~~~~~~~~~~~~~~~~B(\xi)~~~~~~\\ B(\xi)~~~~~~~(|\xi|^{2\gamma}+2)I+C(\xi)~~\\ \end{aligned}\right]. \end{align} $$
Then, we obtain
$\widehat {G}(t)=e^{-A(\xi )t}\widehat G_0(\xi )$
, where
$\widehat G_0(\xi )$
is the Fourier transform of
$G_0(x)$
,
$\widehat G_0=\widehat G(\cdot ,0)=(\hat u_0,\hat w_0)^T$
.
In order to derive the mild solution of the system (1.3) clearly, we denote that
$$ \begin{align} U(x,t)=\left(\begin{aligned} u(x,t)\\w(x,t)\\ \end{aligned}\right), \ \ U_0=\left(\begin{aligned} u(x,0)\\w(x,0)\\ \end{aligned}\right)=\left(\begin{aligned} u_{0}\\w_{0}\\ \end{aligned}\right), \end{align} $$
and
$$ \begin{align} u\otimes U=\left(\begin{aligned} u\otimes u\\u\otimes w\\ \end{aligned}\right), \ \ \mathbb P\nabla\cdot(u\otimes U)\triangleq \left(\begin{aligned} \mathbb P\nabla\cdot(u\otimes u)\\ \nabla\cdot(u\otimes w)\\ \end{aligned}\right). \end{align} $$
Then by the Duhamel principle, Fourier transform and inverse Fourier transform, the solution of system (1.3) can be reduced to finding a solution U of the following integral equation:
$$ \begin{align} U(t)=G(t)U_0-\int_{0}^{t}G(t-\tau)\mathbb P\nabla\cdot(u\otimes U)(\tau)d\tau, \end{align} $$
where
$G(t)$
denotes the semigroup corresponding to the system (1.3). The semigroup
$G(t)$
shall be frequently used to prove the locally well-posedness and global well-posedness with small initial data belonging to the critical Fourier–Besov–Morrey spaces. To this end we state the main results as follows.
Theorem 1.1 Let
$0\leq \lambda <3$
,
$1\leq q\leq \infty $
,
$1\leq p<\infty $
, and
$(u_{0},w_{0})\in \mathcal {F\dot {N}}^{4-2\gamma +\frac {\lambda -3}{p}}_{p,\lambda ,q}(\mathbb R^3)$
. Suppose that
$\gamma $
and q satisfy that
$\frac {1}{2}< \gamma < \frac 52+\frac {\lambda -3}{2p}$
and
$1\leq q\leq \infty $
. Then, there exists a constant
$\varepsilon>0$
so that for any initial data satisfying
$\nabla \cdot u_0=0$
and
$$ \begin{align*} \|(u_0,w_0)\|_{\mathcal{F\dot{N}}^{4-2\gamma+\frac{\lambda-3}{p}}_{p,\lambda,q}(\mathbb R^3)}<\varepsilon, \end{align*} $$
then the system (1.1) has a unique global mild solution such that
$$ \begin{align*}(u,w)\in C\left([0,\infty);\mathcal{F\dot{N}}^{4-2\gamma+\frac{\lambda-3}{p}}_ {p,\lambda,q}\right)\cap \mathcal L^1\left([0,\infty);\mathcal{F\dot{N}}^{4+\frac{\lambda-3}{p}}_{p,\lambda,q}\right)\cap \mathcal L^\infty\left([0,\infty);\mathcal{F\dot{N}}^{4-2\gamma+\frac{\lambda-3}{p}}_{p,\lambda,q}\right)\end{align*} $$
and
$$ \begin{align*}\|(u,w)\|_{\mathcal L^1([0,\infty);\mathcal {F\dot{N}}^{4+\frac{\lambda-3}{p}}_{p,\lambda,q})}+\|(u,w)\|_{\mathcal L^\infty([0,\infty);\mathcal {F\dot{N}}^{4-2\gamma+\frac{\lambda-3}{p}}_{p,\lambda,q})}<2\varepsilon.\end{align*} $$
Remark 1.1 Under the conditions of Theorem 1.1, if
$\gamma =\frac 12$
, the above global well-posedness of the system (1.1) can also be established, provided we choose
$q=1$
.
Theorem 1.2 Let
$0\leq \lambda <3$
,
$1\leq q\leq \infty $
,
$1< p< \infty $
and
$\frac {1}{2}< \gamma \leq 1$
, there exists a positive time
$T=T(u_0,w_0)$
, depending on the profile of
$(u_{0},w_{0})$
, for any initial data satisfying
$(u_{0},w_{0})\in \mathcal {F\dot {N}}^{4-2\gamma +\frac {\lambda -3}{p}}_{p,\lambda ,q}(\mathbb R^3)$
and
$\nabla \cdot u_0=0$
, then the system (1.1) has a unique local mild solution such that
$$ \begin{align*} (u,w)\in C\left([0,T);\mathcal{F\dot{N}}^{4-2\gamma+\frac{\lambda-3}{p}}_ {p,\lambda,q}\right)\cap\mathcal L^{2r}\left([0,T);\mathcal{F\dot{N}}^{4+ \frac{\lambda-3}{p}-\frac{2\gamma}{(2r)'}}_{p,\lambda,q}\right), \end{align*} $$
where
$(2r)'$
is the conjugate of
$2r$
satisfying
$\frac {1}{2r}+\frac {1}{(2r)'}=1$
and
$1<\frac {2\gamma }{(2r)'}<\frac 52+\frac {\lambda -3}{2p}$
.
We outline the main steps in this article. In Section 2, we first recall the Littlewood–Paley decomposition and definitions of the Morrey spaces and homogeneous Fourier–Besov–Morrey spaces, then present some lemmas which play an important role in the proofs of the theorems. In Section 3, we finish the linear and bilinear estimates in Fourier–Besov–Morrey spaces, which will be frequently used to prove theorem. In Section 4, we prove the Theorems 1.1 and 1.2.
2 Preliminaries
The results presented in this section are based on homogeneous Littlewood–Paley decomposition in the Fourier variables, and we introduce Fourier transform and the Littlewood–Paley decomposition, for more details refer to [Reference Triebel18]. Given
$f(x)\in \mathcal {S}(\mathbb {R}^{3})$
, the Schwartz class of rapidly decreasing functions, define the Fourier transform as
$$ \begin{align*} \hat{f}(\xi)=\mathcal{F}f(\xi)=(2\pi)^{-3/2}\int_{\mathbb{R}^{d}}e^{-ix\cdot\xi}f(x)\mathrm{d}x, \end{align*} $$
and its inverse Fourier transform:
$$ \begin{align*} \check{f}(\xi)=\mathcal{F}^{-1}f(\xi)=(2\pi)^{-3/2}\int_{\mathbb{R}^{d}}e^{ix\cdot\xi}f(\xi)\mathrm{d}\xi. \end{align*} $$
Let
$\varphi $
,
$\chi $
be two nonnegative functions in
$\mathcal {S}(\mathbb R^3)$
satisfying
$$ \begin{align*} \mathrm{supp}\varphi\subset\{\xi\in\mathbb{R}^{3}:\frac34\leq |\xi|\leq\frac83\}, \ \ \ \sum_{j\in \mathbb Z}\varphi(2^{-j}\xi)=1,\ \ \ \ for\ \xi\neq 0, \end{align*} $$
$$ \begin{align*} \mathrm{supp} \chi\subset\{\xi\in\mathbb R^3:|\xi|\leq\frac{4}{3}\},\ \ \ \chi(\xi)+\sum_{j\geq0}\varphi(2^{-j}\xi)=1, \ \ \ \ for\ \xi\in\mathbb{R}^{3}. \end{align*} $$
The homogeneous dyadic blocks
$\dot {\Delta }_j$
and
$\dot {S}_j$
are defined for all
$j\in \mathbb Z$
and
$u\in \mathcal {S'}(\mathbb R^3)$
by
$$ \begin{align*}\dot{\Delta}_{j}u=\mathcal{F}^{-1}[\varphi(2^{-j}\xi)\hat u]=\mathcal{F}^{-1}(\varphi_j\hat u),\end{align*} $$
$$ \begin{align*}\dot{S}_{j}u=\sum_{k\leq j-1}\dot{\Delta}_{k}u.\end{align*} $$
From the definition, we have that
$$ \begin{align*} \dot\Delta_{j}\dot\Delta_{k}u=0, \ \ \ |j-k|\geq 2 \ \ \ and \ \ \ \dot\Delta_{j}(\dot S_{k-1}u\dot\Delta_{k}u)=0, \ \ \ |j-k|\geq 5. \end{align*} $$
The following Bony’s decomposition (see the definition in [Reference Bony1]) will be applied throughout the article:
$$ \begin{align*}uv=\dot T_uv+\dot T_vu+R(u,v),\end{align*} $$
where
$$ \begin{align*} \dot T_uv=\sum_{j\in\mathbb Z}\dot S_{j-1}u\dot\Delta_{j}v, \ \ \ \dot R(u,v)=\sum_{j\in\mathbb Z}\dot\Delta_{j}u\tilde\Delta_{j}v, \ \ \ \tilde\Delta_{j}v=\sum_{j'=j-1}^{j+1}\dot\Delta_{j'}v. \end{align*} $$
Now, we give the definition of the Morrey space which are a complement to the
$L^p$
spaces.
Definition 2.1 [Reference Kato11, Reference Taylor17]
For
$1\leq p<\infty $
,
$0\leq \lambda < 3$
, the Morrey spaces
$M^\lambda _p=M^\lambda _p(\mathbb R^3)$
is defined as the set of function
$f\in L^p_{loc}(\mathbb R^3)$
such that
$$ \begin{align*} \|f\|_{M^\lambda_p}=\sup_{x_0\in\mathbb R^3}\sup_{r>0}r^{-\frac\lambda p}\|f\|_{L^p(B(x_0,r))}<\infty, \end{align*} $$
where
$B(x_0,r)$
denotes the ball in
$\mathbb R^3$
with center
$x_0$
and radius r.
Lemma 2.1 [Reference Kato11, Reference Taylor17]
If
$1\leq p_1,p_2,p_3<\infty $
and
$0\leq \lambda _1,\lambda _2,\lambda _3<3$
with
$\frac {1}{p_3}=\frac {1}{p_1}+\frac {1}{p_2}$
and
$\frac {\lambda _3}{p_3}=\frac {\lambda _1}{p_1}+\frac {\lambda _2}{p_2}$
, then we have the Hölder inequality
$$ \begin{align} \|fg\|_{M^{\lambda_3}_{p_3}}\leq\|f\|_{M^{\lambda_1}_{p_1}}\|g\|_{M^{\lambda_2}_{p_2}}. \end{align} $$
Also, for
$1\leq p<\infty $
and
$0\leq \lambda < 3$
, we have the Young inequality
$$ \begin{align} \|\varphi\ast g\|_{M^{\lambda}_{p}}\leq\|\varphi\|_{L^1}\|g\|_{M^{\lambda}_{p}}, \end{align} $$
for all
$\varphi \in L^1$
and
$g\in M^\lambda _p$
.
Lemma 2.2 [Reference Ferreira and Lima7]
Let
$1\leq p_2 \leq p_1<\infty $
,
$0\leq \lambda _1,\lambda _2<3$
,
$\frac {3-\lambda _1}{p_1}\leq \frac {3-\lambda _2}{p_2}$
, and let
$\gamma $
be a multi-index. If
$\mathrm {supp}\hat {f}\subset \{|\xi |\leq A2^j\}$
for,
$A>0$
then there is a constant
$C>0$
independent of f and j such that
$$ \begin{align} \|(i\xi)^{\gamma}\hat{f}\|_{M^{\lambda_2}_{p_2}}\leq C2^{j|\gamma|+j(\frac{3-\lambda_2}{p_2}-\frac{3-\lambda_1}{p_1})}\|\hat{f}\|_{M^{\lambda_1}_{p_1}}. \end{align} $$
Lemma 2.3 [Reference Kato10, Reference Yuan and Ma20]
Let X be a Banach space with norm
$\|\cdot \|_X$
and
$ B: X \times X \mapsto X$
be a bounded bilinear operator satisfying
$$ \begin{align*}\|B(u,v)\|_{ X}\leq \eta\|u\|_{ X}\|v\|_ X\end{align*} $$
for all
$u,v\in X$
and a constant
$\eta>0$
. If there exists a constant
$\varepsilon>0$
such that
$\|y\|_X<\varepsilon <\frac 1{4\eta }$
, then the equation
$x=y+B(x,x)$
exists a unique solution x in the ball
$\overline B(0,2\varepsilon )$
. Moreover, the solution depends continuously on y in the sense: if
$\|y'\|_X<\varepsilon $
,
$x'=y'+B(x',x')$
and
$\|x'\|_X\leq 2\varepsilon $
, then
$$ \begin{align*}\|x-x'\|_X\leq\frac{1}{1-4\varepsilon\eta}\|y-y'\|_X.\end{align*} $$
Definition 2.2 (Homogeneous Fourier–Besov–Morrey spaces)
Let
$s\in \mathbb R$
,
$1\leq p<\infty $
,
$1\leq q\leq \infty $
and
$0\leq \lambda <3$
, the spaces
$\mathcal {F \dot N}^{s}_{p,\lambda ,q}(\mathbb R^3)$
denotes a set of all
$u\in \mathcal Z'(\mathbb R^3)$
such that
$$ \begin{align*}\|u\|_{\mathcal{F \dot N}^{s}_{p,\lambda,q}(\mathbb R^3)} =\left\{\sum_{j\in\mathbb Z}2^{jqs}\|\widehat{\dot{\Delta}_{j} u}\|_{M^{\lambda}_p}^{q} \right\}^{\frac{1}{q}}<\infty~~~\mathrm{for}~q<\infty,\end{align*} $$
with appropriate modifications for
$l^q$
-norm when
$q=\infty $
. The space
$\mathcal Z'(\mathbb R^3)$
denotes the topological dual of the space
$\mathcal Z'(\mathbb R^3)=\{f\in \mathcal S(\mathbb R^3);\partial ^\alpha \hat {f} (0)=0\ \mathrm {for\ every\ multi}{-} \mathrm{index\ \alpha } \}$
, and can be identified to the quotient spaces
$\mathcal S'(\mathbb R^3)/\mathcal P$
, where
$\mathcal P$
represents the set of all polynomials on
$\mathbb R^3$
.
Lemma 2.4 We have the equivalence of norms that
$\|\Lambda ^{s} u\|_{\mathcal {F\dot {N}}^{s'} _{p,\lambda ,q}}\leq C\|u\|_{\mathcal {F\dot {N}}^{s+s'}_{p,\lambda ,q}}$
, for
$s\in \mathbb R$
, where
$\Lambda =(-\Delta )^{\frac 12}$
and
$C>0$
is a pure constant .
Proof Using the definition of
$\mathcal {F\dot {N}}^{s'}_{p,\lambda ,q}(\mathbb R^3)$
, and
$|\xi |\approx 2^j$
for all
$j\in \mathbb Z$
, we have
$$ \begin{align*} \begin{aligned} \|\Lambda^s u\|_{\mathcal{F\dot{N}}^{s'}_{p,\lambda,q}} &= \left\{\sum_{j\in\mathbb Z}2^{jqs'}\|\widehat{\dot\Delta_j \Lambda^s u}\|^q_{M^\lambda_p}\right\}^{\frac1q}= \left\{\sum_{j\in\mathbb Z}2^{jqs'}\||\xi|^{s}\widehat{\dot\Delta_j u}\|^q_{M^\lambda_p}\right\}^{\frac1q}\\ &\leq C\left\{\sum_{j\in\mathbb Z}2^{jqs'}2^{jqs}\|\widehat{\dot\Delta_j u}\|^q_{M^\lambda_p}\right\}^{\frac1q}\leq C\|u\|_{\mathcal{F\dot{N}} ^{s'+ s}_{p,\lambda,q}}. \end{aligned}\\[-34pt] \end{align*} $$
Definition 2.3 Let
$s\in \mathbb R$
,
$1\leq p<\infty $
,
$1\leq q,\rho \leq \infty $
,
$0\leq \lambda <3$
and
$T\in (0,\infty ]$
. The space-time norm is defined on
$u(t,x)$
by
$$ \begin{align*}\|u(t,x)\|_{\mathcal L^\rho([0,T);\mathcal{F \dot N}^{s}_{p,\lambda,q})} =\left\{\sum_{j\in\mathbb Z}2^{jqs}\|\widehat{\dot{\Delta}_{j} u}\|_{\mathcal L^\rho([0,T);M^{\lambda}_p)}^{q} \right\}^{\frac{1}{q}},\end{align*} $$
and denote by
$\mathcal L^\rho ([0,T);\mathcal {F \dot N}^{s}_{p,\lambda ,q})$
the set of distributions in the space
$\mathcal S'(\mathbb R\times \mathbb R^3)/\mathcal P$
with finite
$\|\cdot \|_{\mathcal L^\rho ([0,T);\mathcal {F \dot N}^{s}_{p,\lambda ,q})}$
norm.
3 Linear and bilinear estimates
Next, we will prove Theorems 1.1 and 1.2. As preparations, we give the following lemmas, which shall be frequently used to prove the main theorem. Using the same arguments as in [Reference Ferreira and Villamizar-Roa8] and observing the structure of
$A(\xi )$
, it can be shown the following Lemma 3.1.
Lemma 3.1 [Reference Yuan and Ma20]
Let
$A(\xi )$
be defined in (1.7), then we have
$$ \begin{align} \|e^{-A(\xi)t}\|\leq e^{-C(\gamma)|\xi|^{2\gamma} t},\ \forall t>0,\ \xi\in\mathbb R^3. \end{align} $$
For some constant
$C(\gamma )=1-\frac {1}{2\gamma }>0$
independent of t and
$\xi $
, where
$\|\cdot \|$
denotes the maximum value norm.
Lemma 3.2 Let
$0<T\leq \infty $
,
$1\leq p<\infty $
,
$1\leq q\leq \infty $
,
$0\leq \lambda < 3$
and
$U_0\in \mathcal {F\dot {N}}^{4-2\gamma +\frac {\lambda -3}{p}}_{p,\lambda ,q} (\mathbb R^3)$
. Then there exists a constant
$C>0$
such that
$$ \begin{align} &\|G(t)U_0\|_{\mathcal L^1([0,T);\mathcal{F\dot{N}}^{4+\frac{\lambda-3} {p}}_{p,\lambda,q})}\leq C\|U_0\|_{\mathcal{F\dot{N}}^{4-2\gamma+\frac{\lambda-3}{p}}_{p,\lambda,q}}, \end{align} $$
$$ \begin{align} \|G(t)U_0\|_{\mathcal L^\infty([0,T);\mathcal{F\dot{N}}^{4-2\gamma+\frac{\lambda-3} {p}}_{p,\lambda,q})}\leq C\|U_0\|_{\mathcal{F\dot{N}}^{4-2\gamma+\frac{\lambda-3}{p}}_{p,\lambda,q}}, \end{align} $$
$$ \begin{align} \|G(t)U_0\|_{\mathcal L^{2r}([0,T);\mathcal{F\dot{N}}^{4+ \frac{\lambda-3}{p}-\frac{2\gamma}{(2r)'}}_{p,\lambda,q})}\leq C\|U_0\|_{\mathcal{F\dot{N}}^{4-2\gamma+\frac{\lambda-3}{p}}_{p,\lambda,q}}. \end{align} $$
Proof To prove the first inequality (3.15), from Definition 2.3, it is easy to see that
$$ \begin{align*} \begin{aligned} \|G(t)U_0\|_{\mathcal L^1([0,T);\mathcal{F\dot{N}}^{4+\frac{\lambda-3}{p}}_{p,\lambda,q})} &\leq C\left(\sum_{j\in\mathbb Z}2^{jq(4+\frac{\lambda-3}{p})}\|e^{-t2^{2\gamma j}}\widehat{\dot\Delta_j U_0}\|^q_{L^1([0,T);M^\lambda_p)}\right)^{\frac1q}\\ &\leq C\left\{\sum_{j\in\mathbb Z}2^{jq(4+\frac{\lambda-3}{p})}\left(\int^T_0e^{-t2^{2\gamma j}}\|\widehat{\dot\Delta_j U_0}\|_{M^\lambda_p}dt\right)^q\right\}^{\frac1q}\\ &\leq C\left(\sum_{j\in\mathbb Z}2^{jq(4+\frac{\lambda-3}{p})}2^{-2\gamma jq}\|\widehat {\dot\Delta_j U_0}\|^q_{M^\lambda_p}\right)^{\frac1q}\\ &\leq C\|U_0\|_{\mathcal{F\dot{N}}^{4-2\gamma+\frac{\lambda-3}{p}}_{p,\lambda,q}}. \end{aligned} \end{align*} $$
Similarly, for (3.16), using
$e^{-t2^{2\gamma j}}\leq 1$
, we have
$$ \begin{align*} \begin{aligned} \|G(t)U_0\|_{\mathcal L^\infty([0,T);\mathcal{F\dot{N}}^{4-2\gamma+\frac{\lambda-3} {p}}_{p,\lambda,q})} &\leq\left(\sum_{j\in\mathbb Z}2^{jq(4-2\gamma+\frac{\lambda-3}{p})}\|\widehat {\dot\Delta_j U_0}\|^q_{M^\lambda_p}\right)^{\frac1q}\\ &\leq C\|U_0\|_{\mathcal{F\dot{N}}^{4-2\gamma+\frac{\lambda-3}{p}}_{p,\lambda,q}}. \end{aligned} \end{align*} $$
To estimate the inequality (3.17), it suffices to write that
$$ \begin{align*} \begin{aligned} \|G(t)U_0\|_{\mathcal L^{2r}([0,T);\mathcal{F\dot{N}}^{4+ \frac{\lambda-3}{p}-\frac{2\gamma}{(2r)'}}_{p,\lambda,q})} &\leq C\left(\sum_{j\in\mathbb Z}2^{jq(4+\frac{\lambda-3}{p}-\frac{2\gamma}{(2r)'})}\|e^{-t2^{2\gamma j}}\widehat {\dot\Delta_j U_0}\|^q_{L^{2r}([0,T);M^\lambda_p)}\right)^{\frac1q}\\ &\leq C\left\{\sum_{j\in\mathbb Z}2^{jq(4+\frac{\lambda-3}{p}-\frac{2\gamma}{(2r)'})}\left(\int^T_0e^{-2rt2^{2\gamma j}}\|\widehat {\dot\Delta_j U_0}\|^{2r}_{M^\lambda_p}dt\right)^{\frac{q}{2r}}\right\}^{\frac1q}\\ &\leq C\|U_0\|_{\mathcal{F\dot{N}}^{4-2\gamma+\frac{\lambda-3}{p}}_{p,\lambda,q}}, \end{aligned} \end{align*} $$
which finish the proof.
Lemma 3.3 Let
$0<T\leq \infty $
,
$s\in \mathbb R$
,
$1\leq p<\infty $
,
$1\leq q,\rho \leq \infty $
,
$1\leq r\leq \rho $
,
$0\leq \lambda < 3$
and
$f\in \mathcal L^\rho ([0,T); \mathcal {F\dot {N}}^s_{p,\lambda ,q})$
. Then there exists a constant
$C>0$
such that
$$ \begin{align*} \|\int^t_0G(t-\tau)f(\tau)d\tau\|_{\mathcal L^\rho\left([0,T);\mathcal{F\dot{N}}^s_{p,\lambda,q}\right)}\leq C\|f\|_{\mathcal L^r([0,T);{\mathcal{F\dot{N}}^{s-2\gamma(1+\frac1\rho-\frac1r)}_{p,\lambda,q}})}. \end{align*} $$
Proof Set
$1+\frac 1\rho =\frac {1}{\tilde {\rho }}+\frac 1r$
. By Definition 2.3 and the Young’s inequality, we obtain that
$$ \begin{align*} \begin{aligned} &\ \ \ \ \|\int^t_0G(t-\tau)f(\tau)d\tau\|_{{\mathcal L}^{\rho}\left([0,T);\mathcal{F\dot{N}}^s_{p,\lambda,q}\right)}\\ &\leq\left\{\sum_{j\in{\mathbb Z}}2^{jqs}\left(\int^T_0\|\int^t_0 e^{-(t-\tau)2^{2\gamma j}}\widehat{\dot{\Delta}_j f}(\tau)d\tau\|^\rho_{M^\lambda_p}dt\right)^{\frac q\rho}\right\}^{\frac1q}\\ &\leq \left\{\sum_{j\in\mathbb Z}2^{jqs}\left(\int^T_0 e^{-\tilde{\rho}t2^{2\gamma j}}dt\right)^{\frac{q}{\tilde{\rho}}}\|\widehat{\dot\Delta_j f}(\tau)d\tau\|^q_{L^r([0,T);M^\lambda_p)}dt\right\}^{\frac1q}\\ &\leq C\left\{\sum_{j\in\mathbb Z}2^{jq(s-2\gamma(1+\frac1\rho-\frac1r))}\|\widehat{\dot\Delta_j f}(\tau)d\tau\|^q_{L^r([0,T);M^\lambda_p)}dt\right\}^{\frac1q}\\ &\leq C\|f\|_{\mathcal L^r([0,T);{\mathcal{F\dot{N}}^{s-2\gamma(1+\frac1\rho-\frac1r)} _{p,\lambda,q}})}. \end{aligned} \end{align*} $$
Thus the proof is completed.
Lemma 3.4 Let
$0<T\leq \infty $
,
$1\leq p<\infty $
,
$\frac {1}{2}<\gamma <\frac {5+\frac {\lambda -3}{p}}{4-\frac {2}{\sigma }}$
,
$1\leq q\leq \infty $
,
$1\leq \sigma \leq \infty $
and
$0\leq \lambda < 3$
, then there exists a constant
$C>0$
such that
$$ \begin{align*} \begin{aligned} \|uv\|_{\mathcal L^\sigma([0,T);\mathcal{F\dot{N}}^{5-4\gamma+\frac{2\gamma}{\sigma}+\frac{\lambda-3}{p}}_{p,\lambda,q})}&\leq C \|u\|_{\mathcal L^\sigma([0,T);\mathcal{F\dot{N}}^{4-2\gamma+\frac{2\gamma}{\sigma}+\frac{\lambda-3}{p}}_{p,\lambda,q})}\|v\|_{\mathcal L^\infty([0,T);\mathcal{F\dot{N}}^{4-2\gamma+ \frac{\lambda-3}{p}}_{p,\lambda,q})}\\ &+C\|v\|_{\mathcal L^\sigma([0,T);\mathcal{F\dot{N}}^{4-2\gamma+\frac{2\gamma}{\sigma}+\frac{\lambda-3}{p}}_{p,\lambda,q})}\|u\|_{\mathcal L^\infty([0,T);\mathcal{F\dot{N}}^{4-2\gamma+ \frac{\lambda-3}{p}}_{p,\lambda,q})}. \end{aligned} \end{align*} $$
Remark 3.1 The above Lemma 3.4 also hold true for the case
$\gamma =\frac {1}{2}$
, provided we choose
$q=1$
.
Proof Applying Bony’s decomposition, we have
$$ \begin{align*} \begin{aligned} \dot\Delta_j(uv)&=\sum_{|k-j|\leq4}\dot\Delta_j(\dot S_{k-1}u\dot\Delta_kv)+\sum_{|k-j|\leq4}\dot\Delta_j(\dot S_{k-1}v\dot\Delta_ku)+\sum_{k\geq j-3}\dot\Delta_j(\dot\Delta_{k}u\tilde{\dot\Delta}_kv)\\ &:=I_1+I_2+I_3. \end{aligned} \end{align*} $$
By the triangular inequality, one finds that
$$ \begin{align} \begin{aligned} \|uv\|_{\mathcal L^\sigma([0,T);\mathcal{F\dot{N}}^{5-4\gamma+\frac{2\gamma}{\sigma}+\frac{\lambda-3}{p}}_{p,\lambda,q})} &\leq C \left(\sum_{j\in\mathbb Z}2^{jq(5-4\gamma+\frac{2\gamma}{\sigma}+\frac{\lambda-3}{p})}\|\hat I_1\|^q_{L^\sigma([0,T);M^\lambda_p)}\right)^{\frac1q}\\ & \quad+C\left(\sum_{j\in\mathbb Z}2^{jq(5-4\gamma+\frac{2\gamma}{\sigma}+\frac{\lambda-3}{p})}\|\hat I_2\|^q_{L^\sigma([0,T);M^\lambda_p)}\right)^{\frac1q}\\ & \quad +C\left(\sum_{j\in\mathbb Z}2^{jq(5-4\gamma+\frac{2\gamma}{\sigma}+\frac{\lambda-3}{p})}\|\hat I_3\|^q_{L^\sigma([0,T);M^\lambda_p)}\right)^{\frac1q}\\ &:=J_1+J_2+J_3. \end{aligned} \end{align} $$
We firstly evaluate
$J_1$
. By using Young’s inequality (2.12) and Hölder inequality, one finds
$$ \begin{align*} \begin{aligned} &\ \ \ \|\hat I_1\|_{L^\sigma([0,T);M^\lambda_p)}\\ &\leq \sum_{|k-j|\leq4}\|\widehat{\dot S_{k-1}u{\dot\Delta_{k}}v}\ \ \|_{L^\sigma([0,T);M^\lambda_p)}\\ &\leq \sum_{|k-j|\leq4}\|\hat v_k\|_{L^\sigma([0,T);M^\lambda_p)}\sum_{l\leq k-2}2^{l(3+\frac{\lambda-3}{p})}\|\hat u_l\|_{L^\infty([0,T);M^\lambda_p)}\\ &\leq \sum_{|k-j|\leq4}\|\hat v_k\|_{L^\sigma([0,T);M^\lambda_p)}\left(\sum_{l\leq k-2}2^{lq(4-2\gamma+\frac{\lambda-3}{p})}\|\hat u_l\|^q_{L^\infty([0,T);M^\lambda_p)}\right)^{\frac1q}\left(\sum_{l\leq k-2}2^{(2\gamma-1)lq'}\right)^{\frac{1}{q'}}\\ &\leq C \sum_{|k-j|\leq4}2^{(2\gamma-1)k}\|\hat v_k\|_{L^\sigma([0,T);M^\lambda_p)}\|u\|_{\mathcal L^\infty([0,T);\mathcal {F\dot{N}}^{4-2\gamma+\frac{\lambda-3}{p}}_{p,\lambda,q})}. \end{aligned} \end{align*} $$
Taking
$l^q$
-norm, we obtain
$$ \begin{align*} \begin{aligned} J_1&\leq C \left\{\sum_{j\in\mathbb Z}\left(\sum_{|k-j|\leq4}2^{(j-k)(5-4\gamma+\frac{2\gamma}{\sigma}+\frac{\lambda-3}{p})}2^{(4-2\gamma+ \frac{2\gamma}{\sigma}+\frac{\lambda-3}{p})k}\|\hat v_k\|_{L^\sigma([0,T);M^\lambda_p)}\right)^q\right\}^{\frac1q}\|u\|_{\mathcal L^\infty ([0,T);\mathcal{F\dot{N}}^{4-2\gamma+ \frac{\lambda-3}{p}}_{p,\lambda,q})}\\ &\leq C\|v\|_{\mathcal L^\sigma([0,T);\mathcal{F\dot{N}}^{4-2\gamma+\frac{2\gamma}{\sigma}+\frac{\lambda-3}{p}}_{p,\lambda,q})}\|u\|_{\mathcal L^\infty ([0,T);\mathcal{F\dot{N}}^{4-2\gamma+ \frac{\lambda-3}{p}}_{p,\lambda,q})}.\\ \end{aligned} \end{align*} $$
Similarly, we have
$$ \begin{align*} \begin{aligned} J_2\leq C\|u\|_{\mathcal L^\sigma([0,T);\mathcal{F\dot{N}}^{4-2\gamma+\frac{2\gamma}{\sigma}+\frac{\lambda-3}{p}}_{p,\lambda,q})}\|v\|_{\mathcal L^\infty ([0,T);\mathcal{F\dot{N}}^{4-2\gamma+ \frac{\lambda-3}{p}}_{p,\lambda,q})}.\\ \end{aligned} \end{align*} $$
To estimate
$I_3$
, we use a different approach. First, we use the Young’s inequality (2.12) to get
$$ \begin{align*} & \|\hat I_3\|_{L^\sigma([0,T);M^\lambda_p)}\\ &\leq \sum_{k\geq j-3}\sum_{|k-i|\leq1}\|\hat u_k\|_{L^\sigma([0,T);M^\lambda_p)}2^{i(3+\frac{\lambda-3}{p})}\|\hat v_i\|_{L^\infty([0,T);M^\lambda_p)}\\ &\leq C \sum_{k\geq j-3}\left(\sum_{|k-i|\leq1}2^{(i-k)(2\gamma-1)}2^{i(4-2\gamma+\frac{\lambda-3}{p})}\|\hat v_i\|_{L^\infty([0,T);M^\lambda_p)}\right) 2^{(2\gamma-1)k}\|\hat u_k\|_{L^\sigma([0,T);M^\lambda_p)}\\ &\leq C \sum_{k\geq j-3}2^{(2\gamma-1)k}\|\hat u_k\|_{L^\sigma([0,T);M^\lambda_p)}\|v\|_{\mathcal L^\infty ([0,T);\mathcal{F\dot{N}}^{4-2\gamma+ \frac{\lambda-3}{p}}_{p,\lambda,q})}. \end{align*} $$
Then we have
$$ \begin{align*} \begin{aligned} &\ \ \ 2^{j(5-4\gamma+\frac{2\gamma}{\sigma}+\frac{\lambda-3}{p})}\|\hat I_3\|_{L^\sigma([0,T);M^\lambda_p)}\\ &\leq C \|v\|_{\mathcal L^\infty ([0,T);\mathcal{F\dot{N}}^{4-2\gamma+ \frac{\lambda-3}{p}}_{p,\lambda,q})}\sum_{k\geq j-3}2^{(j-k)(5-4\gamma+\frac{2\gamma}{\sigma}+\frac{\lambda-3}{p})}2^{(4-2\gamma+\frac{2\gamma}{\sigma}+\frac{\lambda-3}{p})k}\|\hat u_k\|_{L^\sigma([0,T);M^\lambda_p)}. \end{aligned} \end{align*} $$
Since
$\gamma < \frac {5+\frac {\lambda -3}{p}}{4-\frac {2}{\sigma }}$
, we have
$5-4\gamma +\frac {2\gamma }{\sigma }+\frac {\lambda -3}{p}>0$
. By taking
$l^q$
-norm on both sides in the above estimate, and then we apply Young’s inequality for series to get
$$ \begin{align*} \begin{aligned} J_3\leq C\|u\|_{\mathcal L^\sigma([0,T);\mathcal{F\dot{N}}^{4-2\gamma+\frac{2\gamma}{\sigma}+\frac{\lambda-3}{p}}_{p,\lambda,q})}\|v\|_{\mathcal L^\infty([0,T);\mathcal{F\dot{N}}^{4-2\gamma+ \frac{\lambda-3}{p}}_{p,\lambda,q})}. \end{aligned} \end{align*} $$
Thus the proof is complete.
Lemma 3.5 Let
$0<T\leq \infty $
,
$1< p<\infty $
,
$\gamma>\frac {1}{2}$
,
$1\leq q\leq \infty $
and
$0\leq \lambda < 3$
, and set
$$ \begin{align*}Y=\mathcal L^{2r}([0,T);\mathcal{F\dot{N}}^{4+ \frac{\lambda-3}{p}-\frac{2\gamma}{(2r)'}}_{p,\lambda,q}), \end{align*} $$
there exists a constant
$C=C(p,q)>0$
depending on p, q such that
$$ \begin{align*} \begin{aligned} \|uv\|_{\mathcal L^r([0,T);\mathcal{F\dot{N}}^{5-\frac{4\gamma}{(2r)'}+\frac{\lambda-3}{p}}_{p,\lambda,q})}\leq C \|u\|_Y\|v\|_Y. \end{aligned} \end{align*} $$
where
$(2r)'$
is the conjugate of
$2r$
satisfying
$\frac {1}{2r}+\frac {1}{(2r)'}=1$
for
$1<r<\infty $
. We choose r satisfying
$1-\frac {1}{2\gamma }(\frac 52+\frac {\lambda -3}{2p})<\frac {1}{2r}<1-\frac {1}{2\gamma }$
such that
$1<\frac {2\gamma }{(2r)'}<\frac 52+\frac {\lambda -3}{2p}$
.
Proof Using the same methods as in (3.18), we have
$$ \begin{align*} \begin{aligned} \|uv\|_{\mathcal L^r([0,T);\mathcal{F\dot{N}}^{5-\frac{4\gamma}{(2r)'}+\frac{\lambda-3}{p}}_{p,\lambda,q})} &\leq C \left(\sum_{j\in\mathbb Z}2^{jq(5-\frac{4\gamma}{(2r)'}+\frac{\lambda-3}{p})}\|\hat I_1\|^q_{L^r([0,T);M^\lambda_p)}\right)^{\frac1q}\\ & \quad+C\left(\sum_{j\in\mathbb Z}2^{jq(5-\frac{4\gamma}{(2r)'}+\frac{\lambda-3}{p})}\|\hat I_2\|^q_{L^r([0,T);M^\lambda_p)}\right)^{\frac1q}\\ & \quad +C\left(\sum_{j\in\mathbb Z}2^{jq(5-\frac{4\gamma}{(2r)'}+\frac{\lambda-3}{p})}\|\hat I_3\|^q_{L^r([0,T);M^\lambda_p)}\right)^{\frac1q}. \end{aligned} \end{align*} $$
We evaluate the above three terms separately, first, using Young’s inequality (2.12) and Hölder inequality, one finds
$$ \begin{align*} \begin{aligned} &\ \ \ \|\hat I_1\|_{L^r([0,T);M^\lambda_p)}\\ &\leq \sum_{|k-j|\leq4}\|\hat v_k\|_{L^{2r}([0,T);M^\lambda_p)}\sum_{l\leq k-2}\|\hat u_l\|_{L^{2r}([0,T);L^1)}\\ &\leq \sum_{|k-j|\leq4}\|\hat v_k\|_{L^{2r}([0,T);M^\lambda_p)}\sum_{l\leq k-2}2^{l(3+\frac{\lambda-3}{p})}\|\hat u_l\|_{L^{2r}([0,T);M^\lambda_p)}\\ &\leq \sum_{|k-j|\leq4}\|\hat v_k\|_{L^{2r}([0,T);M^\lambda_p)}\left(\sum_{l\leq k-2}2^{lq(4-\frac{2\gamma}{(2r)'}+\frac{\lambda-3}{p})}\|\hat u_l\|^q_{L^{2r}([0,T);M^\lambda_p)}\right)^{\frac1q}\left(\sum_{l\leq k-2}2^{(\frac{2\gamma}{(2r)'}-1)lq'}\right)^{\frac{1}{q'}}\\ &\leq C \sum_{|k-j|\leq4}2^{(\frac{2\gamma}{(2r)'}-1)k}\|\hat v_k\|_{L^{2r}([0,T);M^\lambda_p)}\|u\|_Y. \end{aligned} \end{align*} $$
Taking
$l^q$
-norm, we obtain
$$ \begin{align*} \begin{aligned} &\left(\sum_{j\in\mathbb Z}2^{jq(5-\frac{4\gamma}{(2r)'}+\frac{\lambda-3}{p})}\|\hat I_1\|^q_{L^r([0,T);M^\lambda_p)}\right)^{\frac1q}\\ &\leq C \left\{\sum_{j\in\mathbb Z}\left(\sum_{|k-j|\leq4}2^{(j-k)(5-\frac{4\gamma}{(2r)'}+\frac{\lambda-3}{p})} 2^{(4-\frac{2\gamma}{(2r)'}+\frac{\lambda-3}{p})k}\|\hat v_k\|_{L^{2r}([0,T);M^\lambda_p)}\right)^q\right\}^{\frac1q}\|u\|_Y\\ &\leq C\|v\|_Y\|u\|_Y.\\ \end{aligned} \end{align*} $$
Similarly, we have
$$ \begin{align*} \begin{aligned} \left(\sum_{j\in\mathbb Z}2^{jq(5-\frac{4\gamma}{(2r)'}+\frac{\lambda-3}{p})}\|\hat I_2\|^q_{L^r([0,T);M^\lambda_p)}\right)^{\frac1q}\leq C\|u\|_Y\|v\|_Y.\\ \end{aligned} \end{align*} $$
To estimate
$I_3$
, we use a different approach. First, we use the Young’s inequality (2.12) to get
$$ \begin{align*} \begin{aligned} &\ \ \ \|\hat I_3\|_{L^r([0,T);M^\lambda_p)}\\ &\leq \sum_{k\geq j-3}\sum_{|k-i|\leq1}\|\hat u_k\|_{L^{2r}([0,T);M^\lambda_p)}2^{i(3+\frac{\lambda-3}{p})}\|\hat v_i\|_{L^{2r}([0,T);M^\lambda_p)}\\ &\leq C \sum_{k\geq j-3}\left(\sum_{|k-i|\leq1}2^{(1-\frac{2\gamma}{(2r)'})(k-i)}2^{i(4-\frac{2\gamma}{(2r)'}+ \frac{\lambda-3}{p})}\|\hat v_i\|_{L^{2r}([0,T);M^\lambda_p)}\right) 2^{(\frac{2\gamma}{(2r)'}-1)k}\|\hat u_k\|_{L^{2r}([0,T);M^\lambda_p)}\\ &\leq C \sum_{k\geq j-3}2^{(\frac{2\gamma}{(2r)'}-1)k}\|\hat u_k\|_{L^{2r}([0,T);M^\lambda_p)}\|v\|_Y. \end{aligned} \end{align*} $$
Then we have
$$ \begin{align*} \begin{aligned} &\ \ \ 2^{jq(5-\frac{4\gamma}{(2r)'}+\frac{\lambda-3}{p})}\|\hat I_3\|_{L^r([0,T);M^\lambda_p)}\\ &\leq C \|v\|_Y\sum_{k\geq j-3}2^{(j-k)(5-\frac{4\gamma}{(2r)'}+\frac{\lambda-3}{p})}2^{(4-\frac{2\gamma}{(2r)'}+\frac{\lambda-3}{p})k}\|\hat u_k\|_{L^{2r}([0,T);M^\lambda_p)}. \end{aligned} \end{align*} $$
Since
$1< p<\infty $
and
$1<\frac {2\gamma }{(2r)'}<\frac 52+\frac {\lambda -3}{2p}$
, we have
$5-\frac {4\gamma }{(2r)'}+\frac {\lambda -3}{p}>0$
. By taking
$l^q$
-norm on both sides in the above estimate, and then we apply Young’s inequality for series to get
$$ \begin{align*} \begin{aligned} \left(\sum_{j\in\mathbb Z}2^{jq(5-\frac{4\gamma}{(2r)'}+\frac{\lambda-3}{p})}\|\hat I_3\|^q_{L^r([0,T);M^\lambda_p)}\right)^{\frac1q}\lesssim\|u\|_Y\|v\|_Y.\\ \end{aligned} \end{align*} $$
Thus the proof is complete.
4 Proof of Theorems 1.1 and 1.2
To ensure the existence of a global mild solution with small initial data of the system (1.1), we will use the linear and bilinear estimates that we have established in Sections 3.
Proof of Theorem 1.1
Let
$0<T\leq \infty $
,
$1\leq p<\infty $
,
$1\leq q\leq \infty $
,
$0\leq \lambda < 3$
, and set
$$ \begin{align*}X:=\mathcal L^1\left([0,T);\mathcal{F\dot{N}}^{4+\frac{\lambda-3}{p}}_{p,\lambda,q}\right)\cap \mathcal L^\infty\left([0,T);\mathcal{F\dot{N}}^{4-2\gamma+\frac{\lambda-3}{p}}_{p,\lambda,q}\right),\end{align*} $$
with the norm
$$ \begin{align*}\|u\|_X=\|u\|_{\mathcal L^1([0,T);\mathcal{F\dot{N}}^{4+\frac{\lambda-3}{p}}_{p,\lambda,q})}+ \|u\|_{\mathcal L^\infty([0,T);\mathcal{F\dot{N}}^{4-2\gamma+\frac{\lambda-3}{p}}_ {p,\lambda,q})}. \end{align*} $$
Then, we begin with the integral equation
$$ \begin{align} U(t)=G(t)U_0-\int_{0}^{t}G(t-\tau)\mathbb P\nabla\cdot(u\otimes U)(\tau)d\tau, \end{align} $$
and we consider the bilinear operator B given by
$$ \begin{align} B(U_1,U_2)=\int_{0}^{t}G(t-\tau)\mathbb P\nabla\cdot(u_1\otimes U_2)(\tau)d\tau, \end{align} $$
where
$$ \begin{align*} U_1(x,t)=\left(\begin{aligned} u_1(x,t)\\ w_1(x,t)\\ \end{aligned}\right), \ \ U_2(x,t)=\left(\begin{aligned} u_2(x,t)\\ w_2(x,t)\\ \end{aligned}\right). \end{align*} $$
By Lemma 3.3, we have
$$ \begin{align*}\|\int_{0}^{t}G(t-\tau)\mathbb P\nabla\cdot(u_1\otimes U_2)d\tau\|_{\mathcal L^\infty([0,T);\mathcal{F\dot{N}}^{4-2\gamma+\frac{\lambda-3}{p}}_ {p,\lambda,q})}\leq C\|\mathbb P\nabla\cdot(u_1\otimes U_2)\|_{\mathcal L^1([0,T);\mathcal{F\dot{N}}^{4-2\gamma+\frac{\lambda-3}{p}}_ {p,\lambda,q})},\end{align*} $$
and
$$ \begin{align*}\|\int_{0}^{t}G(t-\tau)\mathbb P\nabla\cdot(u_1\otimes U_2)d\tau\|_{\mathcal L^1([0,T);\mathcal{F\dot{N}}^{4+\frac{\lambda-3}{p}}_{p,\lambda,q})}\leq C\|\mathbb P\nabla\cdot(u_1\otimes U_2)\|_{\mathcal L^1([0,T);\mathcal{F\dot{N}}^{4-2\gamma+\frac{\lambda-3}{p}}_ {p,\lambda,q})}.\end{align*} $$
Then, according to Lemmas 2.4, 3.3, and 3.4, we get
$$ \begin{align} \begin{aligned} \|B(U_1,U_2)\|_X &\leq C\|\mathbb P\nabla\cdot(u_1\otimes U_2)\|_{\mathcal L^1([0,T);\mathcal{F\dot{N}}^{4-2\gamma+\frac{\lambda-3}{p}}_ {p,\lambda,q})}\\ &\leq C \|u_1\|_{\mathcal L^{\infty}([0,T);\mathcal{F\dot{N}}^{4-2\gamma+ \frac{\lambda-3}{p}}_{p,\lambda,q})}\|U_2\|_{\mathcal L^{1}([0,T);\mathcal{F\dot{N}}^{4+ \frac{\lambda-3}{p}}_{p,\lambda,q})}\\ &\ \ \ +C\|U_2\|_{\mathcal L^{\infty}([0,T);\mathcal{F\dot{N}}^{4-2\gamma+ \frac{\lambda-3}{p}}_{p,\lambda,q})}\|u_1\|_{\mathcal L^{1}([0,T);\mathcal{F\dot{N}}^{4+ \frac{\lambda-3}{p}}_{p,\lambda,q})}\\ &\leq C_2\|U_1\|_X\|U_2\|_X. \end{aligned} \end{align} $$
Lemma 3.2 yields
$$ \begin{align} \|G(t)U_0\|_X\leq C_1\|U_0\|_{\mathcal{F\dot{N}}^{4-2\gamma+\frac{\lambda-3}{p}}_ {p,\lambda,q}}. \end{align} $$
If
$\|U_0\|_{\mathcal {F\dot {N}}^{4-2\gamma +\frac {\lambda -3}{p}}_ {p,\lambda ,q}}< \frac {1}{4C_1C_2}$
, then by Lemma 2.3, there has a unique global solution
$U\in X$
satisfying
$$ \begin{align*} \|U\|_X\leq 2C_1\|U_0\|_{\mathcal{F\dot{N}}^{4-2\gamma+\frac{\lambda-3}{p}}_ {p,\lambda,q}}. \end{align*} $$
Thus the proof of the global existence is complete.
As for continuity, by using the definition of the Fourier–Besov–Morrey spaces, we have
$$ \begin{align*} \begin{aligned} &\ \ \ \|u(t_1)-u(t_2)\|^q_{\mathcal{F\dot{N}}^{4-2\gamma+\frac{\lambda-3}{p}}_ {p,\lambda,q}}\\ &\leq\sum_{j\leq N}2^{jq(4-2\gamma+\frac{\lambda-3}{p})}\|\hat u_j(t_1)-\hat u_j(t_2)\|^q_{M^\lambda}+2\sum_{j>N}2^{jq(4-2\gamma+\frac{\lambda-3}{p})}\|\hat u_j(t)\|^q_{L^\infty([0,T);M^\lambda_p)}, \end{aligned} \end{align*} $$
where we denote
$\hat u_j=\varphi _j\hat u$
. For any small constant
$\varepsilon>0$
, let N be large enough so that
$$ \begin{align*}\sum_{j>N}2^{jq(4-2\gamma+\frac{\lambda-3}{p})}\|\hat u_j(t)\|^q_{L^\infty([0,T);M^\lambda_p)}\leq\frac{\varepsilon}{4}.\end{align*} $$
Using the same arguments as in [Reference Zhao, Liu and Cui21], we get
$$ \begin{align} &\ \ \ \sum_{j\leq N}2^{jq(4-2\gamma+\frac{\lambda-3}{p})}\|\hat u_j(t_1)-\hat u_j(t_2)\|^q_{M^\lambda_p}\\ &\leq C |t_1-t_2|^q\sum_{j\leq N}2^{jq(4-2\gamma+\frac{\lambda-3}{p})}\sup_{t_1\leq t\leq t_2}\|\widehat {(\partial_tu)}_j\|^q_{M^\lambda_p}\nonumber\\ &\leq C |t_1-t_2|^q\sum_{j\leq N}2^{jq(4-2\gamma+\frac{\lambda-3}{p})}\nonumber\\&\ \ \ \times\sup_{t_1\leq t\leq t_2}\left(\|\widehat{(-\Delta)^\gamma} u_j \|^q_{{M^\lambda_p}}+\|\widehat{\mathbb P\nabla\cdot(u_j\otimes u_j)} \|^q_{M^\lambda_p}+\|\widehat{\nabla\times w_j}\|^q_{M^\lambda_p}\right)\nonumber\\ &\leq C |t_1-t_2|^q\left(2^{2\gamma qN}\|u \|^q_{\mathcal L^\infty([0,T);\mathcal{F\dot N}_{p,\lambda,q}^{4-2\gamma+\frac{\lambda-3}{p}})}+2^{2\gamma qN}\|\nabla\cdot(u\otimes u) \|^q_{\mathcal L^\infty([0,T);\mathcal{F\dot N}_{p,\lambda,q}^{4-4\gamma+\frac{\lambda-3}{p}})}\right)\nonumber\\ &\ \ \ +C|t_1-t_2|^q2^{Nq}\|w\|^q_{\mathcal L^\infty([0,T);\mathcal{F\dot N}_{p,\lambda,q}^{4-2\gamma+\frac{\lambda-3}{p}})}.\nonumber \end{align} $$
Applying Lemma 2.4 and taking
$\sigma =\infty $
in Lemma 3.4, we have
$$ \begin{align} \begin{aligned} &\ \ \ \|\nabla\cdot(u\otimes u) \|^q_{\mathcal L^\infty([0,T);\mathcal{F\dot N}_{p,\lambda,q}^{4-4\gamma+\frac{\lambda-3}{p}})}\\ &\leq C\|u\otimes u \|^q_{\mathcal L^\infty([0,T);\mathcal{F\dot N}_{p,\lambda,q}^{5-4\gamma+\frac{\lambda-3}{p}})}\\ &\leq C\|u \|^{2q}_{\mathcal L^\infty([0,T);\mathcal{F\dot N}_{p,\lambda,q}^{5-2\gamma+\frac{\lambda-3}{p}})}. \end{aligned} \end{align} $$
Substituting (4.24) into (4.23) to obtain that
$$ \begin{align*} \begin{aligned} &\ \ \ \sum_{j\leq N}2^{jq(4-2\gamma+\frac{\lambda-3}{p})}\|\hat u_j(t_1)-\hat u_j(t_2)\|^q_{M^\lambda_p}\\ &\leq C |t_1-t_2|^q\left(2^{2\gamma qN}\|u \|^q_{\mathcal L^\infty([0,T);\mathcal{F\dot N}_{p,\lambda,q}^{4-2\gamma+\frac{\lambda-3}{p}})}+2^{2\gamma qN}\|u \|^{2q}_{\mathcal L^\infty([0,T);\mathcal{F\dot N}_{p,\lambda,q}^{5-2\gamma+\frac{\lambda-3}{p}})}\right)\\ &\ \ \ +C|t_1-t_2|^q2^{Nq}\|w\|^q_{\mathcal L^\infty([0,T);\mathcal{F\dot N}_{p,\lambda,q}^{4-2\gamma+\frac{\lambda-3}{p}})}\\ &\leq C |t_1-t_2|^q\left(\|u_0\|^q_{\mathcal L^\infty([0,T);\mathcal{F\dot N}_{p,\lambda,q}^{4-2\gamma+\frac{\lambda-3}{p}})}+\|u_0 \|^{2q}_{\mathcal L^\infty([0,T);\mathcal{F\dot N}_{p,\lambda,q}^{5-2\gamma+\frac{\lambda-3}{p}})}\|w_0\|^q_{\mathcal L^\infty([0,T);\mathcal{F\dot N}_{p,\lambda,q}^{4-2\gamma+\frac{\lambda-3}{p}})}\right)\\ & \leq\Big(\frac{\varepsilon}{2}\Big)^q\\ \end{aligned} \end{align*} $$
provided
$|t_1-t_2|<\delta $
for some a small enough
$\delta $
. Thus,
$\|u(t_1)-u(t_2)\|_{\mathcal {F\dot {N}}^{4-2\gamma +\frac {\lambda -3}{p}}_ {p,\lambda ,q}}\le \varepsilon $
, we obtain the continuity of u in time t. Similarly, we use the same discussion to get the continuity of w in time t. Hence
$(u,w)\in \mathcal C \left ([0,T);\mathcal {F\dot N}_{p,\lambda ,q}^{4-2\gamma +\frac {\lambda -3}{p}}\right )$
. Thus the proof of the Theorem 1.1 is complete.
Proof of Theorem 1.2
Now, we will use Lemma 3.5 to prove the local existence. We set
$$ \begin{align*} Y=\mathcal L^{2r}([0,T);\mathcal{F\dot{N}}^{4+ \frac{\lambda-3}{p}-\frac{2\gamma}{(2r)'}}_{p,\lambda,q}), \end{align*} $$
Similarly, according to Lemmas 2.4, 3.3, and 3.5, we get
$$ \begin{align} \begin{aligned} \|B(U_1,U_2)\|_Y &\leq C\|\mathbb P\nabla\cdot(u_1\otimes U_2)\|_{\mathcal L^r([0,T);\mathcal{F\dot{N}}^{4-\frac{4\gamma}{(2r)'}+ \frac{\lambda-3}{p}}_{p,\lambda,q})}\\ &\leq C\|u_1U_2\|_{\mathcal L^r([0,T);\mathcal{F\dot{N}}^{5-\frac{4\gamma}{(2r)'}+\frac{\lambda-3}{p}}_{p,\lambda,q})}\\ &\leq C\|U_1\|_Y\|U_2\|_Y. \end{aligned} \end{align} $$
Lemma 3.2 yields
$$ \begin{align} \|G(t)U_0\|_Y\leq C\|U_0\|_{\mathcal{F\dot{N}}^{4-2\gamma+\frac{\lambda-3}{p}}_ {p,\lambda,q}}. \end{align} $$
We divide the initial data
$U_0$
into two terms:
$$ \begin{align*} \begin{aligned} U_0=\mathcal F^{-1}(\chi_{B(0,\delta)}\widehat U_0)+\mathcal F^{-1}(\chi_{B^C(0,\delta)}\widehat U_0):=U_{0,1}+U_{0,2}, \end{aligned} \end{align*} $$
where
$\delta =\delta (U_0)>0$
is a real number to be determined later. Since
$U_{0}\in \mathcal {F\dot {N}}^{4-2\gamma +\frac {\lambda -3}{p}}_ {p,\lambda ,q}$
, by (4.26) we choose
$\delta $
large enough satisfying
$$ \begin{align*}\|G(t)U_{0,2}\|_Y\leq \frac{\varepsilon}{2},\end{align*} $$
thus, we get
$$ \begin{align*}\|G(t)U_0\|_Y\leq \|G(t)U_{0,1}\|_{Y}+\frac{\varepsilon}{2}.\end{align*} $$
For the first term
$U_{0,1}$
, applying
$l^q$
-norm to it and using the fact that
$|\xi |\approx 2^j\le \delta $
for every
$j\in \mathbb Z$
, we have
$$ \begin{align*} \begin{aligned} \|G(t)U_{0,1}\|_Y &\leq\left\{\sum_{j\in\mathbb Z}2^{jq(4+\frac{\lambda-3}{p}-\frac{2\gamma} {(2r)'})}\|e^{-t|\xi|^{2\gamma}}\widehat{\dot\Delta_j U_{0,1}}\|^q_{L^{2r}([0,T);M^\lambda_p)}\right\}^{\frac1q}\\ &\leq \left\{\sum_{j\in\mathbb Z}2^{jq(4+\frac{\lambda-3}{p}-\frac{2\gamma} {(2r)'})}\|\widehat{\dot\Delta_j U_{0}}\|^q_{L^{2r}([0,T);M^\lambda_p)}\right\}^{\frac1q}\\ &\leq C_3\delta^{\frac{2\gamma}{2r}}T^{\frac{1}{2r}}\|U_0\|_{\mathcal{F\dot{N}}^{4-2\gamma+\frac{\lambda-3}{p}}_ {p,\lambda,q}}. \end{aligned} \end{align*} $$
Therefore, if we choose T such that
$$ \begin{align} C_3\delta^{\frac{2\gamma}{2r}}T^{\frac{1}{2r}}\|U_0\|_{\mathcal{F\dot{N}}^{4-2\gamma+\frac{\lambda-3}{p}}_ {p,\lambda,q}}\leq\frac\varepsilon 2, \end{align} $$
then we have
$$ \begin{align*} \|G(t)U_{0}\|_Y\leq \varepsilon. \end{align*} $$
Applying Lemma 2.3, we get a fixed point in the closed ball
$\overline B(0,2\varepsilon )=\{x\in Y : \ \|x\|_{Y}\leq 2\varepsilon \}$
. Thus for arbitrary
$U_0$
in
$\mathcal {F\dot {N}}^{4-2\gamma +\frac {\lambda -3}{p}}_ {p,\lambda ,q}$
, (4.19) has a unique local solution in Y on
$[0,T)$
, where
$$ \begin{align*} T\leq\left(\frac{\varepsilon}{2C_3\delta^{\frac{2\gamma}{2r}}\|U_0\|_{\mathcal{F\dot{N}}^{4-2\gamma+\frac{\lambda-3}{p}}_ {p,\lambda,q}}}\right)^{2r}. \end{align*} $$
We can use the same discussion to get the continuity of
$(u,w)$
in time t. For the integral equation
$$ \begin{align*} U(t)=G(t)U_0-\int_{0}^{t}G(t-\tau)\mathbb P\nabla\cdot(u\otimes U)(\tau)d\tau, \end{align*} $$
by Lemma 3.2, we have that
$$ \begin{align*}\|G(t)U_0\|_{\mathcal L^{\infty}([0,T);{\mathcal{F\dot{N}}^{4+ \frac{\lambda-3}{p}-2\gamma} _{p,\lambda,q}})}\leq C\|U_0\|_{\mathcal{F\dot{N}}^{4-2\gamma+\frac{\lambda-3}{p}}_ {p,\lambda,q}}. \end{align*} $$
Then, we estimate the bilinear part. By Definition 2.3, Lemma 3.5, and the Young’s inequality, set
$1+\frac 1\infty =\frac {1}{r'}+\frac {1}{r}$
, we obtain that
$$ \begin{align*} \begin{aligned} &\ \ \ \ \|\int^t_0G(t-\tau)\mathbb P\nabla\cdot(u\otimes U)(\tau)d\tau\|_{\mathcal L^\infty([0,T); \mathcal{F\dot{N}}^{4+ \frac{\lambda-3}{p}-2\gamma}_{p,\lambda,q})}\\ &\leq \left\{\sum_{j\in\mathbb Z}2^{jq(4+ \frac{\lambda-3}{p}-2\gamma)}\left(\int^T_0 e^{-r't2^{2\gamma j}}dt\right)^{\frac{q}{r'}}\|\widehat{\dot\Delta_j \mathbb P\nabla\cdot(u\otimes U)}(\tau)d\tau\|^q_{L^{r}([0,T);M^\lambda_p)}dt\right\} ^{\frac1q}\\ &\leq C\left\{\sum_{j\in\mathbb Z}2^{jq(4+ \frac{\lambda-3}{p}-2\gamma-2\gamma(1-\frac{1}{r}))}\|\widehat{\dot\Delta_j \mathbb P\nabla\cdot(u\otimes U)}(\tau)d\tau\|^q_{L^{r}([0,T);M^\lambda_p)}dt\right\}^{\frac1q}\\ &\leq C||u\otimes U||_{\mathcal L^r([0,T);\mathcal{F\dot{N}}^{5-\frac{4\gamma}{(2r)'}+\frac{\lambda-3}{p}}_{p,\lambda,q})}\\ &\leq C||U||_{\mathcal L^{2r}([0,T);{\mathcal{F\dot{N}}^{4+ \frac{\lambda-3}{p}-\frac{2\gamma}{(2r)'}} _{p,\lambda,q}})}||U||_{\mathcal L^{2r}([0,T);{\mathcal{F\dot{N}}^{4+ \frac{\lambda-3}{p}-\frac{2\gamma}{(2r)'}} _{p,\lambda,q}})}. \end{aligned} \end{align*} $$
Therefore,
$(u,w)\in \mathcal L^{\infty }([0,T);{\mathcal {F\dot {N}}^{4+ \frac {\lambda -3}{p}-2\gamma } _{p,\lambda ,q}})$
. Same as the proof of the Theorem 1.1, we have the continuity of
$(u,w)$
in time t. Hence
$(u,w)\in \mathcal C \left ([0,T);\mathcal {F\dot N}_{p,\lambda ,q}^{4+ \frac {\lambda -3}{p}-2\gamma }\right )$
. Thus the proof of the Theorem 1.2 is complete.
Data availability statement
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Conflict of interest
The authors declare no conflict of interest exist.
