Global well-posedness of $3$-D anisotropic Navier-Stokes system with small unidirectional derivative

TL;DR

This paper establishes the global well-posedness of the 3-D anisotropic Navier-Stokes system with horizontal dissipation, assuming small unidirectional derivatives of initial data in a specific scaling invariant space.

Contribution

It extends previous results to the anisotropic case with only horizontal dissipation, proving global existence under smallness conditions on the unidirectional derivative.

Findings

Global unique solution exists under small unidirectional derivative condition.

The result applies to initial data in the Besov space 0,.

The approach generalizes prior isotropic Navier-Stokes results to anisotropic systems.

Abstract

In \cite{LZ4}, the authors proved that as long as the one-directional derivative of the initial velocity is sufficiently small in some scaling invariant spaces, then the classical Navier-Stokes system has a global unique solution. The goal of this paper is to extend this type of result to the 3-D anisotropic Navier-Stokes system $(ANS)$ with only horizontal dissipation. More precisely, given initial data $u_0=(u_0^\h,u_0^3)\in \cB^{0,\f12},$ $(ANS)$ has a unique global solution provided that $|D_\h|^{-1}\pa_3u_0$ is sufficiently small in the scaling invariant space $\cB^{0,\f12}.$

Equations

$$(ANS)\quad\left\{\begin{array}[]{l}\displaystyle\partial_{t}u+u\cdot\nabla u-\Delta_{\rm h}u=-\nabla p,\qquad(t,x)\in\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{+}\times\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3},\\ \displaystyle{\mathop{\rm div}\nolimits}\,u=0,\\ \displaystyle u|_{t=0}=u_{0},\end{array}\right.$$

$$\frac{\Vert a{\Vert }_{H^{s,s^{\prime }}}^2\text{buildrel}def}{={\int }_{R^3}|{\xi }_{\mathrm{h}}{|}^{2s}|{\xi }_3{|}^{2s^{\prime }}|\hat{a}(\xi ){|}^{2}d\xi <\infty with{\xi }_{\mathrm{h}}=({\xi }_1,{\xi }_2).}$$

$$\Vert u_0{\Vert }_{L^2}^{\epsilon }\Vert u_0{\Vert }_{H^{0,\frac{1}{2}+\epsilon }}^{1-\epsilon }\le c$$

$$(NS)\quad\left\{\begin{array}[]{l}\displaystyle\partial_{t}u+u\cdot\nabla u-\Delta u=-\nabla p,\qquad(t,x)\in\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{+}\times\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3},\\ \displaystyle{\mathop{\rm div}\nolimits}\,u=0,\\ \displaystyle u|_{t=0}=u_{0},\end{array}\right.$$

$$u_{\lambda}(t,x)\buildrel\hbox{\footnotesize def}\over{=}\lambda u(\lambda^{2}t,\lambda x)\quad\hbox{and}\quad u_{0,\lambda}(x)\buildrel\hbox{\footnotesize def}\over{=}\lambda u_{0}(\lambda x),$$

$$\begin{split}&\Delta_{k}^{\rm h}a\buildrel\hbox{\footnotesize def}\over{=}{\mathcal{F}}^{-1}(\varphi(2^{-k}|\xi_{\rm h}|)\widehat{a}),\quad\Delta_{\ell}^{\rm v}a\buildrel\hbox{\footnotesize def}\over{=}{\mathcal{F}}^{-1}(\varphi(2^{-\ell}|\xi_{3}|)\widehat{a}),\\ &S^{\rm h}_{k}a\buildrel\hbox{\footnotesize def}\over{=}{\mathcal{F}}^{-1}(\chi(2^{-k}|\xi_{\rm h}|)\widehat{a}),\quad\ S^{\rm v}_{\ell}a\buildrel\hbox{\footnotesize def}\over{=}{\mathcal{F}}^{-1}(\chi(2^{-\ell}|\xi_{3}|)\widehat{a}),\end{split}$$

$$\displaystyle\mathop{\rm Supp}\nolimits\ \varphi\subset\Bigl{\{}\tau\in\mathop{\mathbb{R}\kern 0.0pt}\nolimits\,:\,\frac{3}{4}\leq|\tau|\leq\frac{8}{3}\Bigr{\}}\quad\mbox{and}\quad\forall\tau>0\,,\ \sum_{j\in\mathop{\mathbb{Z}\kern 0.0pt}\nolimits}\varphi(2^{-j}\tau)=1;$$

$$\displaystyle\mathop{\rm Supp}\nolimits\ \chi\subset\Bigl{\{}\tau\in\mathop{\mathbb{R}\kern 0.0pt}\nolimits\,:\,|\tau|\leq\frac{4}{3}\Bigr{\}}\quad\mbox{and}\quad\forall\tau\in\mathop{\mathbb{R}\kern 0.0pt}\nolimits\,,\ \chi(\tau)+\sum_{j\geq 0}\varphi(2^{-j}\tau)=1.$$

$$\frac{\Vert a{\Vert }_{{\mathcal{B}}^{0,\frac{1}{2}}}\text{buildrel}def}{={\sum }_{\ell \in Z}{2}^{\frac{\ell }{2}}\Vert {\Delta }_{\ell }^{\mathrm{v}}a{\Vert }_{L^2(R^3)}<\infty .}$$

$$\|u^{\rm h}_{0}\|_{{\mathcal{B}}^{0,\frac{1}{2}}}\exp\bigl{(}C\|u^{3}_{0}\|_{{\mathcal{B}}^{0,\frac{1}{2}}}^{4}\bigr{)}\leq c_{0}$$

$$u_{0}^{\varepsilon}(x)=\sin\Bigl{(}\frac{x_{1}}{\varepsilon}\Bigr{)}\left(0,-\partial_{3}\varphi,\partial_{2}\varphi\right)$$

$$\|a\|_{{\mathcal{B}}^{-\frac{1}{2},\frac{1}{2}}_{4}}\buildrel\hbox{\footnotesize def}\over{=}\sum_{\ell\in\mathop{\mathbb{Z}\kern 0.0pt}\nolimits}2^{\frac{\ell}{2}}\Bigl{(}\bigl{(}\sum_{k=\ell-1}^{\infty}2^{-k}\|\Delta^{\rm h}_{k}\Delta^{\rm v}_{\ell}a\|_{L^{4}_{\rm h}(L^{2}_{\rm v})}^{2}\bigr{)}^{\frac{1}{2}}+\|S^{\rm h}_{\ell-1}\Delta^{\rm v}_{\ell}a\|_{L^{2}}\Bigr{)}<\infty.$$

$$\|u^{\rm h}_{0}\|_{{\mathcal{B}}^{-\frac{1}{2},\frac{1}{2}}_{4}}\exp\bigl{(}C\|u^{3}_{0}\|_{{\mathcal{B}}^{-\frac{1}{2},\frac{1}{2}}_{4}}^{4}\bigr{)}\leq c_{0}$$

$$\Vert u_0{\Vert }_{{\dot{B}}_{p,\infty }^{-1+\frac{3}{p}}}\le c\nu$$

$$\|\partial_{3}u_{0}\|_{H^{-\frac{1}{2},0}}^{2}\exp\bigl{(}CA_{\delta}(u^{\rm h}_{0})+CB_{\delta}(u_{0})\bigr{)}\leq\varepsilon_{0},$$

$$\displaystyle\begin{split}B_{\delta}(u_{0})\buildrel\hbox{\footnotesize def}\over{=}&\|u_{0}\|_{H^{-\delta,0}}^{\frac{1}{2}}\|u_{0}\|_{H^{\delta,0}}^{\frac{1}{2}}\|\partial_{3}u_{0}\|_{H^{-\delta,0}}^{\frac{1}{2}}\|\partial_{3}u_{0}\|_{H^{\delta,0}}^{\frac{1}{2}}\exp\bigl{(}CA_{\delta}(u^{\rm h}_{0})\bigr{)}\quad\hbox{and}\quad\\ A_{\delta}(u_{0}^{\rm h})\buildrel\hbox{\footnotesize def}\over{=}&\biggl{(}\frac{\|\nabla_{\rm h}u^{\rm h}_{0}\|_{L^{\infty}_{\rm v}(L^{2}_{\rm h})}^{2}\|u^{\rm h}_{0}\|_{L^{\infty}_{\rm v}(B^{-\delta}_{2,\infty})_{\rm h}}^{\frac{2}{\delta}}}{\|u^{\rm h}_{0}\|_{L^{\infty}_{\rm v}(L^{2}_{\rm h})}^{\frac{2}{\delta}}}+\|u^{\rm h}_{0}\|_{L^{\infty}_{\rm v}(L^{2}_{\rm h})}^{2}\Bigr{)}\cdot\exp\Big{(}C_{\delta}(1+\|u^{\rm h}_{0}\|_{L^{\infty}_{\rm v}(L^{2}_{\rm h})}^{4})\biggr{)}\end{split}$$

$$u_0^{\epsilon }(x)=(v_0^{\mathrm{h}}+\epsilon w_0^{\mathrm{h}},w_0^3)(x_{\mathrm{h}},\epsilon x_3)with{\mathrm{div}}_{\mathrm{h}}{v}_0^{\mathrm{h}}=0=\mathrm{div}{w}_0$$

$$\begin{split}\|\Lambda^{-1}_{\rm h}\partial_{3}u_{0}\|_{{\mathcal{B}}^{0,\frac{1}{2}}}\exp\Bigl{(}L\bigl{(}1+\|u_{0}^{3}\|_{{\mathcal{B}}^{-\frac{1}{2},\frac{1}{2}}_{4}}^{4}\bigr{)}\exp\bigl{(}M\mathfrak{A}_{N}^{4}\bigl{(}\|u^{\rm h}_{0}\|_{{\mathcal{B}}^{0,\frac{1}{2}}}\bigr{)}\bigr{)}\Bigr{)}\leq\varepsilon_{0},\end{split}$$

$$\|\Lambda^{-1}_{\rm h}\partial_{3}u_{0}\|_{{\mathcal{B}}^{0,\frac{1}{2}}}\exp\Bigl{(}L\bigl{(}1+\|u_{0}^{3}\|_{{\mathcal{B}}^{-\frac{1}{2},\frac{1}{2}}_{4}}^{4}\bigr{)}\exp\left(\exp\bigl{(}M\|u^{\rm h}_{0}\|_{L^{2}}\|\partial_{3}u^{\rm h}_{0}\|_{L^{2}}\bigr{)}\right)\Bigr{)}\leq\varepsilon_{0},$$

$$\Vert u_0^3{\Vert }_{{\mathcal{B}}_4^{-\frac{1}{2},\frac{1}{2}}}\lesssim \Vert u_0^3{\Vert }_{{\mathcal{B}}^{0,\frac{1}{2}}},$$

$$\begin{split}\|\Lambda^{-1}_{\rm h}\partial_{3}u_{0}\|_{{\mathcal{B}}^{0,\frac{1}{2}}}\exp\Bigl{(}L\bigl{(}1+\|u_{0}^{3}\|_{{\mathcal{B}}^{0,\frac{1}{2}}}^{4}\bigr{)}\exp\bigl{(}M\mathfrak{A}_{N}^{4}\bigl{(}\|u^{\rm h}_{0}\|_{{\mathcal{B}}^{0,\frac{1}{2}}}\bigr{)}\bigr{)}\Bigr{)}\leq\varepsilon_{0},\end{split}$$

$$\|\Lambda^{-1}_{\rm h}\partial_{3}u_{0}\|_{{\mathcal{B}}^{0,\frac{1}{2}}}\exp\Bigl{(}L\bigl{(}1+\|u_{0}^{3}\|_{{\mathcal{B}}^{0,\frac{1}{2}}}^{4}\bigr{)}\exp\left(\exp\bigl{(}M\|u^{\rm h}_{0}\|_{L^{2}}\|\partial_{3}u^{\rm h}_{0}\|_{L^{2}}\bigr{)}\right)\Bigr{)}\leq\varepsilon_{0}.$$

$$\Vert {\Lambda }_{\mathrm{h}}^{-1}{\partial }_3{u}_0{\Vert }_{{\mathcal{B}}^{0,\frac{1}{2}}}=\Vert ({\Lambda }_{\mathrm{h}}^{-1}{\partial }_3{u}_0^{\mathrm{h}},-{\Lambda }_{\mathrm{h}}^{-1}{\mathrm{div}}_{\mathrm{h}}{u}_0^{\mathrm{h}}){\Vert }_{{\mathcal{B}}^{0,\frac{1}{2}}}.$$

$$u_{0}^{\varepsilon}(x)=\bigl{(}\varepsilon(-\ln\varepsilon)^{\delta}w^{\rm h}_{0},(-\ln\varepsilon)^{\delta}w^{3}_{0}\bigr{)}(x_{\rm h},\varepsilon x_{3})\quad\hbox{with}\quad\delta\in]0,1/4[$$

$$u_{0}^{\varepsilon}(x)=\bigl{(}v^{\rm h}_{0}+\varepsilon(-\ln\varepsilon)^{\delta}w^{\rm h}_{0},(-\ln\varepsilon)^{\delta}w^{3}_{0}\bigr{)}(x_{\rm h},\varepsilon x_{3})\quad\hbox{with}\quad\delta\in]0,1/4[$$

$$\Vert e^{i{x}_1/\epsilon }\varphi (x){\Vert }_{{\mathcal{B}}_4^{-\frac{1}{2},\frac{1}{2}}}\le C{\epsilon }^{\frac{1}{2}}.$$

$$u_{0}^{\varepsilon}(x)=(v^{\rm h},0)(x_{\rm h},\varepsilon x_{3})+(-\ln\varepsilon)^{\delta}\sin(x_{1}/\varepsilon)\bigl{(}0,-\varepsilon^{\frac{1}{2}}\partial_{3}\phi(x_{\rm h},\varepsilon x_{3}),\varepsilon^{-\frac{1}{2}}\partial_{2}\phi(x_{\rm h},\varepsilon x_{3})\bigr{)}$$

$$\begin{array}{rl}\text{mbox}if\mathrm{Supp}\hat{a}\subset 2^k{\mathbf{B}}_{\mathrm{h}}& \Rightarrow \Vert {\partial }_{x_{\mathrm{h}}}^{\alpha }a{\Vert }_{L_{\mathrm{h}}^{p_1}(L_{\mathrm{v}}^{q_1})}\lesssim 2^{k\left(|\alpha |+\frac{2}{p_2}-\frac{2}{p_1}\right)}\Vert a{\Vert }_{L_{\mathrm{h}}^{p_2}(L_{\mathrm{v}}^{q_1})};\\ \text{mbox}if\mathrm{Supp}\hat{a}\subset 2^{\ell }{\mathbf{B}}_{\mathrm{v}}& \Rightarrow \Vert {\partial }_{x_3}^{\beta }a{\Vert }_{L_{\mathrm{h}}^{p_1}(L_{\mathrm{v}}^{q_1})}\lesssim 2^{\ell \left(\beta +\frac{1}{q_2}-\frac{1}{q_1}\right)}\Vert a{\Vert }_{L_{\mathrm{h}}^{p_1}(L_{\mathrm{v}}^{q_2})};\\ \text{mbox}if\mathrm{Supp}\hat{a}\subset 2^k{\mathcal{C}}_{\mathrm{h}}& \Rightarrow \Vert a{\Vert }_{L_{\mathrm{h}}^{p_1}(L_{\mathrm{v}}^{q_1})}\lesssim 2^{-kN}\underset{|\alpha |=N}{\sup }\Vert {\partial }_{x_{\mathrm{h}}}^{\alpha }a{\Vert }_{L_{\mathrm{h}}^{p_1}(L_{\mathrm{v}}^{q_1})};\\ \text{mbox}if\mathrm{Supp}\hat{a}\subset 2^{\ell }{\mathcal{C}}_{\mathrm{v}}& \Rightarrow \Vert a{\Vert }_{L_{\mathrm{h}}^{p_1}(L_{\mathrm{v}}^{q_1})}\lesssim 2^{-\ell N}\Vert {\partial }_{x_3}^{N}a{\Vert }_{L_{\mathrm{h}}^{p_1}(L_{\mathrm{v}}^{q_1})}.\end{array}$$

$$\frac{\Vert a{\Vert }_{{\tilde{L}}_T^p({\mathcal{B}}^{0,\frac{1}{2}})}\text{buildrel}def}{={\sum }_{\ell \in Z}{2}^{\frac{\ell }{2}}\Vert {\Delta }_{\ell }^{\mathrm{v}}a{\Vert }_{L_T^p(L^2(R^3))}.}$$

$$\frac{\Vert a{\Vert }_{{\mathcal{B}}^{0,\frac{1}{2}}(T)}\text{buildrel}def}{=\Vert a{\Vert }_{{\tilde{L}}_T^{\infty }({\mathcal{B}}^{0,\frac{1}{2}})}+\Vert {\nabla }_{\mathrm{h}}a{\Vert }_{{\tilde{L}}_T^2({\mathcal{B}}^{0,\frac{1}{2}})}.}$$

$$\begin{array}{rl}\Vert a{\Vert }_{{\mathcal{B}}^{0,\frac{1}{2}}}=& \sum _{\ell \le N}{2}^{\frac{\ell }{2}}\Vert {\Delta }_{\ell }^{\mathrm{v}}a{\Vert }_{L^2}+\sum _{\ell >N}{2}^{\frac{\ell }{2}}\Vert {\Delta }_{\ell }^{\mathrm{v}}a{\Vert }_{L^2}\\ \le & \sum _{\ell \le N}{2}^{\frac{\ell }{2}}\Vert {\Delta }_{\ell }^{\mathrm{v}}a{\Vert }_{L^2}+\sum _{\ell >N}{2}^{-\frac{\ell }{2}}\Vert {\partial }_3{\Delta }_{\ell }^{\mathrm{v}}a{\Vert }_{L^2}\\ \lesssim & 2^{\frac{N}{2}}\Vert a{\Vert }_{L^2}+2^{-\frac{N}{2}}\Vert {\partial }_{3}a{\Vert }_{L^2}.\end{array}$$

Full text

Keywords: Anisotropic Navier-Stokes system, Littlewood-Paley theory, well-posedness

AMS Subject Classification (2000): 35Q30, 76D03

1. Introduction

In this paper, we investigate the global well-posedness of the following $3$-D anisotropic Navier-Stokes system:

[TABLE]

where $\Delta_{\rm h}\buildrel\hbox{\footnotesize def}\over{=}\partial_{1}^{2}+\partial_{2}^{2},$ $u$ designates the velocity of the fluid and $p$ the scalar pressure function which guarantees the divergence free condition of the velocity field.

Systems of this type appear in geophysical fluid dynamics (see for instance [5, 18]). In fact, meteorologists often modelize turbulent diffusion by putting a viscosity of the form: $-\mu_{\rm h}\Delta_{\rm h}-\mu_{3}\partial_{3}^{2}$, where $\mu_{\rm h}$ and $\mu_{3}$ are empirical constants, and $\mu_{3}$ is usually much smaller than $\mu_{\rm h}$. We refer to the book of Pedlovsky [18], Chap. $4$ for a complete discussion about this model.

Considering system $(ANS)$ has only horizontal dissipation, it is reasonable to use functional spaces which distinguish horizontal derivatives from the vertical one, for instance, the anisotropic Sobolev space defined as follows:

Definition 1.1**.**

For any $(s,s^{\prime})$ in $\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2}$, the anisotropic Sobolev space $H^{s,s^{\prime}}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3})$ denotes the space of homogeneous tempered distribution $a$ such that

[TABLE]

Mathematically, Chemin et al. [4] first studied the system $(ANS).$ In particular, Chemin et al. [4] and Iftimie [13] proved that $(ANS)$ is locally well-posed with initial data in $L^{2}\cap H^{0,\frac{1}{2}+\varepsilon}$ for some $\varepsilon>0$, and is globally well-posed if in addition

[TABLE]

for some sufficiently small constant $c$.

Notice that just as the classical Navier-Stokes system

[TABLE]

the system $(ANS)$ has the following scaling invariant property:

[TABLE]

which means that if $u$ is a solution of $(ANS)$ with initial data $u_{0}$ on $[0,T]$, $u_{\lambda}$ determined by (1.2) is also a solution of $(ANS)$ with initial data $u_{0,\lambda}$ on $[0,T/\lambda^{2}]$.

It is easy to observe that the smallness condition (1.1) in [4] is scaling invariant under the scaling transformation (1.2), nevertheless, the norm of the space $H^{0,\frac{1}{2}+\varepsilon}$ is not. To work $(ANS)$ with initial data in the critical spaces, we first recall the following anisotropic dyadic operators from [2]:

[TABLE]

where $\xi_{\rm h}=(\xi_{1},\xi_{2}),$ ${\mathcal{F}}a$ or $\widehat{a}$ denotes the Fourier transform of $a$, while ${\mathcal{F}}^{-1}a$ designates the inverse Fourier transform of $a$, $\chi(\tau)$ and $\varphi(\tau)$ are smooth functions such that

[TABLE]

Definition 1.2**.**

We define ${\mathcal{B}}^{0,\frac{1}{2}}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3})$ to be the set of homogenous tempered distribution $a$ so that

[TABLE]

The above space was first introduced by Iftimie in [12] to study the global well-posedness of the classical 3-D Navier-Stokes system with initial data in the anisotropic functional space. The second author [16] proved the local well-posedness of $(ANS)$ with any solenoidal vector field $u_{0}\in{\mathcal{B}}^{0,\frac{1}{2}}$ and also the global well-posedness with small initial data in ${\mathcal{B}}^{0,\frac{1}{2}}.$ This result corresponds to the Fujita-Kato’s theorem ([11]) for the classical Navier-Stokes system. Moreover, the authors [17, 19] proved the the global well-posedness of $(ANS)$ with initial data $u_{0}=(u_{0}^{\rm h},u_{0}^{3})$ satisfying

[TABLE]

for some $c_{0}$ sufficiently small.

Although the norm of $B^{0,\frac{1}{2}}$ is scaling invariant under the the scaling transformation (1.2), yet we observe that the solenoidal vector field

[TABLE]

is not small in the space $B^{0,\frac{1}{2}}$ no matter how small $\varepsilon$ is. In order to find a space so that the norm of $u_{0}^{\varepsilon}(x)$ given by (1.5) is small in this space for small $\varepsilon,$ Chemin and the third author [8] introduced the following Besov-Soblev type space with negative index:

Definition 1.3**.**

We define the space ${\mathcal{B}}^{-\frac{1}{2},\frac{1}{2}}_{4}$ to be the set of a homogenous tempered distribution $a$ so that

[TABLE]

Chemin and the third author [8] proved the global well-posedness of $(ANS)$ with initial data being small in the space ${\mathcal{B}}^{-\frac{1}{2},\frac{1}{2}}_{4}.$ In particular, this result ensures the global well-posedness of $(ANS)$ with initial data $u_{0}^{\varepsilon}(x)$ given by (1.5) as long as $\varepsilon$ is sufficiently small. Furthermore the second and third authors [17] proved the global well-posedness of $(ANS)$ provided that the initial data $u_{0}=(u_{0}^{\rm h},u_{0}^{3})$ satisfies

[TABLE]

for some $c_{0}$ sufficiently small. We remark that this result corresponds to Cannone, Meyer and Planchon’s result in [3] for the classical Nvaier-Stokes system, where the authors proved that if the initial data satisfies

[TABLE]

for some $p$ greater than $3$ and some constant $c$ small enough, then $(NS)$ is globally well-posed. The end-point result in this direction is due to Koch and Tataru [14] for initial data in the space of  $\partial BMO.$

On the other hand, motivated by the study of the global well-posedness of the classical Navier-Stokes system with slowly varying initial data [6, 7, 9], the first and third authors proved the following theorem for $(NS)$ in [15]:

Theorem 1.1**.**

Let $u_{0}=(u^{h}_{0},u_{0}^{3})\in H^{-\delta,0}\cap H^{2}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3})$ for some $\delta\in]0,1[$ with $\partial_{3}u_{0}^{\rm h}\in H^{-\delta,0}\cap H^{-\frac{1}{2},0}$. There exists a universal small positive constant $\varepsilon_{0}$ such that if

[TABLE]

where

[TABLE]

then $(NS)$ has a unique global solution u\in C\bigl{(}\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{+};H^{\frac{1}{2}}\bigr{)}\cap L^{2}\bigl{(}\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{+};H^{\frac{3}{2}}\bigr{)}.

We remark that Theorem 1.1 ensures the global well-posedness of $(NS)$ with initial data

[TABLE]

for $\varepsilon\leq\varepsilon_{0},$ which was first proved in [6]. We mention that the proof of Theorem 1.1 requires a regularity criteria in [10], which can only be proved for the classical Navier-Stokes system so far.

Motivated by [15] and [17, 19], here we are going to study the global well-posedness of $(ANS)$ with initial data $u_{0}$ satisfying $\partial_{3}u_{0}$ being sufficiently small in some critical spaces.

The main result of this paper states as follows:

Theorem 1.2**.**

Let $\Lambda_{\rm h}^{-1}$ be a Fourier multiplier with symbol $|\xi_{\rm h}|^{-1},$ let $u_{0}\in{\mathcal{B}}^{0,\frac{1}{2}}$ be a solenoidal vector field with $\Lambda_{\rm h}^{-1}\partial_{3}u_{0}\in{{\mathcal{B}}^{0,\frac{1}{2}}}.$ Then there exist some sufficiently small positive constant $\varepsilon_{0}$ and some universal positive constants $L,M,N$ so that for \mathfrak{A}_{N}\bigl{(}\|u^{\rm h}_{0}\|_{{\mathcal{B}}^{0,\frac{1}{2}}}\bigr{)} given by (3.5) if

[TABLE]

$(ANS)$ has a unique global solution $u=\mathfrak{v}+e^{t\Delta_{h}}\begin{pmatrix}0\\ u^{3}_{0,{\rm hh}}\end{pmatrix}$ with $\mathfrak{v}\in C([0,\infty[\,;{\mathcal{B}}^{0,\frac{1}{2}})$ and $\nabla_{\rm h}\mathfrak{v}\in L^{2}([0,\infty[\,;{\mathcal{B}}^{0,\frac{1}{2}}),$ where $u_{0,{\rm hh}}^{3}\buildrel\hbox{\footnotesize def}\over{=}\sum_{k\geq\ell-1}\Delta_{k}^{\rm h}\Delta_{\ell}^{\rm v}u_{0}^{3}.$

We remark that all the norms of $u^{0}$ in (1.9) is scaling invariant under the scaling transformation (1.2). With regular initial data, we may write explicitly the constant \mathfrak{A}_{N}\bigl{(}\|u^{\rm h}_{0}\|_{{\mathcal{B}}^{0,\frac{1}{2}}}\bigr{)}. For instance,

Corollary 1.1**.**

Let $u_{0}\in L^{2}$ be a solenoidal vector field with $\partial_{3}u_{0}\in L^{2}$ and $\Lambda_{\rm h}^{-1}\partial_{3}u_{0}\in{{\mathcal{B}}^{0,\frac{1}{2}}}.$ Then there exist some sufficiently small positive constant $\varepsilon_{0}$ and some universal positive constants $L,M$ so that if

[TABLE]

$(ANS)$ has a unique global solution $u$ as in Theorem 1.2. **

Remark 1.1**.**

We have several remarks in order as follows about Theorem 1.2:

• (a)

It follows from **[8]** that

[TABLE]

so that the smallness condition (1.9) and (1.10) can also be formulated as

[TABLE]

and

[TABLE]

• (b)

Due to ${\mathop{\rm div}\nolimits}\,u_{0}=0,$ we find

[TABLE]

Therefore the smallness condition (1.9) is of a similar type as (1.4). Yet Roughly speaking, (1.9) requires only $\partial_{3}u^{\rm h}_{0}$ and ${\mathop{\rm div}_{\rm h}\nolimits}\,u^{\rm h}_{0}$ to be small in some scaling invariant space, but without any restriction on ${\mathop{\rm curl}_{\rm h}\nolimits}\,u^{\rm h}_{0}$. Thus the smallness condition (1.9) is weaker than (1.4).

• (c)

Let $w_{0}$ be a smooth solenoidal vector field, we observe that the following data

[TABLE]

satisfies (1.4) for $\varepsilon$ sufficiently small.

While since our smallness condition (1.12) does not have any restriction on ${\mathop{\rm curl}\nolimits}\,u_{0}^{\rm h},$ for any smooth vector field $v^{\rm h}_{0}$ satisfying ${\mathop{\rm div}_{\rm h}\nolimits}\,v^{h}_{0}=0,$ we find

[TABLE]

satisfy (1.12) for any $\varepsilon$ sufficiently small. Therefore Theorem 1.2 ensures the global well-posedness of $(ANS)$ with initial data given by (1.13). Compared with (1.8), which corresponds to $\delta=1$ in (1.13), this type of result is new even for the classical Navier-Stokes system.

• (d)

Given $\phi\in{\mathcal{S}}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3})$, we deduce from Proposition 1.1 in **[8]** that

[TABLE]

As a result, we find that for any $\delta\in]0,1/4[$, the following class of initial data

[TABLE]

satisfies the smallness condition (1.11) for small enough $\varepsilon$, and hence the data given by (1.14) can also generate unique global solution of $(ANS)$.

• (e)

Since all the results that work for the anisotropic Navier-Stokes system $(ANS)$ should automatically do for the classical Navier-Stokes system $(NS),$ Theorem 1.2 holds also for $(NS)$.

Let us end this section with some notations that will be used throughout this paper.

Notations: Let $A,B$ be two operators, we denote $[A;B]=AB-BA,$ the commutator between $A$ and $B,$ for $a\lesssim b$, we means that there is a uniform constant $C,$ which may be different in each occurrence, such that $a\leq Cb$. We shall denote by $(a|b)_{L^{2}}$ the $L^{2}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3})$ inner product of $a$ and $b.$ $\left(d_{j}\right)_{j\in\mathop{\mathbb{Z}\kern 0.0pt}\nolimits}$ designates a generic elements on the unit sphere of $\ell^{1}(\mathop{\mathbb{Z}\kern 0.0pt}\nolimits)$, i.e. $\sum_{j\in\mathop{\mathbb{Z}\kern 0.0pt}\nolimits}d_{j}=1$. Finally, we denote $L^{r}_{T}(L^{p}_{\rm h}(L^{q}_{\rm v}))$ the space $L^{r}([0,T];L^{p}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits_{x_{1}}\times\mathop{\mathbb{R}\kern 0.0pt}\nolimits_{x_{2}};L^{q}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits_{x_{3}}))),$ and $\nabla_{\rm h}\buildrel\hbox{\footnotesize def}\over{=}(\partial_{x_{1}},\partial_{x_{2}}),$ ${\mathop{\rm div}\nolimits}\,_{\rm h}=\partial_{x_{1}}+\partial_{x_{2}}$.

2. Littlewood-Paley Theory

In this section, we shall collect some basic facts on anisotropic Littlewood-Paley theory. We first recall the following anisotropic Bernstein inequalities from [8, 16]:

Lemma 2.1**.**

Let ${\bf B}_{{\rm h}}$ (resp. ${\bf B}_{\rm v}$) a ball of $\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2}_{{\rm h}}$ (resp. $\mathop{\mathbb{R}\kern 0.0pt}\nolimits_{\rm v}$), and ${\mathcal{C}}_{{\rm h}}$ (resp. ${\mathcal{C}}_{\rm v}$) a ring of $\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2}_{{\rm h}}$ (resp. $\mathop{\mathbb{R}\kern 0.0pt}\nolimits_{\rm v}$); let $1\leq p_{2}\leq p_{1}\leq\infty$ and  $1\leq q_{2}\leq q_{1}\leq\infty.$ Then there holds

[TABLE]

Definition 2.1**.**

For any $p\in[1,\infty],$, let us define the Chemin-Lerner type norm

[TABLE]

In particular, we denote

[TABLE]

We remark that the inhomogeneous version of the anisotropic Sobolev space $H^{0,1}$ can be continuously imbedded into ${\mathcal{B}}^{0,\frac{1}{2}}.$ Indeed for any integer $N$, we deduce from Lemma 2.1 that

[TABLE]

Taking the integer $N$ so that $2^{N}\sim\|\partial_{3}a\|_{L^{2}}\|a\|_{L^{2}}^{-1}$ in the above inequality leads to

[TABLE]

Along the same line, we have

[TABLE]

To overcome the difficulty that one can not use Gronwall’s inequality in the Chemin-Lerner type norms, we recall the following time-weighted Chemin-Lerner norm from [17]:

Definition 2.2**.**

Let $f(t)\in L^{1}_{\rm{loc}}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits_{+})$, $f(t)\geq 0$. We define

[TABLE]

In order to take into account functions with oscillations in the horizontal variables, we recall the following anisotropic Besov type space with negative indices from [8]:

Definition 2.3**.**

For any $p\in[1,\infty],$ we define

[TABLE]

In particular, we denote

[TABLE]

In the sequel, for $a\in{\mathcal{B}}^{-\frac{1}{2},\frac{1}{2}}_{4},$ we shall frequently use the following decomposition:

[TABLE]

Lemma 2.2** (Lemma $2.5$ in [8]).**

For any $a\in{{\mathcal{B}}^{-\frac{1}{2},\frac{1}{2}}_{4}},$ there holds

[TABLE]

Definition 2.4**.**

Let us define

[TABLE]

In view of the 2-D interpolation inequality that

[TABLE]

we find

[TABLE]

Similarly, we have

[TABLE]

Before preceding, let us recall Bony’s decomposition for the vertical variable from [1]:

[TABLE]

Sometimes we shall also use Bony’s decomposition for the horizontal variables.

Let us now apply the above basic facts on Littlewood-Paley theory to prove the following proposition:

Proposition 2.1**.**

For any $a\in{{\mathcal{B}}^{-\frac{1}{2},\frac{1}{2}}_{4}}(T),$ there holds

[TABLE]

Proof.

In view of (2.3) and Definition 2.3, we get, by applying (2.6), that

[TABLE]

Then it remains to prove (2.8) for $a_{\rm hh}.$ Indeed in view of Definition 2.4, we write

[TABLE]

Applying Bony’s decomposition for the horizontal variables yields

[TABLE]

We observe that

[TABLE]

Whereas we get, by using Young’s inequality, that

[TABLE]

As a result, it comes out

[TABLE]

and

[TABLE]

Along the same line, we can prove that the second term in (2.9) shares the same estimate. This ensures that (2.8) holds for $a_{\rm hh}.$ We thus complete the proof of the proposition. ∎

3. Sketch of the proof

Motivated by the study of the global large solutions to the classical 3-D Navier-Stokes system with slowly varying initial data in one direction ([6, 7, 9, 15]), here we are going to decompose the solution of $(ANS)$ as a sum of a solution to the two-dimensional Navier-Stokes system with a parameter and a solution to the three-dimensional perturbed anisotropic Navier-Stokes system. We point out that compared with the references [6, 7, 9, 15], here the 3-D solution to the perturbed anisotropic Navier-Stokes system will not be small. Indeed only its vertical component is not small. In order to deal with this part, we are going to appeal to the observation from [17, 19], where the authors proved the global well-posedness to 3-D anisotropic Navier-Stokes system with the horizontal components of the initial data being small (see the smallness conditions (1.4) and (1.6)).

For $u^{\rm h}=(u^{1},u^{2}),$ we first recall the two-dimensional Biot-Savart’s law:

[TABLE]

where ${\mathop{\rm curl}_{\rm h}\nolimits}\,u^{\rm h}\buildrel\hbox{\footnotesize def}\over{=}\partial_{1}u^{2}-\partial_{2}u^{1}$ and ${\mathop{\rm div}_{\rm h}\nolimits}\,u^{\rm h}\buildrel\hbox{\footnotesize def}\over{=}\partial_{1}u^{1}+\partial_{2}u^{2}.$

In particular, let us decompose the horizontal components $u_{0}^{\rm h}$ of the initial velocity $u_{0}$ of $(ANS)$ as the sum of $u^{\rm h}_{0,\mathop{\rm curl}\nolimits}$ and $u^{\rm h}_{0,\mathop{\rm div}\nolimits}.$ And we consider the following 2-D Navier-Stokes system with a parameter:

[TABLE]

Concerning the system (3.2), we have the following a priori estimates:

Proposition 3.1**.**

Let $\bar{u}^{\rm h}_{0}\in{\mathcal{B}}^{0,\frac{1}{2}}$ with $\Lambda_{\rm h}^{-1}\partial_{3}\bar{u}^{\rm h}_{0}\in{\mathcal{B}}^{0,\frac{1}{2}}.$ Then (3.2) has a unique global solution so that for any time $t>0$, there hold

[TABLE]

and

[TABLE]

where

[TABLE]

and $N$ is taken so large that \bigl{\|}\bar{u}^{\rm h}_{0,N}\bigr{\|}_{{\mathcal{B}}^{0,\frac{1}{2}}} is sufficiently small.

The proof of Proposition 3.1 will be presented in Section 4.

Remark 3.1**.**

Under the assumptions that $\bar{u}^{\rm h}_{0}\in L^{2}$ with $~{}\partial_{3}\bar{u}^{\rm h}_{0}\in L^{2}$ and $\Lambda_{\rm h}^{-1}\partial_{3}\bar{u}^{\rm h}_{0}\in{\mathcal{B}}^{0,\frac{1}{2}},$ we have the following alternative estimates for (3.3) and (3.4)

[TABLE]

and

[TABLE]

We shall present the proof in Remark 4.1.

We notice that

[TABLE]

which satisfies ${\mathop{\rm div}\nolimits}\,v_{0}=0,$ and yet $v_{0}$ is not small according to our smallness condition (1.9).

Before proceeding, let us recall the main idea of the proof to Theorem 1.1 in [15]. The authors [15] first constructed $(\bar{u}^{\rm h},\bar{p})$ via the system (3.2). Then in order to get rid of the large part of the initial data $v_{0},$ given by (3.8), the authors introduced a correction velocity, $\widetilde{u},$ through the system

[TABLE]

With $\bar{u}^{\rm h}$ and $\widetilde{u}$ being determined respectively by the systems (3.2) and (3.9), the authors [15] decompose the solution $(u,p)$ to the classical Navier-Stokes system $(NS)$ as

[TABLE]

The key estimate for $v$ states as follows:

Proposition 3.2**.**

Let $u=(u^{\rm h},u^{3})\in C([0,T^{\ast}[;H^{\frac{1}{2}})\cap L^{2}(]0,T^{\ast}[;H^{\frac{3}{2}})$ be a Fujita-Kato solution of $(NS).$ We denote $\omega\buildrel\hbox{\footnotesize def}\over{=}\partial_{1}v^{2}-\partial_{2}v^{1}$ and

[TABLE]

Then under the assumption (1.7), there exists some positive constant $\eta$ such that

[TABLE]

Then in order to complete the proof of Theorem 1.1, the authors [15] invoked the following regularity criteria for the classical Navier-Stokes system:

Theorem 3.1** (Theorem 1.5 of [10]).**

Let $u\in C([0,T^{\ast}[;H^{\frac{1}{2}})\cap L^{2}(]0,T^{\ast}[;H^{\frac{3}{2}})$ be a solution of $(NS)$. If the maximal existence time $T^{\ast}$ is finite, then for any $(p_{i,j})$ in $]1,\infty[^{9}$, one has

[TABLE]

We remark that Theorem 3.1 only works for the classical 3-D Navier-Stokes system. Therefore the above procedure to prove Theorem 1.1 can not be applied to construct the global solutions to the 3-D anisotropic Navier-Stokes system.

On the other hand, we remark that the main observation in [17, 19] is that: by using ${\mathop{\rm div}\nolimits}\,u=0,$ $(ANS)$ can be equivalently reformulated as

[TABLE]

so that al least seemingly the $u^{3}$ equation is a linear one. And this explains in some sense why there is no size restriction for $u_{0}^{3}$ in (1.4) and (1.6).

Motivated by [17, 19], for $\bar{u}^{\rm h}$ being determined by the systems (3.2), we decompose the solution $u$ of $(ANS)$ as $u=\begin{pmatrix}\bar{u}^{\rm h}\\ 0\end{pmatrix}+v$. It is easy to verify that the remainder term $v$ satisfies

[TABLE]

We notice that under the smallness condition (1.9), the horizontal components, $v_{0}^{\rm h},$ are small in the critical space ${\mathcal{B}}^{0,\frac{1}{2}}.$ Then the crucial ingredient used in the proof of Theorem 1.2 is that the horizontal components $v^{\rm h}$ of the remainder velocity keeps small for any positive time.

Due to the additional difficulty caused by the fact that $u_{0}^{3}$ belongs to the Sobolev-Besov type space with negative index, as in [8], we further decompose $v^{3}$ as

[TABLE]

And then $w$ solves

[TABLE]

Proposition 3.3**.**

Let $v$ be a smooth enough solution of (3.14) on $[0,T^{*}[$. Then there exists some positive constant $C$ so that for any $t\in]0,T^{\ast}[,$ we have

[TABLE]

and

[TABLE]

The proof of the estimates (3.17) and (3.18) will be presented respectively in Sections 5 and 6. Now let us admit the above Propositions 3.1 and 3.3 temporarily, and continue our proof of Theorem 1.2.

Proof of Theorem 1.2.

It is well-known that the existence of global solutions to a nonlinear partial differential equations can be obtained by first constructing the approximate solutions, and then performing uniform estimates and finally passing to the limit to such approximate solutions. For simplicity, here we just present the a priori estimates for smooth enough solutions of $(ANS).$

Let $u$ be a smooth enough solution of $(ANS)$ on $[0,T^{\ast}[$ with $T^{\ast}$ being the maximal time of existence. Let $\bar{u}^{\rm h}$ and $v$ be determined by (3.2) and (3.14) respectively. Thanks to (3.1) and Proposition 3.1, we first take $L,M,N$ large enough and $\varepsilon_{0}$ small enough in (1.9) so that

[TABLE]

We now define

[TABLE]

Then thanks to (3.19) and Proposition 3.3, for $t\leq T^{\star},$ we find

[TABLE]

and

[TABLE]

It follows from Lemma 2.2 and Proposition 2.1 that

[TABLE]

Whereas we deduce from (2.6) and Proposition 3.1 that

[TABLE]

By inserting the above two inequalities to (3.22) and using (3.3), we obtain that for $t\leq T^{\star}$

[TABLE]

Then we deduce that for $t\leq T^{\star},$

[TABLE]

Inserting the above estimates into (3.21) gives rise to

[TABLE]

for $t\leq T^{\star}.$ Therefore, if we take $L,M,N$ large enough and $\varepsilon_{0}$ small enough in (1.9), we deduce from (3.24) that

[TABLE]

(3.25) contradicts with (3.20). This in turn shows that $T^{\star}=T^{\ast}.$ (3.23) along with (3.25) shows that $T^{\ast}=\infty.$ Moreover, thanks to (3.15), we have $\mathfrak{v}\buildrel\hbox{\footnotesize def}\over{=}u-e^{t\Delta_{h}}\begin{pmatrix}0\\ u^{3}_{0,{\rm hh}}\end{pmatrix}\in C([0,\infty[\,;{\mathcal{B}}^{0,\frac{1}{2}})$ with $\nabla_{\rm h}\mathfrak{v}\in L^{2}([0,\infty[\,;{\mathcal{B}}^{0,\frac{1}{2}}).$ This completes the proof of our Theorem 1.2. ∎

Proof of Corollary 1.1.

Under the assumptions that $u^{\rm h}_{0}\in L^{2}$ with $~{}\partial_{3}u^{\rm h}_{0}\in L^{2}$ and $\Lambda_{\rm h}^{-1}\partial_{3}u^{\rm h}_{0}\in{\mathcal{B}}^{0,\frac{1}{2}},$ we deduce from (3.1), (3.4) and (3.7) that

[TABLE]

Then by repeating the argument from (3.19) to (3.24), we conclude the proof of Corollary 1.1. ∎

4. Estimates of the 2-D solution $\bar{u}^{\rm h}$

The goal of this section is to present the proof of Proposition 3.1. Let us start the proof by the following lemma, which is in the spirit of Lemma 3.1 of [6].

Lemma 4.1**.**

Let $a^{\rm h}=(a^{1},a^{2})$ be a smooth enough solution of

[TABLE]

Then for any $t>0$ and any fixed $x_{3}\in\mathop{\mathbb{R}\kern 0.0pt}\nolimits,$ there holds

[TABLE]

and

[TABLE]

Proof.

By taking $L^{2}_{\rm h}$ inn-product of (4.1) with $a^{\rm h}$ and using ${\mathop{\rm div}_{\rm h}\nolimits}\,a^{\rm h}=0,$ we obtain (4.2).

While by applying $\partial_{3}$ to (4.1) and then taking $L^{2}_{\rm h}$ inner product of the resulting equation with $\partial_{3}a^{\rm h},$ we find

[TABLE]

Due to ${\mathop{\rm div}_{\rm h}\nolimits}\,a^{\rm h}=0,$ we get, by applying (2.4), that

[TABLE]

Applying Young’s inequality yields

[TABLE]

Inserting the above estimate into (4.4) gives rise to

[TABLE]

Applying Gronwall’s inequality and using (4.2), we achieve

[TABLE]

which leads to (4.3). This completes the proof of this lemma. ∎

Let us now present the proof of Proposition 3.1.

Proof of Proposition 3.1.

For any positive integer $N,$ and $\bar{u}^{\rm h}_{0,N}$ being given by (3.5), we split the solution $\bar{u}^{\rm h}$ to (3.2) as

[TABLE]

with $\bar{u}^{\rm h}_{1}$ and $\bar{u}^{\rm h}_{2}$ being determined respectively by

[TABLE]

and

[TABLE]

We first deduce from Lemma 4.1 that

[TABLE]

which together with (2.2) ensures that

[TABLE]

Next we handle the estimate of $\bar{u}^{\rm h}_{2}$. To do it, for any $\kappa>0,$ we denote

[TABLE]

Then by multiplying \exp\Bigl{(}-\kappa\int_{0}^{t}f^{\rm h}(t^{\prime})\,dt^{\prime}\Bigr{)} to the $\bar{u}^{\rm h}_{2}$ equation in (4.7), we write

[TABLE]

Applying the operator $\Delta_{\ell}^{\rm v}$ to the above equation and taking $L^{2}$ inner product of the resulting equation with $\Delta_{\ell}^{\rm v}\bar{u}^{\rm h}_{2,\kappa},$ and then using integration by parts, we get

[TABLE]

The estimate of the second line of (4.10) relies on the following lemma, whose proof will be postponed in the Appendix A:

Lemma 4.2**.**

Let $a,b,c\in{\mathcal{B}}^{0,\frac{1}{2}}(T)$ and $\mathfrak{f}(t)\buildrel\hbox{\footnotesize def}\over{=}\|a(t)\|_{{\mathcal{B}}_{4}^{0,\frac{1}{2}}}^{4}$. Then for any smooth homogeneous Fourier multiplier, $A(D),$ of degree zero and any $\ell\in\mathop{\mathbb{Z}\kern 0.0pt}\nolimits$, there hold

[TABLE]

Moreover, for non-negative function $\mathfrak{g}\in L^{\infty}(0,T),$ one has

[TABLE]

By applying (4.13) with $a=c=\bar{u}^{\rm h}_{2},$ $b=\nabla_{h}\bar{u}^{\rm h}_{2,\kappa}$ and \mathfrak{f}=\exp\Bigl{(}-\kappa\int_{0}^{t}f^{\rm h}(t^{\prime})\,dt^{\prime}\Bigr{)}, we get

[TABLE]

Whereas due to (2.5), one has

[TABLE]

By applying (4.12) with $a=\bar{u}^{\rm h}_{1},~{}b=\bar{u}^{\rm h}_{2,\kappa},~{}c=\nabla_{\rm h}\bar{u}^{\rm h}_{2,\kappa}$, we infer

[TABLE]

Then we get, by first integrating (4.10) over $[0,t]$ and inserting (4.14) and (4.15) into the resulting inequality, that

[TABLE]

Multiplying the above inequality by $2^{\ell}$ and taking square root of the resulting inequality, and then summing up the inequalities for $\ell\in\mathop{\mathbb{Z}\kern 0.0pt}\nolimits,$ we arrive at

[TABLE]

In particular, taking $2\kappa=C^{2}$ in the above inequality gives rise to

[TABLE]

On the other hand, in view of (3.5), we can take $N$ so large that

[TABLE]

Then a standard continuity argument shows that, for any time $t>0$, there holds

[TABLE]

Due to the definition of $\bar{u}^{\rm h}_{2,\lambda}$ given by (4.9), one has

[TABLE]

which together with (4.8) and (4.18) implies that

[TABLE]

By combining (4.8) with (4.19), we obtain (3.3).

It remains to prove (3.4). In order to do, for any $\gamma>0,$ we denote

[TABLE]

Then by multiplying $\exp\left(-\gamma\int_{0}^{t}g^{\rm h}(t^{\prime})\,dt^{\prime}\right)$ to the $\bar{u}^{\rm h}$ equation in (3.2), we write

[TABLE]

Applying the operator $\Delta_{\ell}^{\rm v}\Lambda_{\rm h}^{-1}\partial_{3}$ to the above equation and then taking $L^{2}$ inner product of the resulting equation with $\Delta_{\ell}^{\rm v}\Lambda_{\rm h}^{-1}\partial_{3}\bar{u}^{\rm h}_{\gamma},$ we get

[TABLE]

Noting that $\Lambda_{\rm h}^{-1}{\mathop{\rm div}_{\rm h}\nolimits}\,$ is a bounded Fourier multiplier, we get, by using (4.11) with $a=\bar{u}^{\rm h},~{}b=\partial_{3}\bar{u}^{\rm h}_{\gamma}$ and $c=\Lambda_{\rm h}^{-1}\partial_{3}\bar{u}^{\rm h}_{\gamma},$ that

[TABLE]

By integrating (4.21) over $[0,t]$ and then inserting the above estimate into the resulting inequality, we find

[TABLE]

Multiplying the above inequality by $2^{\ell}$ and taking square root of the resulting inequality, and then summing up the inequalities for $\ell\in\mathop{\mathbb{Z}\kern 0.0pt}\nolimits,$ we arrive at

[TABLE]

In particular, taking $2\gamma=C^{2}$ in the above inequality gives rise to

[TABLE]

Then a similar derivation from (4.18) to (4.19) leads to

[TABLE]

which together with (3.3) ensures (3.4). This completes the proof of this proposition. ∎

Remark 4.1**.**

For smoother initial data $\bar{u}^{\rm h}_{0},$ we may write explicitly the constant \mathfrak{A}_{N}\bigl{(}\|\bar{u}^{\rm h}_{0}\|_{{\mathcal{B}}^{0,\frac{1}{2}}}\bigr{)} in (3.3). For instance, if $\bar{u}^{\rm h}_{0}\in L^{2}$ with $~{}\partial_{3}\bar{u}^{\rm h}_{0}\in L^{2}$ and $\Lambda_{\rm h}^{-1}\partial_{3}\bar{u}^{\rm h}_{0}\in{\mathcal{B}}^{0,\frac{1}{2}}$, we deduce from Lemma 4.1 that

[TABLE]

which together with (2.2) and

[TABLE]

ensures (3.6). By virtue of (3.6) and (4.22), we deduce (3.7).

5. The estimate of the horizontal components $v^{\rm h}$

The goal of this section is to present the proof of (3.17), namely, we are going to deal with the estimate to the horizontal components of the remainder velocity determined by (3.14).

In order to do so, let $u$ be a smooth enough solution of $(ANS)$ on $[0,T^{\ast}[,$ let $\bar{u}^{\rm h},v_{F}$ and $w$ be determined respectively by (3.2), (3.15) and (3.16), for any constant $\lambda>0$, we denote

[TABLE]

and similar notations for $\bar{u}^{\rm h}_{\lambda},~{}p_{\lambda},~{}\bar{p}_{\lambda}$ and $v^{\rm h}_{\lambda/2}$.

By multiplying \exp\Bigl{(}-\lambda\int_{0}^{t}f(t^{\prime})\,dt^{\prime}\Bigr{)} to the $v^{\rm h}$ equation of (3.14), we get

[TABLE]

Applying $\Delta_{\ell}^{\rm v}$ to the above equation and taking $L^{2}$ inner product of the resulting equation with $\Delta_{\ell}^{\rm v}v^{\rm h}_{\lambda}$, and then integrating the equality over $[0,t],$ we obtain

[TABLE]

where

[TABLE]

We mention that since our system (3.14) has only horizontal dissipation, it is reasonable to distinguish the terms above with horizontal derivatives from the ones with vertical derivative. Next let us handle term by term above.

$\bullet$ The estimates of ${\rm I}_{1}$ to ${\rm I}_{4}.$

We first get, by using (4.11) with $a=\bar{u}^{\rm h},~{}b=\nabla_{\rm h}v^{\rm h}_{\lambda}$ and $c=v^{\rm h}_{\lambda},$ that

[TABLE]

Applying (4.13) with $a=v^{\rm h},~{}b=\nabla_{\rm h}v^{\rm h},~{}c=v^{\rm h}$ and \mathfrak{g}(t)=\exp\bigl{(}-\lambda\int_{0}^{t}f(t^{\prime})\,dt^{\prime}\bigr{)} yields

[TABLE]

To handle ${\rm I}_{3},$ by using integration by parts, we write

[TABLE]

Applying (4.11) with $a=\bar{u}^{\rm h},~{}b={\mathop{\rm div}_{\rm h}\nolimits}\,v^{\rm h}_{\lambda}$ and $c=v^{\rm h}_{\lambda}$ gives

[TABLE]

Whereas applying (4.12) with $a=\bar{u}^{\rm h},~{}b=v^{\rm h}_{\lambda}$ and $c=\nabla_{\rm h}v^{\rm h}_{\lambda}$ yields

[TABLE]

As a result, it comes out

[TABLE]

While by applying (4.11) with $a=v^{3},~{}b=\partial_{3}\bar{u}^{\rm h}_{\lambda},~{}c=v^{\rm h}_{\lambda}$, and using the fact that

[TABLE]

we find

[TABLE]

$\bullet$ The estimates of ${\rm I}_{5}$.

The estimate of ${\rm I}_{5}$ is much more complicated, since there is no vertical dissipation in $(ANS)$. To overcome this difficulty, we first use Bony’s decomposition in vertical variable (2.7) to write

[TABLE]

Following [8, 16], we get, by using a standard commutator’s process, that

[TABLE]

By applying commutator’s estimate (see Lemma $2.97$ in [2]), we find

[TABLE]

Due to $\partial_{3}v^{3}=-{\mathop{\rm div}_{\rm h}\nolimits}\,v^{\rm h},$ we get, by applying (2.4), that

[TABLE]

Next, since the support to the Fourier transform of $\sum_{|\ell^{\prime}-\ell|\leq 5}(S^{\rm v}_{\ell^{\prime}-1}v^{3}-S^{\rm v}_{\ell-1}v^{3})$ is contained in $\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2}\times\cup_{|\ell^{\prime}-\ell|\leq 5}2^{\ell^{\prime}}{\mathcal{C}}_{\rm v},$ we get, by applying Lemma 2.1, that

[TABLE]

from which, we infer

[TABLE]

As a result, it comes out

[TABLE]

Finally, by using integration by parts and $\partial_{3}v^{3}=-{\mathop{\rm div}_{\rm h}\nolimits}\,v^{\rm h}$ again, we find

[TABLE]

On the other hand, by applying Lemma 2.1 once again, we find

[TABLE]

Observing that

[TABLE]

we infer

[TABLE]

which together with (5.7) ensures that

[TABLE]

$\bullet$ The estimates of ${\rm I}_{6}.$

We first get, by taking the space divergence operators, ${\mathop{\rm div}\nolimits}\,$ and ${\mathop{\rm div}_{\rm h}\nolimits}\,,$ to $(ANS)$ and (3.2) respectively, that

[TABLE]

so that thanks to the fact that

[TABLE]

we write

[TABLE]

Accordingly, we decompose ${\rm I}_{6}$ as

[TABLE]

where

[TABLE]

Noticing that $\nabla_{\rm h}(-\Delta)^{-1}{\mathop{\rm div}_{\rm h}\nolimits}\,$ is a bounded Fourier multiplier. Then along the same line to the estimate of ${\rm I}_{1}$ to ${\rm I}_{4},$ we achieve

[TABLE]

However, ${\rm I}_{6,2}$ can not be handled along the same line to that of ${\rm I}_{5}$, since the symbol of the operator $\nabla_{\rm h}(-\Delta)^{-1}{\mathop{\rm div}_{\rm h}\nolimits}\,$ depends not only on $\xi_{3}$, but also on $\xi_{\rm h},$ which makes it impossible for us to deal with the commutator’s estimate. Fortunately, the appearance of the operator $(-\Delta)^{-1}$ can absorb the vertical derivative. Indeed, by using integration by parts, and the divergence-free condition of $v,$ we write

[TABLE]

Since both $\nabla_{\rm h}(-\Delta)^{-1}\partial_{3}$ and $\nabla_{\rm h}(-\Delta)^{-1}{\mathop{\rm div}_{\rm h}\nolimits}\,$ are bounded Fourier multiplier, we get, by applying Lemma 4.2, that

[TABLE]

To handle ${\rm I}_{6,3}$, we use ${\mathop{\rm div}\nolimits}\,v={\mathop{\rm div}_{\rm h}\nolimits}\,\bar{u}^{\rm h}=0$ to write

[TABLE]

Applying (4.11) with $A(D)=\nabla_{\rm h}(-\Delta)^{-1}\partial_{3},a=v^{3},b={\mathop{\rm div}_{\rm h}\nolimits}\,v^{\rm h}_{\lambda}$ and $c=v^{\rm h}_{\lambda}$ yields

[TABLE]

The remaining terms in ${\rm I}_{6,3}$ can be handled along the same line to that of ${\rm I}_{6,1}$ and ${\rm I}_{6,2}.$ As a consequence, we obtain

[TABLE]

To deal with ${\rm I}_{6,4}$, it is crucial to observe that

[TABLE]

Then due to the fact that $\sum_{i,j=1}^{2}\nabla_{\rm h}\partial_{3}(-\Delta)^{-1}(-\Delta_{\rm h})^{-1}\partial_{i}\partial_{j}$ is a bounded Fourier multiplier, we get, by applying (4.11) with $a=\bar{u}^{\rm h},b=\partial_{3}\bar{u}^{\rm h}_{\lambda},c=v^{\rm h}_{\lambda},$ that

[TABLE]

By summing up (5.10-5.13), we arrive at

[TABLE]

Now we are in a position to complete the proof of (3.17).

Proof of (3.17).

By inserting the estimates (5.3-5.6), (5.8) and (5.14) into (5.2), we achieve

[TABLE]

Multiplying the above inequality by $2^{\ell+1}$ and taking square root of the resulting inequality, and then summing up the inequalities over $\mathop{\mathbb{Z}\kern 0.0pt}\nolimits,$ we find

[TABLE]

It follows from Young’s inequality that

[TABLE]

Inserting the above inequality into (5.15) and taking $\lambda$ so that $\sqrt{2\lambda}=C$, we obtain

[TABLE]

which together with the following consequence of (5.1) that

[TABLE]

gives rise to (3.17). ∎

6. The estimate of the vertical component $v^{3}$

The purpose of this section is to present the proof of (3.18). Compared with [17], where the third component of the velocity field can be estimated in the standard Besov spaces, here due to the additional terms like $\bar{u}^{\rm h}\cdot\nabla_{\rm h}v$ appears in (3.14), we will have to use the weighted Chemin-Lerner norms once again. Indeed for any constant $\mu>0$, we denote

[TABLE]

And similar notations for $v_{\mu},~{}\bar{u}^{\rm h}_{\mu},$ and $~{}p_{\mu}$.

By multiplying $\bar{\mathfrak{g}}(t)$ to (3.16), we write

[TABLE]

By applying $\Delta_{\ell}^{\rm v}$ to the above equation and taking $L^{2}$ inner product of the resulting equation with $\Delta_{\ell}^{\rm v}w_{\mu}$, and then integrating the equality over $[0,t],$ we obtain

[TABLE]

where

[TABLE]

Let us handle term by term above.

$\bullet$ The estimates of ${\rm II}_{1}$ and ${\rm II}_{2}$

We first get, by applying (4.11) with $a=\bar{u}^{\rm h},~{}b=\nabla_{\rm h}w_{\mu}$ and $c=w_{\mu},$ that

[TABLE]

Whereas by applying a modified version of (4.13) with $a=v^{\rm h},~{}b=\nabla_{\rm h}w_{\mu},~{}c=w_{\mu}$ and \mathfrak{g}(t)=\exp\bigl{(}-\mu\int_{0}^{t}\hbar(t^{\prime})\,dt^{\prime}\bigr{)}, we find

[TABLE]

$\bullet$ The estimate of ${\rm II}_{3}$

The estimate of ${\rm II}_{3}$ relies on the following lemma, the proof of which will be postponed in the Appendix A.

Lemma 6.1**.**

Let $a,~{}c\in{\mathcal{B}}^{0,\frac{1}{2}}(T)$ and $b\in{{\mathcal{B}}^{-\frac{1}{2},\frac{1}{2}}_{4}}(T).$ Then for any smooth homogeneous Fourier multiplier, $A(D),$ of degree zero and any $\ell\in\mathop{\mathbb{Z}\kern 0.0pt}\nolimits$, there hold

[TABLE]

and

[TABLE]

Remark 6.1**.**

Indeed the proof of Lemma 6.1 shows that $\|b\|_{{{\mathcal{B}}^{-\frac{1}{2},\frac{1}{2}}_{4}}(T)}$ in (6.5) and (6.6) can be replaced by $\|b\|_{{\mathcal{B}}^{0,\frac{1}{2}}(T)}.$

Let us admit this lemma temporarily, and continue our estimate of ${\rm II}_{3}$. By using integration by parts, we write

[TABLE]

Applying (6.6) with $a={\mathop{\rm div}_{\rm h}\nolimits}\,v^{\rm h}_{\mu},~{}b=v_{F}$ and $c=w_{\mu}$ yields

[TABLE]

Whereas by applying (6.5) with $a=v^{\rm h}_{\mu},~{}b=v_{F}$ and $c=\nabla_{\rm h}w_{\mu},$ we obtain

[TABLE]

Inserting the above two estimates into (6.7) and using (2.6), we achieve

[TABLE]

$\bullet$ The estimate of ${\rm II}_{4}$

Due to ${\mathop{\rm div}_{\rm h}\nolimits}\,\bar{u}^{\rm h}=0,$ by using integration by parts, we write

[TABLE]

By applying Bony’s decomposition (2.7), we get

[TABLE]

We first observe that

[TABLE]

applying Hölder’s inequality and Proposition 2.1 gives

[TABLE]

Along the same line, we find

[TABLE]

As a result, it comes out

[TABLE]

$\bullet$ The estimates of ${\rm II}_{5}$

Due to $\partial_{3}v^{3}=-{\mathop{\rm div}_{\rm h}\nolimits}\,v^{\rm h}$ and $v^{3}=w+v_{F},$ we write

[TABLE]

Then applying (6.6) gives rise to

[TABLE]

$\bullet$ The estimates of ${\rm II}_{6}$

The estimate of ${\rm II}_{6}$ can be handled similarly as ${\rm I}_{6}$. Indeed in view of (5.9), we write

[TABLE]

Accordingly, we have the decomposition ${\rm II}_{6}=\sum_{i=1}^{5}{\rm II}_{6,i}$ with

[TABLE]

It is easy to observe from the estimate of ${\rm I}_{6,1}$ that

[TABLE]

While by using $\partial_{3}v^{3}=-{\mathop{\rm div}_{\rm h}\nolimits}\,v^{\rm h}$ and integration by parts, we write

[TABLE]

It follows from (6.5) and $v^{3}=v_{F}+w$ that

[TABLE]

Whereas by using a modified version of (4.13), we infer

[TABLE]

Therefore, we obtain

[TABLE]

Whereas applying (6.6) with $a=\partial_{3}\bar{u}^{\rm h},~{}b=v^{3}_{\mu}$ and $c=w_{\mu}$ leads to

[TABLE]

On the other hand, again due to ${\mathop{\rm div}\nolimits}\,v=0,$ we write

[TABLE]

Noticing that $(-\Delta)^{-1}\partial_{3}^{2}$ is a bounded Fourier operator, we observe that ${\rm II}_{6,4}$ shares the same estimate as $\sum_{i=1}^{5}{\rm II}_{i}$ given before, that is

[TABLE]

Finally since $(-\Delta)^{-1}\partial_{i}\partial_{j}$ is a bounded Fourier operator, we get, by applying (4.11) with $a=\bar{u}^{\rm h},~{}b=\partial_{3}\bar{u}^{\rm h}_{\mu},~{}c=w_{\mu}$, that

[TABLE]

By summing (6.12-6.16), we arrive at

[TABLE]

Let us now complete the proof of (3.18).

Proof of (3.18).

By inserting the estimates (6.3), (6.4), (6.9-6.11) and (6.17) into (6.2), and then multiplying $2^{\ell+1}$ to the resulting inequality, and finally taking square root and then summing up the resulting inequalities over $\mathop{\mathbb{Z}\kern 0.0pt}\nolimits,$ we obtain

[TABLE]

Applying Young’s inequality gives

[TABLE]

and

[TABLE]

As a result, it comes out

[TABLE]

Taking $\mu$ in the above inequality so that $\sqrt{2\mu}=C$ gives rise to

[TABLE]

On the other hand, in view of the definition of $u^{3}_{0,{\rm l}{\rm h}}$, there holds for any $\ell\in\mathop{\mathbb{Z}\kern 0.0pt}\nolimits$ that

[TABLE]

which indicates

[TABLE]

Inserting the above estimate into (6.18) and repeating the argument from (4.18) to (4.19), we conclude the proof of (3.18). ∎

Appendix A The proof of Lemmas 4.2 and 6.1

In this section, we present the proof of Lemmas 4.2 and 6.1.

Proof of Lemma 4.2.

By applying Bony’s decomposition in the vertical variable (2.7) to $a\otimes b$, we write

[TABLE]

Considering the support properties to the Fourier transform of the terms in $T^{\rm v}_{a}b$, and noting that $A(D)$ is a smooth homogeneous Fourier multiplier of degree zero, we find

[TABLE]

It follows from Lemma 2.1 and Definition 2.4 that

[TABLE]

This together with Definition 2.2 ensures that

[TABLE]

Along the same line, we get, by applying (2.5), that

[TABLE]

On the other hand, once again considering the support properties to the Fourier transform of the terms in $R^{\rm v}({a},b),$ we find

[TABLE]

Yet it follows from Lemma 2.1 that

[TABLE]

As a result, by virtue of Definition 2.2, we obtain

[TABLE]

Similarly, thanks to (2.5), one has

[TABLE]

Combining (A.2) with (A.4) gives (4.11). And (4.13) follows from (A.3) and (A.5).

It remains to prove (4.12). Indeed similar to the proof of (A.2), we write

[TABLE]

from which and Definition 2.2, we infer

[TABLE]

While we deduce from Definition 2.4 that

[TABLE]

Whereas we get, by applying triangle inequality and Lemma 2.1, that

[TABLE]

This in turn shows that

[TABLE]

which together with (A.6) ensures (4.12). This completes the proof of Lemma 4.2. ∎

Proof of Lemma 6.1.

Let $Q_{1}$ be given by (LABEL:fone4). We first get, by a similar derivation of (A.2), that

[TABLE]

which together with Proposition 2.1 implies that

[TABLE]

While for $Q_{2}$ given by (LABEL:fone4), we get, by a similar derivation of (A.4), that

[TABLE]

from which and Proposition 2.1, we infer

[TABLE]

This together with (LABEL:fone4) and (A.7) ensures (6.5).

The inequality (6.6) can be proved similarly. As a matter of fact, we observe that

[TABLE]

and

[TABLE]

Then (6.6) follows from Proposition 2.1. This completes the proof of this lemma. ∎

Acknowledgments. M. Paicu was partially supported by the Agence Nationale de la Recherche, Project IFSMACS, Grant ANR-15-CE40-0010. P. Zhang is partially supported by NSF of China under Grants 11371347 and 11688101, Morningside Center of Mathematics of The Chinese Academy of Sciences and innovation grant from National Center for Mathematics and Interdisciplinary Sciences.