Global strong solutions to 3-D Navier-Stokes system with strong dissipation in one direction

TL;DR

This paper proves the global existence of strong solutions for the 3-D Navier-Stokes equations with anisotropic dissipation, under conditions where vertical or horizontal viscosity is sufficiently large relative to initial data.

Contribution

It establishes global well-posedness for 3-D Navier-Stokes with anisotropic viscosity, considering different viscous coefficients in vertical and horizontal directions.

Findings

Global strong solutions exist when vertical or horizontal viscosity is large enough.

Anisotropic smallness condition on initial data ensures well-posedness.

Results extend understanding of Navier-Stokes with directional dissipation.

Abstract

We consider three dimensional incompressible Navier-Stokes equation $(NS)$ with different viscous coefficient in the vertical and horizontal variables. In particular, when one of these viscous coefficients is large enough compared to the initial data, we prove the global well-posedness of this system. In fact, we obtain the existence of a global strong solution to $(NS)$ when the initial data verify an anisotropic smallness condition which takes into account the different roles of the horizontal and vertical viscosity.

Equations

$$\{\begin{array}{l}{\partial }_{t}u+u\cdot \nabla u-{\nu }_{\mathrm{h}}{\Delta }_{\mathrm{h}}u-{\nu }_{\mathrm{v}}{\partial }_3^{2}u=-\nabla p\text{in}[0,T]\times R^3,\\ \mathrm{div}u=0,\\ u{|}_{t=0}=u_0,\end{array}$$

$$u_0^{ϵ}={ϵ}^{-\alpha }\sin (\frac{x_3}{ϵ})(-{\partial }_2\phi (x),{\partial }_1\phi (x),0),$$

$${\dot{H}}^{\frac{1}{2}}(R^3)\hookrightarrow L^3(R^3)\hookrightarrow {\dot{B}}_{p,\infty }^{-1+\frac{3}{p}}(R^3)\hookrightarrow {\text{BMO}}^{-1}(R^3)\hookrightarrow {\dot{B}}_{\infty ,\infty }^{-1}(R^3),$$

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

$$\|w_{0}\|_{H^{\frac{1}{2}}(\mathbb{T}^{3})}\exp\bigg{(}\frac{\|u_{0}\|_{L^{2}(\mathbb{T}^{2}_{\rm h})}^{2}}{\nu^{2}}\bigg{)}\leq c\nu$$

$$u_{0}^{N}=\bigl{(}Nu_{\rm h}(x_{h})\cos(Nx_{3}),-\mathop{\rm div}\nolimits_{\rm h}u_{h}(x_{h})sin(Nx_{3})\bigr{)},$$

$$u_0^{\epsilon }(x)=(v_0^{\mathrm{h}}+\epsilon w_0^{\mathrm{h}},w_0^3)(x_{\mathrm{h}},\epsilon x_3)$$

$$\nu_{\rm v}\geq C_{1}\bigl{(}M_{0}+M_{0}^{\frac{1}{4}}\bigr{)}\quad\hbox{with}\quad M_{0}\buildrel\hbox{\footnotesize def}\over{=}\|\omega_{0}\|_{L^{\frac{3}{2}}}^{\frac{3}{2}}+\|\nabla u^{3}_{0}\|_{{H^{-\frac{1}{2},0}}}^{2}\quad\hbox{and}\quad\omega_{0}\buildrel\hbox{\footnotesize def}\over{=}\partial_{1}u_{0}^{2}-\partial_{2}u_{0}^{1},$$

$$\bigl{\|}\omega(t)\bigr{\|}_{L^{\frac{3}{2}}}^{\frac{3}{2}}+\|\nabla u^{3}(t)\|_{{H^{-\frac{1}{2},0}}}^{2}+\int_{0}^{t}\left(\bigl{\|}\nabla_{\nu_{\rm v}}\omega_{\frac{3}{4}}\bigr{\|}_{L^{2}}^{2}+\bigl{\|}\nabla_{\nu_{\rm v}}\nabla v^{3}\bigr{\|}_{{H^{-\frac{1}{2},0}}}^{2}\right)\,dt^{\prime}\leq 2M_{0},$$

$$\begin{split}\Bigl{(}{\mathcal{A}}_{0}^{\frac{1}{2}}\bigl{(}1+{\mathcal{A}}_{0}^{\frac{1}{2}}\bigr{)}\bigl{(}\nu_{\rm h}^{-\frac{7}{8}}+\nu_{\rm h}^{\frac{1}{8}}\bigr{)}\nu_{\rm v}^{-\frac{1}{8}}+&\varepsilon^{1-\theta}\nu_{\rm h}^{-1+\frac{\theta}{2}}\nu_{\rm v}^{-\frac{\theta}{2}}\Bigr{)}{\mathcal{A}}_{\delta,\nu_{h}}(\bar{u}^{\rm h}_{0})\leq c_{0}\nu_{\rm h}\quad\hbox{with}\quad\\ &\qquad{\mathcal{A}}_{0}\buildrel\hbox{\footnotesize def}\over{=}\|v_{0}\|_{{\mathcal{B}}^{\frac{1}{2},0}}\bigl{(}\|v_{0}\|_{B^{0,\frac{1}{2}}}+\|v_{0}\|_{{\mathcal{B}}^{\frac{1}{2},0}}\bigr{)}\end{split}$$

$$u_{0,\varepsilon}(x_{\rm h},x_{3})=\bigl{(}\bar{u}^{\rm h}_{0}(x_{\rm h},\varepsilon x_{3}),0\bigr{)}+v_{0}(x)$$

$$\{\begin{array}{l}{\partial }_{t}u+u\cdot \nabla u-{\nu }_{\mathrm{v}}{\partial }_3^{2}u=-\nabla p\text{mbox}for(t,x)\in R^{+}\times \Omega ,\\ \mathrm{div}u=0,\\ u{|}_{t=0}=u_0,\\ u{|}_{x_3=0}=u{|}_{x_3=1}=0,\end{array}$$

$$\Vert u_0{\Vert }_{L^2}+\Vert u_0{\Vert }_{{\mathcal{B}}_{\mathrm{h}}^2}\le c{\nu }_{\mathrm{v}},$$

$$\frac{[f{]}_{\epsilon }(t,x_{\mathrm{h}},x_3)\text{buildrel}def}{=f(t,x_{\mathrm{h}},\epsilon x_3).}$$

$$\{\begin{array}{l}{\partial }_t{\bar{u}}^{\mathrm{h}}+{\bar{u}}^{\mathrm{h}}\cdot {\nabla }_{\mathrm{h}}{\bar{u}}^{\mathrm{h}}-{\nu }_{\mathrm{h}}{\Delta }_{\mathrm{h}}{\bar{u}}^{\mathrm{h}}-{\nu }_{\mathrm{v}}{\epsilon }^2{\partial }_z^2{\bar{u}}^{\mathrm{h}}=-{\nabla }_{\mathrm{h}}{p}^{\mathrm{h}}\text{mbox}for(t,x_{\mathrm{h}},z)\in R^{+}\times R^3,\\ {\mathrm{div}}_{\mathrm{h}}{\bar{u}}^{\mathrm{h}}=0,\\ {\bar{u}}^{\mathrm{h}}{|}_{t=0}={\bar{u}}_0^{\mathrm{h}}.\end{array}$$

$$p^{\mathrm{h}}=\sum _{i,j=1}^2(-{\Delta }_{\mathrm{h}}{)}^{-1}{\partial }_i{\partial }_j({\bar{u}}^i{\bar{u}}^j).$$

$$\{\begin{array}{l}{\partial }_t{v}_L-{\nu }_{\mathrm{h}}{\Delta }_{\mathrm{h}}{v}_L-{\nu }_{\mathrm{v}}{\partial }_3^2{v}_L=0\text{mbox}for(t,x)\in R^{+}\times R^3,\\ v_L{|}_{t=0}=v_0.\end{array}$$

$$u=v_L+[{\bar{u}}^{\mathrm{h}}{]}_{\epsilon }+Rand\pi =p-p^{\mathrm{h}}.$$

$$\{\begin{array}{l}{\partial }_{t}R+u\cdot \nabla R+R\cdot \nabla (v_L+[{\bar{u}}^{\mathrm{h}}{]}_{\epsilon })-{\nu }_{\mathrm{h}}{\Delta }_{\mathrm{h}}R-{\nu }_{\mathrm{v}}{\partial }_3^{2}R+\nabla \pi =F,\\ withF=(F^{\mathrm{h}},F^{\mathrm{v}})and\\ F^{\mathrm{h}}=-v_L\cdot \nabla (v_L^{\mathrm{h}}+[{\bar{u}}^{\mathrm{h}}{]}_{\epsilon })-[{\bar{u}}^{\mathrm{h}}{]}_{\epsilon }\cdot {\nabla }_{\mathrm{h}}{v}_L^{\mathrm{h}}and\\ F^{\mathrm{v}}=-v_L\cdot \nabla v_L^3-[{\bar{u}}^{\mathrm{h}}{]}_{\epsilon }\cdot {\nabla }_h{v}_L^3-{\partial }_3[p^{\mathrm{h}}{]}_{\epsilon },\\ \mathrm{div}R=0,\\ R{|}_{t=0}=0.\end{array}$$

$$\frac{E(v_0)\text{buildrel}def}{=\Vert v_0{\Vert }_{L^2}^3\Vert {\nabla }_{\mathrm{h}}{v}_0{\Vert }_{L^2}+\Vert v_0{\Vert }_{L^{\infty }}^2\Vert v_0{\Vert }_{L^2}^2.}$$

$$\frac{E(v_{0})}{\sqrt{\nu_{\rm h}\nu_{\rm v}}}\exp\Bigl{(}\frac{\|v_{0}\|_{H^{0,1}}^{4}}{C\nu_{\rm h}^{4}}\Bigr{)}\leq c\nu_{\rm h}^{3},$$

$$u\in C_b([0,\infty [;H^{0,1})\text{and}\nabla u\in L^2(R_{+};H^{0,1}).$$

$$\bigl{(}a\cdot\nabla b|b\bigr{)}_{H^{0,1}}\leq C\|a\|_{H^{0,1}}^{\frac{1}{2}}\|\nabla_{\rm h}a\|_{H^{0,1}}^{\frac{1}{2}}\|b\|_{H^{0,1}}^{\frac{1}{2}}\|\nabla_{\rm h}b\|_{H^{0,1}}^{\frac{3}{2}}.$$

$$\displaystyle\begin{split}\bigl{(}a\cdot\nabla b|b\bigr{)}_{H^{0,1}}=&\bigl{(}a\cdot\nabla b|b\bigr{)}_{L^{2}}+\bigl{(}\partial_{3}(a\cdot\nabla b)|\partial_{3}b\bigr{)}_{L^{2}}\\ =&\bigl{(}a\cdot\nabla\partial_{3}b|\partial_{3}b\bigr{)}_{L^{2}}+\bigl{(}\partial_{3}a\cdot\nabla b|\partial_{3}b\bigr{)}_{L^{2}}\\ =&\bigl{(}\partial_{3}a^{\rm h}\cdot\nabla_{\rm h}b|\partial_{3}b\bigr{)}_{L^{2}}+\bigl{(}\partial_{3}a_{3}\partial_{3}b|\partial_{3}b\bigr{)}_{L^{2}}.\end{split}$$

$$\Vert f{\Vert }_{L_{\mathrm{v}}^{\infty }(L_{\mathrm{h}}^2)}\lesssim \Vert f{\Vert }_{L^2}^{\frac{1}{2}}\Vert {\partial }_{3}f{\Vert }_{L^2}^{\frac{1}{2}},$$

$$\displaystyle\begin{split}\mid\bigl{(}\mathop{\rm div}\nolimits_{\rm h}a^{\rm h}\partial_{3}b|\partial_{3}b\bigr{)}_{L^{2}}\mid\leq&\|\mathop{\rm div}\nolimits_{h}a^{h}\|_{L^{\infty}_{\rm v}(\dot{H}^{-\frac{1}{2}}_{\rm h})}\|(\partial_{3}b)^{2}\|_{L^{1}_{\rm v}(\dot{H}^{\frac{1}{2}}_{\rm h})}\\ \leq&\|\nabla_{\rm h}a\|_{L^{\infty}_{\rm v}(\dot{H}^{-\frac{1}{2}}_{\rm h})}\|\partial_{3}b\|_{L^{2}_{\rm v}(\dot{H}^{\frac{3}{4}}_{\rm h})}^{2}\\ \lesssim&\|a\|_{L^{\infty}_{\rm v}(\dot{H}^{\frac{1}{2}}_{\rm h})}\|\partial_{3}b\|_{L^{2}}^{\frac{1}{2}}\|\nabla_{\rm h}\partial_{3}b\|_{L^{2}}^{\frac{3}{2}}.\end{split}$$

$$\frac{d}{d{x}_3}{\int }_{R^2}|{\xi }_{\mathrm{h}}||\hat{f}({\xi }_{\mathrm{h}},x_3){|}^{2}d{\xi }_{\mathrm{h}}={\int }_{R^2}|{\xi }_{\mathrm{h}}|{\partial }_3|\hat{f}({\xi }_{\mathrm{h}},x_3){|}^{2}d{\xi }_{\mathrm{h}}.$$

$$\Vert f{\Vert }_{L_{\mathrm{v}}^{\infty }({\dot{H}}_{\mathrm{h}}^{\frac{1}{2}})}\lesssim \Vert f{\Vert }_{L^2}^{\frac{1}{2}}\Vert {\nabla }_{\mathrm{h}}{\partial }_{3}f{\Vert }_{L^2}^{\frac{1}{2}}.$$

$$\mid\bigl{(}\mathop{\rm div}\nolimits_{\rm h}a^{\rm h}\partial_{3}b|\partial_{3}b\bigr{)}_{L^{2}}\mid\lesssim\|a\|_{L^{2}}^{\frac{1}{2}}\|\nabla_{\rm h}\partial_{3}a\|_{L^{2}}^{\frac{1}{2}}\|\partial_{3}b\|_{L^{2}}^{\frac{1}{2}}\|\nabla_{\rm h}\partial_{3}b\|_{L^{2}}^{\frac{3}{2}}.$$

$$\displaystyle\begin{split}\mid\bigl{(}\partial_{3}a^{\rm h}\cdot\nabla_{\rm h}b|\partial_{3}b\bigr{)}_{L^{2}}\mid\leq&\|\partial_{3}a^{\rm h}\|_{L^{2}_{\rm v}(L^{4}_{\rm h})}\|\nabla_{\rm h}b\|_{L^{\infty}_{\rm v}(L^{2}_{\rm h})}\|\partial_{3}b\|_{L^{2}_{\rm v}(L^{4}_{\rm h})}\\ \lesssim&\|\partial_{3}a^{\rm h}\|_{L^{2}}^{\frac{1}{2}}\|\nabla_{\rm h}\partial_{3}a^{\rm h}\|_{L^{2}}^{\frac{1}{2}}\|\nabla_{\rm h}b\|_{L^{2}}^{\frac{1}{2}}\|\nabla_{\rm h}\partial_{3}b\|_{L^{2}}^{\frac{1}{2}}\|\partial_{3}b\|_{L^{2}}^{\frac{1}{2}}\|\nabla_{\rm h}\partial_{3}b\|_{L^{2}}^{\frac{1}{2}},\end{split}$$

Full text

1. Introduction

In this paper, we consider 3-D anisotropic incompressible Navier-Stokes system with different vertical and horizontal viscosity. Our goal is to study the role of a large viscous coefficient in one direction that plays in obtaining the global existence of strong solutions to Navier-Stokes system. Let us recall Navier-Stokes equations which describes the evolution of viscous fluid in ${\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{3}:$

[TABLE]

where $\Delta_{\rm h}\buildrel\hbox{\footnotesize def}\over{=}\partial_{x_{1}}^{2}+\partial_{x_{2}}^{2}$ designates the horizontal Laplacian, $\nu_{\rm h}>0$ and $\nu_{\rm v}>0$ are respectively the horizontal and the vertical viscous coefficients, $u=(u_{1},u_{2},u_{3})$ denotes the fluid velocity, and $p$ the scalar pressure function, which guarantees the divergence free condition of the velocity field.

When $\nu_{\rm h}=\nu_{\rm v}=\nu>0,$ (1.1) is exactly the classical Navier-Stokes system. In the sequel, we shall always denote this system by $(NS_{\nu}).$ Whereas when $\nu_{\rm h}=\nu>0$ and $\nu_{\rm v}=0,$ $(\ref{NS})$ reduces to the anisotropic Navier-Stokes system arising from geophysical fluid mechanics (see [7]). We remark that Navier-Stokes system with large vertical viscosity is a usual model to study the evolution of the fluid in a thin domain in the vertical direction (see (2.4) of [21] for instance).

We begin by recalling some important and classical results about Navier-Stokes system. Especially, we shall focus on the conditions which guarantee the global existence of strong solution to $(NS_{\nu}).$

The first important result about the classical Navier-Stokes system was obtained by Leray in the seminar paper [16] in 1933. He proved that given an arbitrary finite energy solenoidal vector field, $(NS_{\nu})$ has a global in time weak solution which verifies the energy inequality. This solution is unique and regular for positive time in ${\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{2},$ but unfortunately the uniqueness and regularities of such solution in three space dimension is still one of most challenging open questions in the field of mathematical fluid mechanics. Fujita-Kato [12] gave a partial answer to the construction of global unique solution to $(NS_{\nu})$. Indeed, the theorem of Fujita-Kato [12] allows to construct local in time unique solution to (1.1) with initial data in the homogeneous Sobolev spaces $\dot{H}^{\frac{1}{2}}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{3})$, or in the Lebsegue space $L^{3}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{3})$ (see [14]). Moreover, if the initial data is small enough compared to the the viscosity, that is, $\|u_{0}\|_{\dot{H}^{\frac{1}{2}}}\leq c\nu$, for some sufficiently small constant $c,$ then the strong solution exists globally in time.

The above result was extended by Cannone, Meyer and Planchon [5] for initial data in Besov spaces with negative index. More precisely, they proved that if the initial data belongs to the Besov space, $\dot{B}^{-1+\frac{3}{p}}_{p,\infty}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{3})$ for some $p\in]3,\infty[$ and its norm is sufficiently small compared to the viscosity, then $(NS_{\nu})$ has a unique global solution. The typical example of such kind of initial data reads

[TABLE]

where $0<\alpha<1$ and $\varphi\in\mathcal{S}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{3};{\mathop{\mathbb{R}\kern 0.0pt}\nolimits})$. We remark that this type of initial data is not small in either $\dot{H}^{\frac{1}{2}}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{3})$ or $L^{3}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{3}).$

The end-point result in this direction is given by Koch and Tataru [15]. They proved that given initial data in the derivatives of BMO space and its norm is sufficiently small compared to the viscosity, then $(NS_{\nu})$ has a unique global solution (one may check [22] for the regularities of such solutions). We remark that for $p\in]3,\infty[$, there holds

[TABLE]

and the norms to the above spaces are sclaing-invariant under the following scaling transformation

[TABLE]

We notice that for any solution $u$ of $(NS_{\nu})$ on $[0,T],$ $u_{\lambda}$ determined by (1.2) is also a solution of $(NS_{\nu})$ on $[0,T/\lambda^{2}].$ We point out that the largest space, which belongs to ${\mathcal{S}}^{\prime}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{3})$ and the norm of which is scaling invariant under (1.2), is $\dot{B}^{-1}_{\infty,\infty}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{3})$. Moreover, Bourgain and Pavlović [4] proved that $(NS_{\nu})$ is actually ill-posed with initial data in $\dot{B}^{-1}_{\infty,\infty}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{3}).$

We mention some examples of large initial data which generate unique global solution to $(NS_{\nu})$. First of all, Raugel and Sell [21] obtained the global well-posedness of $(NS_{\nu})$ in a thin enough domain. This result was extended by Gallagher [13] that $(NS_{\nu})$ has a unique global periodic solution provided that the initial data $u_{0}$ can be split as $u_{0}=v_{0}+w_{0}$, where $v_{0}$ is a bi-dimensional solenoidal vector field in $L^{2}({\mathbb{T}}^{2}_{\rm h})$ and $w_{0}\in H^{\frac{1}{2}}(\mathbb{T}^{3})$ which satisfy

[TABLE]

for some $c$ being sufficiently small.

In 2006, Chemin and Gallagher [8] constructed the following example of initial data which generates a unique global solution to $(NS_{\nu})$, and which is large in $\dot{B}^{-1}_{\infty,\infty}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{3})$ and is strongly oscillatory in one direction,

[TABLE]

where $\|u_{\rm h}\|_{L^{2}(\mathbb{T}^{2}_{h})}\leq C(\ln N)^{\frac{1}{9}}$.

The other interesting class of large initial data which generate unique global solutions to $(NS_{\nu})$ is the so-called slowly varying data,

[TABLE]

which was introduced by Chemin and Gallagher in [9] (see also [10, 11]).

On the other hand, by crucially using the fact that $\mathop{\rm div}\nolimits u=0,$ Zhang [23] and the authors [20] improved Fujita-Kato’s result by requiring only two components of the initial velocity being sufficiently small in some critical Besov space even when $\nu_{\rm h}>0$ and $\nu_{\rm v}=0$ in (1.1).

Liu and the second author [17] first investigated the global well-podesness of (1.1) with $\nu_{\rm h}=1$ and $\nu_{\rm v}$ being sufficiently large. In particular, they proved the following result:

Theorem 1.1**.**

Let $\nu_{\rm h}=1,$ $u_{0}$ satisfy $\mathop{\rm curl}\nolimits u_{0}\in L^{\frac{3}{2}}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{3})$ and $\mathop{\rm div}\nolimits u_{0}=0.$ Then there exists some universal positive constant $C_{1}$ such that if

[TABLE]

(1.1) has a unique global solution u\in C\bigl{(}[0,\infty[,H^{\frac{1}{2}}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{3})\bigr{)}\bigcap L^{2}_{\rm{loc}}\bigl{(}{\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{+};H^{\frac{3}{2}}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{3})\bigr{)} so that for any $t>0,$

[TABLE]

where $\nabla_{\nu_{\rm v}}\buildrel\hbox{\footnotesize def}\over{=}\left(\nabla_{\rm h},\nu_{\rm v}\partial_{3}\right).$

Our goal in this paper is to study the role of one big viscosity that plays in obtaining global existence of strong solutions to (1.1) even the viscous coefficients in other variables are small. One of the consequence of our result below (see Theorem 1.2) ensures that given regular initial data $u_{0},$ if the horizontal viscosity in (1.1) is small like $\nu_{h}=\epsilon$ and the vertical one is big enough like $\frac{\mathfrak{K}}{\epsilon^{15}}$ for some large enough constant $\mathfrak{K},$ then (1.1) has a unique global solution. This case is obviously not covered by Theorem 1.1. Furthermore, here we allow the initial data to have a large and slowly varying part.

The main result of this paper states as follows:

Theorem 1.2**.**

Let  $v_{0}\in{\mathcal{B}}^{\frac{1}{2},0}\cap B^{0,\frac{1}{2}}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{3})$ be a solenoidal vector field and $\bar{u}^{\rm h}_{0}$ be a horizontal solenoidal vector field in $H^{1}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{3})\cap L^{\infty}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}_{\rm v};\dot{H}^{-\delta}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}_{\rm h}))\cap L^{\infty}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}_{\rm v};H^{1}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}_{\rm h}))$ for some $\delta\in]0,1[.$ We also assume that  $\bar{u}^{\rm h}_{0},$  $\partial_{z}\bar{u}^{\rm h}_{0}$ and $\partial_{z}^{2}\bar{u}^{\rm h}_{0}$ belongs to $L^{2}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}_{\rm v};\dot{H}^{-\frac{1}{2}}\cap\dot{H}^{\frac{1}{2}}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}_{\rm h})).$ Then for any $\theta\in[0,1/2[,$ there exists a positive constant ${\mathcal{A}}_{\delta,\nu_{\rm h}}(\bar{u}^{\rm h}_{0}),$ which depends on the norms of $\bar{u}^{\rm h}_{0}$ above, such that if $\nu_{\rm v}\geq\nu_{\rm h}>0$ and $\nu_{\rm v}$ is so large that

[TABLE]

for some $c_{0}$ sufficiently small, the initial data

[TABLE]

generates a unique global solution $u$ to (1.1) in the space $C_{b}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{+};B^{0,\frac{1}{2}}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{3}))$ with $\nabla u\in L^{2}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{+};B^{0,\frac{1}{2}}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{3}))$. **

The definitions of the Besov norms will be presented in Section 3. The exact form of the constant ${\mathcal{A}}_{\delta,\nu_{h}}(\bar{u}^{\rm h}_{0})$ will be given by (5.9).

Remark 1.1**.**

• (1)

We remark that the norms of $v_{0}$ in both ${\mathcal{B}}^{\frac{1}{2},0}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{3})$ and $B^{0,\frac{1}{2}}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{3})$ are scaling invariant under the scaling transformation (1.2).

• (2)

Similar result holds for (1.1) in a regular bounded domain with Dirichlet boundary condition for the velocity field. Indeed the first eigenvalue of the operator, $A\buildrel\hbox{\footnotesize def}\over{=}-\nu_{\rm h}\Delta_{\rm h}-\nu_{\rm v}\partial_{3}^{2}$, is bigger than $\max\left(\nu_{\rm h},\nu_{\rm v}\right)$. On the other hand, Avrin **[1]** proved that as long as $\|u_{0}\|_{D(A^{\frac{1}{2}})}\leq\min\left(\nu_{\rm h},\nu_{\rm v}\right)\lambda_{1}^{\frac{1}{4}},$ (1.1) has a unique global solution.

• (3)

Let us note that in the periodic case if the vertical viscosity, $\nu_{\rm v},$ in (1.1) is big enough, the spectrum of the Laplace on functions with null vertical mean is pushed far from zero, which ensures the exponential decay of the linear solution. So that in order to prove the global well-posedness of (1.1) in $\mathop{\mathbb{T}\kern 0.0pt}\nolimits^{3},$ we need first to decompose the solution into a 2D part and a 3D part with vertical null mean, and then to obtain that the interactions between the 2D part and the 3D part are small. We shall not present detail here.

In the case when $\nu_{\rm h}=0$ in (1.1), that is

[TABLE]

where $\Omega={\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{2}\times]0,1[,$ we have the following global well-posedness result for (1.7):

Theorem 1.3**.**

Given solenoidal vector field $u_{0}\in L^{2}\cap{\mathcal{B}}_{h}^{2}(\Omega),$ there exists a small enough positive constant $c$ so that if

[TABLE]

(1.7) has a unique solution $u\in C_{b}([0,\infty[;L^{2}\cap{\mathcal{B}}_{h}^{2}(\Omega))$ with $\partial_{3}u\in L^{2}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{+};L^{2}\cap{\mathcal{B}}_{h}^{2}(\Omega)).$**

We remark that the boundary condition in (1.7) is quite natural, which corresponds to the Dirichlet boundary condition for 1-D heat equation in a bounded interval. So far, we still do not know how to prove similar version of Theorem 1.3 in the whole space case.

Let us complete this section by the notations in this context.

Let $A,B$ be two operators, we denote $[A;B]=AB-BA,$ the commutator between $A$ and $B$. For $a\lesssim b$, we mean that there is a uniform constant $C,$ which may be different on different lines, such that $a\leq Cb$. We denote by $(a|b)$ the $L^{2}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{3})$ inner product of $a$ and $b,$ $(d_{\ell})_{\ell\in\mathop{\mathbb{Z}\kern 0.0pt}\nolimits}$ will be a generic element of $\ell^{1}(\mathop{\mathbb{Z}\kern 0.0pt}\nolimits)$ so that $\sum_{\ell\in\mathop{\mathbb{Z}\kern 0.0pt}\nolimits}d_{\ell}=1$. For $X$ a Banach space and $I$ an interval of ${\mathop{\mathbb{R}\kern 0.0pt}\nolimits},$ we denote by ${C}(I;\,X)$ the set of continuous functions on $I$ with values in $X,$ and by ${{C}}_{b}(I;\,X)$ the subset of bounded functions of ${C}(I;\,X).$ For $q\in[1,+\infty],$ the notation $L^{q}(I;\,X)$ stands for the set of measurable functions on $I$ with values in $X,$ such that $t\longmapsto\|f(t)\|_{X}$ belongs to $L^{q}(I).$

2. Ideas of the proof and structure of the paper

In what follows, we shall always denote

[TABLE]

In order to deal with $[\bar{u}^{\rm h}_{0}]_{\varepsilon}$ which appears in (1.6), motivated by [9, 11], we construct $(\bar{u}^{\rm h},p^{\rm h})$ through

[TABLE]

By taking $\mathop{\rm div}\nolimits_{\rm h}$ to the momentum equation of (2.1), we find

[TABLE]

To handle $v_{0}$ in (1.6), we construct $v_{L}$ via

[TABLE]

Then the strategy to the proof of Theorem 1.2 will be as follows: we first write

[TABLE]

It follows form (1.1), (2.1) and (2.3) that $(R,\nabla\pi)$ verifies

[TABLE]

We are going to prove that (2.5) has a unique global solution under the smallness condition (1.5).

In order to exploring the main idea to the proof of Theorem 1.2. Let us first assume $\bar{u}^{\rm h}_{0}=0$ in (1.6). Moreover, to avoid technicality, we assume $v_{0}\in H^{1}\cap L^{\infty}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{3}).$ Then we have the following simplified version of Theorem 1.2.

Theorem 2.1**.**

Let $v_{0}\in H^{1}\cap L^{\infty}$ be a solenoidal vector field, we denote

[TABLE]

We assume that $\nu_{\rm v}\geq\nu_{\rm h}>0.$ Then there exist two positive constants $c,C$ so that if

[TABLE]

(1.1) with initial data $v_{0}$ has a unique global solution $u$ with

[TABLE]

Here $H^{0,1}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{3})$ denotes the space of functions $f$ with both $f$ and $\partial_{3}f$ belonging to $L^{2}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{3}).$**

We begin the proof of the above theorem by the following useful lemmas.

Lemma 2.1**.**

Let $a=(a^{\rm h},a_{3})$ be a solenoidal vector field in $H^{0,1}$ with $\nabla_{\rm h}a$ belonging to $H^{0,1}$. Let $b\in H^{0,1}$ with $\nabla_{\rm h}b\in H^{0,1}$. Then one has

[TABLE]

Proof.

We recall that \|a\|_{H^{0,1}}=\bigl{(}\|a\|_{L^{2}}^{2}+\|\partial_{3}a\|_{L^{2}}^{2}\bigr{)}^{\frac{1}{2}}. It is easy to observe that due to $\mathop{\rm div}\nolimits a=0,$ \bigl{(}a\cdot\nabla b|b\bigr{)}_{L^{2}}=0, so that there holds

[TABLE]

Notice that

[TABLE]

and $\partial_{3}a_{3}=-\mathop{\rm div}\nolimits_{\rm h}a^{h}$, we deduce from the law of product in Sobolev spaces that

[TABLE]

On the other hand, we write

[TABLE]

Integrating the above equality over $]-\infty,x_{3}]$ and using Hölder’s inequality, we achieve

[TABLE]

As a result, it comes out

[TABLE]

Whereas observing that

[TABLE]

which together with (2.8) ensures the lemma. $\Box$

The next lemma is concerned with the linear equation (2.3), which tells us the small quantities that will be used in what follows.

Lemma 2.2**.**

Let $v_{0}\in L^{2}$ with $\nabla_{\rm h}v_{0}\in L^{2}$. Let $v_{L}$ be the corresponding solution of (2.3). Then we have

[TABLE]

Proof.

Indeed by applying standard energy method to (2.3), we get

[TABLE]

which implies the lemma. $\Box$

Lemma 2.3**.**

Let $E(v_{0})$ be given by (2.6). Then under the assumptions of Lemma 2.2 and $\nu_{\rm v}\geq\nu_{\rm h}>0,$ one has

[TABLE]

Proof.

We begin by writing that

[TABLE]

Applying Lemma 2.2 and maximal principle for (2.3) gives

[TABLE]

To handle the other term in (2.9), by applying the Sobolev imbedding of $\dot{H}^{\frac{1}{4}}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}_{\rm v})\hookrightarrow L^{4}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}_{\rm v}),$ we obtain

[TABLE]

which together with Lemma 2.2 implies that

[TABLE]

Whereas noticing from Sobolev imbedding Theorem that $\dot{H}^{\frac{2}{3}}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{2}_{h})\hookrightarrow L^{6}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{2}_{h})$, we write

[TABLE]

Taking the $L^{2}_{\rm v}$ norm leads to

[TABLE]

which together with Lemma 2.2 ensures that

[TABLE]

This gives rise to

[TABLE]

Along with (2.9) and (2.10), we complete the proof of the lemma. $\Box$

Proof of the Theorem 2.1.

Let $v_{L}$ be determined by (2.3). We write

[TABLE]

Inserting the above substitution into (1.1) yields

[TABLE]

It follows from classical theory on Navier-Stokes system that (2.11) has a unique solution $R\in C([0,T^{\ast}[;H^{0,1})$ with $\nabla R\in L^{2}(]0,T^{\ast}[;H^{0,1})$ for some maximal existing time $T^{\ast}.$ In the following, we are going to prove that $T^{\ast}=\infty$ under the smallness condition (2.7). For simplicity, we just present the a priori estimate.

By taking $H^{0,1}$ scalar product of the $R$ equation of (2.11) with $R,$ we obtain

[TABLE]

Applying Lemma 2.1 gives

[TABLE]

and

[TABLE]

Then by applying Young’s inequality, $xy\leq\frac{3}{4}x^{\frac{4}{3}}+\frac{1}{4}y^{4},$ we achieve

[TABLE]

Similarly, notice that $H^{\frac{1}{2}}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{2}_{\rm h})\subset L^{4}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{2}_{\rm h}),$ we get

[TABLE]

For the last term in (2.11), we first get, by using integrating by parts, that

[TABLE]

which implies

[TABLE]

Let us denote

[TABLE]

We are going to prove that $T^{\star}=T^{\ast}.$ Otherwise by substituting the above estimates into (2.12), for $t\leq T^{\star},$ we arrive at

[TABLE]

Applying Gronwall’s inequality to (2.14) yields

[TABLE]

from which, Lemmas 2.2 and 2.3, for $t\leq T^{\star},$ we infer

[TABLE]

Then under the smallness condition (2.7), we have

[TABLE]

which contradicts with (2.13). This in turn shows that $T^{\star}=T^{\ast}=\infty.$ Furthermore inserting the estimate (2.15) into (2.14) shows that $\nabla R\in L^{2}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{+};H^{0,1}).$ This completes the proof of Theorem 2.1. $\Box$

The organization of this paper is as follows:

In the third section, we shall recall some basic facts on Littlewood-Paley theory;

In the fourth section, we present the priori estimates for smooth enough solutions of (2.1) and (2.3);

In the fifth section, we prove Theorem  1.2;

In the sixth section, we present the proof of Theorem 1.3;

Finally in the Appendix, we present the proofs of several technical lemmas which have been used in the proof of Theorem  1.2.

3. Basics on Littlewood-Paley theory

Before we present the function spaces we are going to work with in this context, let us briefly recall some basic facts on Littlewood-Paley theory (see e.g. [2]). Let $\varphi$ and $\chi$ be smooth functions supported in $\mathcal{C}\buildrel\hbox{\footnotesize def}\over{=}\{\tau\in{\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{+},\ \frac{3}{4}\leq\tau\leq\frac{8}{3}\}$ and $\mathfrak{B}\buildrel\hbox{\footnotesize def}\over{=}\{\tau\in{\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{+},\ \tau\leq\frac{4}{3}\}$ respectively such that

[TABLE]

For $a\in{\mathcal{S}}^{\prime}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{3}),$ we set

[TABLE]

where $\xi_{\rm h}=(\xi_{1},\xi_{2}),$ $\xi=(\xi_{\rm h},\xi_{3}),$ ${\mathcal{F}}a$ and $\widehat{a}$ denote the Fourier transform of the distribution $a.$ The dyadic operators satisfy the property of almost orthogonality:

[TABLE]

Similar properties hold for $\Delta_{k}^{\rm h}$ and $\Delta_{\ell}^{\rm v}.$

Let us recall the anisotropic Bernstein type lemma from [11, 19] .

Lemma 3.1**.**

Let ${\mathcal{B}}_{{\rm h}}$ (resp. ${\mathcal{B}}_{\rm v}$) 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 hold

[TABLE]

Due to the anisotropic spectral properties of the linear equation (2.3), we need the following anisotropic type Besov norm:

Definition 3.1**.**

Let $s_{1},s_{2}\in{\mathop{\mathbb{R}\kern 0.0pt}\nolimits}$ and $a\in{\mathcal{S}}_{h}^{\prime}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{3}),$ we define the norm

[TABLE]

In particular, when $s_{1}=0$ we denote ${\mathcal{B}}^{0,s_{2}}$ by $B^{0,s_{2}}$ with

[TABLE]

We recall the classical homogeneous anisotropic Sobolev norm as follows

[TABLE]

In order to obtain a better description of the regularizing effect for the transport-diffusion equation, we will use Chemin-Lerner type spaces.

Definition 3.2**.**

Let $p\in[1,+\infty]$ and $T\in]0,+\infty]$. We define the norms of $\widetilde{L}^{p}_{T}({\mathcal{B}}^{s_{1},s_{2}}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{3}))$ and $\widetilde{L}^{p}_{T}(B^{0,s_{2}})$ by

[TABLE]

and

[TABLE]

respectively. **

In particular, when $p=2,$ we have

[TABLE]

In order to study fluid evolving between two parallel plans, namely to prove Theorem 1.3, we also need the following norms:

Definition 3.3**.**

Let $\Omega={\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{2}\times]0,1[,$ $s\in{\mathop{\mathbb{R}\kern 0.0pt}\nolimits}$ and $p\in[1,\infty],$ for $a\in C^{\infty}(\Omega),$ we define

[TABLE]

To overcome the difficulty that one can not use Gronwall’s type argument for the Chemin-Lerner type norms, we need the time-weighted Chemin-Lerner norm introduced by the authors in [20]:

Definition 3.4**.**

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

[TABLE]

Finally we recall the isentropic para-differential decomposition from [3]: let $a$ and  $b$ be in ${\mathcal{S}}^{\prime}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{3})$,

[TABLE]

In what follows, we shall use the anisotropic version of Bony’s decomposition for both horizontal and vertical variables.

As an application of the above basic facts on Littlewood-Paley theory, we present the following product law in the anisotropic Besov spaces.

Lemma 3.2**.**

Let $\tau_{1},\tau_{2}\in]-1/2,1/2]$ with $\tau_{1}+\tau_{2}>0.$ Let $a\in B^{0,\tau_{1}}$ with $\nabla_{\rm h}a\in B^{0,\tau_{1}},$ let $b\in B^{0,\tau_{2}}$ with $\nabla_{\rm h}b\in B^{0,\tau_{2}}.$ Then one has

[TABLE]

Proof.

We first get, by applying Bony’s decomposition to $ab$ in vertical variable, that

[TABLE]

Due to $\tau_{1}\leq\frac{1}{2},$ applying Lemma 3.1 gives

[TABLE]

from which, and the support properties to the Fourier transform of the terms in $T^{\rm v}_{a}b,$ we infer

[TABLE]

Here and in all that follows, we always denote $\left(d_{\ell}\right)_{\ell\in\mathop{\mathbb{Z}\kern 0.0pt}\nolimits}$ to be a generic element of $\ell^{1}(\mathop{\mathbb{Z}\kern 0.0pt}\nolimits)$ so that $\sum_{\ell\in\mathop{\mathbb{Z}\kern 0.0pt}\nolimits}d_{\ell}=1.$ The same estimate holds for $T^{\rm v}_{b}a.$

On the other hand, we deduce from Lemma 3.1 that

[TABLE]

where in the last step, we used the fact that $\tau_{1}+\tau_{2}>0,$ so that

[TABLE]

This completes the proof of the lemma. $\Box$

Remark 3.1**.**

We remark that the law of product (3.6) works also for Chemin-Lerner norms.

4. The a priori estimates of $v_{L}$ and $\bar{u}^{\rm h}$

The goal of this section is to present the a priori estimates of $v_{L}$ and $\bar{u}^{\rm h}.$

Proposition 4.1**.**

Let $v_{0}\in{\mathcal{B}}^{\frac{1}{2},0}\cap B^{0,\frac{1}{2}}$ and $v_{L}$ be the corresponding solution of (2.3). Then we have

[TABLE]

Proof.

We get, by first applying the operator $\Delta_{k}^{\rm h}\Delta_{\ell}^{\rm v}$ to the system (2.3) and then taking $L^{2}$ inner product of the resulting equation with $\Delta_{k}^{\rm h}\Delta_{\ell}^{\rm v}v_{L},$ that

[TABLE]

Integrating the above equality over $[0,t]$ and then taking square root of the resulting equality, we write

[TABLE]

By multiplying $2^{\frac{k}{2}}$ to the above inequality and taking $\ell^{2}$ norm with respect to $k\in\mathop{\mathbb{Z}\kern 0.0pt}\nolimits$ and then taking $\ell^{1}$ norm with respect to $\ell\in\mathop{\mathbb{Z}\kern 0.0pt}\nolimits,$ we achieve

[TABLE]

Whereas it follows from Fourier-Plancherel equality and Lemma 3.1 that

[TABLE]

from which and (4.2), we infer

[TABLE]

This leads to the first inequality of (4.1).

On the other hand, we get, by first applying the operator $\Delta_{\ell}^{\rm v}$ to the system (2.3) and then taking $L^{2}$ inner product of the resulting equation with $2^{\ell}\Delta_{\ell}^{\rm v}v_{L},$ that

[TABLE]

Integrating the above equality over $[0,t]$ and taking square root of the resulting equality, and then taking $\ell^{1}$ norm with respect to $\ell\in\mathop{\mathbb{Z}\kern 0.0pt}\nolimits,$ we obtain the second inequality of (4.1). This completes the proof of the proposition. $\Box$

Lemma 4.1**.**

Let $\bar{u}^{\rm h}_{0}$ and $\nabla_{\rm h}\bar{u}^{\rm h}_{0}$ be in $L^{2}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{3})\cap L^{\infty}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}_{\rm v};L^{2}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{2}_{\rm h})).$ Then (2.1) has a unique global solution so that

[TABLE]

If moreover, $\bar{u}^{\rm h}_{0}\in L^{\infty}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}_{\rm v};\dot{H}^{-\delta}({\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{2}_{\rm h}))$ for some $\delta\in]0,1[,$ then we have

[TABLE]

Proof.

Theorem 1.2 of [11] ensures the global existence of solutions to (2.1). Moreover, (2.4) of [11] gives (4.3). To prove the estimate (4.4), we introduce

[TABLE]

Then in view of (2.1), we write

[TABLE]

It follows from Theorem 1.2 of [11] that

[TABLE]

Yet by virtue of (4.5), we have

[TABLE]

from which and (4.7), we deduce (4.4). $\Box$

Proposition 4.2**.**

Under the assumptions of Lemma 4.1, for any $t>0,$ we have

[TABLE]

where $A_{\nu_{\rm h},\delta}(\bar{u}^{\rm h}_{0})$ is given by (4.4).**

Proof.

Let us denote

[TABLE]

Then by virtue of (2.1), we write

[TABLE]

By applying $\Delta_{\ell}^{\rm v}$ to the above equation and then taking $L^{2}$ inner product of the resulting equation with $\Delta_{\ell}^{\rm v}\bar{u}^{\rm h}_{\lambda},$ we obtain

[TABLE]

By applying Bony’s decomposition (3.5) to $\bar{u}^{\rm h}\otimes\bar{u}^{\rm h}_{\lambda}$ for the vertical variable, one has

[TABLE]

Due to the support properties to the Fourier of the terms in $T^{\rm v}_{\bar{u}^{\rm h}}\bar{u}^{\rm h}_{\lambda},$ we deduce

[TABLE]

By applying Hölder’s inequality and using Definition 3.4, we get

[TABLE]

Along the same line, we have

[TABLE]

which implies

[TABLE]

As a result, it comes out

[TABLE]

By integrating (4.10) over $[0,t]$ and then inserting (4.11) 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 resulting inequality over $\mathop{\mathbb{Z}\kern 0.0pt}\nolimits,$ we achieve

[TABLE]

Taking $\lambda=C^{2}\nu_{\rm h}^{-3}$ in the above inequality leads to

[TABLE]

from which, and (4.9), we infer

[TABLE]

Note that

[TABLE]

we deduce that

[TABLE]

This together with (4.3) and (4.4) ensures (4.8). $\Box$

Proposition 4.3**.**

Let $\bar{u}^{\rm h}$ be a smooth enough solution of (2.1). Then for any $s^{\prime}\in]-1,1[$ and $s\in]-1,0[,$ we have

[TABLE]

and

[TABLE]

where $A_{\nu_{\rm h},\delta}(\bar{u}^{\rm h}_{0})$ is given by (4.4). **

Proof.

In fact, (4.12) and (4.13) follows directly from Lemma 4.2 of [11] and Lemma 4.1. To prove (4.14), we get, by first applying $\partial_{z}^{2}$ to (2.1) and then taking $\dot{H}^{s}_{\rm h}$ inner product of the resulting equation with $\partial_{z}^{2}\bar{u}^{\rm h},$ that

[TABLE]

Applying Lemma 1.1 of [6] yields

[TABLE]

Due to $s\in]-1,0[,$ the law of product in Sobolev spaces implies that

[TABLE]

Along the same line, one has

[TABLE]

Inserting the above estimates into (4.15) and integrating the resulting equality with respect to $z$ gives rise to

[TABLE]

Notice that

[TABLE]

we get, by applying Young’s inequality, that

[TABLE]

Applying Gronwall’s inequality and using (4.13) leads to (4.14). $\Box$

Corollary 4.1**.**

Under the assumptions of Theorem 1.2, for any $\theta\in[0,1/2[,$ we have

[TABLE]

Proof.

Indeed it follows from interpolation inequality in Besov spaces that

[TABLE]

which together with Proposition 4.3 ensures (4.16). $\Box$

5. The proof of Theorem 1.2

The goal of this section is to present the proof of Theorem 1.2.

Proof of Theorem 1.2.

For simplicity, we just present the priori estimates for smooth enough solutions of (2.5). Let $\bar{u}^{\rm h}$ be a smooth enough solution of (2.1). Let $\theta\in[0,1/2[,$ $\lambda_{i},i=1,2,3,$ and $\nabla_{\varepsilon}\buildrel\hbox{\footnotesize def}\over{=}(\nabla_{\rm h},\varepsilon\partial_{3}),$ we denote

[TABLE]

And similar notations for $\pi_{\lambda}$ and $F_{\lambda}.$ Then it follows from (2.5) that

[TABLE]

where $F_{\lambda}=(F_{\lambda}^{\rm h},F^{3}_{\lambda})$ with $(F^{\rm h},F^{3})$ being given by (2.5).

Applying the operator $\Delta_{\ell}^{\rm v}$ to (5.2) and taking $L^{2}$ inner product of the resulting equation with $\Delta_{\ell}^{\rm v}R_{\lambda}$ yields

[TABLE]

The estimate of the above terms relies on the following lemmas:

Lemma 5.1**.**

There holds

[TABLE]

Lemma 5.2**.**

There holds

[TABLE]

Lemma 5.3**.**

There holds

[TABLE]

Lemma 5.4**.**

Let $p^{\rm h}$ be given by (2.2). Then for any $\theta\in[0,1/2[,$ we have

[TABLE]

We shall postpone the proof of the above lemmas in the Appendix.

Let us admit the above Lemmas for the time being and continue to handle the terms in the second line of (5.3). We first get, by using integration by parts, that

[TABLE]

Then applying the law of product, Lemma 3.2, gives

[TABLE]

and

[TABLE]

Along the same line, by using integrating by parts, one has

[TABLE]

It follows from the law of product in anisotropic Besov space, Lemma 3.2, that

[TABLE]

which implies

[TABLE]

Whereas we get, by using integrating by parts, that

[TABLE]

Applying Lemma 5.1 leads to

[TABLE]

Applying Lemma 5.2 gives

[TABLE]

Applying Lemma 5.3 yields

[TABLE]

Finally for any $\theta\in[0,\frac{1}{2}[,$ we get, by applying Lemma 5.4, that

[TABLE]

Let us denote

[TABLE]

Then it follows from Proposition 4.1 that

[TABLE]

By integrating (5.3) over $[0,t]$ and inserting the above estimates into the resulting inequality, and then we take the square root of resulting inequality to achieve

[TABLE]

Multiplying $2^{\frac{\ell}{2}}$ to the above inequality and summing up the resulting inequality for $\ell\in\mathop{\mathbb{Z}\kern 0.0pt}\nolimits$ leads to

[TABLE]

Let us assume that $\nu_{\rm v}\geq\nu_{\rm h}.$ Then by applying Young’s inequality, we obtain

[TABLE]

and

[TABLE]

and

[TABLE]

and

[TABLE]

Inserting the above estimates into (5.5) leads to

[TABLE]

In view of (5.6), we deduce from standard theory of Navier-Stokes system that (2.5) has a unique solution $R\in C([0,T^{\ast}[;B^{0,\frac{1}{2}})$ with $\nabla R\in L^{2}(]0,T^{\ast}[;B^{0,\frac{1}{2}})$ for some maximal existing time $T^{\ast}$. We are going to prove that $T^{\ast}=\infty$ under the smallness condition (1.5). Otherwise, we denote

[TABLE]

and we take

[TABLE]

Then for $t\leq T^{\star},$ we deduce from (5.4), (5.6) and (5.7) that

[TABLE]

which together with (5.1) implies that

[TABLE]

from which and Proposition 4.2 and Corollary 4.1, we deduce that for $t\leq T^{\star},$

[TABLE]

where $A_{\nu_{\rm h},\delta}(\bar{u}^{\rm h}_{0})$ is given by (4.4).

Then under the smallness condition (1.5), we obtain

[TABLE]

This contradicts with (5.7), which in torn shows that $T^{\ast}=T^{\star}=\infty.$ This completes the proof of Theorem 1.2. $\Box$

6. The case of fluid evolving between two parallel plans

The goal of this section is to investigate the global well-posedness of the anisotropic Navier-Stokes system with only vertical viscosity, (1.7). To do it, let us first present the following lemmas:

Lemma 6.1**.**

Let $\Omega={\mathop{\mathbb{R}\kern 0.0pt}\nolimits}^{2}\times]0,1[,$ and $f$ satisfy $f|_{\Omega}=0$ and $\partial_{3}f\in{\mathcal{B}}_{h}^{1}(\Omega)$ (see Definition 3.3). Then one has

[TABLE]

Proof.

We first get, by applying Lemma 3.1, that

[TABLE]

Recalling that Poincar’e inequality holds in the strip $\Omega$ with Dirichlet boundary condition

[TABLE]

So that we obtain

[TABLE]

This completes the proof of the lemma. $\Box$

Lemma 6.2**.**

Let $u=(u^{\rm h},u^{3})$ be a smooth solenoidal vector field on $\Omega.$ Then one has

[TABLE]

Proof.

To estimate the trilinear term \bigl{(}\Delta_{k}^{\rm h}(u\cdot\nabla u)|\Delta_{k}^{\rm h}u\bigr{)}_{L^{2}(\Omega)}, we have to distinguish the terms containing the horizontal derivatives with the term containing the vertical derivative. We write

[TABLE]

We start with the estimate of $I_{\rm h}$. Applying Bony’s decomposition (3.5) to $u^{\rm h}\cdot\nabla_{\rm h}u$ in the horizontal variables gives

[TABLE]

Through a commutative argument, we find

[TABLE]

It follows from standard commutator’s estimate (see [2]) and Lemma 3.1 that

[TABLE]

so that we deduce from Lemma 6.1 that

[TABLE]

The same estimate holds for $I^{2}_{\rm h}(t).$

To handle $I_{\rm h}^{3}(t),$ we first get, by using integrating by parts and $\mathop{\rm div}\nolimits u=0,$ that

[TABLE]

from which, we infer

[TABLE]

Applying (6.1) gives

[TABLE]

As a result, it comes out

[TABLE]

Whereas observing that

[TABLE]

which together with (6.1) and Lemma 6.1 ensures that

[TABLE]

Along with (6.3), we conclude that

[TABLE]

To handle $I_{\rm v}(t),$ we get, by applying Bony’s decomposition (3.5) to $u^{3}\partial_{3}u$ in the horizontal variables, that

[TABLE]

We first observe from Lemma 6.1 that

[TABLE]

Similarly, we get, by applying (6.1), that

[TABLE]

This leads to

[TABLE]

Together with (6.2) and (6.4), we complete the proof of Lemma 6.2. $\Box$

Let now present the proof of Theorem 1.3.

Proof of Theorem 1.3.

For simplicity, we only present the a priori estimates for smooth enough solutions of (1.7). By taking $L^{2}$ inner product of the momentum equation of (1.7) with $u,$ we get

[TABLE]

Integrating the above inequality over $[0,t]$ leads to

[TABLE]

which implies

[TABLE]

On the other hand, by applying the operator $\Delta_{k}^{\rm h}$ to the momentum equation of (1.7) and performing $L^{2}$ inner product of the resulting equation with $\Delta_{k}^{\rm h}u,$ we obtain

[TABLE]

Integrating the above inequality over $[0,t]$ and then applying Lemma 6.2 yields

[TABLE]

By taking square root of the above inequality and multiplying the resulting inequality by $2^{2k},$ and then summing up the resulting inequality for $k\in\mathop{\mathbb{Z}\kern 0.0pt}\nolimits,$ we achieve

[TABLE]

Summing up (6.5) with (6.6) gives rise to

[TABLE]

Thanks to (6.7), we deduce by a standard argument that (1.7) has a unique solution $u\in C([0,T^{\ast}[;L^{2}\cap{\mathcal{B}}_{h}^{2}(\Omega))$ with $\partial_{3}u\in L^{2}(]0,T^{\ast}[;L^{2}\cap{\mathcal{B}}_{h}^{2}(\Omega))$ for some maximal existing time $T^{\ast}.$ We are going to prove that $T^{\ast}=\infty$ under the assumption of (1.8).

Let us denote

[TABLE]

Then for $t\leq T^{\star},$ we deduce from (6.7) that

[TABLE]

In particular, under the assumption of (1.8), for $t\leq T^{\star},$ we have

[TABLE]

as long as $c$ in (1.8) is small enough. This contradicts with the definition of $T^{\star}$ determined by (6.8), which in turn shows that $T^{\star}=T^{\ast}=\infty.$ We complete the proof of Theorem 1.3. $\Box$

Appendix A The proof of Lemmas 5.1 to 5.4

The goal of this section is to present the proof of Lemmas 5.1 to 5.4.

Proof of Lemma 5.1.

Applying Bony’s decomposition (3.5) to $a\otimes[\bar{u}^{\rm h}]_{\varepsilon}$ in the vertical variable yields

[TABLE]

Considering the support properties to the Fourier transform of the terms in $T^{\rm v}_{a}[\bar{u}^{\rm h}]_{\varepsilon},$ we find

[TABLE]

On the other hand, it follows from Lemma 3.1 that

[TABLE]

from which and Definition 3.4, we infer

[TABLE]

for $f_{1}(t)$ being given by (5.1). As a result, it comes out

[TABLE]

Exactly along the same line, we have

[TABLE]

which implies

[TABLE]

Along with (A.1), we complete the proof of Lemma 5.1. $\Box$

Proof of Lemma 5.2.

Applying Bony’s decomposition (3.5) to $[\bar{u}^{\rm h}]_{\varepsilon}\cdot\nabla_{\rm h}v_{L}$ in the vertical variable gives

[TABLE]

We first observe that

[TABLE]

where we used Definition 3.4 in the last step.

Similarly, we have

[TABLE]

then by applying Hölder’s inequality and using Definition 3.4, we obtain

[TABLE]

This completes the proof of Lemma 5.2. $\Box$

Proof of Lemma 5.3.

By applying Bony’s decomposition (3.5) to $v_{L}\cdot[\nabla_{\varepsilon}\bar{u}^{\rm h}]_{\varepsilon}$ in the vertical variable to $v_{L}\cdot[\nabla_{\varepsilon}\bar{u}^{\rm h}]_{\varepsilon},$ we write

[TABLE]

Note that

[TABLE]

Applying Hölder’s inequality gives

[TABLE]

On the other hand, we observe that

[TABLE]

Then along the same line to proof of (A.2), we arrive at

[TABLE]

This together with (A.2) completes the proof of Lemma 5.3. $\Box$

Proof of Lemma 5.4.

In view of (2.2), we get, by applying Bony’s decomposition (3.5) to $[\partial_{z}\bar{u}^{i}]_{\varepsilon}[\bar{u}^{j}]_{\varepsilon}$ in the vertical variable that

[TABLE]

We first observe from Lemma 3.1 that for any $\theta\in[0,1/2[$

[TABLE]

where $|D_{\rm h}|^{\frac{1}{2}}$ denotes Fourier multiplier with symbol $|\xi_{\rm h}|^{\frac{1}{2}}.$ Then noticing the definition of $f_{3}(t)$ given by (5.1), we get, by applying Hölder’s inequality and Definition 3.4, that

[TABLE]

Along the same line, we have

[TABLE]

which together with the fact that $\theta\in[0,1/2[$ implies that

[TABLE]

Then by virtue of (A.3), we complete the proof of Lemma 5.4. $\Box$

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.