Abstract
In Liu and Zhang (Arch Ration Mech Anal 235:1405–1444, 2020), 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 \(u_0=(u_0^\mathrm{h},u_0^3)\in \mathcal {B}^{0,\frac{1}{2}},\) (ANS) has a unique global solution provided that \(|D_\mathrm{h}|^{-1}\partial _3u_0\) is sufficiently small in the scaling invariant space \(\mathcal {B}^{0,\frac{1}{2}}\).
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
In this paper, we investigate the global well-posedness of the following 3-D anisotropic Navier–Stokes system:
where \(\Delta _\mathrm{h}\buildrel {\mathrm{def}}\over =\partial _1^2+\partial _2^2,\)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 model turbulent diffusion by using a viscosity of the form \(-\mu _\mathrm{h}\Delta _\mathrm{h}-\mu _3\partial _3^2\), where \(\mu _\mathrm{h}\) and \(\mu _3\) are empirical constants, and \(\mu _3\) is usually much smaller than \(\mu _\mathrm{h}\). We refer to the book of Pedlovsky [18, Chap. 4], for a complete discussion about this model.
Considering that 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 \((s,s')\) in \({\mathbb {R}}^2\), the anisotropic Sobolev space \(H^{s,s'}({\mathbb {R}}^3)\) denotes the space of homogeneous tempered distribution a such that
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,
for some sufficiently small constant c.
Notice that just as the classical Navier–Stokes system
the system (ANS) has the following scaling invariant property:
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]:
where \(\xi _\mathrm{h}=(\xi _1,\xi _2),\)\(\mathcal {F}a\) 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
Definition 1.2
We define \(\mathcal {B}^{0,\frac{1}{2}}({\mathbb {R}}^3)\) to be the set of homogenous tempered distribution a so that
The above space was first introduced by Iftimie [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 Fujita–Kato’s theorem [11] for the classical Navier–Stokes system. Moreover, the authors [17, 19] proved the global well-posedness of (ANS) with initial data \(u_0=(u_0^\mathrm{h},u_0^3)\) satisfying that
for some \(c_0\) sufficiently small.
Although the norm of \(\mathcal {B}^{0,\frac{1}{2}}\) is scaling invariant under the the scaling transformation (1.2), yet we observe that the solenoidal vector field
is not small in the space \(\mathcal {B}^{0,\frac{1}{2}}\) 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–Sobolev 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
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 \(u_0=(u_0^\mathrm{h},u_0^3)\) satisfies that
for some \(c_0\) sufficiently small. We remark that this result corresponds to Cannone, Meyer and Planchon’s result in [3] for the classical Navier–Stokes system, where the authors proved that if the initial data satisfies that
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 \(\delta \in ]0,1[, u_0=(u_0^\mathrm{h},u_0^3)\in H^{\frac{1}{2}}({\mathbb {R}}^3)\cap B^{0,\frac{1}{2}}_{2,1}({\mathbb {R}}^3)\) with \(u^{\mathrm{h}}_0\) belonging to \(L^2({\mathbb {R}}^3)\cap L^\infty ({\mathbb {R}}_\mathrm{v}; H^{-\delta }({\mathbb {R}}^2_\mathrm{h}))\cap L^\infty ({\mathbb {R}}_\mathrm{v}; H^3({\mathbb {R}}^2_\mathrm{h})).\) If we assume in addition that \(\partial _3u_0\in H^{-\frac{1}{2},0},\) then there exists a small enough positive constant \(\varepsilon _0\) such that if
(NS) has a unique global solution \(u\in C\bigl ({\mathbb {R}}^+;H ^{\frac{1}{2}}\bigr )\cap L^2\bigl ({\mathbb {R}}^+;H ^{\frac{3}{2}}\bigr ),\) where
and
are scaling invariant under the scaling transformation (1.2).
We remark that Theorem 1.1 ensures the global well-posedness of (NS) with initial data
for \(\varepsilon \leqq \varepsilon _0,\) 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, 17, 19], here we are going to study the global well-posedness of (ANS) with initial data \(u_0\) satisfying \(\partial _3u_0\) being sufficiently small in some critical spaces.
The main result of this paper is as follows:
Theorem 1.2
Let \(\Lambda _\mathrm{h}^{-1}\) be a Fourier multiplier with symbol \(|\xi _\mathrm{h}|^{-1},\) let \(u_0\in \mathcal {B}^{0,\frac{1}{2}}\) be a solenoidal vector field with \(\Lambda _\mathrm{h}^{-1}\partial _3 u_0\in {\mathcal {B}^{0,\frac{1}{2}}}.\) Then there exist some sufficiently small positive constant \(\varepsilon _0\) and some universal positive constants L, M, N so that for \(\mathfrak {A}_N\bigl (\Vert u^\mathrm{h}_0\Vert _{\mathcal {B}^{0,\frac{1}{2}}}\bigr )\) given by (3.5) if
(ANS) has a unique global solution \(u=\mathfrak {v}+e^{t\Delta _h} \begin{pmatrix} 0\\ u^3_{0,\mathrm{hh}} \end{pmatrix}\) with \(\mathfrak {v}\in C([0,\infty [\,;\mathcal {B}^{0,\frac{1}{2}})\) and \(\nabla _{\mathrm{h}}\mathfrak {v}\in L^2([0,\infty [\,;\mathcal {B}^{0,\frac{1}{2}}),\) where \(u_{0,\mathrm{hh}}^3\buildrel {\mathrm{def}}\over =\sum _{k\geqq \ell -1}\Delta _k^\mathrm{h}\Delta _\ell ^\mathrm{v}u_0^3.\)
We remark that all the norms of \(u^0\) in (1.11) is scaling invariant under the scaling transformation (1.2). Especially for the term \(\Vert \Lambda ^{-1}_\mathrm{h}\partial _3 u_0\Vert _{\mathcal {B}^{0,\frac{1}{2}}},\) we do not know how to propagate this regularity for the solutions of 3-D Navier–Stokes system. In the sequel, we shall only propagate this regularity for the solutions of 2-D Navier–Stokes system with a parameter [(see (3.4) and (3.7)]. With regular initial data, we may write explicitly the constant \(\mathfrak {A}_N\bigl (\Vert u^\mathrm{h}_0\Vert _{\mathcal {B}^{0,\frac{1}{2}}}\bigr ).\) For instance, we have
Corollary 1.1
Let \(u_0\in L^2\) be a solenoidal vector field with \(\partial _3u_0\in L^2\) and \(\Lambda _\mathrm{h}^{-1}\partial _3 u_0\in {\mathcal {B}^{0,\frac{1}{2}}}.\) Then there exist some sufficiently small positive constant \(\varepsilon _0\) and some universal positive constants L, M so that if
(ANS) has a unique global solution u as in Theorem 1.2.
Remark 1.1
Several remarks are in order about Theorem 1.2:
-
(a)
It follows from [8] that
$$ \Vert u_0^3\Vert _{\mathcal {B}^{-\frac{1}{2},\frac{1}{2}}_4}\lesssim \Vert u_0^3\Vert _{\mathcal {B}^{0,\frac{1}{2}}},$$so that the smallness condition (1.11) and (1.12) can also be formulated as
$$\begin{aligned} \begin{aligned} \Vert \Lambda ^{-1}_\mathrm{h}\partial _3 u_0\Vert _{\mathcal {B}^{0,\frac{1}{2}}} \exp \Bigl (L\bigl (1+\Vert u_0^3\Vert _{\mathcal {B}^{0,\frac{1}{2}}}^4\bigr ) \exp \bigl (M\mathfrak {A}_N^4\bigl (\Vert u^\mathrm{h}_0\Vert _{\mathcal {B}^{0,\frac{1}{2}}}\bigr )\bigr )\Bigr ) \leqq \varepsilon _0, \end{aligned} \end{aligned}$$(1.13)and
$$\begin{aligned} \Vert \Lambda ^{-1}_\mathrm{h}\partial _3 u_0\Vert _{\mathcal {B}^{0,\frac{1}{2}}} \exp \Bigl (L\bigl (1+\Vert u_0^3\Vert _{\mathcal {B}^{0,\frac{1}{2}}}^4\bigr ) \exp \left( \exp \bigl (M\Vert u^{\mathrm{h}}_0\Vert _{L^2}\Vert \partial _3u^{\mathrm{h}}_0\Vert _{L^2}\bigr )\right) \Bigr ) \leqq \varepsilon _0. \end{aligned}$$(1.14) -
(b)
Due to \(\mathrm{div}\,u_0=0,\) we find
$$\begin{aligned} \Vert \Lambda ^{-1}_\mathrm{h}\partial _3 u_0\Vert _{\mathcal {B}^{0,\frac{1}{2}}}= \Vert (\Lambda ^{-1}_\mathrm{h}\partial _3u^{\mathrm{h}}_0,-\Lambda ^{-1}_\mathrm{h}{{\mathrm{div}}_{\mathrm{h}}}\,u^{\mathrm{h}}_0)\Vert _{\mathcal {B}^{0,\frac{1}{2}}}. \end{aligned}$$Therefore the smallness condition (1.11) is of a similar type as (1.4). Yet roughly speaking, (1.11) requires only \(\partial _3 u^{\mathrm{h}}_0\) and \({{\mathrm{div}}_{\mathrm{h}}}\,u^{\mathrm{h}}_0\) to be small in some scaling invariant space, but without any restriction on \({{\mathrm{curl}}_{\mathrm{h}}}\,u^{\mathrm{h}}_0\). Thus the smallness condition (1.11) is weaker than (1.4).
-
(c)
Let \(w_0\) be a smooth solenoidal vector field, we observe that the data
$$\begin{aligned} u_0^\varepsilon (x)=\bigl (\varepsilon (-\ln \varepsilon )^\delta w^\mathrm{h}_0, (-\ln \varepsilon )^\delta w^3_0\bigr )(x_\mathrm{h},\varepsilon x_3) \quad \hbox {with}\quad \delta \in ]0,1/4[ \end{aligned}$$satisies (1.4) for \(\varepsilon \) sufficiently small.
While since our smallness condition (1.14) does not have any restriction on \({\mathrm{curl}\,}u_0^\mathrm{h},\) for any smooth vector field \(v^\mathrm{h}_0\) satisfying \({{\mathrm{div}}_{\mathrm{h}}}\,v^h_0=0,\) we find
$$\begin{aligned} u_0^\varepsilon (x)=\bigl (v^{\mathrm{h}}_0+\varepsilon (-\ln \varepsilon )^\delta w^\mathrm{h}_0, (-\ln \varepsilon )^\delta w^3_0\bigr )(x_\mathrm{h},\varepsilon x_3) \quad \hbox {with}\quad \delta \in ]0,1/4[ \end{aligned}$$(1.15)satisfies (1.14) for any \(\varepsilon \) sufficiently small. Therefore Theorem 1.2 ensures the global well-posedness of (ANS) with initial data given by (1.15). Compared with (1.10), which corresponds to \(\delta =0\) in (1.15), this type of result is new even for the classical Navier–Stokes system.
-
(d)
Given \(\phi \in \mathcal {S}({\mathbb {R}}^3)\), we deduce from Proposition 1.1 in [8] that
$$\begin{aligned} \Vert e^{ix_1/\varepsilon }\phi (x)\Vert _{{\mathcal {B}^{-\frac{1}{2},\frac{1}{2}}_4}} \leqq C\varepsilon ^{\frac{1}{2}}. \end{aligned}$$As a result, we find that for any \(\delta \in ]0,1/4[\), the following class of initial data:
$$\begin{aligned}&u_0^\varepsilon (x)=(v^\mathrm{h}, 0)(x_\mathrm{h},\varepsilon x_3) +(-\ln \varepsilon )^\delta \sin (x_1/\varepsilon )\nonumber \\&\bigl (0,-\varepsilon ^{\frac{1}{2}}\partial _3\phi (x_\mathrm{h},\varepsilon x_3), \varepsilon ^{-\frac{1}{2}}\partial _2\phi (x_\mathrm{h},\varepsilon x_3)\bigr ), \end{aligned}$$(1.16)satisfies the smallness condition (1.13) for small enough \(\varepsilon \), and hence the data given by (1.16) 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 \(a\leqq Cb\). We shall denote by \((a|b)_{L^2}\) the \(L^2({\mathbb {R}}^3)\) inner product of a and b. \(\left( d_j\right) _{j\in {\mathbb {Z}}}\) designates a generic elements on the unit sphere of \(\ell ^1({\mathbb {Z}})\), i.e. \(\sum _{j\in {\mathbb {Z}}}d_j=1\). Finally, we denote \(L^r_T(L^p_\mathrm{h}(L^q_\mathrm{v}))\) the space \(L^r([0,T]; L^p({\mathbb {R}}_{x_1}\times {\mathbb {R}}_{x_2}; L^q({\mathbb {R}}_{x_3}))),\) and \(\nabla _\mathrm{h}\buildrel {\mathrm{def}}\over =(\partial _{x_1},\partial _{x_2}),\)\(\mathrm{div}\,_\mathrm{h}= \partial _{x_1}+\partial _{x_2}\).
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 \(\mathbf{B}_{\mathrm{h}}\) (resp. \(\mathbf{B}_\mathrm{v}\)) a ball of \({\mathbb {R}}^2_{\mathrm{h}}\) (resp. \({\mathbb {R}}_\mathrm{v}\)), and \(\mathcal {C}_{\mathrm{h}}\) (resp. \(\mathcal {C}_\mathrm{v}\)) a ring of \({\mathbb {R}}^2_{\mathrm{h}}\) (resp. \({\mathbb {R}}_\mathrm{v}\)); let \(1\leqq p_2\leqq p_1\leqq \infty \) and \(1\leqq q_2\leqq q_1\leqq \infty .\) Then it holds that
Definition 2.1
For any \(p\in [1,\infty ],\), let us define the Chemin–Lerner type norm
In particular, we denote
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
Taking the integer N so that \(2^{N}\sim \Vert \partial _3 a\Vert _{L^2}\Vert a\Vert _{L^2}^{-1}\) in the above inequality leads to
Along the same lines, we have
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 \(f(t)\in L^1_\mathrm{{loc}}({\mathbb {R}}_+)\), \(f(t)\geqq 0\). We define
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
In particular, we denote
In the sequel, for \(a\in \mathcal {B}^{-\frac{1}{2},\frac{1}{2}}_4,\) we shall frequently use the following decomposition:
Lemma 2.2
(Lemma 2.5 in [8]) For any \(a\in {\mathcal {B}^{-\frac{1}{2},\frac{1}{2}}_4},\) it holds that
Definition 2.4
Let us define
In view of the 2-D interpolation inequality that
we find
Similarly, we have
Before preceding, let us recall Bony’s decomposition for the vertical variable from [1]:
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 \(a\in {\mathcal {B}^{-\frac{1}{2},\frac{1}{2}}_4}(T),\) it holds that
Proof
In view of (2.3) and Definition 2.3, we get, by applying (2.6), that
Then it remains to prove (2.8) for \(a_\mathrm{hh}.\) Indeed in view of Definition 2.4, we write
Applying Bony’s decomposition for the horizontal variables yields
We observe that
Whereas we get, by using Young’s inequality, that
As a result, it turns out that
and
Along the same lines, we can prove that the second term in (2.9) shares the same estimate. This ensures that (2.8) holds for \(a_\mathrm{hh}.\) We thus complete the proof of the proposition. \(\quad \square \)
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 \(u^\mathrm{h}=(u^1,u^2),\) we first recall the two-dimensional Biot–Savart’s law:
where \({{\mathrm{curl}}_{\mathrm{h}}}\,u^\mathrm{h}\buildrel {\mathrm{def}}\over =\partial _1u^2-\partial _2u^1\) and \({{\mathrm{div}}_{\mathrm{h}}}\,u^{\mathrm{h}}\buildrel {\mathrm{def}}\over =\partial _1u^1+\partial _2u^2.\)
In particular, let us decompose the horizontal components \(u_0^\mathrm{h}\) of the initial velocity \(u_0\) of (ANS) as the sum of \(u^\mathrm{h}_{0,{\mathrm{curl}}}\) and \( u^\mathrm{h}_{0,{\mathrm{div}}}\), and let us consider the following 2-D Navier–Stokes system with a parameter:
Concerning the system (3.2), we have the following a priori estimates:
Proposition 3.1
Let \(\bar{u}^\mathrm{h}_0\in \mathcal {B}^{0,\frac{1}{2}} \) with \(\Lambda _\mathrm{h}^{-1}\partial _3\bar{u}^\mathrm{h}_0\in \mathcal {B}^{0,\frac{1}{2}}.\) Then (3.2) has a unique global solution so that for any time \(t>0\), it holds that
and
where
and N is taken so large that \(\bigl \Vert \bar{u}^\mathrm{h}_{0,N}\bigr \Vert _{\mathcal {B}^{0,\frac{1}{2}}}\) is sufficiently small.
The proof of Proposition 3.1 will be presented in Section 4.
Remark 3.1
Under the assumptions that \(\bar{u}^\mathrm{h}_0\in L^2\) with \(~\partial _3\bar{u}^\mathrm{h}_0\in L^2\) and \(\Lambda _\mathrm{h}^{-1}\partial _3\bar{u}^\mathrm{h}_0\in \mathcal {B}^{0,\frac{1}{2}},\) we have the following alternative estimates for (3.3) and (3.4):
and
We shall present the proof right after (4.7).
We notice that
which satisfies \(\mathrm{div}\,v_0=0,\) and yet \(v_0\) is not small according to our smallness condition (1.11).
Before proceeding, let us recall the main idea of the proof to Theorem 1.1 in [15]. The authors [15] first constructed \((\bar{u}^\mathrm{h}, \bar{p})\) 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
With \(\bar{u}^\mathrm{h}\) 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
The key estimate for v is as follows:
Proposition 3.2
Let \(u=(u^\mathrm{h},u^3)\in C([0,T^*[; H^{\frac{1}{2}})\cap L^2(]0,T^*[; H^{\frac{3}{2}})\) be a Fujita–Kato solution of (NS). We denote \(\omega \buildrel {\mathrm{def}}\over =\partial _1 v^2-\partial _2v^1\) and
Then under the assumption (1.7), there exists some positive constant \(\eta \) such that
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 \(u\in C([0,T^*[; H^{\frac{1}{2}})\cap L^2(]0,T^*[; H^{\frac{3}{2}})\) be a solution of (NS). If the maximal existence time \(T^*\) is finite, then for any \((p_{i,j})\) in \(]1,\infty [^9\), one has
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 cannot 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 \(\mathrm{div}\,u=0,\) (ANS) can be equivalently reformulated as
so that at least, seemingly, the \(u^3\) equation is a linear one; this explains why there is no size restriction for \(u_0^3\) in (1.4) and (1.6).
Motivated by [17, 19], for \(\bar{u}^\mathrm{h}\) being determined by the systems (3.2), we decompose the solution u of (ANS) as \(u=\begin{pmatrix} \bar{u}^\mathrm{h}\\ 0 \end{pmatrix}+v\). It is easy to verify that the remainder term v satisfies
We notice that under the smallness condition (1.11), the horizontal components, \(v_0^\mathrm{h},\) 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 \(v^\mathrm{h}\) 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
Then w solves
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 \(t\in ]0,T^*[,\) we have
and
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 \([0, T^*[\) with \(T^*\) being the maximal time of existence. Let \(\bar{u}^\mathrm{h}\) 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 (1.11) so that
We now define
Then, thanks to (3.19) and Proposition 3.3, for \(t\leqq T^\star ,\) we find
and
It follows from Lemma 2.2 and Proposition 2.1 that
whereas we deduce from (2.6) and Proposition 3.1 that
By inserting the above two inequalities to (3.22) and using (3.3), we obtain that, for \(t\leqq T^\star \),
Then we deduce that for \(t\leqq T^\star ,\)
Inserting the above estimates into (3.21) gives
for \(t\leqq T^\star .\) Therefore, if we take L, M, N large enough and \(\varepsilon _0\) small enough in (1.11), we deduce from (3.24) that
(3.25) contradicts (3.20). This in turn shows that \(T^\star =T^*.\) (3.23) along with (3.25) shows that \(T^*=\infty .\) Moreover, thanks to (3.15), we have \(\mathfrak {v}\buildrel {\mathrm{def}}\over =u-e^{t\Delta _h} \begin{pmatrix} 0\\ u^3_{0,\mathrm{hh}} \end{pmatrix}\in C([0,\infty [\,;\mathcal {B}^{0,\frac{1}{2}})\) with \(\nabla _{\mathrm{h}}\mathfrak {v}\in L^2([0,\infty [\,;\mathcal {B}^{0,\frac{1}{2}}).\) This completes the proof of our Theorem 1.2. \(\quad \square \)
Proof of Corollary 1.1
Under the assumptions that \(u^{\mathrm{h}}_0\in L^2\) with \(~\partial _3u^{\mathrm{h}}_0\in L^2\) and \(\Lambda _\mathrm{h}^{-1}\partial _3u^{\mathrm{h}}_0\in \mathcal {B}^{0,\frac{1}{2}},\) we deduce from (3.1), (3.4) and (3.7) that
Then by repeating the argument from (3.19) to (3.24), we conclude the proof of Corollary 1.1. \(\quad \square \)
4 Estimates of the 2-D Solution \(\pmb {\bar{u}^\mathrm{h}}\)
The goal of this section is to present the proof of Proposition 3.1. Let us start the proof with the following lemma, which is in the spirit of Lemma 3.1 of [6]:
Lemma 4.1
Let \(a^\mathrm{h}=(a^1,a^2)\) be a smooth enough solution of
Then for any \(t>0\) and any fixed \(x_3\in {\mathbb {R}},\) it holds that
and
Proof
By taking \(L^2_\mathrm{h}\) inn-product of (4.1) with \(a^\mathrm{h}\) and using \({{\mathrm{div}}_{\mathrm{h}}}\,a^\mathrm{h}=0,\) we obtain (4.2).
While by applying \(\partial _3\) to (4.1) and then taking \(L^2_\mathrm{h}\) inner product of the resulting equation with \(\partial _3a^\mathrm{h},\) we find
Due to \({{\mathrm{div}}_{\mathrm{h}}}\,a^\mathrm{h}=0,\) we get, by applying (2.4), that
Applying Young’s inequality yields
Inserting the above estimate into (4.4) gives
Applying Gronwall’s inequality and using (4.2), we achieve
which leads to (4.3). This completes the proof of this lemma. \(\quad \square \)
Let us now present the proof of Proposition 3.1.
Proof of Proposition 3.1
For any positive integer N, and \(\bar{u}^\mathrm{h}_{0,N}\) being given by (3.5), we split the solution \(\bar{u}^\mathrm{h}\) to (3.2) as
with \(\bar{u}^\mathrm{h}_1\) and \(\bar{u}^\mathrm{h}_2\) being determined, respectively, by
and
Indeed for smoother initial data \(\bar{u}^\mathrm{h}_0,\) we may write explicitly the constant \(\mathfrak {A}_N\bigl (\Vert \bar{u}^\mathrm{h}_0\Vert _{\mathcal {B}^{0,\frac{1}{2}}}\bigr )\) in (3.3). For instance, if \(\bar{u}^\mathrm{h}_0\in L^2\) with \(~\partial _3\bar{u}^\mathrm{h}_0\in L^2\) and \(\Lambda _\mathrm{h}^{-1}\partial _3\bar{u}^\mathrm{h}_0\in \mathcal {B}^{0,\frac{1}{2}}\), we deduce from Lemma 4.1 that
which, together with (2.2) and
ensures (3.6). By virtue of (3.6) and (4.22), we deduce (3.7).
In general, we first deduce from Lemma 4.1 that
which, together with (2.2), ensures that
Next we handle the estimate of \(\bar{u}^\mathrm{h}_2\). To do this, for any \(\kappa >0,\) we denote
Then by multiplying \(\exp \Bigl (-\kappa \int \nolimits _0^t f^\mathrm{h}(t')\,\mathrm{d}t'\Bigr )\) to the \(\bar{u}^\mathrm{h}_2\) equation in (4.7), we write
Applying the operator \(\Delta _\ell ^\mathrm{v}\) to the above equation and taking \(L^2\) inner product of the resulting equation with \(\Delta _\ell ^\mathrm{v}\bar{u}^\mathrm{h}_{2,\kappa },\) and then using integration by parts, we get
\(\square \)
The estimate of the second line of (4.10) relies on the following lemma, whose proof will be postponed until the “Appendix A”:
Lemma 4.2
Let \(a,b,c\in \mathcal {B}^{0,\frac{1}{2}}(T)\) and \(\mathfrak {f}(t)\buildrel {\mathrm{def}}\over =\Vert a(t)\Vert _{\mathcal {B}_4^{0,\frac{1}{2}}}^4\). Then for any smooth homogeneous Fourier multiplier, A(D), of degree zero and any \(\ell \in {\mathbb {Z}}\), it holds that
Moreover, for non-negative function \(\mathfrak {g}\in L^\infty (0,T),\) one has
By applying (4.13) with \(a=c=\bar{u}^\mathrm{h}_2,\)\(b=\nabla _h\bar{u}^\mathrm{h}_{2}\) and \(\mathfrak {g}= \exp \Bigl (-\kappa \int \nolimits _0^t f^\mathrm{h}(t')\,\mathrm{d}t'\Bigr )\), we get
Whereas due to (2.5), one has
By applying (4.12) with \(a=\bar{u}^\mathrm{h}_1,~b=\bar{u}^\mathrm{h}_{2,\kappa },~ c=\nabla _{\mathrm{h}}\bar{u}^\mathrm{h}_{2,\kappa }\), we infer
Then we get, by first integrating (4.10) over [0, t] and inserting (4.14) and (4.15) into the resulting inequality, that
Multiplying the above inequality by \(2^\ell \) and taking square root of the resulting inequality, and then summing up the inequalities for \(\ell \in {\mathbb {Z}},\) we arrive at
In particular, taking \(2\kappa =C^2\) in the above inequality gives
On the other hand, in view of (3.5), we can take N so large that
Then a standard continuity argument shows that, for any time \(t>0\), it holds that
Due to the definition of \(\bar{u}^\mathrm{h}_{2,\lambda }\) given by (4.9), one has
which, together with (4.8) and (4.18), implies that
By combining (4.8) with (4.19), we obtain (3.3).
It remains to prove (3.4). In order to do, this for any \(\gamma >0,\) we denote
Then, by multiplying \(\exp \left( -\gamma \int \nolimits _0^t g^\mathrm{h}(t')\,\mathrm{d}t'\right) \) to the \(\bar{u}^\mathrm{h}\) equation in (3.2), we write
Applying the operator \(\Delta _\ell ^\mathrm{v}\Lambda _\mathrm{h}^{-1}\partial _3\) to the above equation and then taking \(L^2\) inner product of the resulting equation with \(\Delta _\ell ^\mathrm{v}\Lambda _\mathrm{h}^{-1}\partial _3\bar{u}^\mathrm{h}_\gamma ,\) we get
Noting that \(\Lambda _\mathrm{h}^{-1}{{\mathrm{div}}_{\mathrm{h}}}\,\) is a bounded Fourier multiplier, we get, by using (4.11) with \(a=\bar{u}^\mathrm{h},~b= \partial _3\bar{u}^\mathrm{h}_\gamma \) and \(c=\Lambda _\mathrm{h}^{-1}\partial _3\bar{u}^\mathrm{h}_\gamma ,\) that
By integrating (4.21) over [0, t] and then inserting the above estimate into the resulting inequality, we find
Multiplying the above inequality by \(2^\ell \) and taking square root of the resulting inequality, and then summing up the inequalities for \(\ell \in {\mathbb {Z}},\) we arrive at
In particular, taking \(2\gamma =C^2\) in the above inequality gives
Then a similar derivation from (4.18) to (4.19) leads to
which together with (3.3), ensures (3.4). This completes the proof of this proposition. \(\quad \square \)
5 The Estimate of the Horizontal Components \(\pmb {v^\mathrm{h}}\)
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 this, let u be a smooth enough solution of (ANS) on \([0,T^*[,\) let \(\bar{u}^\mathrm{h}, v_F\) and w be determined respectively by (3.2), (3.15) and (3.16), for any constant \(\lambda >0\), we denote
and similar notations for \(\bar{u}^\mathrm{h}_\lambda ,~p_\lambda ,~\bar{p}_\lambda \) and \(v^{\mathrm{h}}_{\lambda /2}\).
By multiplying \(\exp \Bigl (-\lambda \int \nolimits _0^tf(t')\,\mathrm{d}t'\Bigr )\) to the \(v^{\mathrm{h}}\) equation of (3.14), we get
Applying \(\Delta _{\ell }^{\mathrm{v}}\) to the above equation and taking \(L^2\) inner product of the resulting equation with \(\Delta _{\ell }^{\mathrm{v}}v^{\mathrm{h}}_\lambda \), and then integrating the equality over [0, t], we obtain
where
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 the above term by term.
\(\bullet \) The estimates of \(\underline{\mathrm{I}_1}\) to \(\underline{\mathrm{I}_4}.\)
We first get, by using (4.11) with \(a=\bar{u}^\mathrm{h},~b=\nabla _{\mathrm{h}}v^{\mathrm{h}}_\lambda \) and \(c=v^{\mathrm{h}}_\lambda ,\) that
Applying (4.13) with \(a=v^{\mathrm{h}},~b=\nabla _{\mathrm{h}}v^{\mathrm{h}},~c=v^{\mathrm{h}}\) and \(\mathfrak {g}(t)=\exp \bigl (-\lambda \int \nolimits _0^tf(t')\,\mathrm{d}t'\bigr )\) yields
To handle \(\mathrm{I}_3,\) by using integration by parts, we write
Applying (4.11) with \(a=\bar{u}^\mathrm{h},~b={{\mathrm{div}}_{\mathrm{h}}}\,v^{\mathrm{h}}_\lambda \) and \(c=v^{\mathrm{h}}_\lambda \) gives
Whereas applying (4.12) with \(a=\bar{u}^\mathrm{h},~b=v^{\mathrm{h}}_\lambda \) and \(c=\nabla _{\mathrm{h}}v^{\mathrm{h}}_\lambda \) yields
As a result, it turns out that
While by applying (4.11) with \(a=v^3,~b=\partial _3 \bar{u}^\mathrm{h}_\lambda ,~c=v^{\mathrm{h}}_\lambda \), and using the fact that
we find
\(\bullet \)The estimates of \(\underline{\mathrm{I}_5}\).
The estimate of \(\mathrm{I}_5\) 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
Following [8, 16], we get, by using a standard commutator’s process, that
By applying the commutator’s estimate (see Lemma 2.97 in [2]), we find
Due to \(\partial _3v^3=-{{\mathrm{div}}_{\mathrm{h}}}\,v^\mathrm{h},\) we get, by applying (2.4), that
Next, since the support to the Fourier transform of \(\sum _{|\ell '-\ell |\leqq 5} (S^\mathrm{v}_{\ell '-1}v^3-S^\mathrm{v}_{\ell -1}v^3)\) is contained in \({\mathbb {R}}^2\times \cup _{|\ell '-\ell |\leqq 5}2^{\ell '}\mathcal {C}_\mathrm{v},\) we get, by applying Lemma 2.1, that
from which we infer
Finally, by using integration by parts and \(\partial _3v^3=-{{\mathrm{div}}_{\mathrm{h}}}\,v^\mathrm{h}\) again, we find that
As a result, it turns out that
On the other hand, by applying Lemma 2.1 once again, we find that
Observing that
we infer
which, together with (5.7), ensures that
\(\bullet \)The estimates of \(\underline{\mathrm{I}_6.}\)
We first get, by taking the space divergence operators, \(\mathrm{div}\,\) and \({{\mathrm{div}}_{\mathrm{h}}}\,,\) to (ANS) and (3.2) respectively, that
so that thanks to the fact that
we write
Accordingly, we decompose \(\mathrm{I}_6\) as
where
Noticing that \(\nabla _{\mathrm{h}}(-\Delta )^{-1}{{\mathrm{div}}_{\mathrm{h}}}\,\) is a bounded Fourier multiplier. Then along the same line to the estimate of \(\mathrm{I}_1\) to \(\mathrm{I}_4,\) we achieve
However, \(\mathrm{I}_{6,2}\) can not be handled along the same line to that of \(\mathrm{I}_5\), since the symbol of the operator \(\nabla _{\mathrm{h}}(-\Delta )^{-1}{{\mathrm{div}}_{\mathrm{h}}}\,\) depends not only on \(\xi _3\), but also on \(\xi _\mathrm{h},\) 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
Since both \(\nabla _{\mathrm{h}}(-\Delta )^{-1}\partial _3\) and \(\nabla _{\mathrm{h}}(-\Delta )^{-1}{{\mathrm{div}}_{\mathrm{h}}}\,\) are bounded Fourier multiplier, we get, by applying Lemma 4.2, that
To handle \(\mathrm{I}_{6,3}\), we use \(\mathrm{div}\,v={{\mathrm{div}}_{\mathrm{h}}}\,\bar{u}^\mathrm{h}=0\) to write
Applying (4.11) with \(A(D)=\nabla _{\mathrm{h}}(-\Delta )^{-1}\partial _3, a=v^3, b={{\mathrm{div}}_{\mathrm{h}}}\,v^\mathrm{h}_\lambda \) and \(c=v^\mathrm{h}_\lambda \) yields
The remaining terms in \(\mathrm{I}_{6,3}\) can be handled along the same lines as to those of \(\mathrm{I}_{6,1}\) and \(\mathrm{I}_{6,2}.\) As a consequence, we obtain
To deal with \(\mathrm{I}_{6,4}\), it is crucial to observe that
Then due to the fact that \(\sum _{i,j=1}^2\nabla _{\mathrm{h}}\partial _3(-\Delta )^{-1}(-\Delta _\mathrm{h})^{-1} \partial _i\partial _j\) is a bounded Fourier multiplier, we get, by applying (4.11) with \(a=\bar{u}^\mathrm{h},b=\partial _3\bar{u}^\mathrm{h}_\lambda ,c=v^{\mathrm{h}}_\lambda ,\) that
By summing up (5.10–5.13), we arrive at
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
Multiplying the above inequality by \(2^{\ell +1} \) and taking square root of the resulting inequality, and then summing up the inequalities over \({\mathbb {Z}},\) we find that
It follows from Young’s inequality that
Inserting the above inequality into (5.15) and taking \(\lambda \) so that \(\sqrt{2\lambda }=C\), we obtain
which, together with the following consequence of (5.1):
gives rise to (3.17). \(\quad \square \)
6 The Estimate of the Vertical Component \(\pmb {v^3}\)
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 \(\bar{u}^\mathrm{h}\cdot \nabla _\mathrm{h}v\) that appears in (3.14), we will have to use the weighted Chemin–Lerner norms once again. Indeed for any constant \(\mu >0\), we denote
and similar notations for \(v_\mu ,~\bar{u}^\mathrm{h}_\mu ,\) and \(~p_\mu \).
By multiplying \(\bar{\mathfrak {g}}(t)\) to (3.16), we write
By applying \(\Delta _{\ell }^{\mathrm{v}}\) to the above equation and taking \(L^2\) inner product of the resulting equation with \(\Delta _{\ell }^{\mathrm{v}}w_\mu \), and then integrating the equality over [0, t], we obtain
where
Let us handle the above term by term.
\(\bullet \) The estimates of \(\underline{\mathrm{II}_1}\) and \(\underline{\mathrm{II}_2}\)
We first get, by applying (4.11) with \(a=\bar{u}^\mathrm{h},~b=\nabla _{\mathrm{h}}w_\mu \) and \(c=w_\mu ,\) that
whereas by applying a modified version of (4.13) with \(a=v^{\mathrm{h}},~b=\nabla _{\mathrm{h}}w_\mu ,~c=w_\mu \) and \(\mathfrak {g}(t)=\exp \bigl (-\mu \int \nolimits _0^t\hbar (t')\,\mathrm{d}t'\bigr ),\) we find
\(\bullet \)The estimate of \(\underline{\mathrm{II}_3}\)
The estimate of \(\mathrm{II}_3\) relies on the following lemma, the proof of which will be postponed until 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 \(\ell \in {\mathbb {Z}}\), it holds that
and
Remark 6.1
Indeed the proof of Lemma 6.1 shows that \(\Vert b\Vert _{{\mathcal {B}^{-\frac{1}{2},\frac{1}{2}}_4}(T)}\) in (6.5) and (6.6) can be replaced by \(\Vert b\Vert _{\mathcal {B}^{0,\frac{1}{2}}(T)}.\)
Let us admit this lemma temporarily, and continue our estimate of \(\mathrm{II}_3\). By using integration by parts, we write
Applying (6.6) with \(a={{\mathrm{div}}_{\mathrm{h}}}\,v^\mathrm{h}_\mu ,~b=v_F\) and \(c=w_\mu \) yields
whereas by applying (6.5) with \(a=v^\mathrm{h}_\mu ,~b=v_F\) and \(c=\nabla _{\mathrm{h}}w_\mu ,\) we obtain
Inserting the above two estimates into (6.7) and using (2.6), we achieve
\(\bullet \)The estimate of \(\underline{\mathrm{II}_4}\)
Due to \({{\mathrm{div}}_{\mathrm{h}}}\,\bar{u}^\mathrm{h}=0,\) by using integration by parts, we write
By applying Bony’s decomposition (2.7), we get
We first observe that
and applying Hölder’s inequality and Proposition 2.1 gives
Along the same lines, we find
As a result, it turns out that
\(\bullet \)The estimates of \(\underline{\mathrm{II}_5}\)
Due to \(\partial _3v^3=-{{\mathrm{div}}_{\mathrm{h}}}\,v^\mathrm{h}\) and \(v^3=w+v_F,\) we write
Then applying (6.6) gives rise to
\(\bullet \)The estimates of \(\underline{\mathrm{II}_6}\)
The estimate of \(\mathrm{II}_6\) can be handled similarly as \(\mathrm{I}_6\). Indeed in view of (5.9), we write
Accordingly, we have the decomposition \(\mathrm{II}_6=\sum _{i=1}^5\mathrm{II}_{6,i}\) with
It is easy to observe from the estimate of \(\mathrm{I}_{6,1}\) that
Mean while, by using \(\partial _3v^3=-{{\mathrm{div}}_{\mathrm{h}}}\,v^\mathrm{h}\) and integration by parts, we write
It follows from (6.5) and \(v^3=v_F+w\) that
whereas by using a modified version of (4.13), we infer
Therefore, we obtain
whereas applying (6.6) with \(a=\partial _3\bar{u}^\mathrm{h},~b=v^3_\mu \) and \(c=w_\mu \) leads to
On the other hand, again due to \(\mathrm{div}\,v=0,\) we write
Noticing that \((-\Delta )^{-1}\partial _3^2\) is a bounded Fourier operator, we observe that \(\mathrm{II}_{6,4}\) shares the same estimate as \(\sum _{i=1}^5\mathrm{II}_{i}\) given before, that is,
Finally since \((-\Delta )^{-1}\partial _i\partial _j\) is a bounded Fourier operator, we get, by applying (4.11) with \(a=\bar{u}^\mathrm{h},~b=\partial _3\bar{u}^\mathrm{h}_\mu ,~c=w_\mu \), that
By summing (6.12–6.16), we arrive at
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 \({\mathbb {Z}},\) we obtain
Applying Young’s inequality gives
and
As a result, we have
Taking \(\mu \) in the above inequality so that \(\sqrt{2\mu }=C\) gives rise to
On the other hand, in view of the definition of \(u^3_{0,\mathrm{l}\mathrm{h}}\), it holds for any \(\ell \in {\mathbb {Z}}\) that
which indicates that
Inserting the above estimate into (6.18) and repeating the argument from (4.18) to (4.19), we conclude the proof of (3.18). \(\quad \square \)
References
Bony, J.-M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sci. École Norm. Sup. 14, 209–246, 1981
Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, 343. Springer, Berlin 2011
Cannone, M., Meyer, Y., Planchon, F.: Solutions autosimilaires des équations de Navier–Stokes. Séminaire “Équations aux Dérivées Partielles" de l’École polytechnique, Exposé VIII, 1993–1994
Chemin, J.-Y., Desjardins, B., Gallagher, I., Grenier, E.: Fluids with anisotropic viscosity. M2AN Math. Model. Numer. Anal. 34, 315–335, 2000
Chemin, J.-Y., Desjardins, B., Gallagher, I., Grenier, E.: Mathematical Geophysics. An introduction to rotating fluids and the Navier–Stokes equations. Oxford Lecture Series in Mathematics and its Applications, 32, The Clarendon Press, Oxford University Press, Oxford, 2006
Chemin, J.-Y., Gallagher, I.: Large global solutions to the Navier–Stokes equations, slowly varying in one direction. Trans. Am. Math. Soc. 362, 2859–2873, 2010
Chemin, J.-Y., Gallagher, I., Zhang, P.: Sums of large global solutions to the incompressible Navier–Stokes equations. J. Reine Angew. Math. 681, 65–82, 2013
Chemin, J.-Y., Zhang, P.: On the global wellposedness to the \(3\)-D incompressible anisotropic Navier–Stokes equations. Commun. Math. Phys. 272, 529–566, 2007
Chemin, J.-Y., Zhang, P.: Remarks on the global solutions of \(3\)-D Navier–Stokes system with one slow variable. Commun. Partial Differ. Equ. 40, 878–896, 2015
Chemin, J.-Y., Zhang, P.: On the critical one component regularity for 3-D Navier–Stokes system. Ann. Sci. Éc. Norm. Supér. (4) 49, 131–167, 2016
Fujita, H., Kato, T.: On the Navier–Stokes initial value problem I. Arch. Ration. Mech. Anal. 16, 269–315, 1964
Iftimie, D.: The resolution of the Navier–Stokes equations in anisotropic spaces. Rev. Mat. Iberoam. 15, 1–36, 1999
Iftimie, D.: A uniqueness result for the Navier–Stokes equations with vanishing vertical viscosity. SIAM J. Math. Anal. 33, 1483–1493, 2002
Koch, H., Tataru, D.: Well-posedness for the Navier–Stokes equations. Adv. Math. 157, 22–35, 2001
Liu, Y., Zhang, P.: Global solutions of 3-D Navier–Stokes system with small unidirectional derivative. Arch. Ration. Mech. Anal. 235, 1405–1444, 2020
Paicu, M.: Équation anisotrope de Navier–Stokes dans des espaces critiques. Rev. Mat. Iberoam. 21, 179–235, 2005
Paicu, M., Zhang, P.: Global solutions to the \(3\)-D incompressible anisotropic Navier–Stokes system in the critical spaces. Commun. Math. Phys. 307, 713–759, 2011
Pedlosky, J.: Geophysical Fluid Dynamics. Springer, Berlin 1979
Zhang, T.: Erratum to: Global wellposed problem for the \(3\)-D incompressible anisotropic Navier–Stokes equations in an anisotropic space. Commun. Math. Phys. 295, 877–884, 2010
Acknowledgements
We would like to thank the referee for valuable comments for the improvement of the original submission. 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 11688101 and 11371347, Morningside Center of Mathematics of The Chinese Academy of Sciences and innovation grant from National Center for Mathematics and Interdisciplinary Sciences. All the authors are supported by the K. C. Wong Education Foundation.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by F. Lin.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix A. The Proof of Lemmas 4.2 and 6.1
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
Considering the support properties to the Fourier transform of the terms in \(T^\mathrm{v}_{a} b\), and noting that A(D) is a smooth homogeneous Fourier multiplier of degree zero, we find
It follows from Lemma 2.1 and Definition 2.4 that
This together with Definition 2.2 ensures that
Along the same lines, we get, by applying (2.5), that
On the other hand, once again considering the support properties to the Fourier transform of the terms in \(R^\mathrm{v}({a}, b),\) we find
It follows from Lemma 2.1 however that
As a result, by virtue of Definition 2.2, we obtain
Similarly, thanks to (2.5), one has
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). Similarly to the proof of (A.2), we write
from which, along with Definition 2.2, we infer
We deduce from Definition 2.4 that
whereas we get, by applying the triangle inequality and Lemma 2.1, that
This in turn shows that
which, together with (A.6), ensures (4.12). This completes the proof of Lemma 4.2.
\(\square \)
Proof of Lemma 6.1
Let \(Q_1\) be given by (A.1). We first get, by a similar derivation of (A.2), that
which, together with Proposition 2.1, implies that
For \(Q_2\) given by (A.1), we get, by a similar derivation of (A.4), that
from which, with Proposition 2.1, we infer
This, together with (A.1) and (A.7), ensures (6.5).
The inequality (6.6) can be proved similarly. As a matter of fact, we observe that
and
Then (6.6) follows from Proposition 2.1. This completes the proof of this lemma.
\(\square \)
Rights and permissions
About this article
Cite this article
Liu, Y., Paicu, M. & Zhang, P. Global Well-Posedness of 3-D Anisotropic Navier–Stokes System with Small Unidirectional Derivative. Arch Rational Mech Anal 238, 805–843 (2020). https://doi.org/10.1007/s00205-020-01555-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-020-01555-x