Large time behavior of solutions to 3-D MHD system with initial data near equilibrium

TL;DR

This paper proves that solutions to the 3-D incompressible MHD system with small perturbations near equilibrium decay over time, confirming a conjecture that energy dissipation is independent of magnetic resistivity.

Contribution

It provides a rigorous mathematical justification for the energy dissipation conjecture in 3-D MHD systems without magnetic diffusion for small initial data.

Findings

Global existence of solutions for small perturbations

Decay of velocity and magnetic field difference at explicit rates

Decay rate matches that of the linear system, indicating optimality

Abstract

In \cite{ChCa}, Califano and Chiuderi conjectured that the energy of incompressible Magnetic hydrodynamical system is dissipated at a rate that is independent of the ohmic resistivity. The goal of this paper is to mathematically justify this conjecture in three space dimension provided that the initial magnetic field and velocity is a small perturbation of the equilibrium state $(e_3,0).$ In particular, we prove that for such data, 3-D incompressible MHD system without magnetic diffusion has a unique global solution. Furthermore, the velocity field and the difference between the magnetic field and $e_3$ decay to zero in both $L^\infty$ and $L^2$ norms with explicit rates. We point out that the decay rate in the $L^2$ norm is optimal in sense that this rate coincides with that of the linear system. The main idea of the proof is to exploit H$\ddot{o}$rmander's version of Nash-Moser iteration scheme, which is very much motivated by the seminar papers \cite{Kl80, Kl82, Kl84} by Klainerman on the long time behavior to the evolution equations.

Equations

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

$${\int }_R(b_0-e_3)(Z(t,\alpha ))dt=0\text{mbox}forall\alpha \in R^d\times \{0\}$$

$$\displaystyle\left\{\begin{array}[]{l}\displaystyle\frac{dZ(t,\alpha)}{dt}=b_{0}(Z(t,\alpha)),\\ \displaystyle Z(t,\alpha)|_{t=0}=\alpha.\end{array}\right.$$

$$\Vert {\partial }_{x_2}^{k}b(t){\Vert }_{L^2}+\Vert {\partial }_{x_2}^{k}u(t){\Vert }_{L^2}\le C⟨t{⟩}^{-\frac{1+2k}{4}+ϵ}\text{mbox}foranyϵ\in ]0,1/2[andk=0,1,2,$$

$$\Vert u(t){\Vert }_{H^2}+\Vert b(t)-e_3{\Vert }_{H^2}\le C⟨t{⟩}^{-\frac{1}{4}}.$$

$$\Vert u_0{\Vert }_{W^{N_0,1}}+\Vert u_0{\Vert }_{H^{N_0}}\le c_{0}and\epsilon \le {\epsilon }_0,$$

$$\begin{array}{rl}& \Vert u(t){\Vert }_{W^{2,\infty }}\le C_{\kappa }⟨t{⟩}^{-\frac{5}{4}+\kappa },\Vert b(t)-e_3{\Vert }_{W^{2,\infty }}\le C_{\kappa }⟨t{⟩}^{-\frac{3}{4}+\kappa }and\\ & \Vert u(t){\Vert }_{H^2}+\Vert b(t)-e_3{\Vert }_{H^2}\le C⟨t{⟩}^{-\frac{1}{2}},\Vert \nabla u(t){\Vert }_{L^2}\le C⟨t{⟩}^{-1}.\end{array}$$

$$\{\begin{array}{rl}& Y_{tt}-\Delta Y_t-{\partial }_3^{2}Y=\mathbf{f}\text{in}{R}^{+}\times R^3,\\ & Y{|}_{t=0}=Y^{(0)},Y_t{|}_{t=0}=Y^{(1)}.\end{array}$$

$${\lambda }_1(\xi )=-\frac{|\xi {|}^2}{2}+\sqrt{\frac{|\xi {|}^4}{4}-{\xi }_3^2}and{\lambda }_2(\xi )=-\frac{|\xi {|}^2}{2}-\sqrt{\frac{|\xi {|}^4}{4}-{\xi }_3^2}.$$

$$\displaystyle\lambda_{1}(\xi)=-\frac{2\xi_{3}^{2}}{|\xi|^{2}\bigl{(}1+\sqrt{1-\frac{4\xi_{3}^{2}}{|\xi|^{4}}}\bigr{)}}\to-1\quad\mbox{as}\quad|\xi|\to\infty$$

$$\begin{split}X(t,y)=&y+\int_{0}^{t}u(t^{\prime},X(t^{\prime},y))dt^{\prime}\buildrel\hbox{\footnotesize def}\over{=}y+Y(t,y),\quad\boldsymbol{u}(t,y)\buildrel\hbox{\footnotesize def}\over{=}u(t,X(t,y)),\\ {\boldsymbol{b}}(t,y)\buildrel\hbox{\footnotesize def}\over{=}&b(t,X(t,y)),\quad\boldsymbol{p}(t,y)\buildrel\hbox{\footnotesize def}\over{=}p(t,X(t,y)),\quad{\mathcal{A}}\buildrel\hbox{\footnotesize def}\over{=}\left(Id+\nabla_{y}Y\right)^{-1}\quad\hbox{and}\quad\nabla_{Y}\buildrel\hbox{\footnotesize def}\over{=}{\mathcal{A}}^{t}\nabla_{y}.\end{split}$$

$$\left\{\begin{array}[]{l}\displaystyle{\boldsymbol{b}}(t,y)=\partial_{b_{0}}X(t,y),\quad\nabla_{Y}\cdot{\boldsymbol{b}}=0,\\ \displaystyle Y_{tt}-\Delta_{y}Y_{t}-\partial_{b_{0}}^{2}Y=\partial_{b_{0}}b_{0}+g,\\ \displaystyle Y_{|t=0}=Y^{(0)}=0,\qquad{Y_{t}}_{|t=0}=Y^{(1)}=u_{0}(y),\end{array}\right.$$

$$\begin{split}&g=\mbox{div}_{y}\bigl{[}(\mathcal{A}\mathcal{A}^{t}-Id)\nabla_{y}Y_{t}\bigr{]}-\mathcal{A}^{t}\nabla_{y}\boldsymbol{p},\quad\partial_{b_{0}}\buildrel\hbox{\footnotesize def}\over{=}b_{0}\cdot\nabla_{y},\quad\hbox{and}\quad\\ &(\Delta_{x}p)(t,X(t,y))=\sum_{i,j=1}^{3}\nabla_{Y^{i}}\nabla_{Y^{j}}\bigl{(}\partial_{b_{0}}X^{i}\partial_{b_{0}}X^{j}-Y^{i}_{t}Y^{j}_{t}\bigr{)}(t,y).\end{split}$$

$$\text{mbox}supp(b_0(x_{\mathrm{h}},\cdot )-e_3)\subset [0,K]and{b}_0^3\ne 0.$$

$$\left\{\begin{array}[]{l}\displaystyle\frac{dy_{1}}{dy_{3}}=\frac{b_{0}^{1}}{b_{0}^{3}}(y_{1},y_{2},y_{3}),\quad y_{1}|_{y_{3}=0}=w_{1},\\ \displaystyle\frac{dy_{2}}{dy_{3}}=\frac{b_{0}^{2}}{b_{0}^{3}}(y_{1},y_{2},y_{3}),\quad y_{2}|_{y_{3}=0}=w_{2},\\ \displaystyle y_{3}=w_{3},\end{array}\right.$$

$$\begin{split}&z_{1}=w_{1},\quad z_{2}=w_{2},\quad z_{3}=w_{3}+\int_{0}^{w_{3}}\Bigl{(}\frac{1}{b_{0}^{3}(y(w))}-1\Bigr{)}\,dw_{3}^{\prime}.\end{split}$$

$$\begin{split}&\partial_{b_{0}}f(y)=\frac{\partial f(y(w(z)))}{\partial z_{3}},\quad\hbox{and}\quad\nabla_{y}=\nabla_{Z}={{\mathcal{B}}}^{t}(z)\nabla_{z}\quad\hbox{with}\quad{{\mathcal{B}}}(z)=\Bigl{(}\frac{\partial y(w(z))}{\partial z}\Bigr{)}^{-1}.\end{split}$$

$$\displaystyle\begin{split}{{\mathcal{B}}}(z)=\Bigl{(}\frac{\partial y(w(z))}{\partial z}\Bigr{)}^{-1}=&\Bigl{(}\frac{\partial y(w(z))}{\partial w}\times\frac{\partial w(z)}{\partial z}\Bigr{)}^{-1}\\ =&\Bigl{(}\frac{\partial w(z)}{\partial z}\Bigr{)}^{-1}\Bigl{(}\frac{\partial y(w(z))}{\partial w}\Bigr{)}^{-1}=\Bigl{(}\frac{\partial z}{\partial w}\Bigr{)}\Bigl{(}\frac{\partial y(w(z))}{\partial w}\Bigr{)}^{-1}.\end{split}$$

$$\begin{split}\Bigl{(}\frac{\partial y(w)}{\partial w}\Bigr{)}=&\begin{pmatrix}1&0&\frac{b_{0}^{1}}{b_{0}^{3}}\\ 0&1&\frac{b_{0}^{2}}{b_{0}^{3}}\\ 0&0&1\end{pmatrix}+\begin{pmatrix}\int_{0}^{w_{3}}\frac{\partial}{\partial y_{1}}\bigl{(}\frac{b_{0}^{1}}{b_{0}^{3}}\bigr{)}dy_{3}^{\prime}&\int_{0}^{w_{3}}\frac{\partial}{\partial y_{2}}\bigl{(}\frac{b_{0}^{1}}{b_{0}^{3}}\bigr{)}dy_{3}^{\prime}&0\\ \int_{0}^{w_{3}}\frac{\partial}{\partial y_{1}}\bigl{(}\frac{b_{0}^{2}}{b_{0}^{3}}\bigr{)}dy_{3}^{\prime}&\int_{0}^{w_{3}}\frac{\partial}{\partial y_{2}}\bigl{(}\frac{b_{0}^{2}}{b_{0}^{3}}\bigr{)}dy_{3}^{\prime}&0\\ 0&0&0\end{pmatrix}\begin{pmatrix}\frac{\partial y_{1}}{\partial w_{1}}&\frac{\partial y_{1}}{\partial w_{2}}&\frac{\partial y_{1}}{\partial w_{3}}\\ \frac{\partial y_{2}}{\partial w_{1}}&\frac{\partial y_{2}}{\partial w_{2}}&\frac{\partial y_{2}}{\partial w_{3}}\\ \frac{\partial y_{3}}{\partial w_{1}}&\frac{\partial y_{3}}{\partial w_{2}}&\frac{\partial y_{3}}{\partial w_{3}}\end{pmatrix}\\ \buildrel\hbox{\footnotesize def}\over{=}&A_{1}(y(w))+A_{2}(y(w))\Bigl{(}\frac{\partial y(w)}{\partial w}\Bigr{)},\end{split}$$

$$\Bigl{(}\frac{\partial y(w)}{\partial w}\Bigr{)}=\bigl{(}Id-A_{2}(y(w))\bigr{)}^{-1}A_{1}(y(w)).$$

$$\Bigl{(}\frac{\partial z(w)}{\partial w}\Bigr{)}=\begin{pmatrix}1&0&0\\ 0&1&0\\ \int_{0}^{w_{3}}\frac{\partial}{\partial w_{1}}\bigl{(}\frac{1}{b_{0}^{3}(y(w))}\bigr{)}dw_{3}^{\prime}&\int_{0}^{w_{3}}\frac{\partial}{\partial w_{2}}\bigl{(}\frac{1}{b_{0}^{3}(y(w))}\bigr{)}dy_{3}^{\prime}&\frac{1}{b_{0}^{3}}\end{pmatrix}\buildrel\hbox{\footnotesize def}\over{=}A_{3}(w).$$

$$\begin{split}&y(w)=(y_{\rm h}(w_{\rm h},w_{3}),w_{3}),\quad w(z)=(z_{\rm h},w_{3}(z)),\quad\hbox{and}\quad y(w(z))=\bigl{(}y_{\rm h}(z_{\rm h},w_{3}(z)),w_{3}(z)\bigr{)},\\ &{{\mathcal{B}}}(z)=A_{3}(w(z))A_{1}^{-1}(y(w(z))\bigl{(}Id-A_{2}(w(z))\bigr{)},\end{split}$$

$$\left\{\begin{array}[]{l}\displaystyle Y_{tt}-\Delta_{z}Y_{t}-\partial_{z_{3}}^{2}Y=\bigl{(}\nabla_{Z}\cdot\nabla_{Z}-\Delta_{z})Y_{t}+\partial_{z_{3}}b_{0}(y(w(z)))+g(y(w(z))),\\ \displaystyle Y_{|t=0}=Y_{0}=0,\qquad{Y_{t}}_{|t=0}=Y_{1}(z)=u_{0}(y(w(z))),\end{array}\right.$$

$$\begin{split}\widetilde{Y}(z)=&\eta(z_{3})\Bigl{(}\int_{-1}^{z_{3}}\bigl{(}e_{3}-b_{0}(y(w(z_{\rm h},z_{3}^{\prime})))\bigr{)}\,dz_{3}^{\prime}-\int_{-1}^{K+1}\bigl{(}e_{3}-b_{0}(y(w(z_{\rm h},z_{3}^{\prime})))\bigr{)}\,dz_{3}^{\prime}\Bigr{)},\end{split}$$

$$\partial_{z_{3}}\tilde{Y}(z)=e_{3}-b_{0}(y(w(z))),\quad\hbox{and}\quad\partial_{z_{3}}\bigl{(}\partial_{z_{3}}\tilde{Y}+b_{0}(y(w(z)))\bigr{)}=0.$$

$$\left\{\begin{array}[]{l}\displaystyle\bar{Y}_{tt}-\Delta_{z}\bar{Y}_{t}-\partial_{z_{3}}^{2}\bar{Y}=f,\\ \displaystyle{\bar{Y}}_{|t=0}={\bar{Y}}^{(0)}=-\tilde{Y},\qquad{\bar{Y_{t}}}_{|t=0}=Y^{(1)},\end{array}\right.$$

$$\begin{split}\mathcal{A}=&\left(Id+{{\mathcal{B}}}^{t}\nabla_{z}\widetilde{Y}+{{\mathcal{B}}}^{t}\nabla_{z}\bar{Y}\right)^{-1},\quad\hbox{and}\quad\\ f=&{{\mathcal{B}}}^{t}\nabla_{z}\cdot\bigl{[}(\mathcal{A}\mathcal{A}^{t}-Id){{\mathcal{B}}}^{t}\nabla_{z}\bar{Y}_{t}\bigr{]}+{{\mathcal{B}}}^{t}\nabla_{z}\cdot({{\mathcal{B}}}^{t}\nabla_{z}\bar{Y}_{t}\bigr{)}-\Delta_{z}\bar{Y}_{t}-({{\mathcal{B}}}\mathcal{A})^{t}\nabla_{z}\boldsymbol{p},\\ \nabla_{z}\boldsymbol{p}=&-\nabla_{z}\Delta_{z}^{-1}\mbox{div}_{z}\bigl{(}\det({\mathcal{B}}^{-1})({\mathcal{B}}\mathcal{A}\mathcal{A}^{t}{\mathcal{B}}^{t}-Id)\nabla_{z}\boldsymbol{p}\bigr{)}\\ &-\nabla_{z}\Delta_{z}^{-1}\mbox{div}_{z}\bigl{(}(\det({\mathcal{B}}^{-1})Id-Id)\nabla_{z}\boldsymbol{p}\bigr{)}\\ &+\nabla_{z}\Delta_{z}^{-1}\mbox{div}_{z}\Bigl{(}{\mathcal{B}}\mathcal{A}\mbox{div}_{z}\left(\det({\mathcal{B}}^{-1}){\mathcal{B}}\mathcal{A}\bigl{(}\partial_{3}\bar{Y}\otimes\partial_{3}\bar{Y}-\bar{Y}_{t}\otimes\bar{Y}_{t}\bigr{)}\right)\Bigr{)}.\end{split}$$

$$\displaystyle\|f\|_{W^{N,p}}\buildrel\hbox{\footnotesize def}\over{=}\sum_{|\alpha|\leq N}\|D_{y}^{\alpha}f\|_{L^{p}}\quad\hbox{and}\quad\|u\|_{L^{p};k,N}\buildrel\hbox{\footnotesize def}\over{=}\sup_{t>0}(1+t)^{k}\|u(t)\|_{W^{N,p}}.$$

$$\begin{split}&{|\kern-1.07639pt|\kern-1.07639pt|f|\kern-1.07639pt|\kern-1.07639pt|}_{N}\buildrel\hbox{\footnotesize def}\over{=}\|f\|_{W^{N,1}},\qquad\|f\|_{N}\buildrel\hbox{\footnotesize def}\over{=}\|f\|_{H^{N}},\qquad\quad\ \ |f|_{N}\buildrel\hbox{\footnotesize def}\over{=}\|f\|_{W^{N,\infty}}\\ &\text{and}\qquad\qquad\|u\|_{k,N}\buildrel\hbox{\footnotesize def}\over{=}\|u\|_{L^{2};k,N},\qquad|u|_{k,N}\buildrel\hbox{\footnotesize def}\over{=}\|u\|_{L^{\infty};k,N}.\end{split}$$

$${|||({\bar{Y}}^{(0)},Y^{(1)})|||}_{L_0}+\Vert ({\bar{Y}}^{(0)},Y^{(1)}){\Vert }_{L_0}\le \eta and\epsilon \le {\epsilon }_0.$$

Full text

Keywords: MHD system, Nash-Moser iteration scheme, Littlewood-Paley theory, Besov spaces.

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

1. Introduction

In this paper, we investigate the large time behavior of the global smooth solutions to the following three-dimensional incompressible magnetic hydrodynamical (or MHD in short) system with initial data being sufficiently close to the equilibrium state $(e_{3},0):$

[TABLE]

where $b=(b^{1},b^{2},b^{3})$ denotes the magnetic field, $u=(u^{1},u^{2},u^{3})$ and $p$ stand for the velocity and scalar pressure of the fluid respectively. This MHD system (1.1) with zero diffusivity in the magnetic field equation can be applied to model plasmas when the plasmas are strongly collisional, or the resistivity due to these collisions are extremely small. One may check the references [5, 13, 14, 22] for more explanations to this system.

Whether there is dissipation or not for the magnetic field of (1.1) is a very important problem from physics of plasmas. The heating of high temperature plasmas by MHD waves is one of the most interesting and challenging problems of plasma physics especially when the energy is injected into the system at the length scales which are much larger than the dissipative ones. It has been conjectured that in the two-dimensional MHD system, energy is dissipated at a rate that is independent of the ohmic resistivity [7]. In other words, the diffusivity for the magnetic field equation can be zero yet the whole system may still be dissipative. The goal of this paper is to rigorously justify this conjecture in three space dimension provided that the initial data of (1.1) is a small perturbation of the equilibrium state $(e_{3},0).$

Concerning the well-posedness issue of the system (1.1), Chemin et al [12] proved the local well-posedness of (1.1) with initial data in the critical Besov spaces. Lin and the second author [24] proved the global well-posedness to a modified three-dimensional MHD system with initial data sufficiently close to the equilibrium state (see [25] for a simplified proof). Lin, Xu and the second author [23] established the global well-posedness of (1.1) in 2-D provided that the initial data is near the equilibrium state $(e_{d},0)$ and the initial magnetic field, $b_{0},$ satisfies sort of admissible condition, namely

[TABLE]

with $Z(t,\alpha)$ being determined by

[TABLE]

Similar result in three space dimension was proved by Xu and the second author in [30].

In the 2-D case, the restriction (1.2) was removed by Ren, Wu, Xiang and Zhang in [27] by carefully exploiting the divergence structure of the velocity field. Moreover, the authors proved that

[TABLE]

where $\langle{t}\rangle\buildrel\hbox{\footnotesize def}\over{=}\left(1+t^{2}\right)^{\frac{1}{2}}.$ A more elementary existence proof was also given by Zhang in [31]. Very recently, Abidi and the second author removed the restriction (1.2) in [1] for the 3-D MHD system. Moreover, if the initial magnetic field equals to $e_{3}$ and with other technical assumptions, this solution decays to zero according to

[TABLE]

Note that (1.5) corresponding to the critical case of (1.4), that is, $\epsilon=0$ in (1.4).

This idea of considering the global well-posedness of MHD system with initial data close to the equilibrium sate $(e_{d},0)$ goes back to the work by Bardos, Sulem and Sulem [2] for the global well-posedness of ideal incompressible MHD system. In general, it is not known whether or not classical solutions of (1.1) can develop finite time singularities even in two dimension. In the case when there is full magnetic diffusion in (1.1), one may check [15] for its local well-posedness in the classical Sobolev spaces, and [28] for the global well-posedness of such a system in two space dimension. With mixed partial dissipation and additional magnetic diffusion in the two-dimensional MHD system, Cao and Wu [8] (see also [9]) proved that such a system is globally well-posed for any data in $H^{2}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{2}).$ Lately He, Xu and Yu [16] (see also [6] and [29]) justified the vanishing viscosity limit of the full diffusive MHD system to the solution constructed by Bardos et al in [2] for the ideal MHD system.

The main result of this paper states as follows:

Theorem 1.1**.**

Let $e_{3}=(0,0,1),$ $b_{0}=e_{3}+\varepsilon\phi$ with $\phi=(\phi_{1},\phi_{2},\phi_{3})\in C_{c}^{\infty}$ and $\mathop{\rm div}\nolimits\phi=0,$ let $u_{0}\in W^{N_{0},1}\cap H^{N_{0}}$ for some integer $N_{0}$ sufficiently large. Then there exist sufficiently small positive constants $\varepsilon_{0},c_{0}$ such that if

[TABLE]

(1.1) has a unique global solution $(b,u)$ so that for any $T>0,$ $b\in C^{2}([0,T]\times\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3}),$ $u\in C^{2}([0,T]\times\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3})$. Moreover, for some $\kappa>0,$ there hold

[TABLE]

Let us remark that the above theorem recovers the global well-posedness result of the system (1.1) in [1]. Moreover, the bigger the integer $N_{0},$ the smaller the positive constant $\kappa.$ The main idea of the proof here works in both two space dimension and in three space dimension. The $L^{\infty}$ decay rates of the solution in (1.7) are completely new. The $L^{2}$ decay rates of the solution are optimal in the sense that these decay rates coincide with those of the linearized system (see Propositions 2.1 and 2.5 below), which greatly improves the rate given by (1.5). We can also work on the decay rates for the higher order derivatives of the solutions. But we choose not to pursue on this direction here.

2. Structure and strategies of the proof

2.1. Lagrangian formulation of (1.1)

As observed in the previous references ([23, 30]), the linearized system of (1.1) around the equilibrium state $(e_{3},0)$ reads

[TABLE]

It is easy to calculate that this system has two different eigenvalues

[TABLE]

The Fourier modes corresponding to $\lambda_{2}(\xi)$ decays like $e^{-t|\xi|^{2}}$. Whereas the decay property of the Fourier modes corresponding to $\lambda_{-}(\xi)$ varies with directions of $\xi$ as

[TABLE]

only in the $\xi_{3}$ direction. This simple analysis shows that the dissipative properties of the system (2.1) may be more complicated than that for the linearized system of isentropic compressible Navier-Stokes system (see [11] for instance). Moreover, it is well-known that it is in general impossible to propagate the anisotropic regularities for the transport equation. This motivates us to use the Lagrangian formulation of the system (1.1).

Let us now recall the Lagrangian formulation of (1.1) from [1]. Let $(b,u)$ be a smooth enough solution of (1.1), we define

[TABLE]

Then $(Y,\boldsymbol{b},\boldsymbol{u},\boldsymbol{p})$ solves

[TABLE]

where

[TABLE]

In what follows, we assume that

[TABLE]

Due to the difficulty of the variable coefficients for the linearized system of (2.4), we shall use Frobenius Theorem type argument to find a new coordinate system $\{z\}$ so that $\partial_{b_{0}}=\partial_{z_{3}}.$ Toward this, let us define

[TABLE]

and

[TABLE]

Then we have

[TABLE]

It is easy to observe that

[TABLE]

Yet it follows from (2.7) that

[TABLE]

which gives

[TABLE]

While it is easy to observe that

[TABLE]

As a consequence, we obtain

[TABLE]

with the matrices $A_{1},A_{2},A_{3}$ being determined by (2.10) and (2.12) respectively.

For simplicity, let us abuse the notation that $Y(t,z)=Y(t,y(w(z))).$ Then the system (2.4) becomes

[TABLE]

for $g$ given by (2.4). Since $\partial_{z_{3}}b_{0}(y(w(z)))$ in the source term is a time independent function, we now introduce a smooth cut-off function $\eta(z_{3})$ with \eta(z_{3})=\left\{\begin{array}[]{l}\displaystyle 0,\,\,z_{3}\geq 2+K,\\ \displaystyle 1,\,\,-1\leq z_{3}\leq 1+K,\\ \displaystyle 0,\,\,z_{3}\leq-2,\end{array}\right. and a correction term $\tilde{Y}$ so that $Y=\tilde{Y}+\bar{Y}$ and

[TABLE]

which satisfies

[TABLE]

Then in view of (2.23), (2.24) and (2.30) of [1], $\bar{Y}$ solves

[TABLE]

with

[TABLE]

2.2. The proof of Theorem 1.1

Before presenting the main result for the system (2.17-2.18), let us first introduce notations of the norms: For $f:\mathop{\mathbb{R}\kern 0.0pt}\nolimits_{y}^{3}\to\mathop{\mathbb{R}\kern 0.0pt}\nolimits,$ $u:\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{+}\times\mathop{\mathbb{R}\kern 0.0pt}\nolimits_{y}^{3}\to\mathop{\mathbb{R}\kern 0.0pt}\nolimits$, and $p\in[1,+\infty]$, $N\in\mathop{\mathbb{N}\kern 0.0pt}\nolimits$, we denote

[TABLE]

In particular, when $p=1,$ $p=2$ and $p=\infty,$ we simplify the notations as

[TABLE]

Theorem 2.1**.**

There exist an integer $L_{0}$ and small constants $\eta,\varepsilon_{0}>0$ such that if

[TABLE]

Then the system (2.17) has a unique global solution $\bar{Y}\in C^{2}([0,\infty);C^{N_{1}-4}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3}))$, where $N_{1}=[(L_{0}-12)/2]$. Furthermore, for any $\kappa>0,$ there hold

[TABLE]

and

[TABLE]

Admitting Theorem 2.1 for the time being, let us now turn to the proof of Theorem 1.1.

Proof of Theorem 1.1.

Indeed, in view of (2.3), one has

[TABLE]

with $\widetilde{Y}(z)$ and $\bar{Y}(t,z)$ being determined by (2.15) and (2.17) respectively.

Whereas in view of (2.10), (2.12) and (2.13), we get, by a similar proof of Lemma 4.3 of [1] that for any $N\in\mathop{\mathbb{N}\kern 0.0pt}\nolimits,$

[TABLE]

So that under the assumptions of (1.6), there holds (2.20). Then Theorem 2.1 ensures that the system (2.17-2.18) has a unique global classical solution $\bar{Y}\in C^{2}([0,\infty);C^{N_{1}-4}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3})),$ which verifies (2.21) and (2.22). In particular, it follows from (2.15) and (2.21) that

[TABLE]

which together with (2.23) ensures that $\boldsymbol{u}\in C^{2}([0,\infty)\times\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3})$ and $\boldsymbol{b}\in C^{2}([0,\infty)\times\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3}).$ Furthermore due to

[TABLE]

we deduce from (2.3) that $u\in C^{2}([0,\infty)\times\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3})$ and $b\in C^{2}([0,\infty)\times\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3})$ which verifies the system (1.1) thanks to the derivation at the beginning of Subsection 2.1.

On the other hand, by virtue of (2.16), we have

[TABLE]

which together with (2.21), (2.22) and (2.23) implies that there holds (1.7). This completes the proof of Theorem 1.1. ∎

2.3. Strategies of the proof to Theorem 2.1

Observing from the calculations in [1] that under the assumptions of Theorem 1.1, the matrix ${\mathcal{B}}$ given by (2.13) is sufficiently close to the identity matrix in the norms of $W^{N_{0},1}$ and $H^{N_{0}}$ as long as $\varepsilon$ is sufficiently small. To avoid cumbersome calculation, here we just prove Theorem 2.1 for the system (2.1) with

[TABLE]

which corresponds to ${\mathcal{B}}=Id$ in (2.17). The general case follows along the same line.

Let us remark that the system (2.1) is not scaling, rotation and Lorentz invariant, so that Klainerman’s vector field method ([21]) can not be applied here. Yet the ideas developed by Klainerman in the seminar papers [18, 19, 20] can be well adapted for this system. We now recall the classical result on the global well-posedness to some evolutionary system from [19]. Let us consider the following system

[TABLE]

where ${\mathcal{L}}\buildrel\hbox{\footnotesize def}\over{=}\sum_{|\alpha|\leq\gamma}a_{\alpha}D_{x}^{\alpha}$ with $a_{\alpha}$ being $r\times r$ matrices with constant entries. Under the assumptions that

• (1)

${\mathcal{L}}$ satisfies a dissipative condition of the following type: there exists a positive definite $r\times r$ matrix $A$ such that

[TABLE]

for any $f\in C_{c}^{\infty};$

• (2)

Let $\Gamma(t)u_{0}$ be the solution of

[TABLE]

There is a differential matrix $P$ such that

[TABLE]

for any $u_{0}\in W^{d,1}\cap L^{\infty}$ that satisfies $Pu_{0}=0;$

• (3)

$AF_{u_{t}},AF_{u_{x_{i}}},$ $i=1,\cdots,d,$ are symmetric matrices and $F_{u_{t}}$ is independent of $u_{t}.$ Moreover

[TABLE]

• (4)

$p$ is an integer and $F$ is a smooth function so that there holds

[TABLE]

Klainerman proved in [19] the following celebrated theorem:

Theorem 2.2** (Theorem 1 of [19]).**

There exist an integer $N_{0}>0$ and a small constant $\eta>0$ such that if

[TABLE]

(2.26) has a unique solution $u\in C^{1}([0,T];C^{\gamma})$ for any $T>0.$ Moreover, the solution behaves, for $t$ large, like

[TABLE]

for some small $\varepsilon>0.$ Also

[TABLE]

Let us remark that due to the appearance of the double Riesz transform in the expression of $f$ in (2.25), the source term $f$ in (2.1) can not satisfy the growth condition (2.27); secondly, even if we can assume the source term $f$ is in quadratic growth of $(Y_{t},\partial_{3}Y),$ that corresponds to $p=1$ in (2.27), the growth rate obtained in (3.2) below does not meet the requirement of (2.28). This makes it impossible to apply Theorem 2.2 for the system (2.1). Yet by considering the specific anisotropic structure of the system (2.1), we can still succeed in applying Nash-Moser scheme to establish the global existence as well as the large time behavior of solutions to (2.1-2.25).

Now we outline the proof of Theorem 2.1. According to the strategy in [18, 19, 20], the first step is to study the decay properties of the linear system:

[TABLE]

Proposition 2.1**.**

Let ${\mathcal{Y}}(t)$ be a smooth enough solution of (2.31). Given $\delta\in[0,1]$, $N\in\mathop{\mathbb{N}\kern 0.0pt}\nolimits$, there exist $C_{\delta,N},C_{N}>0$ such that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

We emphasize here the estimates of (2.32) and (2.33) are of anisotropic type, which means that the decay rates of the partial derivatives of the solution to (2.31) are different, which is consistent with the heuristic discussions at the beginning of Section 2. Moreover, the estimate of (2.32) is valid for $\delta=0.$ Similar estimates as (2.34) and (2.35) were not proved in [18, 19, 20]. They are purely due to the special structure of the linearized system (2.31).

With the above proposition, we next turn to the decay estimates for the solutions of the following inhomogeneous equation of (2.31)

[TABLE]

Proposition 2.2**.**

Let $\delta\in[0,1/4[$ and $\theta\in[1,\infty[.$ We assume that $g(t)=0$ if $t\geq\theta$. Then the solution $Y$ to (2.36) verifies for any $N\geq 0$,

[TABLE]

where

[TABLE]

where

[TABLE]

The proof of the above propositions will be presented in Section 3.

The goal of Section 4 is to calculate the linearized system of (2.1), which reads

[TABLE]

where $f^{\prime}(Y;X)=f_{0}^{\prime}(Y;X)+f_{1}^{\prime}(Y;X)+f_{2}^{\prime}(Y;X)$, and $f_{0}^{\prime}(Y;X),$ $f_{1}^{\prime}(Y;X)$ and $f_{2}^{\prime}(Y;X)$ are determined respectively by (4.6) and (4.7). Furthermore, the second derivative of $f^{\prime\prime}(Y;X,W)$ will be presented in Subsection 4.2.

In Section 5, we shall derive the $\dot{W}^{2\delta,1}\cap\dot{W}^{N+4,1}$ and $\dot{H}^{N+1}$ estimates for the source term $f^{\prime}(Y;X)$ in the linearized system (2.40), which will be used to derive the decay estimates for the solutions of (2.40). The main result reads

Proposition 2.3**.**

Let the functionals, $f_{0}^{\prime}(Y;X),f_{1}^{\prime}(Y;X),f_{2}^{\prime}(Y;X),$ be given by (4.6) and (4.7) respectively, and the norm ${|\kern-1.07639pt|\kern-1.07639pt|\cdot|\kern-1.07639pt|\kern-1.07639pt|}_{\delta,N}$ be given by (2.39). Then under the assumptions that $\delta>0,$ and

[TABLE]

for some $\delta_{1}>0$ sufficiently small, we have

[TABLE]

and

[TABLE]

where the function $\mathfrak{f}_{1}(\mathfrak{x},\mathfrak{y})$ is given by

[TABLE]

Proposition 2.4**.**

Under the assumption of Proposition 2.3, we have

[TABLE]

and

[TABLE]

where the functional $\mathfrak{f}_{2}(\mathfrak{x},\mathfrak{y})$ is given by

[TABLE]

Let us remark that Riesz transform does not map continuously from $L^{1}$ to $L^{1}.$ Nevertheless due to (4.8) and (4.9), we can not avoid estimates of this type. To overcome this difficulty, a natural replacement of $\dot{W}^{s,1}$ will be the Besov space $\dot{B}^{s}_{1,1},$ which satisfies

[TABLE]

We now recall the precise definition of the Besov norms from [3] for instance.

Definition 2.1**.**

Let us consider a smooth function $\varphi$ on $\mathop{\mathbb{R}\kern 0.0pt}\nolimits,$ the support of which is included in $[3/4,8/3]$ such that

[TABLE]

Let us define

[TABLE]

Let $(p,r)$ be in $[1,+\infty]^{2}$ and $s$ in $\mathop{\mathbb{R}\kern 0.0pt}\nolimits$. We define the Besov norm by

[TABLE]

We remark that in the special case when $p=r=2,$ the Besov spaces $\dot{B}^{s}_{p,r}$ coincides with the classical homogeneous Sobolev spaces $\dot{H}^{s}$. Moreover, we have the following product laws (see Corollary 2.54 of [3]):

[TABLE]

for $s>0$, $(p,r)\in[1,+\infty]^{2}$. Due to the product law (2.48), we need the index $\delta$ to be positive in Proposition 2.3.

In Section 6, we investigate energy estimates for the solutions of the linearized equation (2.40).

Theorem 2.3**.**

Let $Y$ be a smooth enough vector field and $X$ be a smooth solution to the linearized equation (2.40). We assume that $Y$ satisfies (2.41) and

[TABLE]

Then for any $\varepsilon>0$, we have

[TABLE]

where

[TABLE]

and

[TABLE]

We notice that when we perform the energy estimates for the derivatives of the solutions to (2.40), we are not able to treat the term \nabla\cdot\big{(}({\mathcal{A}}{\mathcal{A}}^{t}-Id)\nabla X_{t}\big{)}, which appears in $f_{0}^{\prime}(Y;X)$ (see (4.6)), as a source term. Instead, we need to rewrite (2.40) as

[TABLE]

where $\widetilde{f}^{\prime}(Y;X)=\widetilde{f}_{0}^{\prime}(Y;X)+f_{1}^{\prime}(Y;X)+f_{2}^{\prime}(Y;X)$ with $f_{m}^{\prime}(Y;X),m=1,2,$ given by (4.7), and $\widetilde{f}_{0}^{\prime}(Y;X)$ by

[TABLE]

With the energy estimates obtained in Theorem 2.3, we can work on the time-weighted energy estimate for the solutions of (2.40).

Corollary 2.1**.**

Under the assumptions of Theorem 2.3, we have

[TABLE]

and for $N\geq 1$,

[TABLE]

Proposition 2.5**.**

Under the assumptions of Theorem 2.3, we have for $N\geq 0$,

[TABLE]

We emphasize that the decay estimates (2.57) can not be obtained by energy estimate. In fact, we will have to exploit anisotropic Littlewood-Paley analysis and the dissipative properties of the linear system (2.1). The proof of Proposition 2.5 will be presented in Section 7, which is of independent interest.

Let us summarize that under the assumptions (2.41), (2.49) and if we assume moreover

[TABLE]

we have the following energy estimates: for $N\geq 0$, (we make the convention $\|u\|_{k,-1}=0$)

[TABLE]

In Section 8, we shall present the estimates to the nonlinear source term $f(Y)$ given by (2.25). The purpose of Section 9 is concerned with the related estimates for the second derivatives, $f^{\prime\prime}(Y;X,W),$ of the nonlinear functional $f(Y)$, computed in Section 4.2.

With the preparations in the previous sections, we can now exploit Nash-Moser iteration scheme to prove Theorem 2.1. In order to do so, we first recall some basic properties of the smoothing operator from [18, 19]. Let $\chi(t)\in C^{\infty}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits;[0,1])$ be such that

[TABLE]

Define for $\theta\geq 1$, the (cutoff-in-time) operator

[TABLE]

Then we have

[TABLE]

and

[TABLE]

For $\theta^{\prime}\geq 1$, we define the usual mollifying operator $S^{(2)}(\theta^{\prime})$ in the space variables by

[TABLE]

where $\varphi\in{\mathcal{S}}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3})$ satisfies

[TABLE]

so that

[TABLE]

We then have

[TABLE]

as well as

[TABLE]

Define the operator

[TABLE]

Then it follows that

[TABLE]

Moreover, due to

[TABLE]

one has

[TABLE]

provided that $k\geq s\geq 0,\ N\geq M\geq 0.$

Let us denote

[TABLE]

for $f$ given by (2.25). Then we can write (2.1) equivalently as

[TABLE]

We aim to solve (2.65) via Nash-Moser iteration scheme in Section 10.

Let us define $Y_{0}$ via

[TABLE]

Inductively, assume that we already determine $Y_{p}$. In order to define $Y_{p+1}$, we introduce a mollified version of $\Phi^{\prime}(Y_{p})$ as follows

[TABLE]

where $S_{p}$ is the smoothing operator defined by

[TABLE]

where $S(\theta,\theta^{\prime})$ is defined in (2.62) and $\bar{\varepsilon}>0$ is a small constant to be chosen later on. Then it follows from (2.63) and (2.64) that

[TABLE]

and

[TABLE]

for $k\geq s\geq 0$, $N\geq M\geq 0,$ where the norm ${|\kern-1.07639pt|\kern-1.07639pt|\cdot|\kern-1.07639pt|\kern-1.07639pt|}_{L_{t}^{1}(\delta,N)}$ is given by (2.39).

Remark 2.1**.**

According to Remark 4.1 below, we can write

[TABLE]

where

[TABLE]

where the functionals $F_{0}^{\prime}$, $F^{\prime}$ will be presented in Remark 4.1.

Following Hörmander’s version of Nash-Moser Scheme ([17]) (see also Klainerman’s seminar papers [18, 19]), we define

[TABLE]

where $L_{p}^{-1}$ is a right inverse operator of $L_{p}$ with zero initial data, that is: $X=L_{p}^{-1}g_{p}$ solves

[TABLE]

In order to prove the convergence of the scheme, we define

[TABLE]

from which, we infer

[TABLE]

As a result, it comes out

[TABLE]

To achieve that the above limit as $p\to\infty$ is equal to $-\Phi(Y_{0})$, we set

[TABLE]

The last relation defines $g_{p}$ as follows

[TABLE]

Remark 2.2**.**

By virtue of Remarks 2.1, 4.1, 4.2, using a Taylor formula to (2.71), we have

[TABLE]

where $f^{\prime\prime}$ should be understood in the way explained in Remark 4.2. Then we have

[TABLE]

Let us fix the small constants: $\varepsilon$, $\bar{\varepsilon}$, $\delta>0,$ so that

[TABLE]

Let us take

[TABLE]

and $N_{0}\in\mathop{\mathbb{N}\kern 0.0pt}\nolimits$ is chosen such that

[TABLE]

In Section 10, we shall inductively prove the following statements:

Proposition 2.6**.**

Let $\delta_{1}>0$ be determined by Propositions 2.3, 2.4, 8.1, 8.2, 9.1, 9.2 and Theorem 2.3. Then for the constants $\beta,\gamma,N_{0},\varepsilon,\bar{\varepsilon}$ and $\delta$ given by (2.76-2.78), for any $0\leq N\leq N_{0},$ we have

[TABLE]

and

[TABLE]

and

[TABLE]

Recall the convention that $\|u\|_{k,-1}=0$. We shall deduce the following propositions from Proposition 2.6.

Proposition 2.7**.**

Under the assumptions of Proposition 2.6, we have, for $N\geq 0$,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Proposition 2.8**.**

Let $e_{p},$ $g_{p}$ and $R_{N,\theta}(g)$ be given by (2.71), (2.74) and (2.38) respectively. Let $\alpha\buildrel\hbox{\footnotesize def}\over{=}\frac{1}{2}-\delta-\bar{\varepsilon}>0$. Then there hold

(1) Estimates for $e_{p}$.

[TABLE]

(2) Estimates for $g_{p+1}$.

[TABLE]

(3) Estimates for $R_{N,\theta_{p+1}}(g_{p+1})$.

[TABLE]

The following interpolation lemma will be crucial in the proof of the above propositions, whose proof is exactly the same as that of Lemma 6.1 of [18], which we omit the details here.

Lemma 2.1** (Interpolation lemma).**

Let $p\in[1,+\infty]$, $\theta\geq 1$ and $\bar{\varepsilon}>0$, which satisfy

[TABLE]

Assume that $u\in C^{\infty}([0,+\infty)\times\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{n})$ satisfies

[TABLE]

Then for all $0\leq k\leq k_{0}$, $0\leq N\leq N_{0}$,

[TABLE]

Finally with the previous propositions, we shall prove the convergence of the approximate solutions constructed by (2.69) in Subsection 10.4, and this completes the proof of Theorem 2.1.

3. Decay estimates of the linear equation

3.1. Decay estimates for the solution operator

Following the strategy in [18, 19], we first investigate the decay properties of the solutions to the linear equation (2.31) with ${\mathcal{Y}}_{0}=0$ and ${\mathcal{Y}}_{1}=Y_{1}.$ By taking Fourier transform to (2.31) with respect to $y$ variables and solving the resulting ODE, we write

[TABLE]

where $\lambda_{1}(\xi)$ and $\lambda_{2}(\xi)$ are given by (2.2).

Proposition 3.1**.**

Given $\delta\in[0,1[$ and $N\in\mathop{\mathbb{N}\kern 0.0pt}\nolimits$, there exists $C_{\delta,N}>0$ such that there holds

[TABLE]

Proof.

The estimate (3.2) for general $N\in\mathop{\mathbb{N}\kern 0.0pt}\nolimits$ follows from the case when $N=0.$ Due to the anisotropic properties of the eigenvalues $\lambda_{1}(\xi),\lambda_{2}(\xi),$ we shall split the frequency space into two parts: $\{\xi\in\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3}:\ |\xi|^{2}\geq 2|\xi_{3}|\ \}$ and $\{\xi\in\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3}:\ |\xi|^{2}<2|\xi_{3}|\ \}.$ When $|\xi|^{2}\geq 2|\xi_{3}|$, let us denote $\alpha(\xi)\buildrel\hbox{\footnotesize def}\over{=}\sqrt{\frac{|\xi|^{4}}{4}-\xi_{3}^{2}}.$ Then we have

[TABLE]

and we write

[TABLE]

When $|\xi|^{2}<2|\xi_{3}|$, let us denote $\beta(\xi)\buildrel\hbox{\footnotesize def}\over{=}\sqrt{\xi_{3}^{2}-\frac{|\xi|^{4}}{4}}.$ Then we have

[TABLE]

and we write

[TABLE]

Next we handle the estimate of (3.2) term by term below.

$\bullet$Estimates of $\|\partial_{3}{\mathcal{Y}}(t)\|_{L^{\infty}}$ and $\|\partial_{3}^{2}{\mathcal{Y}}(t)\|_{L^{\infty}}.$**

In view of (3.1), we deduce that

[TABLE]

It is easy to observe that

[TABLE]

and

[TABLE]

Exactly along the same line, we have

[TABLE]

This proves

[TABLE]

The estimate of $I_{2}$ is much simpler. By virtue of (3.5), we have

[TABLE]

As a result, we achieve

[TABLE]

Along the same line to the proof of (3.8), we infer

[TABLE]

so that for $t$ large enough, there holds

[TABLE]

This gives rise to

[TABLE]

$\bullet$Estimate of $|\partial_{t}{\mathcal{Y}}(t)|_{L^{\infty}}.$**

It follows from (3.1) that

[TABLE]

So that one has

[TABLE]

It is easy to observe that for any $\delta\in[0,1[,$

[TABLE]

and

[TABLE]

This leads to

[TABLE]

While similar to estimate of (3.6) and (3.7), we infer

[TABLE]

and

[TABLE]

Hence by virtue of (3.10), we obtain

[TABLE]

$\bullet$Estimate of $\|{\mathcal{Y}}(t)\|_{L^{\infty}}.$**

Note that

[TABLE]

we find

[TABLE]

Similarly, we have

[TABLE]

As a result, by virtue of (3.3) and (3.4), it comes out

[TABLE]

(3.8) together with (3.9), (3.11) and (3.12) imply the estimate (3.2) for $N=0.$ ∎

Lemma 3.1**.**

For $N\in\mathop{\mathbb{N}\kern 0.0pt}\nolimits$, there exists $C_{N}>0$ such that for $t>0$,

[TABLE]

Proof.

The two inequalities of (3.13) follows from the claim that

[TABLE]

(1) When $|\xi|^{2}\geq 2|\xi_{3}|$, we separate the proof of (3.14) into the following two cases:

• •

If $\frac{\sqrt{3}}{4}|\xi|^{2}\leq|\xi_{3}|\leq\frac{1}{2}|\xi|^{2}$, we deduce from (3.10) that

[TABLE]

As a result, it comes out

[TABLE]

• •

If $|\xi_{3}|\leq\frac{\sqrt{3}}{4}|\xi|^{2}$, then $\frac{|\xi|^{2}}{4}\leq\alpha(\xi)\leq\frac{|\xi|^{2}}{2},$ we deduce from (3.10) that

[TABLE]

so that there holds

[TABLE]

(2) When $|\xi|^{2}>2|\xi_{3}|$, we infer from (3.10) that

[TABLE]

which implies

[TABLE]

To prove the second estimate of (3.14), we divide further the region, $\{|\xi|^{2}>2|\xi_{3}|\},$ into two parts.

• •

If $|\xi|^{2}\leq\sqrt{3}|\xi_{3}|$, then we have $\frac{|\xi_{3}|}{2}\leq\beta(\xi)\leq|\xi_{3}|,$ and it follows from (3.4) that

[TABLE]

• •

When $\sqrt{3}|\xi_{3}|<|\xi|^{2}\leq 2|\xi_{3}|$, we have

[TABLE]

By summarizing the above estimates, we obtain the second estimate of (3.14). This completes the proof of Lemma 3.1. ∎

3.2. Energy estimates for the linear equation

Lemma 3.2**.**

Let ${\mathcal{Y}}(t)$ be a smooth enough solution of the linear equation (2.31) with initial data $({\mathcal{Y}}_{0},{\mathcal{Y}}_{1})$. Then for any $N\in\mathop{\mathbb{N}\kern 0.0pt}\nolimits$, there exists $C_{N}>0$ such that there hold (2.33) and (2.34).**

Proof.

Taking the $L^{2}$-inner product of the equation (2.31) with ${\mathcal{Y}}_{t}$ and ${\mathcal{Y}}_{t}-\frac{1}{4}\Delta{\mathcal{Y}}-\Delta{\mathcal{Y}}_{t}$, respectively, we get

[TABLE]

and

[TABLE]

Integrating the above equalities with respect to $t$ gives rise to

[TABLE]

[TABLE]

This proves (2.33) for $N=0$. The general case with $N>0$ follows similarly.

To show (2.34), we first get, by taking the $H^{N}$-inner product of the equation (2.31) with ${\mathcal{Y}}_{t}$, that

[TABLE]

So that for any nonegative $f(t)\in C^{1}([0,\infty[)$, we have

[TABLE]

Taking $f(t)=\langle t\rangle$ and integrating the resulting equality over $[0,t]$, we find

[TABLE]

Yet it from (2.33) that

[TABLE]

which together with (3.15) ensures (2.34). ∎

Recall that ${\mathcal{Y}}(t)=\Gamma(t)Y_{1}$ is the solution to (2.31) with initial data $({\mathcal{Y}}_{0},{\mathcal{Y}}_{1})=(0,Y_{1})$, so that one can deduce estimates for the operator $\Gamma$ from the energy estimates (2.33) and (2.34): Indeed combining (3.13) with (2.33) gives

[TABLE]

Let us remark that

[TABLE]

summarizing (2.33), (3.16) and (3.17) then leads to

Corollary 3.1**.**

For $N\geq 0$, there exists $C_{N}>0$ such that

[TABLE]

where $\Gamma(t)$ is the solution operator given by (3.1).**

Now we are in a position to complete the proof of Proposition 2.1.

Proof of Proposition 2.1.

(2.33) and (2.34) are already proved by Lemma 3.2. So it remains to deal with the estimates of (2.32) and (2.35). As a matter of fact, according to the definition of the solution operator $\Gamma(t)$ given by (3.1), we have

[TABLE]

from which and (3.2), we infer that for any $\delta\in]0,1[$ and for $N\in\mathop{\mathbb{N}\kern 0.0pt}\nolimits$,

[TABLE]

While notice that $\partial_{t}^{2}\Gamma(t){\mathcal{Y}}_{0}=\Delta\partial_{t}\Gamma(t){\mathcal{Y}}_{0}+\partial_{3}^{2}\Gamma(t){\mathcal{Y}}_{0},$ we get, by applying (3.2) once again, that

[TABLE]

Inserting the above estimates into (3.20) leads to (2.32).

Finally notice that $\Delta\partial_{t}^{2}\Gamma(t){\mathcal{Y}}_{0}=\Delta^{2}\partial_{t}\Gamma(t){\mathcal{Y}}_{0}+\Delta\partial_{3}^{2}\Gamma(t){\mathcal{Y}}_{0}.$ Then by virtue of (3.16), we deduce

[TABLE]

This proves (2.35), and thus we complete the proof of Proposition 2.1. ∎

3.3. Decay estimates for the inhomogeneous equation

Proof of Proposition 2.2.

In view of (3.1), we get, by applying Duhamel’s principle to (2.36), that

[TABLE]

In what follows, we shall present the proof of (2.37) term by term.

$\bullet$Decay estimate of $\partial_{3}Y.$**

We first separate the integral in (3.21) as

[TABLE]

We deduce from (3.8) that

[TABLE]

While it follows from the second inequality in (3.18) that

[TABLE]

Hence we achieve

[TABLE]

$\bullet$Decay estimate of $Y_{t}.$**

Noticing that $\Gamma(0)=0$, we have

[TABLE]

It follows from (3.11) that

[TABLE]

Whereas it follows from the third inequality in (3.18) that

[TABLE]

As a result, it comes out

[TABLE]

$\bullet$Decay estimate of $Y.$**

As in the previous steps, we first split the integral (3.21) into two parts. For the integral from [math] to $t/2$, we use (3.12) to deduce that

[TABLE]

For the integral from $t/2$ to $t$, we apply the first inequality of (3.18) to get

[TABLE]

Hence we obtain

[TABLE]

By summarizing the estimates (3.22), (3.23) and (3.24), we complete the proof of (2.37). ∎

4. The derivatives of $f$ given by (2.25)

4.1. Computation of $f^{\prime}(Y;X)$

The goal of this subsection is to derive the linearized equations of the system (2.1-2.25). We first decompose the pressure function $\boldsymbol{p}$ given by (2.25) as $\boldsymbol{p}=\boldsymbol{p}_{1}+\boldsymbol{p}_{2}$ with

[TABLE]

Let us denote

[TABLE]

Then the functional $f$ given by (2.25) can be decomposed as $f_{0}-f_{1}+f_{2}$.

Before proceeding, let us recall that for a map $f:{\mathcal{U}}\to{\mathcal{Y}}$, where ${\mathcal{U}}$ is an open set of ${\mathcal{X}}$ and ${\mathcal{X}}\buildrel\hbox{\footnotesize def}\over{=}C^{\infty}([0,\infty[\times\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3};\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3})$, the differentiation of $f$ at $Y\in{\mathcal{U}}$ along the direction $X\in{\mathcal{X}}$ is defined as

[TABLE]

For $f\in C^{\infty}([0,+\infty)\times\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3};M_{3\times 3}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits))$, $g\in C^{\infty}([0,+\infty)\times\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3};\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3})$, we have

[TABLE]

Then for ${\mathcal{A}}(Y)=(Id+\nabla Y)^{-1}$, we have

[TABLE]

and thus

[TABLE]

As a result, we deduce that

[TABLE]

For $m=1,2$, we have

[TABLE]

Moreover, it follows from (4.1) that

[TABLE]

And similarly, it follows from (4.2) that

[TABLE]

The linearized equation of (2.1-2.25) then reads as (2.40).

Remark 4.1**.**

Let $V\in C^{\infty}([0,+\infty)\times\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3};M_{3\times 3}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits))$ and $U\in C^{\infty}([0,+\infty)\times\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3};\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3})$, we denote $h(V)\buildrel\hbox{\footnotesize def}\over{=}(Id+V)^{-1}$, and

[TABLE]

Then $f_{0},f_{1},f_{2}$ defined by (4.3) can be written as

[TABLE]

and hence $f_{0}^{\prime},f_{1}^{\prime}$ and $f_{2}^{\prime}$ read

[TABLE]

where the functionals $F^{\prime}_{0,U}(U,V),$ $F^{\prime}_{0,V}(U,V),$ $F^{\prime}_{U}(U,V)$ and $F^{\prime}_{V}(U,V)$ are given by

[TABLE]

and

[TABLE]

4.2. Computation of $f^{\prime\prime}(Y;X,W)$

In order to estimate the error arisen in the Nash-Moser iteration scheme, we need the second derivatives of $f.$ Toward this, let us recall the product rule

[TABLE]

It is easy to observe from (4.4) that

[TABLE]

Then applying the product rule (4.10) and (4.4) gives

[TABLE]

Recall that $f_{0}$ is given by (4.3), we deduce from (4.10) that

[TABLE]

Similarly for $f_{m}(Y)={\mathcal{A}}^{t}\nabla\boldsymbol{p}_{m}$, $m=1,2$, we have

[TABLE]

Then in view of (4.8), (4.9), to obtain the expression of $f_{m}^{\prime\prime}(Y;X,W),m=1,2,$ it remains to calculate $\boldsymbol{p}_{m}^{\prime\prime}(Y;X,W),$ $m=1,2.$ Indeed, it follows from (4.1), (4.2) and (4.10) that

[TABLE]

and

[TABLE]

Remark 4.2**.**

In view of Remark 4.1, $f_{m}^{\prime\prime}$ can be written as follows

[TABLE]

where $F^{\prime\prime}_{0,UU}(U,V)\dot{U}_{1}\cdot\dot{U}_{2}=0$,

[TABLE]

and

[TABLE]

where

[TABLE]

[TABLE]

and

[TABLE]

5. The estimates of $f^{\prime}(Y;X)$

5.1. The estimate of ${|\kern-1.07639pt|\kern-1.07639pt|f^{\prime}(Y;X)|\kern-1.07639pt|\kern-1.07639pt|}_{\delta,N}$

The main result of this subsection is listed in Proposition 2.3. As we explained in the Section 2, the main idea is to use the norm of the homogeneous Besov spaces $\dot{B}^{s}_{1,1}$ to replace the norm of the classical Sobolev spaces $\dot{W}^{s,1}.$ In order to do so, we need not only the product law (2.48) but also the following one.

Lemma 5.1**.**

For any $s>0,$ there holds

[TABLE]

Proof.

We first get, by applying Bony’s decomposition [4] that

[TABLE]

Due to the support properties to the Fourier transform of the terms in $T_{a}b,$ we have

[TABLE]

where $(d_{j})_{j\in{\mathop{\mathbb{Z}\kern 0.0pt}\nolimits}}$ is a non-negative generic element of $\ell^{1}({\mathop{\mathbb{Z}\kern 0.0pt}\nolimits})$ so that $\sum_{j\in{\mathop{\mathbb{Z}\kern 0.0pt}\nolimits}}d_{j}=1.$

Along the same line, we also have

[TABLE]

and

[TABLE]

where in the last step, we used the fact that $s>0.$ By summing up the above inequalities, we arrive at (5.1). ∎

Notice that ${\mathcal{A}}(\nabla Y)=(Id+\nabla Y)^{-1}$, we write

[TABLE]

So that under the assumption of (2.41), for $s>0$, we get, by applying (2.48), that

[TABLE]

Along the same line, we get, by applying (5.1), that

[TABLE]

5.1.1. Estimate of $\|f_{0}^{\prime}(Y;X)\|_{\dot{B}^{s}_{1,1}}$

In view of (4.6), we have

[TABLE]

It follows from (2.41) and (5.1) that

[TABLE]

While applying (5.3) gives

[TABLE]

yet it follows from (2.48) and (5.1) that

[TABLE]

so that there holds

[TABLE]

The same estimate holds for $\|{\mathcal{A}}{\mathcal{A}}^{t}(-\nabla X)^{t}{\mathcal{A}}^{t}\nabla Y_{t}\|_{\dot{B}^{s+1}_{1,1}}$. As a result, we obtain

[TABLE]

5.1.2. Estimate of $\|f_{m}^{\prime}(Y;X)\|_{\dot{B}^{s}_{1,1}}$, $m=1,2$

In view of (4.7), we have

[TABLE]

Applying (5.3) gives

[TABLE]

While applying (2.48) and (5.1) leads to

[TABLE]

which yields

[TABLE]

Hence it comes out

[TABLE]

It remains to handle the estimates of

[TABLE]

$\bullet$Estimate of $\|\nabla\boldsymbol{p}_{m}\|_{0}.$

We first deduce from (4.1) that

[TABLE]

Due to the assumption (2.41), one has

[TABLE]

so that we infer

[TABLE]

Similarly, we have

[TABLE]

$\bullet$Estimates of $\|\nabla\boldsymbol{p}_{m}\|_{\dot{B}^{s}_{2,1}}$ for $s>0.$

We start with the estimate of $\|\nabla\boldsymbol{p}_{m}\|_{\dot{B}^{\frac{3}{2}}_{2,1}}.$ Indeed by (4.1), one has

[TABLE]

from which, (2.41) and the product law (2.48), we infer

[TABLE]

As a result, by virtue of (2.41), it comes out

[TABLE]

In general, for $s>0$, we deduce from (4.1) that

[TABLE]

from which and (2.41), we infer

[TABLE]

Yet it follows from the product law (5.2) that

[TABLE]

which together with (2.41) and (5.8) ensures that

[TABLE]

Exactly along the same line, we have

[TABLE]

$\bullet$Estimate of $\|\nabla\boldsymbol{p}_{m}^{\prime}(Y;X)\|_{0}.$

We first deduce from (4.8) that

[TABLE]

We observe that

[TABLE]

Yet it follows by a similar derivation of (5.6) that

[TABLE]

so that

[TABLE]

Let us handle the remaining terms in (5.12). Indeed with the assumption (2.41), a direct calculation shows that

[TABLE]

Substituting the above estimates into (5.12) leads to

[TABLE]

The same procedure gives rise to

[TABLE]

$\bullet$Estimate of $\|\nabla\boldsymbol{p}_{m}^{\prime}(Y;X)\|_{\dot{B}^{s}_{1,1}}$ with $s>0$

For any $s>0,$ we deduce from (4.8) that

[TABLE]

It follows from (5.1) that

[TABLE]

And applying (5.2) and (5.1) gives

[TABLE]

and

[TABLE]

Exactly along the same line, we find

[TABLE]

and

[TABLE]

Substituting the above estimates into (5.17) and using the estimates (5.6), (5.8), (5.9) and (5.14), we obtain

[TABLE]

The same procedure gives rise to

[TABLE]

Inserting the estimates (5.6), (5.8), (5.9), (5.14) and (5.18) into (5.5) for $m=1$ yields

[TABLE]

While by inserting the estimates (5.7), (5.10), (5.11), (5.15), (5.16) and (5.19) into (5.5) for $m=2,$ we obtain

[TABLE]

Let us now complete the proof of Proposition 2.3.

Proof of Proposition 2.3.

Note that for $s_{1}<s<s_{2}$ and $\alpha=\frac{s_{2}-s}{s_{2}-s_{1}},$ one has

[TABLE]

In particular, for $s>0$, this yields

[TABLE]

On the other hand, recall (2.39), we deduce from (5.4) that

[TABLE]

which together with (5.22) ensures (2.42). Along the same line, we deduce (2.43) and (2.44) from (5.20) and (5.21) respectively. This completes the proof of Proposition 2.3. ∎

5.2. The estimate of $\||D|^{-1}f^{\prime}(Y;X)\|_{N}$

The purpose of this subsection is to prove Proposition 2.4. We split its proof into the following steps:

5.2.1. The estimate of $\||D|^{-1}f_{0}^{\prime}(Y;X)\|_{N}$

We first deduce from (4.6) that

[TABLE]

Applying Moser type inequality and using (2.41) gives

[TABLE]

Substituting the above estimates into (5.23) leads to (2.45).

5.2.2. $L^{2}$-estimates for $f_{m}^{\prime}(Y;X)$

We shall divide the proof of (2.46) and (2.47) into the following steps:

(i) Estimates of $\||D|^{-1}f_{m}^{\prime}(Y;X)\|_{0}.$

By virtue of (4.7), we have

[TABLE]

It follows from the law of product in Besov spaces and the imbedding: $L^{\frac{6}{5}}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3})\hookrightarrow\dot{H}^{-1}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3}),$ that

[TABLE]

from which (5.13) and (5.15), we infer

[TABLE]

Similarly, we get, by applying the law of product in Besov spaces, that

[TABLE]

To deal with the estimate of $\|\nabla\boldsymbol{p}_{m}^{\prime}(Y;X)\|_{\dot{H}^{-1}},$ we deduce from (4.8) and a similar derivation of (5.25) that

[TABLE]

which together with (2.41) ensures that

[TABLE]

Exactly along the same line, we deduce from (4.9) that

[TABLE]

Inserting the above estimates into (5.24) leads to

[TABLE]

(ii) Estimates of $\|f_{m}^{\prime}(Y;X)\|_{\dot{H}^{k}}$ for $k\geq 0$ By (4.7) we have

[TABLE]

$\bullet$ Estimates for $\|{\mathcal{A}}^{t}(\nabla X)^{t}{\mathcal{A}}^{t}\nabla\boldsymbol{p}_{m}\|_{\dot{H}^{k}}.$

We get, by applying Moser type inequalities, that

[TABLE]

Here and in all that follows, we always denote $D^{k}=\sum_{|\alpha|=k}\partial^{\alpha}.$

In view of (4.1), applying Moser type inequalities yields

[TABLE]

from which and (2.41), we infer

[TABLE]

While it is easy to observe that

[TABLE]

which together with (5.13) ensures that

[TABLE]

and hence, we obtain

[TABLE]

By the same procedure, we can show that

[TABLE]

and

[TABLE]

Furthermore, there hold

[TABLE]

$\bullet$ Estimates of $\|{\mathcal{A}}^{t}\nabla\left(\boldsymbol{p}_{m}^{\prime}(Y;X)\right)\|_{\dot{H}^{k}}.$

Applying Moser type inequality gives

[TABLE]

Yet in view of (4.8), we have

[TABLE]

It follows from a similar derivation of (5.31) that

[TABLE]

And we get, by applying Moser type inequality, that

[TABLE]

and

[TABLE]

and

[TABLE]

and finally

[TABLE]

As a result, by virtue of (5.14), it comes out

[TABLE]

Substituting the above estimate and (5.14) into (5.35) for $m=1$ shows that $\|{\mathcal{A}}^{t}\nabla\left(\boldsymbol{p}_{1}^{\prime}(Y;X)\right)\|_{\dot{H}^{k}}$ shares the same estimate as above.

Similarly, we can show that

[TABLE]

Substituting the above estimate and (5.16) into (5.35) for $m=2$ shows that $\|{\mathcal{A}}^{t}\nabla\left(\boldsymbol{p}_{2}^{\prime}(Y;X)\right)\|_{\dot{H}^{k}}$ shares the same estimate as above.

Let us now turn to the estimates of $\|f_{1}^{\prime}(Y;X)\|_{\dot{H}^{k}}$ and $\|f_{2}^{\prime}(Y;X)\|_{\dot{H}^{k}}.$ As a matter of fact, by inserting (5.31) and (5.36) into (5.30) for $m=1,$ we achieve

[TABLE]

Similarly by inserting (5.32) and (5.37) into (5.30) for $m=2,$ we obtain

[TABLE]

Now we are in a position to complete the proof of Proposition 2.4.

Proof of Proposition 2.4.

It remains to prove (2.46) and (2.47). Indeed, combining (5.28) with (5.38), we obtain (2.46). While combining (5.29) with (5.39) leads to (2.47). This completes the proof of Proposition 2.4. ∎

6. Energy estimates for the linearized equation

The goal of this section is to present the proof of Theorem 2.3.

6.1. First-order energy estimates

Let us first carry out the estimate of ${\mathcal{E}}_{0}(t)$ (2.50).

$\bullet$The estimate of $\|\nabla X\|_{0}.$

We first get, by taking $L^{2}$ inner product of (2.40) with $X$, that

[TABLE]

And it follows by taking $L^{2}$ inner product of (2.40) with $(-\Delta)^{-1}X_{t}$ that

[TABLE]

Summing up the above equality with $\frac{1}{4}\times$(6.1) yields

[TABLE]

It is easy to observe that

[TABLE]

and

[TABLE]

and

[TABLE]

Hence in view of (4.6), under the assumption of (2.41), by taking $\delta_{1}$ so small that $C\delta_{1}\leq\frac{1}{4},$ we obtain

[TABLE]

While by virtue of (5.28) and (5.29), we have

[TABLE]

Inserting (6.3) and (6.4) into (6.2) gives rise to

[TABLE]

by applying the assumption (2.49).

On the other hand, since ${\mathcal{A}}{\mathcal{A}}^{t}$ is a positive definite matrix $(|{\mathcal{A}}{\mathcal{A}}^{t}-Id|_{0}\leq C\delta_{1}\leq\frac{1}{4})$, it holds that

[TABLE]

so that one has

[TABLE]

$\bullet$The estimate of $\|X_{t}\|_{0}.$

Multiplying (2.40) by $X_{t}$ and integrating the resulting equality over $\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3}$, we get

[TABLE]

In view of (4.6), we infer

[TABLE]

While it follows from (5.28) to (5.29) that

[TABLE]

As a result, thanks to the assumption (2.49), it comes out

[TABLE]

$\bullet$The estimate of $\|\nabla X_{t}\|_{0}.$

By taking $L^{2}$ inner product of (2.40) with $-\Delta X_{t}$ gives

[TABLE]

It is easy to observe from (2.41) and (4.6) that

[TABLE]

Then by substituting the estimates (6.9), (5.38) and (5.39) into (6.8) and using the assumptions (2.41) and (2.49), we obtain

[TABLE]

which implies

[TABLE]

$\bullet$ The estimate of $\|\nabla X\|_{\dot{H}^{1}}.$

In this step, we shall use the equivalent formulation, (2.53), of (2.40). We first get, by taking $L^{2}$ inner product of (2.53) with -\nabla\cdot\bigl{(}{\mathcal{A}}{\mathcal{A}}^{t}\nabla X\bigr{)}, that

[TABLE]

By using integration by parts, one has

[TABLE]

Since ${\mathcal{A}}{\mathcal{A}}^{t}$ is a positive definitive matrix, we infer

[TABLE]

Yet under the assumption of (2.41), it is easy to observe from (2.54) that

[TABLE]

Whereas it follows from (5.38), (5.39) that

[TABLE]

Inserting the above estimates into (6.11) yields

[TABLE]

Let us denote

[TABLE]

Then by summing up the inequalities (6.5), (6.7), (6.10) and $\frac{1}{48}\times$(6.12), we obtain,

[TABLE]

Notice that

[TABLE]

and

[TABLE]

so that we deduce from from (6.6) and (6.13) that

[TABLE]

Hence for any $\varepsilon>0$, we deduce from (6.14) that

[TABLE]

Applying Gronwall’s inequality yields for any $\varepsilon>0$ that

[TABLE]

which together with (6.15) ensures the first inequality of (2.50).

6.2. Higher-order energy estimates.

In this subsection, we shall derive the estimates for

[TABLE]

We first get, by taking the $\dot{H}^{k+1}$-inner product of (2.40) with $X_{t},$ that

[TABLE]

which implies

[TABLE]

Yet in view of (4.6), it follows from Moser type inequality that

[TABLE]

from which, (5.38), (5.39) and the assumption (2.49), we infer

[TABLE]

Inserting (6.20) into (6.18), and using the assumption (2.41) so that $|\nabla Y|_{0}\leq\delta_{1}$, we deduce that

[TABLE]

Secondly, by taking the $\dot{H}^{k}$-inner product of (2.53) with -\nabla\cdot\bigl{(}{\mathcal{A}}{\mathcal{A}}^{t}\nabla X\bigr{)}, we obtain

[TABLE]

By using integration by parts, one has

[TABLE]

and

[TABLE]

so that it comes out

[TABLE]

Similarly, again by using integration by parts, one has

[TABLE]

Since $|{\mathcal{A}}{\mathcal{A}}^{t}-Id|_{0}\leq C\delta_{1}\leq\frac{1}{4}$ due to (2.41), applying Moser type inequality gives

[TABLE]

and

[TABLE]

so that there holds

[TABLE]

Inserting the above estimates into (6.22) gives rise to

[TABLE]

We remark that

[TABLE]

Moreover, in view of (2.54), we have

[TABLE]

which together with (5.38) and (5.39) ensures that

[TABLE]

Inserting the above inequalities to (6.23) yields

[TABLE]

Let us introduce

[TABLE]

Then it follows from (6.24) that

[TABLE]

with $\dot{E}_{k+1}(t)$ being given by (6.17).

Hence by summing up (6.21) and (6.27), and then integrating the resulting inequality over $[0,t]$ and using (6.29), we achieve

[TABLE]

where ${\mathcal{E}}_{0}(t)$ is given by (2.50) and $\gamma_{\varepsilon,k+1}(Y)$ by (2.52). Applying Gronwall’s inequality to (6.30) and using (2.50), we obtain

[TABLE]

from which and (6.30), we infer

[TABLE]

Summing up the above inequality with respect to $k$ leads to (2.51). This completes the proof of Theorem 2.3.

Now let us turn to the proof of Corollary 2.1.

Proof of Corollary 2.1.

By summing up (6.7) and (6.10), and then multiplying the resulting inequality by $\langle{t}\rangle$ and integrating the above inequality over $[0,t]$, we find

[TABLE]

(2.55) then follows from (2.50).

Similarly, we get, by multiplying (6.21) by $\langle{t}\rangle$, then integrating the inequality over $[0,t]$ and taking the square root of the resulting inequality, that

[TABLE]

(2.56) then follows from (2.51) and (2.55), and this completes the proof of Corollary 2.1. ∎

7. Energy decay for $\nabla X_{t}$

The main idea to prove Proposition 2.5 is to use the following proposition:

Proposition 7.1**.**

Let $X$ be a smooth enough solution of

[TABLE]

on $[0,T].$ Then under the assumption that

[TABLE]

we have, for any $t\in[0,T]$ and any $\varepsilon>0,$ that

[TABLE]

Moreover, we have for $k\in\mathop{\mathbb{N}\kern 0.0pt}\nolimits$,

[TABLE]

Admitting this proposition for the time being, we present the proof of Proposition 2.5.

Proof of Proposition 2.5.

In our situation (2.40),

[TABLE]

We infer from (5.23), (2.46), (2.47) that for $k\geq 0$,

[TABLE]

where $\gamma_{\varepsilon,k+2}(Y)$ is given in (2.52). Proposition 2.5 then follows from Proposition 7.1, (7.5), Corollary 2.1 and the fact that $\|D^{k}\nabla Y\|_{L_{t}^{\infty}(\dot{B}^{\frac{3}{2}}_{2,1})}\leq\|\nabla Y\|_{0,k+2}$. ∎

In order to prove Proposition 7.1, we need to exploit the tool of anisotropic Littlewood-Paley analysis. Similar to the dyadic operators $\Delta_{j},$ and $S_{j}$ given by Definition 2.1, let us recall the dyadic operators in the $x_{3}$ variable

[TABLE]

Let us also recall the following anisotropic type Besov norm from [24, 23]:

Definition 7.1**.**

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

[TABLE]

In particular, when $r=2,$ we denote \|a\|_{\dot{H}^{s_{1},s_{2}}}\buildrel\hbox{\footnotesize def}\over{=}\|a\|_{{\mathcal{B}}^{s_{1},s_{2}}_{2}}=\bigl{\|}|D|^{s_{1}}|D_{x_{3}}|^{s_{2}}a\bigr{\|}_{L^{2}}. **

In order to obtain a better description of the regularizing effect for the transport-diffusion equation, we will use anisotropic version of Chemin-Lerner type norm (see [3] for instance).

Definition 7.2**.**

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

[TABLE]

with the usual change if $r=\infty$.**

For the convenience of the readers, we recall the following Bernstein type lemma from [3, 10, 26]:

Lemma 7.1**.**

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

If the support of $\widehat{a}$ is included in $2^{k}\mathfrak{B}_{{\rm h}}$, then

[TABLE]

If the support of $\widehat{a}$ is included in $2^{\ell}\mathfrak{B}_{{\rm v}}$, then

[TABLE]

If the support of $\widehat{a}$ is included in $2^{k}{\mathcal{C}}_{{\rm h}}$, then

[TABLE]

If the support of $\widehat{a}$ is included in $2^{\ell}{\mathcal{C}}_{{\rm v}}$, then

[TABLE]

Let us now turn to the proof of Proposition 7.1.

Proof of Proposition 7.1.

The proof of this lemma is motivated by the proof of Proposition 4.1 of [23, 30]. By applying the operator $\Delta_{j}\Delta_{\ell}^{\rm v}$ to (7.1) and then taking the $L^{2}$ inner product of the resulting equation with $\Delta_{j}\Delta_{\ell}^{\rm v}X_{t},$ we write

[TABLE]

Along the same line, one has

[TABLE]

Notice that

[TABLE]

so that there holds

[TABLE]

By summing up (7.7) with $\frac{1}{4}$ of (7.8), we obtain

[TABLE]

where

[TABLE]

It is easy to observe that

[TABLE]

Now according to the heuristic analysis presented at the beginning of Section 2, we split the frequency analysis into the following two cases:

$\bullet$ When $j\leq\frac{\ell+1}{2}$

In this case, one has

[TABLE]

and Lemma 7.1 implies that

[TABLE]

Hence it follows from (7.9) that

[TABLE]

which in particular implies that

[TABLE]

and

[TABLE]

Now let us turn to the estimate of $\|\Delta_{j}\Delta_{\ell}^{\rm v}f\|_{L^{1}_{t}(L^{2})}.$ Indeed it follows by the law of product in the anisotropic Besov spaces (see Lemma 3.3 of [30]) that

[TABLE]

where we used the fact that $\dot{B}^{\frac{3}{2}}_{2,1}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3})\hookrightarrow{\mathcal{B}}^{1,\frac{1}{2}}_{1}$ (one may check Lemma 3.2 of [24, 30] for details). Hence we obtain

[TABLE]

where $(c_{j,\ell})_{j,\ell\in{\mathop{\mathbb{Z}\kern 0.0pt}\nolimits}^{2}}$ is a generic element of $\ell^{2}({\mathop{\mathbb{Z}\kern 0.0pt}\nolimits}^{2})$ so that $\sum_{j,\ell\in{\mathop{\mathbb{Z}\kern 0.0pt}\nolimits}^{2}}c_{j,\ell}^{2}=1.$

Whereas it follows from Lemma 7.1 and (7.11) that

[TABLE]

By virtue of (7.13), we have

[TABLE]

Along the same line, we have

[TABLE]

While it is easy to observe from Lemma 7.1 that

[TABLE]

and

[TABLE]

Substituting the above estimates into (7.15) leads to

[TABLE]

for all $(j,\ell)$ satisfying $j\leq\frac{\ell+1}{2}.$

$\bullet$When $j>\frac{\ell+1}{2}$

In this case, we have

[TABLE]

and Lemma 7.1 implies that

[TABLE]

Then we deduce from (7.9) that

[TABLE]

which implies that

[TABLE]

and

[TABLE]

On the other hand, we get, by taking $L^{2}$ inner product of (7.1) with $\Delta_{j}\Delta_{\ell}^{\rm v}X_{t},$ that

[TABLE]

from which, Lemma 7.1, we infer

[TABLE]

so that there hold

[TABLE]

And then we deduce from (7.19) that for $j>\frac{\ell+1}{2}$

[TABLE]

Moreover, in this case, it follows from Lemma 7.1 and (7.18) that

[TABLE]

from which and a similar proof of (7.17), we infer

[TABLE]

Here we used the fact $j\geq\ell-N_{0}$ for some fixed integer $N_{0}$ in the operator $\Delta_{j}\Delta_{\ell}^{\rm v}.$

By virtue of (7.20) and (7.22), we get, by a similar derivation of (7.17) that (7.17) holds for all $(j,\ell)\in{\mathop{\mathbb{Z}\kern 0.0pt}\nolimits}^{2}.$ Furthermore, in view of (7.12)-(7.21), we obtain for all $(j,\ell)\in{\mathop{\mathbb{Z}\kern 0.0pt}\nolimits}^{2},$ that

[TABLE]

Inserting (7.14) into (7.23) gives rise to

[TABLE]

In particular, by taking $\delta_{1}$ to be sufficiently small in (7.2), we conclude that

[TABLE]

Along the same line, we deduce from (7.17) that

[TABLE]

So that by taking $\delta_{1}$ is small enough in (7.2), we obtain

[TABLE]

which leads to (7.3).

The proof of the general estimates (7.4) follows along the same line. Indeed for any $k\geq 1,$ we have

[TABLE]

from which and a similar derivation of (7.24), we inductively infer that

[TABLE]

Hence by applying the interpolation inequality that

[TABLE]

and the assumption (7.2), we obtain

[TABLE]

While it follows from a similar derivation of (7.25) that

[TABLE]

Thus (7.4) follows (7.27) and the argument in (7.26). This completes the proof of Proposition 7.1. ∎

8. Estimates of the source term $f(Y)$

In this section, we shall present the estimates to the nonlinear source term $f(Y)$ determined by (2.25).

$\bullet$The estimate of ${|\kern-1.07639pt|\kern-1.07639pt|f(Y)|\kern-1.07639pt|\kern-1.07639pt|}_{\delta,N}$

Proposition 8.1**.**

Let the functionals $f_{0},f_{1},f_{2}$ be given in (4.3) and the norm $\||\cdot\||_{\delta,N}$ by (2.39). Then under the assumption of (2.41), we have

[TABLE]

Proof.

As in Section 4, we shall deal with the estimate of $f(Y)$ by the norm of the homogeneous Besov space $\dot{B}^{s}_{1,1}$ instead of the one in the homogeneous Sobolev space $\dot{W}^{s,1}.$ Indeed in view of (4.3), we get, by applying the law of product, (5.1), that for $s>0$,

[TABLE]

(8.1) then follows from the above inequality and the interpolation inequality (5.22). Along the same line, we deduce from (4.3) that

[TABLE]

Yet it follows from (4.1) that

[TABLE]

which together with (5.6) implies

[TABLE]

As a result, it comes out

[TABLE]

Similarly, we have

[TABLE]

(8.2) and (8.3) then follow from the above estimates and the interpolation inequality (5.22). This completes the proof of Proposition 8.1. ∎

$\bullet$The estimate of $\||D|^{-1}f(Y)\|_{N+1}$

Proposition 8.2**.**

Under the asme assumptions of Proposition 8.1, we have

[TABLE]

Proof.

In view of (4.3), we get, by applying Moser type inequality, that

[TABLE]

which gives (8.4). While again by (4.3) and the law of product in Besov spaces, one has

[TABLE]

yet it follows from (4.1) that

[TABLE]

from which and the assumption (2.41), we infer

[TABLE]

Similarly, we have

[TABLE]

For $N\geq 0$, we deduce from (4.3) that

[TABLE]

And it follows from (4.1) that

[TABLE]

which together with (2.41) and (5.6) ensures that

[TABLE]

As a result, it comes out

[TABLE]

The same procedure for $f_{2}(Y)$ yields

[TABLE]

(8.5) and (8.6) follow from (8.7)-(8.10). This completes the proof of Proposition 8.2. ∎

9. Estimates of $f^{\prime\prime}(Y;X,W)$

The purpose of this section is to present the related estimates to the second derivatives, $f^{\prime\prime}(Y;X,W),$ of the nonlinear functional $f(Y)$ given by (2.25).

9.1. The estimate of $\||D|^{-1}f^{\prime\prime}(Y;X,W)\|_{N}$

Proposition 9.1**.**

Let $f_{0}^{\prime\prime},f_{1}^{\prime\prime},f_{2}^{\prime\prime}$ be given by (4.13) and (4.14) respectively. Then under the assumption of (2.41), we have

[TABLE]

and

[TABLE]

where the functional $\mathfrak{f}_{3}(\mathfrak{x},\mathfrak{y},\mathfrak{z})$ is given by

[TABLE]

Remark 9.1**.**

We mention that in the above inequalities, it is crucial to estimate the vector, $X,$ by $L^{\infty}$-norm. In Section 10, we shall deal with the estimate of the error term

[TABLE]

where the variable, $(1-S_{p})Y_{p},$ is “small” in the $L^{\infty}$-norm, but only “bounded” in $L^{2}$-norm.

Let us start the proof of Proposition 9.1 by the following lemma:

Lemma 9.1**.**

Under the assumption of (2.41), one has

[TABLE]

and

[TABLE]

Proof.

Indeed the estimates for ${\mathcal{A}}^{\prime}(Y;X),$ $({\mathcal{A}}{\mathcal{A}}^{t}-Id)^{\prime}(Y;X)$ and ${\mathcal{A}}^{\prime\prime}(Y;X,W)$ can be deduced by applying Moser type inequality to (4.4), (4.5) and (4.11). Whereas in view of (4.10), we have

[TABLE]

Thus the estimate for $({\mathcal{A}}{\mathcal{A}}^{t}-Id)^{\prime\prime}(Y;X,W)$ follows. ∎

Proof of Proposition 9.1.

We divide the proof of this proposition into the following steps.

$\bullet$The estimate of $\||D|^{-1}f_{0}^{\prime\prime}(Y;X,W)\|_{N}$

We first deduce from Moser type inequality and Lemma 9.1 that

[TABLE]

and

[TABLE]

and

[TABLE]

Hence thanks to (4.13), we obtain (9.1).

Next, we shall only present the estimates for $f_{1}^{\prime\prime}$, the one for $f_{2}^{\prime\prime}$ follows along the same line. According to (4.14), we write

[TABLE]

$\bullet$The estimate of $\|f_{1}^{\prime\prime}(Y;X,W)\|_{\dot{H}^{-1}}$

(i) $\dot{H}^{-1}$ estimate of $I_{1}$. By virtue of Sobolev embedding: $L^{\frac{6}{5}}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3})\hookrightarrow\dot{H}^{-1}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3}),$ and (5.13), we infer

[TABLE]

(ii) $\dot{H}^{-1}$ estimate of $I_{2}.$ Similar to the estimate of $I_{1},$ we have

[TABLE]

Yet it follows from (4.8) that

[TABLE]

which together with (5.13) and Lemma 9.1 implies that

[TABLE]

As a result, it comes out

[TABLE]

(iii) $\dot{H}^{-1}$ estimate of $I_{3}.$ Note that

[TABLE]

While in view of (4.8), one has

[TABLE]

which together with (5.13) and Lemma 9.1 ensures that

[TABLE]

And we thus obtain

[TABLE]

(iv) $\dot{H}^{-1}$ estimate of $I_{4}.$ We first get, by applying the law of product in Besov spaces, that

[TABLE]

Thanks to (4.15), we get, by applying Sobolev embedding: $L^{\frac{6}{5}}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3})\hookrightarrow\dot{H}^{-1}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3}),$ that

[TABLE]

from which, (2.41), (5.13), Lemma 9.1, (9.6) and (9.7), we infer

[TABLE]

Therefore, we obtain

[TABLE]

By summarizing the estimates of $I_{1},I_{2},I_{3},I_{4}$, we achieve

[TABLE]

$\bullet$The estimate of $\|f_{m}^{\prime\prime}(Y;X,W)\|_{N}$

(i) $H^{N}$ estimate of $I_{1}.$ In view of (9.5), we deduce from (5.13), (5.33) and Lemma 9.1 that

[TABLE]

(ii) $H^{N}$ estimate of $I_{2}.$ By virtue of (5.36), we get, by applying Moser type inequality, that

[TABLE]

(iii) $H^{N}$ estimate of $I_{3}.$ Applying Moser type inequality gives

[TABLE]

Yet it follows from (4.8) that

[TABLE]

from which, (5.33), (9.7), we infer

[TABLE]

Together with (9.7), we deduce that

[TABLE]

(iv) $H^{N}$ estimate of $I_{4}.$ We first get, by applying Moser type inequality, that

[TABLE]

We first deal with the estimate of $\|\nabla\boldsymbol{p}_{1}^{\prime\prime}(Y;X,W)\|_{0}.$ Indeed by (4.15), we have

[TABLE]

from which, (5.36) and (9.7), we deduce that

[TABLE]

In general, along the same line to the proof of (9.11). we get, by using the estimates (5.33), (5.36), (9.7), (9.8), (9.10) and (9.11), that

[TABLE]

The same estimate holds for $\|I_{4}\|_{N}.$ By summing up the estimates of$I_{1},I_{2},I_{3}$ and $I_{4}$, we achieve

[TABLE]

Then (9.2) follows from (9.9) and (9.13). Exactly along the same line, we can prove (9.3), and we omit the details here. This complete the proof of Proposition 9.1. ∎

9.2. The estimate of ${|\kern-1.07639pt|\kern-1.07639pt|f^{\prime\prime}(Y;X,W)|\kern-1.07639pt|\kern-1.07639pt|}_{\delta,N}$

Proposition 9.2**.**

Let $f_{m}^{\prime\prime}$, $m=0,1,2$ be given in (4.13) and (4.14) , the norm ${|\kern-1.07639pt|\kern-1.07639pt|\cdot|\kern-1.07639pt|\kern-1.07639pt|}_{\delta,N}$ be given by (2.39). Then under the assumption of (2.41), we have

[TABLE]

and

[TABLE]

where the functional $\mathfrak{f}_{4}(\mathfrak{x},\mathfrak{y},\mathfrak{z})$ is given by

[TABLE]

Lemma 9.2**.**

Let $s>0.$ Then under the assumption of (2.41), one has

[TABLE]

and

[TABLE]

Proof.

Indeed by virtue of (4.4), we get, by applying the law of product (2.48), that

[TABLE]

Similarly, we deduce estimates for $({\mathcal{A}}{\mathcal{A}}^{t}-Id)^{\prime}(Y;X)$ from (4.5) and complete the proof of (9.17). While it follows from (4.11) and the law of product (5.1) that

[TABLE]

which yields (9.18).

Finally it is easy to observe from the law of product and the previous estimates that

[TABLE]

and

[TABLE]

so that we can deduce (9.19) from (9.4). ∎

Proof of Proposition 9.2.

Again we divide the proof of this proposition into the following steps:

Step 1. Estimate of $f_{0}^{\prime\prime}(Y;X,W)$. We first deduce from the law of product (5.1) and Lemmas 9.1, 9.2 that

[TABLE]

and

[TABLE]

and

[TABLE]

Hence by virtue of (4.13), we conclude that

[TABLE]

Then (9.14) follows from (9.20) and the interpolation inequality (5.22).

Step 2. Estimate of $f_{m}^{\prime\prime}(Y;X,W)$. Again we only present the estimates of $f_{1}^{\prime\prime}(Y;X,W)$. Recall (9.5), we shall split the estimate of $f_{1}^{\prime\prime}(Y;X,W)$ into the following 4 parts:

(i) Estimate of $I_{1}.$ It follows from (5.9) and (9.18) that

[TABLE]

(ii) Estimate of $I_{2}$. We deduce from (5.18) and (5.36) that

[TABLE]

(iii) Estimate of $I_{3}.$ Similar to the estimate of $I_{2},$ we have

[TABLE]

(iv) Estimate of $I_{4}.$ We first deduce from the law of product (5.1) that

[TABLE]

Thanks to (9.11), it remains to handle the estimate of \|\nabla\big{(}\boldsymbol{p}_{m}^{\prime\prime}(Y;X,W)\big{)}\|_{\dot{B}^{s}_{1,1}}. As a matter of fact, thanks to (4.15), by applying the laws of product, (2.48) and (5.1), and using the estimates (5.9), (5.18), (5.36), (9.11) and (9.18), we obtain

[TABLE]

The same estimate holds for $\|I_{4}\|_{\dot{B}^{s}_{1,1}}$.

By summing up the above estimates of $I_{1},I_{2},I_{3},I_{4}$, we arrive at

[TABLE]

Then (9.15) follows from (9.22) and interpolation inequality (5.22). Finally, the proof of (9.16) follows along the same line to that of (9.15). We omit the details here. This completes the proof of Proposition 9.2. ∎

10. The proof of Theorem 2.1

The goal of this section is to prove Theorem 2.1 by using Nash-Moser scheme. The key ingredients are the uniform estimates of the approximate solutions obtained in Propositions 2.6, 2.7 and 2.8, which we will prove by induction in what follows.

10.1. The estimates of $Y_{0}$

Recall that $Y_{0}$ solves the linear equation (2.66). Let $\bar{N}_{0}=N_{0}+6$, for $\eta\in]0,1[$, we choose the initial data $(Y^{(0)},Y^{(1)})$ such that (2.20) holds for $L_{0}=N_{0}+12$. Then we get, by applying (2.32) of Proposition 2.1, that

[TABLE]

Note that

[TABLE]

so that we get, by applying (2.33), (2.34) and (2.35) of Proposition 2.1, that

[TABLE]

By virtue of (10.1) and (10.2), we deduce from Proposition 8.2 that

[TABLE]

and

[TABLE]

Similarly, we deduce from Proposition 8.1 and (10.1), (10.2) that

[TABLE]

10.2. The proof of Proposition 2.7 and Proposition 2.8 from Proposition 2.6

Let us assume that

[TABLE]

we are going to prove Proposition 2.7 and Proposition 2.8.

Proof of Proposition 2.7.

Notice from (2.69) that

[TABLE]

which together with (10.1) and (P2, $j$) with $j\leq p$ ensures that for $\frac{1}{2}\leq k\leq 1$, $0\leq N\leq N_{0}$,

[TABLE]

While for $\hat{k}\buildrel\hbox{\footnotesize def}\over{=}\min(k,1)$, $\hat{N}\buildrel\hbox{\footnotesize def}\over{=}\min(N,N_{0}),$ we observe from the property (S I) of smoothing operator $S_{p+1}$ that

[TABLE]

the first inequalities of (I)(i) and (II)(i) of Proposition 2.7 then follow from (10.7).

Along the same line to proof of (10.7), we have

• •

for $1-\delta\leq k\leq\frac{3}{2}-\delta$, $0\leq N\leq N_{0}$,

[TABLE]

• •

for $0\leq k\leq\frac{1}{2}$, $0\leq N\leq N_{0}$,

[TABLE]

Then other inequalities in (I)(i) and (II)(i) of Proposition 2.7 follows.

(I)(ii) and (II)(ii) of Proposition 2.7 follow from property (S I) of the mollifying operator and the following fact

[TABLE]

which is a direct consequence of (P1,$j$) of Proposition 2.6 for $j\leq p$ and (10.2).

Finally let us prove (III) of Proposition 2.7. Indeed it follows from property (S II) of $S_{p+1}$ that

[TABLE]

While due to (2.77) and (2.78), there hold $\frac{1}{2}-\gamma\geq\bar{\varepsilon}$ and $-\gamma+\bar{\varepsilon}N_{0}\geq\bar{\varepsilon}$, so that we can apply (10.7) to deduce that

[TABLE]

Using (10.7) once again gives rise to

[TABLE]

Interpolating between (10.11), (10.12) and (10.13) leads to

[TABLE]

The other two inequalities in (III) of Proposition 2.7 can be proved by the same procedure. This completes the proof of Proposition 2.7. ∎

Let us now turn to the proof of Proposition 2.8.

Proof of Proposition 2.8.

We shall divide the proof of this proposition by the following steps:

Step 1. The Proof of (IV) of Proposition 2.8. The proof of (IV) will be based on the following lemmas:

Lemma 10.1**.**

Let $e_{p,j}^{\prime},e_{p,j}^{\prime\prime},$ for $j=0,1,2,$ be given by (2.75). Then under the assumption of (10.6), one has

[TABLE]

Lemma 10.2**.**

Under the assumption of Lemma 10.1, one has

[TABLE]

Lemma 10.3**.**

Under the assumption of Lemma 10.1, for $0\leq N\leq N_{0}-6,$ there hold

[TABLE]

We shall postpone the proof of the above lemmas in the Appendix A. It is easy to observe that (IV) (i) follows from Lemma 10.1, (IV) (ii) from Lemma 10.2, and (IV) (iii) from Lemma 10.3.

Step 2. The proof of (V) of Proposition 2.8. Recall (2.74) that

[TABLE]

In the sequel, we shall handle term by term above.

$\bullet$Estimates of $S_{p+1}e_{p}$****

It follows from (IV) of Proposition 2.8 and property (S I) that for $k\geq 0$ and $N\geq 0$

[TABLE]

Notice that the operator $S_{p+1}$ contains a cutoff in the variable $t$ of size $\theta_{p+1}$ so that

[TABLE]

$\bullet$Estimates for $(S_{p+1}-S_{p})E_{p}$****

We first deduce from (IV) (i) of Proposition 2.8 that for $0\leq k\leq\alpha$ and $0\leq N\leq N_{0}-2$

[TABLE]

In particular, due to the choice of parameters (2.77), (2.78), there hold

[TABLE]

we deduce from (10.28) and the property (S II) of $1-S_{p}$ that

[TABLE]

On the other hand, for $k\leq\alpha$, $N\leq N_{0}-2$ with $k+\delta-\gamma-\beta+\bar{\varepsilon}(N+3)\geq\bar{\varepsilon}$, we have

[TABLE]

Interpolating between (10.30) and (10.31), we conclude that

[TABLE]

for $0\leq k\leq\alpha$ and $0\leq N\leq N_{0}-2.$ This together with property (S I) of $S_{p}$ ensures that (10.32) holds for any $k\geq 0$, $N\geq 0$.

Similarly we infer from (IV) (ii) of Proposition 2.8 that for $0\leq k\leq\frac{1}{2}-\delta$, $0\leq N\leq N_{0}-2$,

[TABLE]

Then due to (10.29), we deduce from (10.33) and the property (S II) of $1-S_{p}$ that

[TABLE]

On the other hand, for $k\leq\frac{1}{2}-\delta$, $N\leq N_{0}-2$ such that $k+\delta-\gamma-\beta+\bar{\varepsilon}(N+2)\geq\bar{\varepsilon}$, we get

[TABLE]

Interpolating between the inequalities (10.34) and (10.35), we achieve (10.35) for any $0\leq k\leq\frac{1}{2}-\delta$, $0\leq N\leq N_{0}-2$. This together with the property (S I) of $S_{p}$ ensures that (10.35) holds for any $k\geq 0$ and $N\geq 0$.

It follows from (IV) (iii) of Proposition 2.8 that for $N\leq N_{0}-6$,

[TABLE]

which together with the property (S I) and compact support of mollifying operator ensures that for any $N\geq 0$

[TABLE]

$\bullet$Estimates for $(S_{p+1}-S_{p})f(Y_{0})$****

Recall (10.29), we get, by applying (S II) and (10.3), that

[TABLE]

Whereas for $k\leq\frac{1}{2}$ and $N\leq N_{0}$ with $k-\gamma-\beta+\bar{\varepsilon}(N+3)\geq\bar{\varepsilon}$, we deduce from (10.3) that

[TABLE]

Interpolating the above two inequalities gives rise to

[TABLE]

for all $0\leq k\leq\frac{1}{2},0\leq N\leq N_{0}$. This together with the property (S I) of $S_{p+1}$ ensures that

[TABLE]

for all $k\geq 0$ and $N\geq 0$.

Along the same line, it follows from (10.4) that for $k\geq 0,\ N\geq 0$,

[TABLE]

And it follows from (10.5) that if $-\gamma+\bar{\varepsilon}(N+5)\leq-\bar{\varepsilon}$ (implying $N\leq N_{0}$),

[TABLE]

and if $-\gamma+\bar{\varepsilon}(N+5)\geq\bar{\varepsilon}$, one has

[TABLE]

by using (S I) and the fact that $\bar{\varepsilon}(N_{0}+5)\geq\gamma$. Along with (10.26), (10.27), (10.32), (10.35), (10.36), (10.37), (10.38), we complete the proof of (V).

Step 3. The proof of (VI) of Proposition 2.8.****

In the case when $-\gamma+\bar{\varepsilon}(N+5)\geq\bar{\varepsilon}$, we deduce from (V)(i), (V)(ii), (V)(iii) of Proposition 2.8 that

[TABLE]

provided that

[TABLE]

which are satisfied due to (2.77) and (2.76).

On the other hand, since $-\gamma+6\bar{\varepsilon}\leq-\bar{\varepsilon}$, we deduce from (V)(i), (V)(ii) and (V)(iv) of Proposition 2.8 that

[TABLE]

due to (10.39) and $\frac{1}{2}-\gamma\geq\bar{\varepsilon}$. This finishes the proof of (VI) of Proposition 2.8 and hence the whole Proposition 2.8. ∎

10.3. The proof of Proposition 2.6 from Proposition 2.7 and Proposition 2.8

Let us assume in this subsection that

[TABLE]

we are going to prove (P1, $p+1$), (P2, $p+1$), (P3, $p+1$), that is, Proposition 2.6 is valid for $p+1.$

Proof of Proposition 2.6.

We shall divide its proof into the following steps:

Step 1. The proof of (P3, $p+1)$ of Proposition 2.6.

(P3, $p+1$) is a direct consequence of (10.7), (10.8), (10.9), (10.10) and the choices of parameters (see (2.77) and (2.76))

[TABLE]

Step 2. The proof of (P1, $p+1$) of Proposition 2.6.

Recall that $X=X_{p+1}$ solves

[TABLE]

Due to (P3, $p+1$), the hypotheses of Theorem 2.3 and (2.58) are satisfied, so that we can apply the energy estimate (2.59) to the system (10.41). When $N\geq 0$ with $-\gamma+\bar{\varepsilon}(N+1)\geq\bar{\varepsilon}$ and $-\beta+\bar{\varepsilon}N\geq\bar{\varepsilon}$, we deduce from (I) (i), (ii) of Proposition 2.7 that

[TABLE]

Then in this case, we get, by applying the energy estimate (2.59) to the system (10.41) and using (V) (i), (V) (ii) of Proposition 2.8, that

[TABLE]

provided that $\gamma\geq\delta+\varepsilon+3\bar{\varepsilon}$ which is satisfied due to (2.77), (2.76). Along the same line, we have

[TABLE]

By interpolating the inequalities (10.42) and (10.43), we achieve (P1, $p+1$) for $N\geq 0$.

Step 3. The proof of (P2, $p+1$) of Proposition 2.6.

Notice that by definition $S_{p+1}Y_{p+1}=0$ and $g_{p+1}=0$ for $t\geq\theta_{p+1}$. In order to apply Proposition 2.2 to the equation (10.41), it remains to estimate

[TABLE]

given by (2.38).

$\bullet$The estimate of ${|\kern-1.07639pt|\kern-1.07639pt|f^{\prime}(S_{p+1}Y_{p+1};X_{p+1})|\kern-1.07639pt|\kern-1.07639pt|}_{L_{t}^{1}(\delta,N)}$****

It follows from (2.43) that

[TABLE]

from which, and (P1, $p+1$), (II) of Proposition 2.7 and the fact that $\beta\geq 6\bar{\varepsilon}$, we infer

[TABLE]

While for $-\beta+\bar{\varepsilon}(N+5)\geq\bar{\varepsilon}$, it follows from (I) (II) of proposition 2.7 and (P1, $p+1$) that

[TABLE]

$f_{2}^{\prime}(S_{p+1}Y_{p+1};X_{p+1})$** can be handled along the same line.**

For $f_{0}^{\prime}(S_{p+1}Y_{p+1};X_{p+1})$, we deduce from (2.42) that

[TABLE]

Notice that $f_{0}^{\prime}(S_{p+1}Y_{p+1};X_{p+1})$ is supported in $\{0\leq t\leq\theta_{p+1}\}$ so that

[TABLE]

which together with (P1, $p+1$) and (II) of Proposition 2.7 ensures that

[TABLE]

$\bullet$The estimate of $\|\langle t\rangle^{\frac{1}{2}}|D|^{-1}f^{\prime}(S_{p+1}Y_{p+1};X_{p+1})\|_{L_{t}^{2}(H^{N+1})}$****

It follows from (2.46) that

[TABLE]

which together with (II) of Proposition 2.7 and (P1, $p+1$) ensures that

[TABLE]

For $N$ satisfying $-\gamma+\bar{\varepsilon}(N+1)\geq\bar{\varepsilon}$, we deduce from (I) of Proposition 2.7 and (P1, $p+1$) that

[TABLE]

$f_{2}^{\prime}(S_{p+1}Y_{p+1};X_{p+1})$** can be treated similarly.**

For $f_{0}^{\prime}(S_{p+1}Y_{p+1};X_{p+1})$, by virtue of (2.45), we get

[TABLE]

As a result, it comes out

[TABLE]

$\bullet$The Estimate of $\||D|^{-1}f^{\prime}(S_{p+1}Y_{p+1};X_{p+1})\|_{\frac{3}{2}-\delta,N+1}$****

By virtue of (2.46), we have

[TABLE]

Noticing from (2.77) that $\frac{1}{4}-\gamma\geq\bar{\varepsilon},$ so that we get, by applying (II) (i) of Proposition 2.7, that

[TABLE]

As a result, it comes out

[TABLE]

provided that $\frac{1}{3}\gamma+\frac{2}{3}\beta\geq\frac{1}{3}\bar{\varepsilon}$, which is the case due to (2.77) and (2.76).

For $N$ with $-\gamma+\bar{\varepsilon}(N+1)\geq\bar{\varepsilon}$, we deduce from (I) (i) of Corollary 2.7 that

[TABLE]

[TABLE]

which together with (P1, $p+1$) ensures that

[TABLE]

Similar estimates as above holds for $f_{2}^{\prime}.$

To deal with the term $f_{0}^{\prime}(S_{p+1}Y_{p+1};X_{p+1})$, we get, by applying (2.45), that

[TABLE]

Then along the same line to proof of (10.48) and (10.49), we can show that

[TABLE]

and for $N$ with $-\gamma+\bar{\varepsilon}(N+1)\geq\bar{\varepsilon}$, there holds

[TABLE]

**Moreover, we can prove in the same way that **

[TABLE]

Recall (2.38), we get, by summarizing the estimates (10.44), (10.46) and (10.50) that

[TABLE]

provided that

[TABLE]

which is the case here due to (2.77) and (2.76).

While due to (10.53), $-\beta+\bar{\varepsilon}(N_{0}+5)\geq\bar{\varepsilon}$ and $-\gamma+\bar{\varepsilon}(N_{0}+2)\geq\bar{\varepsilon},$ by summarizing the estimates (10.45), (10.47) and (10.51), we achieve

[TABLE]

Now we apply Proposition 2.2 and (VI) of Proposition 2.8 to (10.41), to get

[TABLE]

and

[TABLE]

Interpolating the above two inequalities gives for all $0\leq N\leq N_{0}$,

[TABLE]

Whereas it follows from Sobolev embedding Theorem and (P1, $p+1$) that for any $0\leq N\leq N_{0}$,

[TABLE]

provided that $\beta\geq\gamma+2\bar{\varepsilon}$, which is satisfied due to (2.77).

By interpolating the inequalities (10.54) and (10.55), we arrive at (P2, $p+1$). This completes the proof of Proposition 2.6 for $p+1.$ ∎

10.4. The proof of Theorem 2.1

The goal of this subsection is to prove the convergence of the approximate solutions $\{Y_{p}\}$ constructed via (2.69) in some appropriate norms, which in particular ensures Theorem 2.1.

Proof of Theorem 2.1.

We infer from (2.70), (10.52), (P1) of Proposition 2.6 and (V) of Proposition 2.8, that

[TABLE]

Interpolating the above two inequalities leads to

[TABLE]

Due to the choices of the parameters in (2.77) and (2.76), it follows from (P2) of Proposition 2.6 that

[TABLE]

Similarly, let us take $N_{0}=[1/2\bar{\varepsilon}]+1$ and $N_{1}\buildrel\hbox{\footnotesize def}\over{=}[N_{0}/2]$, we deduce from (P2) of Proposition 2.6 and (10.56) that

[TABLE]

This ensures the existence of $Y\in C^{2}([0,+\infty);C^{N_{1}-4}(\mathop{\mathbb{R}\kern 0.0pt}\nolimits^{3}))$ such that

[TABLE]

and

[TABLE]

which ensures (2.21) and (2.22).

Next we show that $Y$ is the solution to (2.65). As a matter of fact, we first observe from (2.72) and (2.73) that

[TABLE]

which implies

[TABLE]

from which, (10.34), (10.38) and (IV) of Proposition 2.8, we infer

[TABLE]

Next, we show that $\Phi(Y_{p+1})\to\Phi(Y)$ as $p\to+\infty$ in the norm $\|\cdot\|_{1,0}$. Indeed let us denote $\widetilde{\Box}\buildrel\hbox{\footnotesize def}\over{=}\partial_{t}^{2}-\Delta\partial_{t}-\partial_{3}^{2},$ then one has

[TABLE]

Using a Taylor formula, applying (2.45), (2.46), (2.47) and using (10.58), (10.59), we have

[TABLE]

On the other hand, recall from (2.70) that

[TABLE]

then we get, by applying (P1) of Proposition 2.6, (II) of Proposition 2.7 and (V)(ii) of Proposition 2.8, that

[TABLE]

Consequently, we achieve

[TABLE]

We then deduce from (10.61) and (10.62) that

[TABLE]

which together with (10.60) implies $\Phi(Y)=0$. Finally, for each $p$, we have

[TABLE]

therefore,

[TABLE]

and thus $Y$ is the desired classical solution to (2.65). This ends the proof of Theorem 2.1. ∎

Appendix A The proof of Lemmas 10.1, 10.2 and 10.3

The goal of this appendix is to present the proof of Lemmas 10.1, 10.2 and 10.3. Notice that the estimates for $e_{p,2}^{\prime}$, $e_{p,2}^{\prime\prime}$ are the same as (even better than) those for $e_{p,1}^{\prime}$, $e_{p,1}^{\prime\prime}$, so that we only preform the estimates for the latter in what follows.

A.1. The proof of Lemma 10.1

We divide the proof of this lemma by the following steps:

$\bullet$The proof of (10.14)**

In view of (2.75), we get, by applying (9.2) (with $Y\simeq Y_{p}+Y_{p+1}$, $X=W=X_{p}$), that for $N\geq 0$,

[TABLE]

Similar estimate holds for $\|\langle t\rangle|D|^{-1}e_{p,1}^{\prime\prime}\|_{L_{t}^{2}(H^{N+1})}$ with $|\partial_{3}X_{p}|_{\frac{1}{2},l}$ above being replaced by $|\partial_{3}X_{p}|_{1,l}$ and $|\nabla X_{p}|_{0,l}$ by $|\nabla X_{p}|_{\frac{1}{2},l}.$

It follows from (10.7), (10.9) and (10.10) that

[TABLE]

As a result, applying (P1, $p$) and (P2, $p$), it comes out

[TABLE]

Interpolating between the above two inequalities gives rise to

[TABLE]

While for $0\leq N\leq N_{0}-1$ such that $-\gamma+\bar{\varepsilon}(N+1)\geq\bar{\varepsilon}$ and $-\beta+\bar{\varepsilon}N\geq\bar{\varepsilon}$, we deduce from (10.7), (10.9) and (10.10) that

[TABLE]

Therefore for such $N$, there hold

[TABLE]

Interpolating the above two inequalities, we obtain for $0\leq k\leq\frac{1}{2}$, $N\leq N_{0}-1$ such that $-\gamma+\bar{\varepsilon}(N+1)\geq\bar{\varepsilon}$ and $-\beta+\bar{\varepsilon}N\geq\bar{\varepsilon}$,

[TABLE]

Interpolating between (A.3) and (A.5) leads to (10.14).

$\bullet$The proof of (10.15)**

In order to do so, we get by applying (9.2) (with $Y\simeq Y_{p}$, $X=(1-S_{p})Y_{p}$, $W=X_{p}$) that for $N\geq 0$,

[TABLE]

Similar estimate for $\|\langle t\rangle|D|^{-1}e_{p,1}^{\prime}\|_{L_{t}^{2}(H^{N+1})}$ holds with $|(1-S_{p})\partial_{3}Y_{p}|_{\frac{1}{2},l}$ above being replaced by $|(1-S_{p})\partial_{3}Y_{p}|_{1,l}$ and $|(1-S_{p})\nabla Y_{p}|_{\frac{1}{2},l}$ by $|(1-S_{p})\nabla Y_{p}|_{0,l}.$

So that by virtue of (A.1), (A.2) and (III) of Proposition 2.7, we infer that

[TABLE]

Interpolating the above two inequalities, we obtain

[TABLE]

Note that for $N\leq N_{0}-1$ such that $-\beta+\bar{\varepsilon}N\geq\bar{\varepsilon}$ and $-\gamma+\bar{\varepsilon}(N+1)\geq\bar{\varepsilon}$, (A.4) holds. And hence we have

[TABLE]

Interpolating the above inequalities gives for such $N$ and $0\leq k\leq\frac{1}{2}$ that

[TABLE]

Interpolating between (A.6) and (A.7) leads to (10.15).

$\bullet$The proof of (10.16)**

It follows from (9.1) and (2.75) that

[TABLE]

Recall that $\alpha=\frac{1}{2}-\delta-\bar{\varepsilon}<\frac{1}{2}-\delta,$ similar estimate for $\|\langle t\rangle^{\frac{1}{2}+\alpha}|D|^{-1}e_{p,0}^{\prime\prime}\|_{L_{t}^{2}(H^{N+1})}$ holds with $|\nabla X_{p}|_{0,l}$ above replaced by $|\nabla X_{p}|_{\alpha,l}$ and $|\partial_{t}X_{p}|_{1+\bar{\varepsilon},l}$ above by $|\partial_{t}X_{p}|_{\frac{3}{2}-\delta,l}.$

Note from (2.76) that $\gamma\geq\delta+4\bar{\varepsilon}$, so that we deduce from (10.8) that,

[TABLE]

which implies

[TABLE]

Interpolating between the above two inequalities yields

[TABLE]

For $N\leq N_{0}-2$ satisfying $-\gamma+\bar{\varepsilon}(N+1)\geq\bar{\varepsilon}$, we deduce from (10.8) and (10.9) that

[TABLE]

so that for such $N,$ we have

[TABLE]

By interpolating between the above inequalities, we achieve for such $N$ and $0\leq k\leq\alpha$ that

[TABLE]

Interpolating between (A.9) and (A.11) leads to (10.16).

$\bullet$The proof of (10.17)**

Again in view of (2.75), we deduce from (9.1) that

[TABLE]

A similar estimate holds for $\|\langle t\rangle^{\frac{1}{2}+\alpha}|D|^{-1}e_{p,0}^{\prime}\|_{L_{t}^{2}(H^{N+1})}$ with $|(1-S_{p})\nabla Y_{p}|_{0,l}$ above being replaced by $|(1-S_{p})\nabla Y_{p}|_{\alpha,l}$ and $|(1-S_{p})\partial_{t}Y_{p}|_{1+\bar{\varepsilon},l}$ above by $|(1-S_{p})\partial_{t}Y_{p}|_{\frac{3}{2}-\delta,l}.$

Hence by virtue of (A.8), we have

[TABLE]

Interpolating between the above inequalities gives

[TABLE]

Whereas for $N\leq N_{0}-2$ such that $-\gamma+\bar{\varepsilon}(N+1)\geq\bar{\varepsilon}$, we infer from (A.10) that

[TABLE]

Interpolating between the above inequalities, we achieve for such $N$ and $0\leq k\leq\alpha$ that

[TABLE]

Interpolating between (A.12) and (A.13) leads to (10.17). The proof of Lemma 10.1 is complete.

A.2. The proof of Lemma 10.2

As in the previous lemma, we shall divide the proof of this lemma into the following steps:

$\bullet$The proof of (10.18)**

Thanks to (2.75), we get, by applying (9.2) that for $N\geq 0$

[TABLE]

A similar estimate holds for $\||D|^{-1}e_{p,1}^{\prime\prime}\|_{1,N+1}$ with $|\partial_{3}X_{p}|_{1,l}$ and $|\nabla X_{p}|_{\frac{1}{2},l}$ above being replaced by $|\partial_{3}X|_{\frac{1}{2},l}$ and $|\nabla X_{p}|_{0,l}$ respectively.

Hence it follows from (A.1), (P1, $p$) and (P2, $p$) that

[TABLE]

Interpolating the above two inequalities yields

[TABLE]

Whereas for $N\leq N_{0}-1$ with $-\gamma+\bar{\varepsilon}(N+1)\geq\bar{\varepsilon}$ and $-\beta+\bar{\varepsilon}N\geq\bar{\varepsilon}$, (A.4) holds, we infer that

[TABLE]

Interpolating the above two inequalities leads to

[TABLE]

for $0\leq k\leq\frac{1}{2}$, and $N\leq N_{0}-1$ with $-\gamma+\bar{\varepsilon}(N+1)\geq\bar{\varepsilon}$ and $-\beta+\bar{\varepsilon}N\geq\bar{\varepsilon}$.

Interpolating between (A.14) and (A.15) gives rise to (10.18).

$\bullet$The proof of (10.19)**

Applying (9.2) to $e_{p,1}^{\prime}$ determined by (2.75) gives that for $N\geq 0$,

[TABLE]

A similar estimate holds for $\||D|^{-1}e_{p,1}^{\prime}\|_{1,N+1}$ with $|(1-S_{p})\partial_{3}X_{p}|_{1,l}$ and $|(1-S_{p})\nabla X_{p}|_{\frac{1}{2},l}$ above being replaced by $|(1-S_{p})\partial_{3}Y_{p}|_{\frac{1}{2},l}$ and $|(1-S_{p})\nabla X_{p}|_{0,l}$ respectively.

Hence we deduce from (A.1) that

[TABLE]

Interpolating the above two inequalities yields

[TABLE]

For $N\leq N_{0}-1$ satisfying $-\beta+\bar{\varepsilon}N\geq\bar{\varepsilon}$ and $-\gamma+\bar{\varepsilon}(N+1)\geq\bar{\varepsilon}$, (A.4) holds, so that we infer that

[TABLE]

Interpolating the above inequalities leads to

[TABLE]

for $0\leq k\leq\frac{1}{2}$, $N\leq N_{0}-1$ such that $-\beta+\bar{\varepsilon}N\geq\bar{\varepsilon}$ and $-\gamma+\bar{\varepsilon}(N+1)\geq\bar{\varepsilon}$.

We then conclude the proof of (10.19) by interpolating between (A.16) and (A.17).

$\bullet$The proof of (10.20)**

Applying (9.1) $e_{p,0}^{\prime\prime},$ which is determined by (2.75) gives that for $N\geq 0$,

[TABLE]

An similar estimate holds for $\||D|^{-1}e_{p,0}^{\prime\prime}\|_{\frac{3}{2}-\delta,N+1}$ with $|\nabla X_{p}|_{0,l}$ and $|\partial_{t}X_{p}|_{1,l}$ being replaced by $|\nabla X_{p}|_{\frac{1}{2}-\delta,l}$ and $|\partial_{t}X_{p}|_{\frac{3}{2}-\delta,l}$ respectively.

Therefore,

[TABLE]

By interpolating between the above two inequalities, we obtain

[TABLE]

For $N\leq N_{0}-2$ with $-\gamma+\bar{\varepsilon}(N+1)\geq\bar{\varepsilon}$, we get, by applying (A.10), that

[TABLE]

Interpolating the above two inequalities, we obtain

[TABLE]

for $0\leq k\leq\frac{1}{2}-\delta$, $N\leq N_{0}-2$ satisfying $-\gamma+\bar{\varepsilon}(N+1)\geq\bar{\varepsilon}.$

By interpolating the inequalities (A.18) and (A.19), we conclude the proof of (10.20).

$\bullet$The proof of (10.21)**

Again by applying (9.1) $e_{p,0}^{\prime}$ determined by (2.75), we obtain for $N\geq 0$

[TABLE]

A similar estimate holds for $\||D|^{-1}e_{p,0}^{\prime}\|_{\frac{3}{2}-\delta,N+1}$ with $|(1-S_{p})\nabla Y_{p}|_{0,l}$ and $|(1-S_{p})\partial_{t}Y_{p}|_{1,l}$ being replaced by $|(1-S_{p})\nabla Y_{p}|_{\frac{1}{2}-\delta,l}$ and $|(1-S_{p})\partial_{t}Y_{p}|_{\frac{3}{2}-\delta,l}$ respectively.

So that it follows from (A.8) that

[TABLE]

Interpolating the above two inequalities yields

[TABLE]

For $N\leq N_{0}-2$ with $-\gamma+\bar{\varepsilon}(N+1)\geq\bar{\varepsilon}$, we deduce from (A.10) that

[TABLE]

Interpolating the above two inequalities yields

[TABLE]

for $0\leq k\leq\frac{1}{2}-\delta$, $N\leq N_{0}-2$ such that $-\gamma+\bar{\varepsilon}(N+1)\geq\bar{\varepsilon}.$ By interpolating the inequalities (A.20), (A.21), we obtain (10.21).

This completes the proof of Lemma 10.2.

A.3. The proof of Lemma 10.3

We divide the proof of this lemma by the following steps:

$\bullet$The proof of (10.22)**

Applying (9.15) to ${}_{p,1}^{\prime\prime}$ (with $Y\simeq Y_{p}+Y_{p+1}$, $X=W=X_{p}$), which is determined by (2.75), yields that for any $N\geq 0$,

[TABLE]

Note from (2.77) and (2.76) that $\beta\geq 6\bar{\varepsilon},$ so that we deduce from (10.10) that

[TABLE]

As a result, it comes out

[TABLE]

The case when $N\leq N_{0}-6$ with $-\beta+\bar{\varepsilon}(N+5)\geq\bar{\varepsilon},$ it follows from (10.10) that

[TABLE]

hence, we achieve

[TABLE]

(10.22) then follows by interpolating (A.23) and (A.25).

$\bullet$The proof of (10.23)**

By applying (9.15) to $e_{p,1}^{\prime}$ ( with $Y\simeq Y_{p}$, $X=(1-S_{p})Y_{p}$ and $W=X_{p}$) and noticing that

[TABLE]

we get

[TABLE]

Inserting (A.22) into the above inequality for $N=0$ gives

[TABLE]

Whereas for $N\leq N_{0}-6$ such that $-\beta+\bar{\varepsilon}(N+5)\geq\bar{\varepsilon}$, by substituting (A.24) into the above inequality, we achieve

[TABLE]

Then (10.23) follows by interpolating the above two inequalities.

$\bullet$The proof of (10.24)**

Applying (9.14) to $e_{p,0}^{\prime\prime}$ gives,

[TABLE]

Again due to $\beta\geq 7\bar{\varepsilon},$ we deduce from (10.10) that

[TABLE]

As a result, it comes out

[TABLE]

The case when $N\leq N_{0}-6$ with $-\beta+\bar{\varepsilon}(N+5)\geq\bar{\varepsilon},$ it follows from (10.10) that

[TABLE]

so that in this case, we have

[TABLE]

(10.24) follows by interpolating the above inequalities.

$\bullet$The proof of (10.25)**

Applying (9.14) to $e_{p,0}^{\prime}$ gives

[TABLE]

Then as in the proof of (10.24), we deduce that

[TABLE]

and

[TABLE]

for $N\leq N_{0}-6$ with $-\beta+\bar{\varepsilon}(N+5)\geq\bar{\varepsilon}.$ Then (10.25) follows by interpolating the above inequalities.

This ends the proof of Lemma 10.3.

Acknowledgments.* P. Zhang would like to thank Professor Fanghua Lin and Professor Jalal Shatah for profitable discussions. P. Zhang is partially supported by NSF of China under Grant 11371347 and innovation grant from National Center for Mathematics and Interdisciplinary Sciences.*