Existence and stability of periodic solution to the 3D Ginzburg-Landau equation in weighted Sobolev spaces

TL;DR

This paper proves the existence and stability of time-periodic solutions to the 3D Ginzburg-Landau equation under small external forcing within weighted Sobolev spaces.

Contribution

It establishes the existence and stability of periodic solutions for the 3D Ginzburg-Landau equation with odd external force in weighted Sobolev spaces, a novel analytical result.

Findings

Existence of time-periodic solutions under small external force

Stability analysis of the periodic solutions

Application of weighted Sobolev spaces in the analysis

Abstract

We prove the existence of time periodic solution to the 3D Ginzburg-Landau equation in weighted Sobolev spaces. We consider the cubic Ginzburg-Landau equation with an external force $g$ satisfying the oddness condition $g(-x,t)=-g(x,t)$. The existence of the periodic solution is proved for small time-periodic external force. The stability of the time periodic solution is also considered.

Equations

$$\{\begin{array}{l}{u}_t-(1+i)\Delta u=|u{|}^{2}u+g,x\in {\mathbb{R}}^3,t\in \mathbb{R},\\ u(t+T)=u(t),x\in {\mathbb{R}}^3,t\in \mathbb{R}\end{array}$$

$$u_t-(a+i\alpha )\Delta u+(b+i\beta )|u{|}^{\sigma }u+|u{|}^{\delta }(\vec{\lambda }\cdot \nabla u)+|u{|}^{\delta -2}{u}^2(\vec{\mu }\cdot \nabla \bar{u})=0.$$

$$\{\begin{array}{l}g(x,t+T)=g(x,t),x\in {\mathbb{R}}^3,t\in \mathbb{R},\text{ T>0 is a fixed constant, }\\ g(-x,t)=-g(x,t),x\in {\mathbb{R}}^3,t\in \mathbb{R}.\end{array}$$

$$u_t=c_{0}u+(c_0+iϵc_1)u_{xx}-(\frac{c_0}{2}+iϵc_2)|u{|}^{2}u-(\frac{c_0}{2}+iϵc_3)|u{|}^{4}u.$$

$$u_t={\alpha }_{0}u+{\alpha }_1{u}_{xx}+{\alpha }_2|u{|}^{2}u+{\alpha }_3|u{|}^2{u}_x+{\alpha }_4{u}^2{\bar{u}}_x+{\alpha }_5|u{|}^{4}u+f$$

$$\hat{\chi}_{1}(\xi)=\left\{\begin{array}[]{ll}{1}&{\left(|\xi|\leq r_{1}\right),}\\ {0}&{\left(|\xi|\geq r_{\infty}\right)}\end{array}\right.$$

$${\hat{\chi }}_{\infty }(\xi )=1-{\hat{\chi }}_1(\xi )$$

$${\hat{\chi }}_j(\xi )\in C^{\infty }\left({\mathbb{R}}^n\right)and0\le {\hat{\chi }}_j\le 1forj=1,\infty ,$$

$$P_{j}f={\mathcal{F}}^{-1}{\hat{\chi }}_j\mathcal{F}[f]\left(f\in L^2,j=1,\infty \right),$$

$$\hat{f}(\xi )=\mathcal{F}[f](\xi )={\int }_{{\mathbb{R}}^3}f(x)e^{-ix\cdot \xi }\text{mbox}dx\left(\xi \in {\mathbb{R}}^3\right)$$

$${\mathcal{F}}^{-1}[f](x)=(2\pi {)}^{-3}{\int }_{{\mathbb{R}}^3}f(\xi )e^{i\xi \cdot x}\text{mbox}d\xi \left(x\in {\mathbb{R}}^3\right).$$

$$\{\begin{array}{l}{\partial }_t{u}_1+A{u}_1=F_1(u,g),\\ {\partial }_t{u}_{\infty }+A{u}_{\infty }=F_{\infty }(u,g),\end{array}$$

$$L_1^p=\{f\in L^p:\Vert f{\Vert }_{L_1^p}=\Vert (1+|x|)f{\Vert }_{L^p}<\infty .\}$$

$$H_1^k=\{f\in H^k:\Vert f{\Vert }_{H_1^k}=\Vert (1+|x|)f{\Vert }_{H^k}<\infty .\}$$

$$X(a,b)=\left\{u_1:\mathrm{supp}{\hat{u}}_1\subset \{|\xi |\le r_{\infty }\},u_1(-x)=-u_1(x),\Vert u_1{\Vert }_{X(a,b)}<\infty \right\},$$

$$\Vert u_1{\Vert }_{X(a,b)}=\Vert u_1{\Vert }_{H^1(a,b;L^2)}+\Vert x\nabla u_1{\Vert }_{H^1(a,b;L^2)}+\Vert {\partial }_t{u}_1{\Vert }_{L^2(a,b;L_1^2)}.$$

$$Y(a,b)=\left\{u_{\infty }:\mathrm{supp}{\hat{u}}_{\infty }\subset \{|\xi |\ge r_1\},u_{\infty }(-x)=-u_{\infty }(x),\Vert u_{\infty }{\Vert }_{Y(a,b)}<\infty \right\},$$

$$\Vert u_{\infty }{\Vert }_{Y(a,b)}=\Vert u_{\infty }{\Vert }_{C([a,b];H_1^2)}+\Vert u_{\infty }{\Vert }_{L^2(a,b;H_1^3)}+\Vert u_{\infty }{\Vert }_{H^1(a,b;H_1^1)}.$$

$$\Vert \{u_1,u_{\infty }\}{\Vert }_{Z(a,b)}=\Vert u_1{\Vert }_{X(a,b)}+\Vert u_{\infty }{\Vert }_{Y(a,b)}.$$

$$B_{X(a,b)}(r)=\left\{\{u_1,u_{\infty }\}\in X(a,b);\Vert \{u_1,u_{\infty }\}{\Vert }_{X(a,b)}\le r\right\}.$$

$$[g]\equiv \Vert g{\Vert }_{L^2(0,T;L_1^1)}+\Vert g{\Vert }_{L^2(0,T;H_1^1)}.$$

$$\Vert \{u_1,u_{\infty }\}{\Vert }_{Z(0,T)}\le C_0[g].$$

$${\partial }_{t}w-(1+i)\Delta w=2|v_{per}{|}^{2}w+v_{per}^2\bar{w}+|w{|}^{2}w+|w{|}^2{v}_{per}.$$

$$\Vert w_0{\Vert }_{H^1\cap L^1}+[g]\le \eta ,$$

$$\Vert {\nabla }^{l}w(t){\Vert }_{L^2}\le C(1+t{)}^{-\frac{3}{4}-\frac{l}{2}}$$

$$\Vert f{\Vert }_{L^{\infty }({\mathbb{R}}^3)}\le C\Vert \nabla f{\Vert }_{H^1({\mathbb{R}}^3)}$$

$${\Vert {\nabla }^k{f}_1\Vert }_{L^2}+{\Vert f_1\Vert }_{L^p}\le C{\Vert f_1\Vert }_{L^2}\left(f\in L^2\right),$$

$${\Vert f_1\Vert }_{H_1^k}\le C{\Vert f_1\Vert }_{L_1^2}\left(f\in L^2\cap L_1^2\right),$$

$${\Vert \nabla f_1\Vert }_{H_1^k}\le C({\Vert f_1\Vert }_{L^2}+\Vert x\nabla f_1{\Vert }_{L^2})\left(f\in L^2,x\nabla f\in L^2\right),$$

$${\Vert f_1\Vert }_{L_1^2}+{\Vert f_1\Vert }_{L^2}+\Vert x\nabla f_1{\Vert }_{L^2}\le C{\Vert f_1\Vert }_{L_1^1}\left(f\in L^2\cap L_1^1\right).$$

$${\Vert f_{\infty }\Vert }_{L^2}\le C{\Vert \nabla f_{\infty }\Vert }_{L^2}.$$

$${\Vert |x|(\chi \ast f)\Vert }_{L^2}\le C{\Vert f\Vert }_{L^2}+C{\Vert xf\Vert }_{L^2}.$$

Full text

Key words: Periodic solution; Ginzburg-Landau equation; Weighted Sobolev space

1 Introduction

In this paper, we are concerned with the following Ginzburg-Landau equation

[TABLE]

where $u=u(t,x)$ is a complex function, and $g=g(x,t)$ is a given external force.

Equation (1.1) without the external force $g$ is a special form of the following generalized derivative Ginzburg-Landau equation

[TABLE]

The Ginzburg-Landau equation has a long history in the research of phase transitions and superconductivity. In fluid mechanical systems, it is a generic amplitude equation near the onset of instabilities that lead to chaotic dynamics. It is also of particularly interest due to it’s certain connection with the nonlinear Schrödinger equation.

This paper is devoted to the study of the existence of a periodic in time solution to equation (1.1) with the external force $g$ satisfying

[TABLE]

Time periodic problems for the Ginzburg-Landau equation have been extensively investigated in the last decades. Guo-Jing-Lu [7, 8] studied the following cubic-quintic Ginzburg-Landau equation

[TABLE]

The existence and properties of the equilibria of the perturbed system and unperturbed system to (1.4) are investigated in [7]. In [8], spatial quasiperiodic solutions to (1.4) are proved to disappear due to the perturbation and several types of heteroclinic orbits are proposed by mathematical and numerical analysis. The following generalized Ginzburg-Landau equation

[TABLE]

is researched by Guo-Yuan [10, 11]. The almost periodic solution to (1.5) is established in Guo-Yuan [11] provided that the external force $f$ is almost periodic in time and the coefficients satisfy some conditions. The one dimension case was investigated in [10] and periodic solutions in Sobolev spaces was established under some conditions on the coefficients and $f.$ Employing a priori estimate and the Leray-Schauder fixed point theorem, Guo-Jiang [9] showed the existence and uniqueness of the time-periodic solution to the Ginzburg-Landau equation coupled with the BBM equation. Guo-Yuan [12] considered equation (1.2) plus a damping term $u$ and an external force $f$ and established the time periodic solution in the setting that $f$ is periodic in time variable and $u$ is periodic in spatial variable. Equation (1.2) in 3D plus a damping term $u$ and an external force $f$ but without derivative in the nonlinear terms was investigated by Li-Guo [17] and periodic solution in Sobolev spaces was obtained with the help of Faedo-Schauder fix point theorem and the standard compactness arguments. For more results concerning the periodic solution to the Ginzburg-Landau equation, one can refer to [1, 4, 20, 21, 26, 27]. For results with respect to other aspects to the Ginzburg-Landau equation, one can refer to [2, 3, 6, 13, 14, 18, 25].

This paper aims to look for time periodic solution to equation (1.1) with external force $g$ satisfying (1.3) in weighted Sobolev spaces. The idea is due to Kagei-Tsuda [15] and Tsuda [22, 23, 24].

Before going further, we define the following functions

[TABLE]

and

[TABLE]

that satisfy

[TABLE]

where $0<r_{1}<r_{\infty}$ will be fixed so that $(\ref{r})$ holds for $|\xi|\leq r_{\infty}$.

Based on the above two smooth functions, we define operators $P_{1}$ and $P_{\infty}$ on $L^{2}$ by

[TABLE]

where $\mathcal{F}[f]$ and $\hat{f}$ denote the Fourier transform of $f$ that is defined by

[TABLE]

and $\mathcal{F}^{-1}[f]$ is the inverse Fourier transform of $f$ defined by

[TABLE]

Note that $P_{1}$ and $P_{\infty}$ decompose a function into its low and high frequency parts, respectively.

Set $F(u,g)\equiv|u|^{2}u+g$.

Applying $P_{1}$ and $P_{\infty}$ to equation (1.1), respectively, one finds

[TABLE]

where $A=-(1+i)\Delta,$ $u=u_{1}+u_{\infty}$, $u_{j}=P_{j}u$ and $F_{j}=P_{j}F$ for $j=1,\infty.$ If we can find a periodic solution $u_{1}(t)=u_{1}(t+T)$ and $u_{\infty}(t)=u_{\infty}(t+T)$, then we see that $u=u_{1}+u_{\infty}$ is a periodic solution satisfying equation (1.1).

To find a periodic solution to system (1.6), we first construct its local solution satisfying $u_{j}(0)=u_{j}(T)$, then we expand the local solution to a global periodic one. The reason we applying $P_{j}$ to (1.6) is that $|1-e^{(1+i)T\theta}|\geq c\theta$ when $\theta>0$ is sufficiently small and this admits us to obtain some decay in frequency space when $|\xi|$ is small enough.

To state our main results, let us first introduce some notations.

We use $L^{p}(1\leq p\leq\infty)$ and $H^{k}$ to denote the usual Lebesgue space and Sobolev space, respectively. The weighted Lebesgue space $L_{1}^{p}$ is defined by

[TABLE]

The weighted Sobolev space $H_{1}^{k}$ is defined by

[TABLE]

We define solution space to (1.6) in which the low frequency part will live by

[TABLE]

where

[TABLE]

The solution space to (1.6) in which the high frequency part will live is defined by

[TABLE]

where

[TABLE]

We define space $Z(a,b)\equiv X(a,b)\times Y(a,b)$ equipped with the norm

[TABLE]

We denote by $C_{per}(\mathbb{R};X)$ the set of all $T-$periodic continuous functions with values in $X$ equipped with the norm $\|\cdot\|_{C([0,T];X)}$ and by $L_{per}^{2}(\mathbb{R};X)$ the set of all $T-$periodic locally square integrable functions with values in $X$ equipped with the norm $\|\cdot\|_{L^{2}(0,T;X)}$, and so on.

Let $B_{X(a,b)}(r)$ be the unit closed ball of $Z(a,b)$ centered at 0 with radius $r$, that is,

[TABLE]

Set

[TABLE]

Our first result concerning the uniquely existence of the periodic solution to the (1.1) in weighted Sobolev space reads

Theorem 1.1**.**

Let the external force $g\in L_{per}^{2}(\mathbb{R};H^{1}_{1}(\mathbb{R}^{3})\cap L^{1}_{1}(\mathbb{R}^{3}))$ and (1.3) be satisfied by $g$. There exists constants $\delta_{0}>0$ and $C_{0}>0$ such that if $[g]\leq\delta_{0}$, then equation (1.1) admits a periodic solution $u_{per}$ with period $T$ that satisfies $\{u_{1},u_{\infty}\}\in Z_{per}(\mathbb{R})$, where $u_{1}=P_{1}u$ and $u_{\infty}=P_{\infty}u$ and there holds

[TABLE]

Define $W=\{u:\{P_{1}u,P_{\infty}u\}\in Z_{per}(\mathbb{R}),\|\{P_{1}u,P_{\infty}u\}\|_{Z(0,T)}\leq C_{0}\delta_{0}\}.$ Then the uniqueness of periodic solution of (1.1) holds in $W.$

After obtaining the time periodic solution to (1.1), a natural question is to ask whether this periodic solution is stable under small perturbation.

Suppose that $v_{per}$ is the time periodic solution to (1.1) established in Theorem 1.1.

Then the perturbation $w=v-v_{per}$ satisfy

[TABLE]

We investigate the initial value problem of equation (1.7) with initial condition $w(0)=w_{0}.$

The second result concerning with the stability of the time periodic solution obtained in Theorem 1.1 reads

Theorem 1.2**.**

Let the external force $g\in L_{per}^{2}(\mathbb{R};H^{1}_{1}(\mathbb{R}^{3})\cap L^{1}_{1}(\mathbb{R}^{3}))$ and (1.3) be satisfied by $g$. Suppose that $v_{per}$ is the time periodic solution established in Theorem 1.1 and assume $w_{0}\in H^{2}\cap L^{1}.$ If there is a constant $\eta$ small enough such that

[TABLE]

then equation (1.7) admits a unique global solution $w\in C([0,T];H^{2})$ satisfying

[TABLE]

for $t\in[0,\infty]$ and $l=0,1.$

The rest of the paper is organized as follows. In section 2, we collect some useful Lemmas. In section 3, we deal with the low frequency part of (1.6). With help of weighted energy method, the high frequency part of (1.6) is treated in section 4. Section 5 devotes to estimate the nonlinear and nonhomogeneous terms of (1.6). We finally give the proof of Theorems 1.1 and 1.2 in Sections 6 and 7, respectively.

2 Preliminaries

The following useful Lemmas can be found in [15].

Lemma 2.1**.**

We have

[TABLE]

for $f\in H^{2}(\mathbb{R}^{3})$.

Lemma 2.2**.**

For $k$ being a nonnegative integer and $2\leq p\leq\infty,$ suppose that a function $f$ satisfies $\operatorname{supp}\hat{f}\subset\{|\xi|\leq r_{\infty}\}.$ Let $f_{1}=P_{1}f.$ Then it finds

[TABLE]

Lemma 2.3**.**

(i)  For $k$ being a nonnegative integer, one has $\|P_{\infty}f\|_{H^{k}}\leq C\|f\|_{H^{k}}.$

(ii)  Suppose that a function $f_{\infty}$ satisfies $\operatorname{supp}\hat{f}_{\infty}\subset\{|\xi|\geq r_{1}\}$ and $f_{\infty}\in H^{1},$ then it finds

[TABLE]

Lemma 2.4**.**

For $\chi$ being a Schwartz function on $\mathbb{R}^{n}$, there holds

[TABLE]

Lemma 2.5**.**

Let $f_{\infty}$ be satisfying $\operatorname{supp}\hat{f}_{\infty}\subset\{|\xi|\geq r_{1}\}$ and $\|f_{\infty}\|_{H^{1}_{1}}<\infty.$ Then there is a constant $C>0$ independent of $f_{\infty}$ such that

[TABLE]

The following Hardy inequality can be found in [5, 19].

Lemma 2.6**.**

[TABLE]

for $u\in H^{1}(\mathbb{R}^{3}).$

3 Estimates of the low frequency part

This section studies the following low frequency equation

[TABLE]

where $F_{1}$ is assumed to satisfy $\operatorname{supp}\hat{F}_{1}\subset\{|\xi|\leq r_{\infty}\}$.

Formally, the solution of equation (3.1) can be written as

[TABLE]

Let $u_{1}(0)=u_{1}(T)$ in (3.2), we have

[TABLE]

which yields

[TABLE]

provided that $(1-\mbox{e}^{-TA})^{-1}$ exists in some sense.

Substituting (3.4) into (3.2) leads to

[TABLE]

This leads to the following Proposition concerning the solvability of (3.1).

Proposition 3.1**.**

If $F_{1}$ satisfies the following three conditions

$F_{1}(-x)=-F_{1}(x);$** 2. 2.

$\operatorname{supp}\hat{F}_{1}\subset\{|\xi|\leq r_{\infty}\};$** 3. 3.

$F_{1}\in L^{2}(0,T;L^{2}\cap L^{1}_{1}),$**

then $u_{1}(t)$ defined by (3.5) is a solution of (3.1) in $X(0,T)$ and there holds

[TABLE]

To prove Proposition 3.1, we first investigate the properties of $e^{-tA}.$ We have the following Proposition describing the properties of $e^{-tA}.$

Proposition 3.2**.**

We have the following assertions:

If $u_{1}$ satisfies $\operatorname{supp}\hat{u}_{1}\subset\{|\xi|\leq r_{\infty}\}$ and $\|u_{1}\|_{L^{2}}+\|x\nabla u_{1}\|_{L^{2}}<\infty$, then $\operatorname{supp}\widehat{e^{-tA}u_{1}}\subset\{|\xi|\leq r_{\infty}\}$ and one finds

[TABLE]

where in the above three inequalities, $t\in[0,T^{\prime}]$ with $T^{\prime}>0$ any given number and $C$ depends on $T^{\prime}.$ 2. 2.

If $\operatorname{supp}\hat{F}_{1}\subset\{|\xi|\leq r_{\infty}\}$ and $\|F_{1}\|_{L^{2}(0,T;L^{2})}+\|x\nabla F_{1}\|_{L^{2}(0,T;L^{2})}<\infty$, then $\operatorname{supp}\mathcal{F}[\int_{0}^{t}\mbox{e}^{-(t-s)A}F_{1}(s)\mbox{d}s]\subset\{|\xi|\leq r_{\infty}\}$ and there holds

[TABLE]

and if in addition $F_{1}\in L^{2}(0,T;L_{1}^{2}),$ then

[TABLE]

where in the above two inequalities, $C$ depends on $T.$

Proof.

The support properties can be easily verified and the three inequalities can be justified with the help of Plancherel theorem and the support property of $u_{1}.$ We omit the details. 2. 2.

Assertions in 1 and the support property of $F_{1}$ leads to assertions in 2.

We thus complete the proof of Proposition 3.2. ∎

Next, we establish sufficient conditions that claim the existence of $(1-\mbox{e}^{-TA})^{-1}.$

Proposition 3.3**.**

Let the conditions in Proposition 3.1 be satisfied. Then the following equation

[TABLE]

admits a unique solution $u_{1}$ that satisfies $\operatorname{supp}\hat{u}_{1}\subset\{|\xi|\leq r_{\infty}\}$ and

[TABLE]

Consequently, $1-\mbox{e}^{-TA}$ has a bounded inverse $(1-\mbox{e}^{-TA})^{-1}:X_{1}\rightarrow X_{2}$, where $X_{1}=\{F_{1}:\operatorname{supp}\hat{F}_{1}\subset\{|\xi|\leq r_{\infty}\},F_{1}\in L^{2}\cap L_{1}^{1}\}$ and $X_{2}=\{u_{1}:\operatorname{supp}\hat{u}_{1}\subset\{|\xi|\leq r_{\infty}\},u_{1}\in L^{2},x\nabla u_{1}\in L^{2}\}$ and there holds

[TABLE]

Proof.

First, direct computation yields

[TABLE]

provided $\theta>0$ is sufficiently small.

This fact allows us to conclude that there exists a constant $r_{\infty}>0$ such that

[TABLE]

when $|\xi|\leq r_{\infty}.$

Applying Fourier transform to (3.12), one obtains

[TABLE]

Thus, the support property of $u_{1}$ can be easily verified.

Plancherel theorem and (3.13) yield

[TABLE]

where $\hat{F}_{1}(0)=0$ since $F_{1}(-x)=-F_{1}(x).$

Note that $|\partial_{\xi}(1-\mbox{e}^{-T\hat{A}})^{-1}|\leq C\frac{1}{|\xi|^{3}}$. This fact leads to

[TABLE]

We thus complete the proof of Proposition 3.3. ∎

We are now in a position to prove Proposition 3.1.

Proof of Proposition 3.1: We only need to prove (3.6).

Using (3.7), (3.8), (3.10) and Proposition 3.3, one finds

[TABLE]

Employing (3.9) and (3.10) yields

[TABLE]

Invoking (3.10), (3.11) and Lemma 2.2(ii), we obtain

[TABLE]

We thus complete the proof of Proposition 3.1.

4 Estimates of the high frequency part

The following high frequency equation are investigated in this section

[TABLE]

where $F_{\infty}$ is assumed to satisfy $\operatorname{supp}\hat{F}_{\infty}\subset\{|\xi|\geq r_{1}\}$.

Similar to the derivation of (3.5), we formally have

[TABLE]

The following Proposition is concerned with the solvability of (4.1).

Proposition 4.1**.**

Assume that $F_{\infty}\in L^{2}(0,T;X_{\infty}^{1})$ and $F_{\infty}(-x)=-F_{\infty}(x)$, where $X_{\infty}^{k}=\{f:\operatorname{supp}\hat{f}\subset\{|\xi|\geq r_{1}\},f\in H_{1}^{k}\}$. Then $u_{\infty}(t)$ defined by (4.2) is a solution of (4.1) in $Y(0,T)$ and there holds

[TABLE]

To prove Proposition 4.1, we first establish the following weighted energy estimates.

Proposition 4.2**.**

For the smooth solution of equation (4.1), there holds the following estimate

[TABLE]

where $d$ is a positive constant large enough.

Proof.

The standard energy estimates yields

[TABLE]

and

[TABLE]

where we have used integration by parts in the derivation of (4.5).

Combining (4.4), (4.5) and Lemma 2.3(ii), one deduces after choosing $\epsilon$ small enough that

[TABLE]

Applying $\partial_{x}^{\alpha}$ to $(\ref{glhigh})_{1}$ for $0\leq|\alpha|\leq 2$ and dotting the result with $\partial_{x}^{\alpha}\bar{u}_{\infty}$, one obtains

[TABLE]

When $|\alpha|=0,$ we find

[TABLE]

and

[TABLE]

where $|f|_{L^{p}_{1}}=\|xf\|_{L^{p}}.$

When $|\alpha|\geq 1,$ one derives

[TABLE]

and

[TABLE]

where $|f|_{H^{k}_{1}}=\|xf\|_{H^{k}}.$

Substituting the above estimates into (4), we obtain

[TABLE]

Choosing $\epsilon_{2}$ small enough and employing Lemmas 2.3(ii) and 2.5 yield

[TABLE]

Choosing $\epsilon_{1}$ small enough, then adding (4) to $d_{1}\times(\ref{energy3})$ with $d_{1}$ sufficiently large, one deduces

[TABLE]

We thus complete the proof of Proposition 4.2. ∎

Based on the weighted energy estimates provided by Proposition 4.2, we next show the existence of $(1-\mbox{e}^{-TA})^{-1}$ in high frequency.

Proposition 4.3**.**

We claim that:

If $\operatorname{supp}\hat{u}_{\infty 0}\subset\{|\xi|\geq r_{1}\}$ and $u_{\infty 0}\in H_{1}^{2}$, then $\operatorname{supp}\mathcal{F}[\mbox{e}^{-tA}u_{\infty 0}]\subset\{|\xi|\geq r_{1}\}$ and

[TABLE]

for all $t\geq 0.$ 2. 2.

If $\operatorname{supp}\hat{F}_{\infty}\subset\{|\xi|\geq r_{1}\}$ and $F_{\infty}\in L^{2}(0,T;H_{1}^{1})$, then $\operatorname{supp}\mathcal{F}[\mbox{e}^{-tA}F_{\infty}]\subset\{|\xi|\geq r_{1}\}$ and

[TABLE]

for $t\in[0,T]$ with $C$ depending on $T.$ 3. 3.

The spectral radius of $\mbox{e}^{-TA}$ in $X_{\infty}^{2}$ is less than 1. Accordingly, $1-\mbox{e}^{-TA}$ has a bounded inverse $(1-\mbox{e}^{-TA})^{-1}$ on $X_{\infty}$ and there holds

[TABLE]

provided that $u\in X_{\infty}^{2}.$

Proof.

From (4.3) we derive

[TABLE]

Let $F_{\infty}=0$ in (4.9), then one finds

[TABLE]

So there exists $a=d$ such that

[TABLE]

This proves claim 1. 2. 2.

The inequality in claim 2 is proved by setting $u_{\infty 0}=0$ in (4.9). 3. 3.

Let $\operatorname{supp}\hat{u}\subset\{|\xi|\geq r_{1}\}$. From claim 1, we know

[TABLE]

Consequently, we find $\|(\mbox{e}^{-TA})^{j}\|\leq\mbox{e}^{-jat}$. This leads to

[TABLE]

We thus complete the proof of Proposition 4.3. ∎

We are in a position to prove Proposition 4.1.

Proof of Proposition 4.1:

Invoking (4.3) yields

[TABLE]

According to claim 3 and claim 2 of Proposition 4.3, one finds

[TABLE]

Note that $u_{\infty}$ satisfies (4.1). This leads to

[TABLE]

We thus complete the proof of Proposition 4.1.

5 Estimates of the nonlinear and inhomogeneous terms

This section deals with the nonlinear and inhomogeneous terms. We have the following Proposition.

Proposition 5.1**.**

For $u=u_{1}+u_{\infty},$ there holds the following estimates

[TABLE]

Proof.

First, one easily derives from Lemma 2.2 that

[TABLE]

Using interpolation and $\|f\|_{L^{6}(\mathbb{R}^{3})}\leq C\|\nabla f\|_{L^{2}(\mathbb{R}^{3})}$, we obtain

[TABLE]

where in the last inequality, we have invoked Lemma 2.2.

Similarly, it finds

[TABLE]

[TABLE]

and

[TABLE]

Inserting the above estimates of $I_{1}-I_{4}$ to (5), one obtains (5.1).

Employing Lemmas 2.3 and 2.4, we easily find

[TABLE]

Lemma (2.2) again leads to

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Substituting the above estimates of $J_{1}-J_{4}$ into (5), we obtain (5.2).

Repeating the above process for $P_{j}(|u|^{2}u-|v|^{2}v)$ instead of $P_{j}(|u|^{2}u)$ yields (5.3) and (5.4), we omit the details and thus complete the proof of Proposition 5.1. ∎

6 Proof of Theorem 1.1

We will use iteration argument to prove Theorem 1.1.

Let $t\in[0,T]$. We define $u_{1}^{(0)}(t)$ and $u_{\infty}^{(0)}(t)$ as

[TABLE]

We inductively define $u_{1}^{(l)}(t)$ and $u_{\infty}^{(l)}(t)$ for $l\geq 1$ as

[TABLE]

where $u^{(l-1)}=u_{1}^{(l-1)}+u_{\infty}^{(l-1)}.$

It is easy to see that $u_{1}^{(l)}(0)=u_{1}^{(l)}(T)$ and $u_{\infty}^{(l)}(0)=u_{\infty}^{(l)}(T)$ for $l\geq 0.$

We first show the existence of local solution to system (1.6) satisfying $u_{j}(0)=u_{j}(T)(j=0,1).$

Proposition 6.1**.**

Let $\delta_{2}>0$ be a constant small enough and let $[g]\leq\delta_{2}.$ Then there uniquely exists solution $\{u_{1},u_{\infty}\}$ to system (1.6) on $[0,T]$ in $B_{Z(0,T)}(C_{3}[g])$ with $u_{j}(0)=u_{j}(T)(j=0,1)$. The uniqueness holds in $B_{Z(0,T)}(C_{3}\delta_{2}).$

Proof.

From Proposition 3.1, Proposition 4.1, (5.1) and (5.2), we deduce that there is two constants $C_{1},C_{2}>0$ independent of $g$ and $l$ such that there holds

[TABLE]

and such that for all $l\geq 0$ there holds

[TABLE]

Setting $l=0$ in (6.4) and employing (6.3) leads to

[TABLE]

Choosing $\delta_{1}<\sqrt{\frac{1}{C_{1}C_{2}^{3}}}$ yields

[TABLE]

provided $[g]\leq\delta_{1}.$

By induction, one easily concludes

[TABLE]

for all $l\geq 2$ if choosing $\delta_{1}<\sqrt{\frac{1}{C_{1}(C_{2}+1)^{3}}}$.

Next, we easily see that $\bar{u}_{j}^{(l)}=u_{j}^{(l+1)}-u_{j}^{(l)}$ for $j=1,\infty$ satisfy the following equation

[TABLE]

where $u^{(l)}=u_{1}^{(l)}+u_{\infty}^{(l)}.$

Invoking Proposition 3.1, Proposition 4.1, (5.3), (5.4) and (6.7), one can deduce that

[TABLE]

for all $l\geq 1$ if $[g]\leq\delta_{1}.$

Setting $\delta_{2}=\min\{\delta_{1},\frac{1}{2C(1+C_{2})^{2}}\}$. Let $[g]\leq\delta_{2}$. Then $u_{j}^{(l)}(j=1,\infty)$ converges to $u_{j}(j=1,\infty)$ in the sense

[TABLE]

And one can easily verify that $\{u_{1},u_{\infty}\}$ is a solution of (1.6) satisfying $u_{j}(0)=u_{j}(T)(j=1,\infty).$

We thus complete the proof of Proposition 6.1. ∎

Proposition 6.2**.**

Assume that $s\in\mathbb{R}$ and $U_{0}=U_{01}+U_{0\infty},$ where $\operatorname{supp}\hat{U}_{01}\subset\{|\xi|\leq r_{\infty}\}$ , $\|U_{01}\|_{L^{2}}+\|x\nabla U_{01}\|_{L^{2}}<\infty$ and $U_{0\infty}\in X_{\infty}^{2}.$ If there is a sufficiently small constant $\delta_{3}$ such that

[TABLE]

then there is a constant $C_{4}$ such that there uniquely exists a solution $\{U_{1},U_{\infty}\}$ to system (1.6) on $[s,s+T]$ in $B_{Z(s,s+T)}(C_{4}M(U_{01},U_{0\infty},g))$ satisfying $U_{j}|_{t=0}=U_{0j}(j=1,\infty).$ The uniqueness holds in $B_{Z(s,s+T)}(C_{4}\delta_{3}).$

Proof.

Using iteration argument, the proof of Proposition 6.2 is similar to the proof of Proposition 6.1 and we omit the details. ∎

Proof of Theorem 1.1: Having Propositions 6.1 and 6.2 in hand, we can use similar method as that in [15] to prove Theorem 1.1, we present the details here for the reader’s convenience.

First, according to Proposition 6.1, one derives that if $[g]\leq\delta_{2},$ then equation (1.6) admits a unique solution $\{u_{1}^{(0)},u_{\infty}^{(0)}\}\in B_{Z(0,T)}(C_{3}[g])$ satisfying $u_{1}^{(0)}(0)=u_{1}^{(0)}(T)$ and $u_{\infty}^{(0)}(0)=u_{\infty}^{(0)}(T)$. This implies that

[TABLE]

Let $[g]\leq\frac{\delta_{3}}{C_{3}+1}$, then Proposition 6.2 allows us to conclude that equation (1.6) admits a unique solution $\{u_{1}^{(1)},u_{\infty}^{(1)}\}\in B_{Z(T,2T)}(C_{4}(C_{3}+1)[g])$ satisfying $u_{1}^{(1)}(T)=u_{1}^{(0)}(T)=u_{1}^{(0)}(0)$ and $u_{\infty}^{(1)}(T)=u_{\infty}^{(0)}(T)=u_{\infty}^{(0)}(0).$

Next, we define $v_{j}^{(1)}(j=1,\infty)$ and $v^{(1)}$ for $t\in[0,T]$ by

[TABLE]

The it is easy to see that equation (1.6) is satisfied with $v_{j}^{(1)}(j=1,\infty)$ instead of $u_{j}$ and $v^{(1)}$ instead of $u.$ And the initial data are $v_{j}^{(1)}(0)=u_{j}^{(0)}(0).$

Let $[g]\leq\min\{\delta_{2},\frac{C_{4}\delta_{3}}{C_{3}},\frac{\delta_{3}}{C_{3}+1}\}$. Then $v_{j}^{(1)}(t)=u_{j}^{(0)}(t)(j=1,\infty)$ for $t\in[0,T]$ due to uniqueness. This yields $u_{j}^{(1)}(t)=u_{j}^{(0)}(t-T)$ for $t\in[T,2T].$

For $j=1,\infty$, define

[TABLE]

Then for $t\in[0,T],$ one finds $u_{j}(t+T)=u_{j}(t).$ Proposition 6.2 and (6.10) allow us to conclude that there uniquely exists a solution $\{w_{1},w_{\infty}\}\in B_{Z(T/2,3T/2)}(C_{4}(C_{3}+1)[g])$ to equation (1.6) on $[T/2,3T/2]$ satisfying $w_{j}(T/2)=u_{j}^{(0)}(T/2)$ and uniqueness yields $w_{j}=u_{j}$ on $[T/2,3T/2]$. This shows that $\{u_{1},u_{\infty}\}$ satisfies equation (1.6) in $Z(0,2T).$ We then can obtain solution $\{u_{1},u_{\infty}\}$ to (1.6) in $Z_{per}(\mathbb{R})$ satisfying $\|\{u_{1},u_{\infty}\}\|_{Z(0,T)}\leq C_{3}[g]$ by repeating this argument on $[mT,(m+1)T]$ for $m=\pm 1,\pm 2,\cdots.$ The periodic solution of (1.1) is then obtained by setting $u=u_{1}+u_{\infty}.$ The proof of the uniqueness part is due to the iteration argument, we omit the details. We thus complete the proof of Theorem 1.1.

7 Proof of Theorem 1.2

The existence of the unique global solution stated in Theorem 1.2 can be proved using similar method as that in [16], we omit the details and just show the decay rates. The method is due to [19].

Applying $P_{1}$ and $P_{\infty}$ to (1.7), respectively, leads to

[TABLE]

and

[TABLE]

Define

[TABLE]

To prove Theorem 1.2, we first show the following two Lemmas.

For the low frequency part, we will show

Lemma 7.1**.**

There exists a $\epsilon>0$ such that if $[g]\leq\epsilon$ and $N(t)\leq 1$ for $t\in[0,T]$. Then there is a constant $C_{5}$ independent of $T$ such that

[TABLE]

for $t\in[0,T]$.

Proof.

The solution of (7.1) can be written as

[TABLE]

where $w_{10}=P_{1}w_{0}.$

Invoking Plancherel theorem yields

[TABLE]

Using Lemma 2.6, Hölder inequality and Theorem 1.1, it finds

[TABLE]

Applying $\nabla^{l}$ to (7.4) and employing (7.5)(7) leads to

[TABLE]

for $\tau\in[0,t].$ This yields (7.3) and we complete the proof of Lemma 7.1. ∎

For the high frequency part, we will show

Lemma 7.2**.**

There exists a $\epsilon>0$ such that if $[g]\leq\epsilon$ and $N(t)\leq 1$ for $t\in[0,T]$. Then we have

[TABLE]

for $t\in[0,T]$ and some constant $d_{2},d_{3}$ large enough.

Proof.

The standard energy method yields

[TABLE]

where integration by parts and Lemma 2.3(ii) have been employed.

Direct calculation leads to

[TABLE]

Substituting (7) into (7.9) and using Young inequality, one obtains

[TABLE]

(7.11) and Lemma 2.3(ii) yield (7.2). We thus complete the proof of Lemma 7.2. ∎

Proof of Theorem 1.2:

We will show that if there is a constant $\epsilon_{2}>0$ small enough such that $\|w_{0}\|_{H^{1}\cap L^{1}}\leq\epsilon_{2},$ then $N(t)\leq C\|w_{0}\|_{H^{1}\cap L^{1}}$ for $t\in[0,T]$ with $c$ independent of $T.$ A bootstrap argument yields $N(t)\leq D_{0}$ for all $t$ with $D_{0}$ a constant independent of $t.$

First, (7.2) implies

[TABLE]

Setting $\mathcal{A}(t)\equiv(1+t)^{5/2}\int_{0}^{t}\mbox{e}^{-d_{3}(t-\tau)}\|\nabla w_{\infty}\|_{H^{1}}^{2}(\tau)\mbox{d}\tau.$ Then one has from (7.12) that

[TABLE]

Adding (7.13) to (7.3) and choosing $\epsilon$ sufficiently small, we obtain

[TABLE]

On the other hand, we note that there is a constant $C_{6}>0$ such that

[TABLE]

The continuity of $N(t)$ in $t$ yields that there is a $t_{0}>0$ such that

[TABLE]

Set $C_{7}=\max\{\sqrt{C_{5}/2},C_{6}\}.$

We will prove that $N(t)<2C_{7}\|w_{0}\|_{H^{1}\cap L^{1}}$ for all $t\in[0,T]$ if $\epsilon_{2}$ is selected suitably small.

Suppose this is not the case. Then there is a $t_{1}\in(t_{0},T]$ such that

[TABLE]

and

[TABLE]

However, from (7.14), we deduce

[TABLE]

provided we choose $\epsilon_{2}<\epsilon_{3}\equiv\min\left\{\frac{1}{\sqrt{8C_{5}C_{7}^{2}}},\frac{\sqrt{d_{2}^{\prime}}}{\sqrt{8C_{5}C_{7}^{2}}}\right\}$.

The above inequality leads to

[TABLE]

which contradicts with (7.15).

We thus complete the proof of Theorem 1.2.

Acknowledgement

The research of B. Guo is partially supported by the National Natural Science Foundation of China, grant 11731014.