金 珍,萬(wàn) 龍

(1. 南昌工程學(xué)院 理學(xué)院, 江西 南昌 330099； 2. 江西財(cái)經(jīng)大學(xué) 信息管理學(xué)院, 江西 南昌 330013)

一類流體動(dòng)力方程周期解的存在性和唯一性

金 珍1,2,萬(wàn) 龍2*

(1. 南昌工程學(xué)院 理學(xué)院, 江西 南昌 330099； 2. 江西財(cái)經(jīng)大學(xué) 信息管理學(xué)院, 江西 南昌 330013)

研究了一類非齊次流體動(dòng)力方程的周期解的存在性和唯一性.首先采用Galerkin方法構(gòu)造近似時(shí)間周期解序列,然后利用先驗(yàn)估計(jì)和Leray-Schauder不動(dòng)點(diǎn)定理,證明近似時(shí)間周期解序列的收斂性,從而得到了該問(wèn)題時(shí)間周期解的存在性,并且證明在一定條件下該解的唯一性.

流體動(dòng)力方程；周期解；Galerkin方法；Leray-Schauder不動(dòng)點(diǎn)定理

Existence and uniqueness of time periodic solution for the fluid dynamics equation. Journal of Zhejiang University(Science Edition), 2017,44(1):040-046

0 引言及預(yù)備知識(shí)

漂移波是磁化非均勻等離子體中的一支低頻靜電波,經(jīng)常與在托卡馬克中觀察到的低頻密度漲落和能量衰變聯(lián)系在一起.被激發(fā)的漂移波,當(dāng)其振幅為有限時(shí),非線性效應(yīng)將起重要作用.這些充分發(fā)展的漂移波，被認(rèn)為是導(dǎo)致等離子體反常輸運(yùn)的重要原因.HASEGAWA等[1-2]首先導(dǎo)出了描述非線性漂移波的Hasegawa-Mima方程：

?t(1-▽2)φ-kn?yφ=▽·[(ez×▽?duì)?·▽]▽?duì)眨?/p>

(1)

這里：▽·≡▽⊥·是垂直于磁場(chǎng)B0方向的散度算子;kn=?xln n0為常數(shù),n和n0分別是有微擾和無(wú)微擾(但不均勻)時(shí)的密度;φ為漂移波的振幅;ez×▽?duì)帐谴呕入x子體的漂移波波速;z軸即為磁場(chǎng)B0的方向,且ez是單位向量.方程(1)描述了等離子體中非線性漂移波的演化過(guò)程.該方程模型與反映二維不可壓流體的Navier-Stokes方程密切相關(guān).二維耗散的Hasegawa-Mima方程

(u-Δu)t-knuy+γ(u-Δu)=

J(u,Δu)+f(x,y,t)，

(2)是一類簡(jiǎn)單的二維湍流系統(tǒng),其中黏性系數(shù)γ>0,u=u(x,y,t),J(ξ,η)是Jacobi導(dǎo)數(shù)矩陣對(duì)應(yīng)的行列式,即J(ξ,η)=ξxηy-ξyηx.在等離子體中,方程(2)描述漂移波的演化過(guò)程,其中，u是靜電的漲落,kn=?xln n0，n0是本底質(zhì)點(diǎn)密度.在地球物理學(xué)流體中,方程(2)描述了地動(dòng)的瞬時(shí)發(fā)展,被稱為Rossby波的擬地動(dòng)勢(shì)渦度方程,此時(shí)u是地動(dòng)流函數(shù).

1986年,SWATERS[3]對(duì)一類流體動(dòng)力方程的穩(wěn)定性條件和先驗(yàn)估計(jì)問(wèn)題進(jìn)行了研究.1993年,ZHOU等[4]考慮一類廣義的流體動(dòng)力方程

(u-Δu)t+J(u,Δu)+AΔux+

BΔuy+f(u)x+g(u)y=h(u)的周期邊界問(wèn)題和Cauchy問(wèn)題,利用Galerkin方法和積分估計(jì)得到上述問(wèn)題的廣義全局解和古典全局解.1998年,GRAUER[5]給出了一類流體動(dòng)力方程的能量估計(jì).2004年,GUO等[6]討論了二維Hasegawa-Mima方程

?t(u-Δu)+k?yu+J(u,Δu)=0

的Cauchy問(wèn)題,得到全局弱解的存在性和唯一性.接著,張瑞鳳、郭柏靈等[7-11]討論了廣義Hasegawa- Mima方程整體解的存在唯一性、整體吸引子、動(dòng)力學(xué)行為等問(wèn)題.

隨后,BRONSKI等[12]通過(guò)構(gòu)造Lyapunov函數(shù)得到文獻(xiàn)[5]中能量估計(jì)的結(jié)論,但推導(dǎo)過(guò)程比文獻(xiàn)[5]更簡(jiǎn)單.近年來(lái),國(guó)內(nèi)外學(xué)者[13-17]討論了幾類Hasegawa-Mima方程的周期初邊值、初始問(wèn)題全局解的衰減估計(jì)、解的存在性和唯一性等問(wèn)題.

為了更好地研究這類方程的性質(zhì),有必要對(duì)其時(shí)間周期解[18-25]的存在性和唯一性加以論證.

本文考慮了一類具有周期邊界條件和非齊次項(xiàng)的流體動(dòng)力方程：

ut-Δut+J(u,Δu)+αΔux+βux-γΔ3u=f,

(3)

u(x,y,t)=u(x+L,y,t)=u(x,y+L,t),

x∈R,y∈R,t>0,

(4)

u(x,y,t)=u(x,y,t+ω), x∈R,y∈R,t>0,

(5)

筆者將采用Galerkin方法和Leray-Schauder不動(dòng)點(diǎn)定理證明其時(shí)間周期解的存在性.記

u(x,y+L,t)=u(x,y,t)},

u(x,y+L,t)=u(x,y,t)},

Ck(ω,X)={f:[0,∞)→X,f(i)是連續(xù)的ω周期函數(shù),i=0,1,…,k},

(unt-Δunt+αΔunx+βunx-γΔ3un,φj)=

(-J(un,Δun)+f,φj)， j=1,2,…,n.

(6)

為證明近似解un(t)的存在性,定義C1(ω;Hn)→C1(ω;Hn)上的映射Fλ:vn→un為

(unt-Δunt+αΔunx+βunx-γΔ3un,φj)=

(-λJ(vn,Δvn)+f,φj),

(7)

(8)

其中，K>0與L,f,γ有關(guān),但與n,α,β,λ無(wú)關(guān).

1 先驗(yàn)估計(jì)

引理1 設(shè)f∈C1(ω;H-3(Ω)),如果Fλ(un)=un,0≤λ≤1，則存在常數(shù)C1使得

其中，C1與f,γ有關(guān),而與n,L,α,β無(wú)關(guān).

證明 方程Fλ(un)=un的等價(jià)形式是

(unt-Δunt+αΔunx+βunx-γΔ3un,φj)=

(-λJ(un,Δun)+f,φj).

將ajn乘以上式的第j個(gè)方程,對(duì)j從1到n求和,得

(unt-Δunt+αΔunx+βunx-γΔ3un,un)=

(-λJ(un,Δun)+f,un).

(9)

注意到

(αΔunx+βunx,un)=0,

(-γΔ3un,un)=γ‖▽?duì)n‖2,

(J(un,Δun),un)=(unxΔuny-unyΔunx,un)=

(10)

在上式兩端關(guān)于t從0到ω積分,注意到un的周期性,得

由積分中值定理,?t*∈[0,ω)使得

另一方面,由式(10)得

上式兩端關(guān)于t從t*到t(t∈[t*,t*+ω))積分,得

‖un(t)‖2+‖▽un(t)‖2≤‖un(t*)‖2+

因此,由Leray-Schauder不動(dòng)點(diǎn)定理可得方程Fλ(un)=un有解.

推論1 在引理1的條件下,

其中，p∈[2,+∞),C2與n,L,α,β無(wú)關(guān).

引理2 設(shè)f∈C1(ω;H-2(Ω)),則存在常數(shù)C3使得

其中，C3與f,γ有關(guān),而與n,L,α,β無(wú)關(guān).

證明 將-λjajn乘以式(6)的第j個(gè)方程,對(duì)j從1到n求和,得

(unt-Δunt+αΔunx+βunx-γΔ3un,Δun)=

(-J(un,Δun)+f,Δun).

(11)

(αΔunx+βunx,Δun)=0,

(-γΔ3un,Δun)=-γ‖Δ2un‖2,(-J(un,Δun),Δun)=-(unxΔuny-unyΔunx,Δun)=

則由式(11)得

(12)

在上式兩端關(guān)于t從0到ω積分,注意到un的周期性,得

由積分中值定理,得?t*∈[0,ω)使得

另一方面,由式(12)得

上式兩端關(guān)于t從t*到t(t∈[t*,t*+ω))積分,由引理1得

‖Δun(t)‖2≤‖▽un(t*)‖2+‖Δun(t*)‖2+

引理3 設(shè)f∈C1(ω;H-1(Ω)),則存在常數(shù)C5使得

其中C5與n,L,α,β無(wú)關(guān).

證明 將(-λj)2ajn乘以式(6)的第j個(gè)方程,對(duì)j從1到n求和,得

(unt-Δunt+αΔunx+βunx-γΔ3un,Δ2un)=

(-J(un,Δun)+f,Δ2un).

(13)

(αΔunx+βunx,Δ2un)=0,

(-γΔ3un,Δ2un)=γ‖▽?duì)?un‖2,|(-J(un,Δun),Δ2un)|=|(unxΔuny-unyΔunx,Δ2un)|=

2‖▽un‖L∞‖Δun‖‖▽?duì)?un‖≤

(14)

在上式兩端關(guān)于t從0到ω積分,注意到un的周期性,得

由積分中值定理,?t*∈[0,ω)使得

上式兩端關(guān)于t從t*到t(t∈[t*,t*+ω))積分,由引理2得

‖▽?duì)n(t)‖2≤‖Δun(t*)‖2+‖▽?duì)n(t*)‖2+

引理4 設(shè)f∈C1(ω;L2(Ω)),則存在常數(shù)C6使得

其中C6與n,L,α,β無(wú)關(guān).

證明 將(-λj)3ajn乘以式(6)的第j個(gè)方程,對(duì)j從1到n求和,得

(unt-Δunt+αΔunx+βunx-γΔ3un,Δ3un)=

(-J(un,Δun)+f,Δ3un).

(15)

(αΔunx+βunx,Δ3un)=0,

(-γΔ3un,Δ3un)=-γ‖Δ3un‖2|(J(un,Δun),Δ3un)|=|(unxΔuny-unyΔunx,Δ3un)|≤

(16)

在上式兩端關(guān)于t從0到ω積分,注意到un的周期性,得

故?t*∈[0,ω)，使得

上式兩端關(guān)于t從t*到t(t∈[t*,t*+ω))積分,由引理3得

‖Δ2un(t)‖2≤‖▽?duì)n(t*)‖2+‖Δ2un(t*)‖2+

2 高階導(dǎo)數(shù)的先驗(yàn)估計(jì)

本節(jié)將得到方程(3)～(5)近似解的高階導(dǎo)數(shù)的先驗(yàn)估計(jì).簡(jiǎn)單起見(jiàn),記C為只依賴于γ,f,λ1,ω的常數(shù).

證明 將(-λj)2kajn乘以式(6)的第j個(gè)方程,對(duì)j從1到n求和,得

(unt-Δunt+αΔunx+βunx-γΔ3un,Δ2kun)=

(-J(un,Δun)+f,Δ2kun).

(17)

(αΔunx+βunx,Δ2kun)=0,(J(un,Δun),Δ2kun)=(▽?duì)-1(J(un,Δun),▽?duì)un).▽?duì)-1(J(un,Δun)的首項(xiàng)是unx▽?duì)uny-uny▽?duì)unx,其他項(xiàng)是∑unxαyβunxα′yβ′,其中，r=α+β,r′=α′+β′,r+r′=2k+3,2≤r,r′≤2k+1.

又(unx▽?duì)uny-uny▽?duì)unx,▽?duì)un)=0,故

|(J(un,Δun),Δ2kun)|≤C‖Δun‖L∞‖▽?duì)un‖2+

ρ(‖Δkun‖2+‖▽?duì)un‖2)≤C.

(18)

則由式(18)可得存在常數(shù)C,使得

引理6 設(shè)f∈C1(ω;H2k-2(Ω)),則存在常數(shù)C,使得

其中C與n,L,α,β無(wú)關(guān).

證明 將(-λj)2k+1ajn乘以式(6)的第j個(gè)方程,對(duì)j從1到n求和,得

(unt-Δunt+αΔunx+βunx-γΔ3un,Δ2k+1un)=

(-J(un,Δun)+f,Δ2k+1un).

(19)

(αΔunx+βunx,Δ2k+1un)=0,

(-γΔ3un,Δ2k+1un)=-γ‖Δk+2un‖2,

(J(un,Δun),Δ2k+1un)=(Δk(J(un,Δun),Δk+1un),

Δk(J(un,Δun)的首項(xiàng)是unxΔk+1uny-unyΔk+1unx,其他項(xiàng)是∑unxαyβunxα′yβ′,其中，r=α+β,r′=α′+β′,r+r′=2k+4,2≤r,r′≤2k+2.

又(unxΔk+1uny-unyΔk+1unx,Δk+1un)=0,故|(J(un,Δun),Δ2k+1un)|≤C‖Δun‖L∞‖Δk+1un‖2+

C(‖Δk+1un‖2+1)≤

ρ(‖▽?duì)un‖2+‖Δk+1un‖2)≤C.

(20)

則由式(20)可得存在常數(shù)C,使得

引理7 設(shè)f∈C1(ω;H2(Ω)),則存在常數(shù)C,使得

其中C與n,L,α,β無(wú)關(guān).

(unt-Δunt+αΔunx+βunx-γΔ3un,unt)=

(-J(un,Δun)+f,unt).

所以‖unt‖2+‖▽unt‖2=(-αΔunx-βunx+γΔ3un-J(un,Δun)+f,unt).當(dāng)f∈C1(ω;H2(Ω))時(shí),可得‖Δ3un‖2≤C,從而存在K使得

‖unt‖2+‖▽unt‖2≤K‖unt‖≤

(unt-Δunt+αΔunx+βunx-γΔ3un,Δmunt)=

(-J(un,Δun)+f,Δmunt)，

所以，

(-1)m(‖▽munt‖2+‖▽m+1unt‖2)=

(-αΔunx-βunx+γΔ3un-J(un,Δun)+

f,Δmunt)=(-1)m(▽m(-αΔunx-βunx+

γΔ3un-J(un,Δun)+f),▽munt).

故存在常數(shù)C,使得

3 解的整體存在性和唯一性

本節(jié)將得到方程(3)～(5)時(shí)間周期解的存在性和唯一性.

定理1 對(duì)任意的k≥0,若f∈C1(ω;Hk+2(Ω)),則問(wèn)題(3)～(5)存在時(shí)間周期解u(x,y,t)∈L∞(ω;Hk+5(Ω))∩W1,∞(ω;Hk(Ω)).

證明 由引理1～8及標(biāo)準(zhǔn)的緊性原理知，可選取{un(t)}的子序列,仍記為{un(t)}，使得對(duì)任意的k≥0,若f∈C1(ω;Hk+2(Ω)),則有

當(dāng)n→∞時(shí),非線性項(xiàng)

‖J(un,Δun)-J(u,Δu)‖=‖(unxΔuny-

unyΔunx)-(uxΔuy-uyΔux)‖=‖(unx-

ux)Δuny+ux(Δuny-Δuy)-(uny-uy)Δunx-

uy(Δunx-Δux)‖≤‖unx-ux‖‖Δuny‖+

‖ux‖‖Δuny-Δuy‖+‖uny-uy‖·

‖Δunx‖+‖uy‖‖Δunx-Δux‖→0.

由引理1～8，得

ut-Δut+αΔux+βux-γΔ3u+J(u,Δu)=f.

綜上可得定理1成立.

定理2 假設(shè)定理1的條件成立,當(dāng)M3足夠小時(shí),則定理1中得到的問(wèn)題(3)～(5)的時(shí)間周期解是唯一的.

證明 設(shè)u,v是問(wèn)題(3)～(5)的2個(gè)時(shí)間周期解,則

ut-Δut+αΔux+βux-γΔ3u+J(u,Δu)=f,

vt-Δvt+αΔvx+βvx-γΔ3v+J(v,Δv)=f,

兩式相減,并令w=u-v,得

wt-Δwt+αΔwx+βwx-γΔ3w+

J(u,Δu)-J(v,Δv)=0.

(21)

注意到J(u,Δu)-J(v,Δv)=(uxΔuy-uyΔux)-(vxΔvy-

vyΔvx)=wxΔuy-wyΔux+vxΔwy-vyΔwx=

J(w,Δu)+J(v,Δw),

則由式(21)，得

wt-Δwt+αΔwx+βwx-γΔ3w+J(w,Δu)+

J(v,Δw)=0.

將上式與w作內(nèi)積,得

(wt-Δwt+αΔwx+βwx-γΔ3w+

J(w,Δu)+J(v,Δw),w)=0.

(22)

注意到

(αΔwx+βwx,w)=0,

(γΔ3w,w)=γ‖▽?duì)‖2,

(J(w,Δu),w)=0,

(J(v,Δw),w)=?Ω(vxΔwy-vyΔwx)wdxdy=

-?Ω(vxwy-vywx)Δwdxdy=

其中，C與f,γ有關(guān),與α,β無(wú)關(guān).

δ(‖w‖2+‖▽w‖2)≤0.

故‖w(t)‖2+‖▽w(t)‖2≤(‖w(0)‖2+‖▽w(0)‖2)e-δt,?t≥0.

因?yàn)閣=u-v關(guān)于時(shí)間t是周期變化的,則對(duì)于任意正整數(shù)s∈N,有

‖w(t)‖2+‖▽w(t)‖2=‖w(t+sω)‖2+

‖▽w(t+sω)‖2,

所以

‖w(t)‖2+‖▽w(t)‖2≤(‖w(0)‖2+

‖▽w(0)‖2)e-δ(t+sω),

從而得到 ‖w(t)‖2+‖▽w(t)‖2=0.

綜上可得定理2成立.

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JIN Zhen1,2, WAN Long2

(1.CollegeofScience,NanchangInstituteofTechnology,Nanchang330099,China; 2.SchoolofInformationTechnology,JiangxiUniversityofFinanceandEconomics,Nanchang330013,China)

This paper studies the existence and uniqueness of time periodic solution for one type of fluid dynamics equation with inhomogeneous term. Firstly, the approximation sequence of time periodic solution is constructed using the Galerkin method. Next, the approximation sequence is verified to be convergent by means of a priori estimate and Leray-Schauder fixed point theorem. It is shown that there is a time periodic solution when the inhomogeneous term is periodic about time. We also prove that the solution is unique under certain conditions.

fluid dynamics equation; periodic solution; Galerkin method; Leray-Schauder fixed point theorem

2015-01-19.

江西省教育廳科技項(xiàng)目(GJJ150463.GJJ150464)；江西省自然科學(xué)基金資助項(xiàng)目(20151BAB211009,20161BAB201028)；國(guó)家自然科學(xué)青年基金項(xiàng)目(11601198)；南昌工程學(xué)院青年基金項(xiàng)目(2014KJ024).

金 珍(1982-)，ORCID:http://orcid.org/0000-0002-2534-3355,女，碩士，主要從事微分方程及其應(yīng)用研究.

*通信作者，ORCID:http://orcid:org/0000-0001-9770-532X,E-mail:cocu3328@163.com.

10.3785/j.issn.1008-9497.2017.01.006

O 175.2

A

1008-9497(2017)01-040-07