非线性色散方程、波方程的自相似解与低正则性
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摘要
本论文分为两部分。第二、三、四、五章研究一些非线性发展方程的自相似解；第六、七章研究非线性Schr(?)dinger方程及Schr(?)dinger-Boussinesq方程的低正则问题。一般地，考虑偏微分方程这里，α=(α_0，……，α_n)为多重指标，假设方程(1.1)具有这样的代数对称性：存在实数α_i，b，对(?)λ∈(0，∞)，若令则仍有对满足相似结构(1.3)的偏微分方程(1.1)，我们考虑它的具有自相似结构的解，即(相似结构及其在方程中的应用可参见[4]，[20])具体到非线性发展方程(令x_0=t)，设u(x，t)是下列诸类非线性方程的Cauchy问题的解，则分别是Cauchy问题(1.4)，(1.5)将初值函数u_0(x)换成λ~(2/α)u_0(λx)的解，是波动方程的Cauchy问题(1.6)将(u_)(x)，u_1(x))换成(λ~(2/α)u_0(λx)，λ~(1+2/α)u_1(λx))时的解，
定义 0.1(ⅰ)称u(x，t)是Schr(?)dinger方程的Cauchy问题(1.4)或热传导方程的Cauchy问题(1.5)的自相似解，如果(ⅱ)称u(x，t)是波方程的Cauchy问题(1.6)的自相似解，如果研究自相似解的经典方法是通过求解椭圆型方程的解来实现的。例如，就热传导方程
而言，设V(x)是椭圆方程
△v+玉二.:v+里v+v3=0，
ZZ
第一章摘要
(1 .10)
的解，则试二，约=。一去v(t一告幻是(l .9)的自相似解.其中称v是。的波阵面.当。=1时，
常微分方程(1 .10)相对来说容易研究;当n全2时，通过椭圆方程(例如(1.10))来求解自相
似解是件困难的事.然而，借助于调和分析方法可以研究非线性发展方程的自相似解.事实
上，设方程(19)的初始条件
二(x，0)二。。(x)，(1 .11)
满足
入。。(入x)==。。(x)，
且热方程(1.9)的Cauchy问题(1.n)在某个合适的Banach空间中适定，若记(1.9)，(1.U)
的解是。(x，t)，则由唯一性可知。(x，约就是自相似解.
自相似解的适定性问题能否在经典的适定性框架下加以解决?一般地，对抽象Cauchy
问题
“‘+A”=F(“)，。(o)=必(二)，价cX，(1 .12)
适定性的研究是通过其积分方程
·(。)一，+芜‘一‘卜·，‘F(·)(·)、，，。X，
(1 .13)
来实现的.确定基本工作空间的原则是:。一tA在x中生成C0半群，即对v价。X，e一以价任
C(I;X)，I=R或R+.由Minlin一H6rmander乘子定理(见【3」)，对Schr6dinger方程，波动
方程而言，x=H‘(R勺，。cR;对热传导方程而言，X=H”，3〔R，1 到非线性问题时，已有的结果表明，当X是临界或次临界空间时，间题(1.12)或(1.13)在
C(无X)中局部适定.对于Schr6dinger方程、热方程、波动方程而言，齐次临界空间满足
}I。。(x)11分二。=11*荟。。(*x)11方:。，v*>o，
11。。(x)11方。。.;=11入誉。。(*x)l!方二。.p，v入>o，vi<; l}。。(x)11方:。=11*荟。。(入x)日户。。，v入>o，
由此可得s。二晋一景，s。=登一号，8。=晋一孚当。>s。时，分别称H‘，HS)P，H‘为对应
于三类方程的次临界空间，反之则为超临界空间.
取自相似解初始条件的一个特例tL。(x)=}二}一荟，按照经典适定性研究的要求应有
。。(x):方一，。。一要一兰，sohr6d，nger方程，
‘Q
”o(x)。H‘。，p，
2二、_
一一，热万程.
(1 .14)
(1 .15)
一一
几一P
一
以
S
下面仅验证(l .14)不成立.事实上由公。(劫二。圈卜“可知
r凶rZ，+1
11”。(x)11升一jR。l“I’‘“’}“。(“)I，“一_艺22‘。’关，l“。(“)l“‘一co·
J=一。。一
因此经典的结果不能用于形如二。(:)=}:}一景的cauchy间题(l .4)，(l .5)的研究.为此需将
Cauchy问题(1.4)，(1.5)，(1.6)适定性的概念推广，即推广连续依赖的概念，保留唯一性，以
达到求解自相似解的目的。
注意到Cazenave，tVeissler，Ribaud，YOussifi，Peeher等人关于Sehr6dinger方程，热方程，
波动方程的工作113」，!141，{45」，【47]，【49〕，我们在第二章中考虑了复Ginzberg一Landau(CGL)
方程的Cauchy问题
二:一:△二一乞△二+(a+乞乙)}。}。。=O，
。(x，O)=。。(x).
的衰减估计，给出解在空间
(1 .16)
通过建立s:(。)=。“‘△e“△
xs，;={
二任S‘(R几xR+
SUP
七>0
t月(，，，)}}。(x。t)}}方。，， 中的存在唯一性，从而当初始函数。。(x)同时满足
。。(二)=入号二。(*x)，I}又(。)。。I!二。，，<<1，(217)
时，cauchy问题在X，，;中有唯一的自相似解。
定理住1设。任U，“任Ja，，任△，p=瑞石，则存在依赖于‘的常数拭的>0，使
得当
1}凡(t)二。{}x。，;<占(:)，
时，cGL在X、;中有唯一的解试t，x)满足，
1{二(亡)日x，，;<2咨(。).
这里U，Ja，△表示指标的范围
In this dissertation, we use the tool of harmonic analysis to study the problem of self-similar solution and low regularity of wave and dispersive wave equations.
In Chapter 2, Chapter 3, we establish the generalized well-posedness of the Cauchy problem of complex Ginzberg-Landau(CGL) equation, which includes the self-similar solutions. Furthermore, we also prove the self-similar solutions to a group of CGL equations converge to the self-similar solution to the limiting equation under suitable conditions.
In Chapter 4, we give the new nonlinear estimates in spaces of Besov type. Using the estimates, we prove the global well-posedness of Schr dinger equation with general nonlinear term. This extends the known well-poseness results.
In Chapter 5, we study the self-similar solution of nonlinear wave equation. By using the nonlinear estimate in Chapter 4 together with the Strichartz estimates of linear wave equation, we prove the global well-posedness of the self-similar solution of the wave equation with general nonlinear term.
In Chapter 6, the Cauchy problem of Schr(?)dinger equation with the initial data possessing infinite L2 norm is considered. Using the method introduced by Bourgain, we prove the Cauchy problem of Schr(?)dinger equation is global well-posed for a class of initial data as the above. Since the global well-posedness of Schr(?)dinger equation in Hs, s < 0 has never been obtained to our best knowledge, our result is an improvement in some sense.
In Chapter 7, we consider the global well-posedness of two dimensional Schr(?)dinger-Boussinesq coupled system when the initial data belong to Hs, 0 < s < 1. We show the solution is globally wellposed when s > 8/(11), and is locally wellposed when s > -1/4.
定义 0.1(ⅰ)称u(x，t)是Schr(?)dinger方程的Cauchy问题(1.4)或热传导方程的Cauchy问题(1.5)的自相似解，如果(ⅱ)称u(x，t)是波方程的Cauchy问题(1.6)的自相似解，如果研究自相似解的经典方法是通过求解椭圆型方程的解来实现的。例如，就热传导方程
而言，设V(x)是椭圆方程
△v+玉二.:v+里v+v3=0，
ZZ
第一章摘要
(1 .10)
的解，则试二，约=。一去v(t一告幻是(l .9)的自相似解.其中称v是。的波阵面.当。=1时，
常微分方程(1 .10)相对来说容易研究;当n全2时，通过椭圆方程(例如(1.10))来求解自相
似解是件困难的事.然而，借助于调和分析方法可以研究非线性发展方程的自相似解.事实
上，设方程(19)的初始条件
二(x，0)二。。(x)，(1 .11)
满足
入。。(入x)==。。(x)，
且热方程(1.9)的Cauchy问题(1.n)在某个合适的Banach空间中适定，若记(1.9)，(1.U)
的解是。(x，t)，则由唯一性可知。(x，约就是自相似解.
自相似解的适定性问题能否在经典的适定性框架下加以解决?一般地，对抽象Cauchy
问题
“‘+A”=F(“)，。(o)=必(二)，价cX，(1 .12)
适定性的研究是通过其积分方程
·(。)一，+芜‘一‘卜·，‘F(·)(·)、，，。X，
(1 .13)
来实现的.确定基本工作空间的原则是:。一tA在x中生成C0半群，即对v价。X，e一以价任
C(I;X)，I=R或R+.由Minlin一H6rmander乘子定理(见【3」)，对Schr6dinger方程，波动
方程而言，x=H‘(R勺，。cR;对热传导方程而言，X=H”，3〔R，1
C(无X)中局部适定.对于Schr6dinger方程、热方程、波动方程而言，齐次临界空间满足
}I。。(x)11分二。=11*荟。。(*x)11方:。，v*>o，
11。。(x)11方。。.;=11入誉。。(*x)l!方二。.p，v入>o，vi<;
由此可得s。二晋一景，s。=登一号，8。=晋一孚当。>s。时，分别称H‘，HS)P，H‘为对应
于三类方程的次临界空间，反之则为超临界空间.
取自相似解初始条件的一个特例tL。(x)=}二}一荟，按照经典适定性研究的要求应有
。。(x):方一，。。一要一兰，sohr6d，nger方程，
‘Q
”o(x)。H‘。，p，
2二、_
一一，热万程.
(1 .14)
(1 .15)
一一
几一P
一
以
S
下面仅验证(l .14)不成立.事实上由公。(劫二。圈卜“可知
r凶rZ，+1
11”。(x)11升一jR。l“I’‘“’}“。(“)I，“一_艺22‘。’关，l“。(“)l“‘一co·
J=一。。一
因此经典的结果不能用于形如二。(:)=}:}一景的cauchy间题(l .4)，(l .5)的研究.为此需将
Cauchy问题(1.4)，(1.5)，(1.6)适定性的概念推广，即推广连续依赖的概念，保留唯一性，以
达到求解自相似解的目的。
注意到Cazenave，tVeissler，Ribaud，YOussifi，Peeher等人关于Sehr6dinger方程，热方程，
波动方程的工作113」，!141，{45」，【47]，【49〕，我们在第二章中考虑了复Ginzberg一Landau(CGL)
方程的Cauchy问题
二:一:△二一乞△二+(a+乞乙)}。}。。=O，
。(x，O)=。。(x).
的衰减估计，给出解在空间
(1 .16)
通过建立s:(。)=。“‘△e“△
xs，;={
二任S‘(R几xR+
SUP
七>0
t月(，，，)}}。(x。t)}}方。，，
。。(二)=入号二。(*x)，I}又(。)。。I!二。，，<<1，(217)
时，cauchy问题在X，，;中有唯一的自相似解。
定理住1设。任U，“任Ja，，任△，p=瑞石，则存在依赖于‘的常数拭的>0，使
得当
1}凡(t)二。{}x。，;<占(:)，
时，cGL在X、;中有唯一的解试t，x)满足，
1{二(亡)日x，，;<2咨(。).
这里U，Ja，△表示指标的范围
In this dissertation, we use the tool of harmonic analysis to study the problem of self-similar solution and low regularity of wave and dispersive wave equations.
In Chapter 2, Chapter 3, we establish the generalized well-posedness of the Cauchy problem of complex Ginzberg-Landau(CGL) equation, which includes the self-similar solutions. Furthermore, we also prove the self-similar solutions to a group of CGL equations converge to the self-similar solution to the limiting equation under suitable conditions.
In Chapter 4, we give the new nonlinear estimates in spaces of Besov type. Using the estimates, we prove the global well-posedness of Schr dinger equation with general nonlinear term. This extends the known well-poseness results.
In Chapter 5, we study the self-similar solution of nonlinear wave equation. By using the nonlinear estimate in Chapter 4 together with the Strichartz estimates of linear wave equation, we prove the global well-posedness of the self-similar solution of the wave equation with general nonlinear term.
In Chapter 6, the Cauchy problem of Schr(?)dinger equation with the initial data possessing infinite L2 norm is considered. Using the method introduced by Bourgain, we prove the Cauchy problem of Schr(?)dinger equation is global well-posed for a class of initial data as the above. Since the global well-posedness of Schr(?)dinger equation in Hs, s < 0 has never been obtained to our best knowledge, our result is an improvement in some sense.
In Chapter 7, we consider the global well-posedness of two dimensional Schr(?)dinger-Boussinesq coupled system when the initial data belong to Hs, 0 < s < 1. We show the solution is globally wellposed when s > 8/(11), and is locally wellposed when s > -1/4.
引文
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[2] J.J.Benedetto, Harmonic Analysis and Applications, CRC Press, 1996．
[3] J.Bergh, J.Lofstrom, Interpolation Spaces Berlin-Heidelberg New-York, 1976．
[4] G. W. Bluman, J. D. Cole, Similarity Methods for Differential Equations, Springer-Verlag New-York Heidelberg Berlin, 1974．
[5] J.Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I, Geom. Funct. Anal. 3, 1993, 107-156．
[6] J.Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Ⅱ, Geom. Funct. Anal. 3, 1993, 209-262．
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