非线性Boussinesq流的长时间形态
摘要
本文分为两个部分，第一部分研究了在Boussinesq近似下的热对流
方程。主要讨论了时间周期解，指数吸引子和吸引子的上半连续性。
第二部分主要讨论了NLS-Boussinesq耦合组的时间周期解和吸引子的正
则性。
全文共分六章，第一章为绪言，简要介绍了Boussinesq近似下的热
对流方程和NLS-Boussinesq耦合组的物理背景，研究状况以及本文所讨
论的基本内容。第二章研究Boussinesq近似下的热对流方程的时间周
期解。首先利用Larey-Sauchder不动点定理证明近似解的存在性，然后
对方程的近似解作高阶导数（关于空间变量和时间变量）估计，最后利
用紧致性方法证明了在R~d（d=3或4）的有界区域上，当外力较小时，方
程存在时间周期解。同时补充一定条件后，给出了一个唯一性结果。
第三章研究了指数吸引子的存在性。首先证明解算子S（t）是Lipschitz连
续的，其次证明离散的解算子S_＊=S（t_＊）具有挤压性质，利用A.Eden和
C.Foias等人构造的理论得到Boussinesq近似下的热对流方程的指数吸引
子的存在性。第四章研究了吸引子的上半连续性。考虑带扰动项的
方程，利用算子分解技术证明其紧吸引子Α（ε）的存在性，然后在证明
lim_ε→0+dist（Α（ε），Α（0））=0，从而得到Boussinesq近似下热对流方程的吸引
子的上半连续性。
第五，第六章为该文的第二部分，对NLS-Boussinesq耦合组的时间
周期解和吸引子的正则性进行了研究。在第五章中用类似于第二章中
的方法证明了该耦合组的时间周期解的存在性，且指出在小外力的情
况下其周期解是唯一的。第六章研究了NLS-Bousinesq耦合的吸引子的
正则性。对解算子进行分解，构造渐进紧的不变集，得到吸引子Α_0在
空间Ε_0中的存在性，进而证明Α_0也是Ε_1中的吸引子，即Α_0=Α_1。
This dissertation consists of two parts. In one part we consider the ex-
istence of the time periodic solutions, exponential attractors and the upper
semi-continuity of the global attractors for the coupled system of equations
of fluid and temperature in the Boussinesq approximation. In the other part,
we consider the existence of the time periodic solutions and regularity of the
global attractors for the coupled system NLS-Boussinesq.
This dissertation consists of six chapters. In chapter 1, we briefly in-
troduce background in physics and the developments in mathematics for the
coupled system of equations of fluid and temperature in the Boussinesq ap-
proximation and the coupled system NLS-Boussinesq. in which the main work
of the dissertation is also described. In chapter 2, we discuss the time peri-
odic solution of the coupled system of equations of fluid and temperature in
Boussinesq approximation. First, we apply the fixed theorem of the Larey-
Schauder to prove the existence of the approximate solution. Next, we get
the estimates of the higher order derivatives (with respect to spatial variable
and time variable) of the approximate solutions. Finally, we use the method
of standard compactness arguments to get the existence of this system in a
bounded domain ~l (€ Rd, d 3,4), whenever the external force are small.
At the same time, after supplementing some conditions we get the result of
uniqueness. In chapter 3, we study the existence of the exponential attractors
for the coupled system of equations of fluid and temperature in Boussinesq
approximation. We first show that the solution operator S(t) is Lipschitz con-
tinuous, then the discrete solution operator S~ (t*) satisfy the squeezing
property, use the theory given by A. Eden and C. Foias, we get the existence
of the exponential attractors M whose fractal dimension is finite. In chapter 4,
we study the upper semi-continuity of global attractors for the coupled system
of equations fluid and temperature in Boussinesq approximation. Considering
the equations with the singularly pertubed term, and decomposing the solu-
tion operator, we first prove the existence of the global attractors .4(&), then
prove that
lim dist(A(5),A(O)) = 0.
?0+
In chapter 5, using the same arguments in chapter 2 for the coupled
system of NLS-Boussinesq, we prove the existence of the time periodic solution
and point out that the time periodic solution is unique when the external force
are small. In chapter 6, using the technology of decomposing solution operator
and constructing the asymptotic compact invariant set, we get the existence
of the global attractor ..4~ in the space E0. Furthermore, A0 is the global
attractor in the space E0, that is A0 = A0.
方程。主要讨论了时间周期解，指数吸引子和吸引子的上半连续性。
第二部分主要讨论了NLS-Boussinesq耦合组的时间周期解和吸引子的正
则性。
全文共分六章，第一章为绪言，简要介绍了Boussinesq近似下的热
对流方程和NLS-Boussinesq耦合组的物理背景，研究状况以及本文所讨
论的基本内容。第二章研究Boussinesq近似下的热对流方程的时间周
期解。首先利用Larey-Sauchder不动点定理证明近似解的存在性，然后
对方程的近似解作高阶导数（关于空间变量和时间变量）估计，最后利
用紧致性方法证明了在R~d（d=3或4）的有界区域上，当外力较小时，方
程存在时间周期解。同时补充一定条件后，给出了一个唯一性结果。
第三章研究了指数吸引子的存在性。首先证明解算子S（t）是Lipschitz连
续的，其次证明离散的解算子S_＊=S（t_＊）具有挤压性质，利用A.Eden和
C.Foias等人构造的理论得到Boussinesq近似下的热对流方程的指数吸引
子的存在性。第四章研究了吸引子的上半连续性。考虑带扰动项的
方程，利用算子分解技术证明其紧吸引子Α（ε）的存在性，然后在证明
lim_ε→0+dist（Α（ε），Α（0））=0，从而得到Boussinesq近似下热对流方程的吸引
子的上半连续性。
第五，第六章为该文的第二部分，对NLS-Boussinesq耦合组的时间
周期解和吸引子的正则性进行了研究。在第五章中用类似于第二章中
的方法证明了该耦合组的时间周期解的存在性，且指出在小外力的情
况下其周期解是唯一的。第六章研究了NLS-Bousinesq耦合的吸引子的
正则性。对解算子进行分解，构造渐进紧的不变集，得到吸引子Α_0在
空间Ε_0中的存在性，进而证明Α_0也是Ε_1中的吸引子，即Α_0=Α_1。
This dissertation consists of two parts. In one part we consider the ex-
istence of the time periodic solutions, exponential attractors and the upper
semi-continuity of the global attractors for the coupled system of equations
of fluid and temperature in the Boussinesq approximation. In the other part,
we consider the existence of the time periodic solutions and regularity of the
global attractors for the coupled system NLS-Boussinesq.
This dissertation consists of six chapters. In chapter 1, we briefly in-
troduce background in physics and the developments in mathematics for the
coupled system of equations of fluid and temperature in the Boussinesq ap-
proximation and the coupled system NLS-Boussinesq. in which the main work
of the dissertation is also described. In chapter 2, we discuss the time peri-
odic solution of the coupled system of equations of fluid and temperature in
Boussinesq approximation. First, we apply the fixed theorem of the Larey-
Schauder to prove the existence of the approximate solution. Next, we get
the estimates of the higher order derivatives (with respect to spatial variable
and time variable) of the approximate solutions. Finally, we use the method
of standard compactness arguments to get the existence of this system in a
bounded domain ~l (€ Rd, d 3,4), whenever the external force are small.
At the same time, after supplementing some conditions we get the result of
uniqueness. In chapter 3, we study the existence of the exponential attractors
for the coupled system of equations of fluid and temperature in Boussinesq
approximation. We first show that the solution operator S(t) is Lipschitz con-
tinuous, then the discrete solution operator S~ (t*) satisfy the squeezing
property, use the theory given by A. Eden and C. Foias, we get the existence
of the exponential attractors M whose fractal dimension is finite. In chapter 4,
we study the upper semi-continuity of global attractors for the coupled system
of equations fluid and temperature in Boussinesq approximation. Considering
the equations with the singularly pertubed term, and decomposing the solu-
tion operator, we first prove the existence of the global attractors .4(&), then
prove that
lim dist(A(5),A(O)) = 0.
?0+
In chapter 5, using the same arguments in chapter 2 for the coupled
system of NLS-Boussinesq, we prove the existence of the time periodic solution
and point out that the time periodic solution is unique when the external force
are small. In chapter 6, using the technology of decomposing solution operator
and constructing the asymptotic compact invariant set, we get the existence
of the global attractor ..4~ in the space E0. Furthermore, A0 is the global
attractor in the space E0, that is A0 = A0.
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