某些非线性发展方程的整体吸引子
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摘要
本文研究了长短波方程组、Hasegawa-Mima方程和Hirota方程等非线性发展方程的周期边值问题或初值问题的动力学行为，得到了相应问题整体解的存在性、整体吸引子的存在性及其分形维数的有限性估计，构造了相应问题的近似惯性流形。
全文内容共分五章。
第一章，给出了长短波方程组、Hasegawa-Mima方程和Hirota方程等非线性发展方程的物理背景，回顾了已有的部分重要成果，简述了本文主要的研究结论。
第二章，考虑了一类一维广义的长短波方程组的周期边值问题。第二节，利用一致先验估计和Galerkin方法，证明了具有周期边值问题的整体吸引子的存在性，估计了吸引子的维数。第三节，构造了两种具有周期边值问题的近似惯性流形，并得到了其逼近整体吸引子的阶数。
第三章，研究了一类一维广义的长短波方程组的初值问题。第二节，应用Kato关于拟线性演化方程的初值问题的理论，讨论了初值问题解的存在性。第三节，通过引入加权空间，在加权空间进行先验估计，利用加权函数在加权空间上的插值不等式，得到了无界区域上的整体吸引子的存在性。
第四章，考虑了Hasegawa-Mima方程的周期边值问题。第二节，利用一致先验估计和Galerkin方法，得到了二维Hasegawa-Mima方程的周期边值问题的整体光滑解的存在性。第三节，证明了一类二维广义Hasegawa-Mima方程整体光滑解的存在性以及在某种特殊情形下，Hasegawa-Mima方程的解趋近于相应的准地转风方程的弱解。在第二节的基础上，第四节讨论了二维Hasegawa-Mima方程的整体吸引子的存在性，并给出了整体吸引子的维数估计。第五节，通过建立与时间t无关的一致先验估计，证明了三维广义Hasegawa-Mima方程组整体光滑解、整体吸引子的存在性。
第五章，研究了一类具耗散的Hirota方程的周期边值问题。第二节，利用一致先验估计，得到了整体解的存在唯一性。第三节，先构造了该问题在H~2中的整体弱吸引子，然后通过能量方程证明整体弱吸引子实际上是H~2中的整体强吸引子。
本文的主要特点和难点在于作高维问题、非线性方程组问题及其无界区域问题的先验估计时，都遇到了许多困难。因此针对不同的问题，我们采用一系
In this paper, we consider the dynamical behavior for some nonlinear evolution equations, such as Long-Short wave equations, Hasegawa-Mima equation and Hirota equation. The existence of globally smooth solutions, the existence of global attractors, its fractal dimensions and the approximate inertial manifolds for this systems are obtained.
This paper is organized in five chapters.
In chapter 1, we give the physical background for the nonlinear evolution equations, such as Long-Short wave equations, Hasegawa-Mima equation and Hirota equation. We recall some important known results and briefly describe our research results of the present paper.
In chapter 2, we consider a class of generalized Long-Short wave equations in one dimension. In section 2.2, by uniformly a priori estimates and Galerkin method, we prove the existence of the global attractors for the periodic boundary value problem, and we get the estimates of the upper bounds of Hausdorff and fractal dimensions for the global attractors. In section 2.3, we construct the two approximate inertial manifolds for the periodic boundary value problem. We obtain the order of approximation of the manifolds to the global attractors.
In chapter 3, we consider a class of generalized Long-Short wave equations in one dimension. In section 3.2, by Kato's method for abstract quasi-linear evolution equations, we consider the initial value problem. In section 3.3, by introducing weighted space and using the method of priori estimates, by using the weighted function space and the interpolation inequality, we prove the existence of the global attractors for the Long-Short wave equations on an unbounded domain.
In chapter 4, we consider the periodic boundary value problem of the Hasegawa-Mima equation. In section 4.2, by uniformly a priori estimates and Galerkin method, we prove the existence of the globally smooth solution for the two-dimensional Hasegawa-Mima equation. In section 4.3, we also prove the existence of the globally smooth solution for a class of generalized Hasegawa-Mima equa-tion(GHM equation) in two dimension. In some special case, we prove that the
全文内容共分五章。
第一章，给出了长短波方程组、Hasegawa-Mima方程和Hirota方程等非线性发展方程的物理背景，回顾了已有的部分重要成果，简述了本文主要的研究结论。
第二章，考虑了一类一维广义的长短波方程组的周期边值问题。第二节，利用一致先验估计和Galerkin方法，证明了具有周期边值问题的整体吸引子的存在性，估计了吸引子的维数。第三节，构造了两种具有周期边值问题的近似惯性流形，并得到了其逼近整体吸引子的阶数。
第三章，研究了一类一维广义的长短波方程组的初值问题。第二节，应用Kato关于拟线性演化方程的初值问题的理论，讨论了初值问题解的存在性。第三节，通过引入加权空间，在加权空间进行先验估计，利用加权函数在加权空间上的插值不等式，得到了无界区域上的整体吸引子的存在性。
第四章，考虑了Hasegawa-Mima方程的周期边值问题。第二节，利用一致先验估计和Galerkin方法，得到了二维Hasegawa-Mima方程的周期边值问题的整体光滑解的存在性。第三节，证明了一类二维广义Hasegawa-Mima方程整体光滑解的存在性以及在某种特殊情形下，Hasegawa-Mima方程的解趋近于相应的准地转风方程的弱解。在第二节的基础上，第四节讨论了二维Hasegawa-Mima方程的整体吸引子的存在性，并给出了整体吸引子的维数估计。第五节，通过建立与时间t无关的一致先验估计，证明了三维广义Hasegawa-Mima方程组整体光滑解、整体吸引子的存在性。
第五章，研究了一类具耗散的Hirota方程的周期边值问题。第二节，利用一致先验估计，得到了整体解的存在唯一性。第三节，先构造了该问题在H~2中的整体弱吸引子，然后通过能量方程证明整体弱吸引子实际上是H~2中的整体强吸引子。
本文的主要特点和难点在于作高维问题、非线性方程组问题及其无界区域问题的先验估计时，都遇到了许多困难。因此针对不同的问题，我们采用一系
In this paper, we consider the dynamical behavior for some nonlinear evolution equations, such as Long-Short wave equations, Hasegawa-Mima equation and Hirota equation. The existence of globally smooth solutions, the existence of global attractors, its fractal dimensions and the approximate inertial manifolds for this systems are obtained.
This paper is organized in five chapters.
In chapter 1, we give the physical background for the nonlinear evolution equations, such as Long-Short wave equations, Hasegawa-Mima equation and Hirota equation. We recall some important known results and briefly describe our research results of the present paper.
In chapter 2, we consider a class of generalized Long-Short wave equations in one dimension. In section 2.2, by uniformly a priori estimates and Galerkin method, we prove the existence of the global attractors for the periodic boundary value problem, and we get the estimates of the upper bounds of Hausdorff and fractal dimensions for the global attractors. In section 2.3, we construct the two approximate inertial manifolds for the periodic boundary value problem. We obtain the order of approximation of the manifolds to the global attractors.
In chapter 3, we consider a class of generalized Long-Short wave equations in one dimension. In section 3.2, by Kato's method for abstract quasi-linear evolution equations, we consider the initial value problem. In section 3.3, by introducing weighted space and using the method of priori estimates, by using the weighted function space and the interpolation inequality, we prove the existence of the global attractors for the Long-Short wave equations on an unbounded domain.
In chapter 4, we consider the periodic boundary value problem of the Hasegawa-Mima equation. In section 4.2, by uniformly a priori estimates and Galerkin method, we prove the existence of the globally smooth solution for the two-dimensional Hasegawa-Mima equation. In section 4.3, we also prove the existence of the globally smooth solution for a class of generalized Hasegawa-Mima equa-tion(GHM equation) in two dimension. In some special case, we prove that the
引文
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[2] 郭柏灵．粘性消去法和差分格式的粘性[M]．科学出版社．1993．
[3] 郭柏灵．非线性演化方程[M]．上海科技教育出版社．1995．
[4] M. Taylor. Partial Differential Equations[M], Ⅰ-Ⅲ Springer-Verlay, Berlin. 1996.
[5] L. C. Evans. Partial Differential Equations[M], Ⅰ-Ⅱ. American Mathematical Society Providence Rhode island. 1998.
[6] A. Haraux. Nonlinear Evolution Equations-Global Behavior of Solutions [M]. Lecture Notes in Mathematics, Springer-Verlag Berlin Heidelbery. 1981.
[7] R. Temam. Infinite Dimensional Dynamical Systems in Mechanics and Physics[M]. Springer-Verlag. New York. 2000.
[8] 郭柏灵．无穷维动力系统(上、下册)[M]．国防工业出版社．2000．
[9] G．B．惠瑟姆．线性与非线性波[M]．科学出版社．1986．
[10] D. J. Benney. Significant interactions between long and short gravity waves[J]. Stud. Appl. Math., 55 (1976), 93-106.
[11] D. J. Benney. General theory for interactions between short and long waves[J]. Stud. Appl. Math., 56 (1977), 81-94.
[12] M. Tsutsumi, S. Hatano. Well-posedness of the Cauchy problem for the long wave-short wave resonance equations [J]. Nonlinear Analysis, Theory, Methods and Applications. 2(1994), 155-171.
[13] T. Kawahara, N. Sugimoto, T. Kakutani. Nonlinear interaction between short and long capillary-gravity waves[J]. J. Phys. Soc. Japan. 39 (1975), 1379-1386.
[14] V. D. Djordjevic, L. G. Redekopp. On two-dimensional packets of capillary-gravity waves[J]. J. Fluid Mech. 79 (1977), 703-714.
[15] D. Bekiranov, T. Ogawa, G. Ponce. Interaction equations for short and long dispersive waves[J]. Journal of Functional Analysis. 158 (1998), 357-388.
[16] R. H. J. Grimshaw. The modulation of an internal gravity-wave packet and the resonance with the mean motion[J]. Stud. Appl. Math. 56 (1977), 241-266.
[17] M. Funakoshi, M. Oikawa. The resonant interaction between a long internal gravity wave and a surface gravity wave packet[J]. J. Phys. Soc. Japan. 52 (1983), 1982-1995.
[18] V. I. Karpman. On the dynamics of sonic-Langmuir soliton[J]. Physica Scripta. 11 (1975), 263-265.
[19] N. Yajima, M. Oikawa. Formation and interaction of sonic-Langmuir soliton[J], Progr. Theor. Phys. 56 (1976), 1719-1739.
[20] B. L. duo. The global solution for one class of the system of LS nonlinear wave interaction[J]. J. Math. Res. and Exposition. 1 (1987), 69-76.
[21] B. L. duo. The periodic initial value problems and initial value problems for one class of generalized LS type equations[J]. J. Engineering Math. 8(1) (1991), 47-53.
[22] B. L. duo, B. X. Wang. Attractors for the Long-Short wave equations[J]. J. Partial Diff. Eqs., 11 (1998), 361-383.
[23] B. L. duo, B.X. Wang. The global solution and its long time behavior for a class of generalized LS type equations[J]. Prod. Natural Sci. 5 (1996), 533-546.
[24] H.J. Gao. Attractor for weakly damped driven long wave short wave resonance equation[J]. Appl. Math. JCU. 13B (1998), 377-384.
[25] 杜先云，郭柏灵．长短波方程在R~1上的整体吸引子[J]．应用数学学报．28(4)(2005)，723-734．
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