【作者】 王怡；

【导师】 司建国；

【作者基本信息】 山东大学 ， 基础数学， 2012， 博士

【摘要】 本文,我们研究非线性梁方程.梁方程起源于Eulcr-Bernoulli梁方程.研究梁方程的经典梁理论是线性弹性理论的简单化,它提供了一种计算梁的载重和偏转特性的方法.最早在大约1750年被提出,但并没有大规模应用,直到19世纪末Eiffel铁塔和Ferris大转轮的建造.紧跟着这些成功的范例,梁方程及其理论变成工程学的基石并推动了第二次工业革命.目前,梁方程在物理、工程、材料学上都有长足发展.本文主要研究如下非线性梁方程模型：uxxxx+utt=F(t,x,u),其中F为加入的某非线性项.以上非线性梁方程描述了一种没有能量耗散的过程,在适当的坐标下,它可以看做一种无穷维Hamilton系统.这种严格的性质使我们可以把它当做一种Hamilton动力系统来研究,并把一些在有限维系统中成熟的方法和理论推广过来.对于Hamilton系统,人们比较关心相空间的动力学行为.然而,除了少数可积系统外,对轨道的动力学行为并不清楚.从上世纪60年代以来,数学家们转而研究近可积系统,这类系统是除可积系统外非常重要而又简单的系统.研究有限维Hamilton系统的周期解是认识其动力学行为的一个重要课题.对于无穷维近可积Hamilton系统,我们不但要考虑周期解,也要考虑其拟周期解、概周期解.目前研究无穷维Hamilton系统的周期解、拟周期解主要有两种方法.一种方法是Craig-Wayne-Bourgain法,它从Liapunov-Schmidt分解和Newton迭代发展而来.另一种方法就是由经典KAM理论推广而来的无穷维KAM理论,而后者是本文的主要研究方法.经典的KAM理论在上世纪五、六十年代由三位著名数学家Kolmogorov [1], Arnold [2], Moscr [3]建立.该理论是Hamilton系统理论发展的里程碑,它合理解释了太阳系的稳定性,在上世纪九十年代被Wayne [4]和Kuksin [5]推广到了无穷维Hamilton系统,并由Poschcl [6]重新叙述.无穷维KAM理论推广了有限维的经典结论,是处理因“小除数”问题导致的正则性缺失的有效手段.而且它弥补了变分法的不足,得到的解是局部的、摄动的,具有小振幅.无穷维KAM理论及其结论被运用到多种偏微分方程模型拟周期解存在性问题的研究中.比如,含参数的一维非线性Schrodinger方程(NLS)iut-uxx+V(x,ζ)u=f(u),以及含参数的一维非线性波动方程(NLW)utt-uxx+V(x,ζ)u=f(u),其中V和f为满足一定条件的函数,ξ为参数.但是不能直接运用到势能函数V是一些特殊类型函数的方程中,比如V为常数的情况Poschel[7]开发了Birkhoff标准型的方法用来处理在Dirichlet边界条件下非线性波动方程势能函数是常数的情况.这种方法又被Kuksin和Poschel[8]运用到证明非线性Schrodinger方程相应于Dirichlet或Neumann边界条件拟周期解的存在性.类似于波动方程和Schrodinger方程,尤建功和耿建生[9]利用KAM理论和标准型方法证明了铰链边界条件下的梁方程拟周期解的存在性.另外值得一提的是,袁小平[10]利用KAM方法巧妙地解决了一维完全共振的波动方程拟周期解存在性问题.周期边界条件的情况则更为复杂,这是因为Sturm-Liouville算子L=-(?)+V的特征值出现了重根的情况Craig和Wayne[11,12]开发了基于Lyapunov-Schmidt分解和Frohlich和Spencer[13]技术的一种新技术.运用这种技术,他们[11]证明了在周期边界条件下非线性波动方程周期解的存在性.后来,他们的方法被Bourgain[14,15]推广并改进,运用到证明周期边界条件下非线性波动方程和非线性Schrodinger方程拟周期解的存在性问题上.耿建生和尤建功[16,17],梁振国[18]则成功运用KAM方法处理周期边界条件下的偏微分方程.另外,Bricmont,Kupiainen,和schenkel[19]利用重正化群的方法证明了非线性波动方程拟周期的、低维椭圆环面的存在性,这是一种区别与以往的新方法.而高维Hamilton偏微分方程的情况比较困难Bourgain[20]首先证明了2维非线性Schrodinger方程具有小振幅的拟周期解.他[21]改进了自已的方法,并证明高维非线性Schrodinger方程和波动方程具有小振幅的拟周期解.后来,耿建生和尤建功[17,22]证明了高维非线性梁方程和非局部的非线性Schrodinger方程具有小振幅的线性稳定的拟周期解Eliasson和Kuksin[23]证明了高维非线性Schrodinger方程小振幅的线性稳定的拟周期解的存在性.最近,耿建生和尤建功[24]获得了高维cubic Schrodinger方程的拟周期解.本文主要研究含有强迫项的方程的解的存在性问题.当强迫项是周期的时候,周期解的存在性常常用变分方法和Lyapunov-Schmidt分解法解决.对于拟周期解的存在性,Bcrti和Procesi[25]证明了周期强迫下完全共振的波动方程拟周期解的存在性Jiao和王弈倩[26]用标准型和KAM方法证明了拟周期强迫下的Schroinger方程拟周期解的存在性.张敏和司建国[27],司建国[28]分别研究了带有拟周期强迫的非线性波动方程在Dirichlet边界条件下和在周期边界条件下拟周期解的存在性Eliasson和Kuksin[29]解决了在周期边界条件下带拟周期强迫的高维非线性Schrodinger方程拟周期解的存在性问题.本文将研究非线性梁方程utt+uxxxx+μu+εg(ωt,x)u3=0,μ>0, x∈[0,π],和utt+uxxxx+μu+∈φ(t)h(u)=0, μ>0在铰链边界条件下拟周期解的存在性,其中ε和∈为小的正参数,ω=(ω1,ω2,..., ωm)∈[(?),2(?)]m((?)>0)是一个频率向量,函数g(ωt,x)=g(v,x),它对两个变量((?),x)∈Tm×[0,π]都是实解析的,而且对于变量t是拟周期的,非线性项h是实解析的奇函数Φ是一个实解析的关于时间变量t的拟周期函数.这两个问题既有共同之处,也有不同之处.相同之处在于,它们的非线性项中都含有时间变量.不同之处在于第一个方程的非线性项中含有空间变量,而第二方程的非线性项中虽然没有空间变量,但是第二个方程的势能函数不是常数而与时间变量有关.所以这两个问题的处理有所不同.我们采用的主要方法是把方程化为无穷维Hamilton系统的Birkhoff标准型,然后利用现有的无穷维KAM理论的结果找出拟周期解.第一个问题的难点在于证明存在一个辛的解析变换把Hamilton函数化为其Birkhoff正规型.由于摄动项中空间变量的缘故,我们失去了通常而言重要的条件i±j±d±l=0.因此,证明的主要困难有两点.一是“小除数”的测度估计问题,一方面,当估计测度时,条件i±j±d±l=0通常很重要,另一方面,与Schrodinger方程不同,梁方程的特征值不是整数,这些无疑对测度估计增加了难度.另一个困难是如何证明辛坐标变换的解析性,我们建立了一个技术性引理,并通过获取Fourier余弦级数的技巧克服这个困难.然而,对于第二个问题,一个大的难点在于无穷维线性拟周期系统的可约化性问题.因为,我们不得不把势能函数化为常数函数.实际上,这个问题本身就是一个有趣的开问题.本文,我们将建立一个无穷维KAM定理来解决该问题.本文由三章组成,主要内容如下：第一章给出Hamilton系统和KAM理论的相关知识.主要包括三节内容.第一节我们简要介绍非线性梁方程模型的由来和问题的研究背景.第二节我们介绍有限维Hamilton系统的知识,叙述经典的KAM理论,包括辛结构的定义、Darboux定理、Liouville定理、Liouville可积性定理、完全可积系统和近可积系统的定义等,并介绍Birkhoff正规型的相关结论.第三节主要介绍无穷维Hamilton系统和KAM理论,并叙述Kuksin著名的抽象无穷维KAM定理.最后简要叙述相关的研究进展.第二章将研究摄动项具有拟周期强迫且依赖于空间变量的非线性梁方程拟周期解的存在性.首先介绍问题的研究背景,然后给出主要结论.要解决这个问题,需要把偏微分方程化为其Hamilton形式,然后再化成Birkhoff正规型.我们还将介绍一个基本的偏微分方程的无穷维KAM定理,利用该定理和得到的Birkhoff正规型,我们给出主要结果的证明.第三章研究具有拟周期势能的非线性梁方程拟周期解的存在性.首先概述该问题的背景,然后直接给出拟周期解存在性的主要定理.在证明该主要结果之前,我们先把梁方程化为Hamilton形式,然后再研究这个无穷维系统的可约化性.之后,把约化后的Hamilton形式化为其Birkhoff正规型.最后套用已知结论找出拟周期解.更多还原

【Abstract】 In this dissertation, we will research nonlinear beam equations. The beam e-quations originate from the Euler-Bernoulli beam equation. Euler-Bernoulli beam theory is a simplification of the linear theory of elasticity which provides a means of calculating the load-carrying and deflection characteristics of beams. It was first enunciated circa1750, but was not applied on a large scale until the develop-ment of the Eiffel Tower and the Ferris wheel in the late19th century. Following these successful demonstrations, it quickly became a cornerstone of engineering and an enabler of the Second Industrial Revolution. Nowadays, beam equations are developed in many aspects. The model of beam equations in this thesis is uxxxx+utt=F(t,x,u), where F is some nonlinear term.The above equation describes processes without dissipation of energy, name-ly, it is a hamiltonian partial differential equation. This means that, in suitable coordinates it can be seen as an infinite-dimensional hamiltonian system. This property is crucial and allows us to study the problem as a hamiltonian dynamical system trying to extend all the well developed machinery of the finite-dimensional case.One is concerned about the dynamical behaviors of phase spaces considering hamiltonian system. However, except a few integrable systems, people can not be clear about all the dynamical behaviors of orbits. Nearly-integrable systems, which mathematicians have been focusing on researching since1960s, are very important and simple systems besides integrable systems. Searching periodic solutions is one of the important topics for the dynamical behaviors of finite-dimensional hamiltonian systems. For infinite-dimensional nearly-integrable hamil-tonian systems, we not only concern about periodic solutions, but also their quasi-periodic solutions and their almost-periodic solutions. So far there are t-wo main approaches to deal with the periodic and quasi-periodic solutions of infinite-dimensional systems. The first one is the Craig-Wayne-Bourgain method. It is a generalization of the Lyapunov-Schmidt reduction and the Newtonian method. The other techniques, based on super convergent (Newton’s) methods, as the KAM theory or the Nash-Moser Implicit Function Theorem, allow to ex-tend well known methods and results from the finite-dimensional case and are the natural ways to deal with the lack of regularity due to the "small divisors" problem. The classical KAM theory which is constructed by three famous math-ematicians Kolmogorov [1], Arnold [2] and Moscr [3] in the last century, is the landmark of the development of hamiltonian systems. It made the stability of solar system got reasonable explanation. In the later1990’s, the KAM theory was successfully generalized to the infinite-dimensional setting by Wayne [4] and Kuksin [5]. Later, Poschcl [6] restated the result. Such techniques arc somewhat complementary to the variational ones allowing us to obtain periods. However, unlike the variational methods, they arc local, perturbed in nature and, therefore, restricted to equations with weak nonlinearities or, equivalently, to solutions of small amplitudes.Moreover, these results apply to, as typical examples, one-dimensional non-linear Schrodingcr equations (NLS) with parameters iut-uxx+V(x, ζ)u=f(u) and nonlinear wave equations (NLW) utt-uxx+V(x, ζ)u=f(u) with Dirichlet boundary conditions and to specific classes of potential V exclud-ing, in particular, the case V=const. Poschel [7] covered the case of constant potential by exploiting the existence of a Birkhoff normal form for the NLW. This approach was applied in Kuksin and Poschel [8] to the persistence of quasi-periodic solutions for the NLS subject to Dirichlet or Neumann boundary condi-tions. Similar results for the NLW and the nonlinear beam equations (NLB), we refer to Geng and You [9]. It’s worth mentioning that Yuan [10] proved the ex- istence of quasi-periodic solutions for a complete resonant one-dimensional wave equation.The case of periodic boundary conditions is more delicate due to the fact that the eigenvalues of the Sturm-Liouville operator L=-d2/dx2+V are degenerate. Craig and Wayne [11,12] developed new techniques based on the Lyapunov-Schmidt method and techniques by Frohlich and Spencer [13]. They proved in [11] the persistence of periodic solutions of the NLW with periodic boundary conditions. Later, their approach was significantly improved by Bourgain [14,15] who constructed quasi-periodic solutions of the NLW and NLS with periodic boundary conditions. The relative results about periodic boundary conditions also can be found in Geng and You [16,17] and Liang [18] using KAM tech-niques. In addition, Bricmont, Kupiainen, and Schenkel [19] gave a new proof of persistence of quasi-periodic, lower-dimensional elliptic tori of the NLW by using of the renormalization group method.The case of higher-dimensional hamiltonian PDEs is difficult. Bourgain [20] first proved that the2-dimensional nonlinear Schrodinger equations admit small-amplitude quasi-periodic solutions. And he [21] improved his method and proved that the higher-dimensional nonlinear Schrodinger and wave equations admit small-amplitude quasi-periodic solutions. Later, Geng and You [17,22] proved that the higher-dimensional nonlinear beam equations and nonlocal Schrodinger equations admit small-amplitude linearly-stable quasi-periodic solutions. Elias-son and Kuksin [23] proved that the higher-dimensional nonlinear Schrodinger equations admit small-amplitude linearly-stable quasi-periodic solutions. Recent-ly, Geng and You [24] obtained quasi-periodic solutions of higher-dimensional cubic Schrodinger equations.We are interested in the existence of the case with forcing. When forced nonlinearities are periodic, the existence of periodic solutions has been proved by the variational method and the Lyapunov-Schmidt reduction. For the exis-tence of quasi-periodic solutions, Berti and Procesi [25] proved the existence of quasi-periodic solutions with two frequencies of complete resonance for the pe-riodically forced wave equations. Jiao and Wang [26] considered the nonlinear Schrodinger equations with Dirichlet boundary conditions. They [26] showed that the NLS admit quasi-periodic solutions by constructing a KAM theorem. Zhang and Si [27], and Si [28] focused on the existence of quasi-periodic solutions for the quasi-periodically forced NLW with Dirichlet boundary conditions and pe-riodic boundary conditions, respectively. Eliasson and Kuksin [29] studied the d-dimensional nonlinear Schrodinger equations under periodic boundary condi-tions. Conclusions have been madden that the NLS have time-quasi-periodic solutions.In this thesis, we will study the existence of quasi-periodic solutions about the following nonlinear beam equations utt+uxxxx+μu+εg(ωt,x)u3=0,μ>0,x[0,π], and utt+uxxxx+μu+εφ(t)h(u)=0,μ>0subject to the hinged boundary conditions, where ε and ε arc small positive numbers and ω=(ω1,ω2,...,ωm)∈[(?),2(?)]m ((?)>0) is a frequency vector; the function g(ωt,x)=g(v,x),(v,x)∈Tm×[0,π] is real analytic in (v.x), and quasi-periodic in t; the nonlinearity h is a real analytic odd function of the form and φ is a real analytic quasi-periodic function.The two problems are similar but different. Both of their nonlinear terms include the time variable. However, the nonlinear term of the first equation includes the spacial variable, while the nonlinear term of the second one does not and its potential, which is not a constant, has the time variable. Therefore, we deal with the two problems by different methods.Our main method is to transform the hamiltonian systems into their Birkhoff normal forms, then we can use an infinite-dimensional KAM theorem to find out quasi-periodic solutions.In our first problem, it is difficult to prove that there is a symplectic and analytic transformation which can change the hamiltonian functions into their Birkhoff normal forms. For the spacial variable in the perturbation terms, we lose the crucial conditions i±j±d±l=0. So, there are two difficulties in the proof. One is the measure estimate of "small divisor" conditions. On one hand, when estimating a measure, the conditions ij±d±l=0are usually important. On the other hand, different from Schrodinger equations, the eigenvalues of beam equations are not integers, which undoubtedly complicates our proof. Another difficulty is the analyticity of the symplectic transformation of coordinates with-out the conditions of i±j±d±l=0. We construct a technical lemma and use the Fourier Cosine series to overcome this difficulty.However, for the second problem, one big difficulty lies in that the reducibility of infinite-dimensional linear quasi-periodic systems, since we have to reduce the potential functions to constant ones. In fact, this problem itself is interesting and open. In this paper, we construct an infinite-dimensional KAM theorem to solve this problem.This dissertation consists of three chapters and the main contents are as follows:In Chapter1, we introduce hamiltonian systems and the KAM theory. Chap-ter1has three sections. In the first section, the research background and models of the nonlinear beam equations are introduced. In the second section, definitions and notations about hamiltonian systems are shown, including the definition of symplectic structures, Darboux Theorem, Liouville’s Theorem, Liouville’s Theo-rem on integrable systems, definitions of integrable systems and nearly-integrable systems, etc., and some conclusions of Birkhoff normal forms are shown. The infinite-dimensional hamiltonian systems and KAM theory are described in the last section. In the same section, Kuksin’s famous abstract infinite-dimensional KAM theorem is stated.In Chapter2, we mainly study the existence of quasi-periodic solutions for nonlinear beam equations with quasi-periodically forced terms depending on the spacial variable. We first introduce the background of our problem and show the main result of the existence of quasi-periodic solutions. To resolve this problem, we need to transform the PDEs into their hamiltonian forms and their Birkhoff normal forms. Then, we prove the main theorem by introducing and using an infinite-dimensional KAM theorem for PDEs.Chapter3is devoted to research the existence of quasi-periodic solutions for nonlinear beam equations with quasi-periodic potential. Similar to Chapter2, we first introduce the background of the problem and show the main result. Transforming the beam equations into their hamiltonian forms, we can research the reducibility for these infinite-dimensional systems. Changing the reduced hamiltonian forms into their Birkhoff normal forms, we prove the main theorem by introducing and using an infinite-dimensional KAM theorem for PDEs.更多还原