The Existence Of Quasi-periodic Solutions For Nonlinear Beam Equations Witn Quasi-periodic Forcing

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.