非线性弹性杆动力学方程解的存在性和唯一性研究
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摘要
非线性方程解的存在性和唯一性问题的研究是非线性动力学的主要研究内容之一。由于非线性的多样性导致了非线性方程形式的复杂性,目前对非线性方程的求解虽有一些适用面较广的方法,但并不象线性方程那样有一般方法可循,大多数非线性方程不可能或很难求出其解析解,因此,必须在不具体求出方程解析解的情况下,利用数值求解方法或根据方程本身的特点来判断非线性方程解的性质。然而,在非线性方程的数值求解过程中,人们往往不考虑方程的解是否存在和唯一,通常取一个或几个模态来研究方程解的性质,这样并不能保证从无穷维空间到有限维子空间约化的合理性,甚至可能会导致错误的结论,所以研究非线性方程解的存在性和唯一性是保证数值求解合理性的前提和理论基础。为此,本文以Sobolev空间为工具,利用Galerkin法和局部延拓法对非线性弹性杆动力学方程解的存在性和唯一性以及与解相关的问题进行了研究,本文的主要工作如下:
1.对国内外有关非线性微分方程解的存在性和唯一性的研究方法和研究现状进行了综述。
2.在一定的边界条件和初始条件下,利用Galerkin法和局部延拓
The research on the existence and uniqueness of solutions of nonlinear partial differential equations is one of main branches of nonlinear dynamics. At present, since the variety of non-linearity leads to the complexity of forms of non-linear equations, no general method is available for solving non-linear equations like in the linear case although some approaches may be applicable in very broad context. As a result, it is difficult or even impossible to acquire analytic solutions for most non-linear equations. Thus, without concrete analytic solutions to an equation, numerical methods have to be employed or properties of the equation are figured out based on the examination of characteristics of the equation. However, one frequently ignores the existence and uniqueness of solutions in the process of seeking numerical solutions. Instead, one or more modes are chosen to investigate the property of solutions of the equation. In doing so, rationality for simplifying an infinite-dimensional system into a finite -dimensional system cannot be ensured; or even worse, incorrect conclusions may result in. Consequently, the research on the existence and uniqueness of solutions
of nonlinear equations is a prerequisite and theoretic foundation for justifying numerical solutions. In view of this, we carry out some studies on the existence and uniqueness of solutions of the dynamics equations of a nonlinear elastic bar by means of the Sobolev space. Our main work includes the following.1. The current research methods concerning the existence and uniqueness of solutions of nonlinear partial differential equations are summarized and commented.2. Under some certain initial and boundary conditions of the equations for nonlinear viscid beam, by applying Galerkin method, the existence and uniqueness of global weak solution and its continuous dependence on the initial conditions are shown. How to select types of basis functions in the Sobolev space is illustrated. Moreover, the existence and uniqueness of the strong solution and the existence of classical solution of the nonlinear viscid beam equation are proved. Necessary conditions for the existence of classical solution are presented.Since the equations investigated in this thesis are very general, the obtained results contain many others in the literature as special cases.3. By applying Galerkin method, the existence of weak solution of dynamics equations with constant coefficients for physical nonlinear bar of three order is firstly verified.4. Under some certain initial and boundary conditions of the equations of A physical nonlinear bar with one end fixed and another end subjected to an axial exponential velocity tension, An exploration on the existence of weak solution of dynamics equations with variable coefficients for physical nonlinear bar of third order is made.
5. A physical nonlinear bar with one end fixed and another end subjected to an axial exponential velocity tension was studied. Galerkin method was applied to analyze the response and the effect of truncation orders on the computation results.
1.对国内外有关非线性微分方程解的存在性和唯一性的研究方法和研究现状进行了综述。
2.在一定的边界条件和初始条件下,利用Galerkin法和局部延拓
The research on the existence and uniqueness of solutions of nonlinear partial differential equations is one of main branches of nonlinear dynamics. At present, since the variety of non-linearity leads to the complexity of forms of non-linear equations, no general method is available for solving non-linear equations like in the linear case although some approaches may be applicable in very broad context. As a result, it is difficult or even impossible to acquire analytic solutions for most non-linear equations. Thus, without concrete analytic solutions to an equation, numerical methods have to be employed or properties of the equation are figured out based on the examination of characteristics of the equation. However, one frequently ignores the existence and uniqueness of solutions in the process of seeking numerical solutions. Instead, one or more modes are chosen to investigate the property of solutions of the equation. In doing so, rationality for simplifying an infinite-dimensional system into a finite -dimensional system cannot be ensured; or even worse, incorrect conclusions may result in. Consequently, the research on the existence and uniqueness of solutions
of nonlinear equations is a prerequisite and theoretic foundation for justifying numerical solutions. In view of this, we carry out some studies on the existence and uniqueness of solutions of the dynamics equations of a nonlinear elastic bar by means of the Sobolev space. Our main work includes the following.1. The current research methods concerning the existence and uniqueness of solutions of nonlinear partial differential equations are summarized and commented.2. Under some certain initial and boundary conditions of the equations for nonlinear viscid beam, by applying Galerkin method, the existence and uniqueness of global weak solution and its continuous dependence on the initial conditions are shown. How to select types of basis functions in the Sobolev space is illustrated. Moreover, the existence and uniqueness of the strong solution and the existence of classical solution of the nonlinear viscid beam equation are proved. Necessary conditions for the existence of classical solution are presented.Since the equations investigated in this thesis are very general, the obtained results contain many others in the literature as special cases.3. By applying Galerkin method, the existence of weak solution of dynamics equations with constant coefficients for physical nonlinear bar of three order is firstly verified.4. Under some certain initial and boundary conditions of the equations of A physical nonlinear bar with one end fixed and another end subjected to an axial exponential velocity tension, An exploration on the existence of weak solution of dynamics equations with variable coefficients for physical nonlinear bar of third order is made.
5. A physical nonlinear bar with one end fixed and another end subjected to an axial exponential velocity tension was studied. Galerkin method was applied to analyze the response and the effect of truncation orders on the computation results.
引文
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[6] S. Klainerman and G. Ponce. Global small amplitude solutions to nonlinearevolution equations. Comm. Pure Appl. Math., (36)1983, 133-141.
[7] S. Klainerman. Uniform decay estimates and the Lorentz invariance of theclassical wave equation. Comm. Pure Appl. Math., (38)1985, 321-332.
[8] A. Matsumura and T. Nishida. Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids. Commun. Math. Phys., 89(1983), 445-464.
[9] Duan J, Holmes P. On the Cauchy problem of generalized Ginzhurg-Laudau equation. Nonlinear Anal., 1994, 22, 1033-1040.
[10] A. Matsumura and T. Nishida. The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ., 20(1980), 67-104.
[11] A. Matsumura and T. Nishida. Initial boundary value problem for the equations of motion of compressible viscous fluids. Contemporary Mathematics, 17(1983), 109-116.
[12] 黄玉盈等,输液管的非线性振动、分叉与混沌.力学进展,1998,28(1),30-42.
[13] 高经武,蔡中民.物理非线性杆周期激励下的动态响应.力学2000,气象出版社,2000.8,p276-272.
[14] 李大潜,陈韵梅,非线性发展方程,科学出版社,1997.
[15] 陈恕行,洪家兴,偏微分方程近代方法,复旦大学出版社,1988.
[16] 郭柏林,无穷维动力系统(上、下),国防工业出版社,2000.
[17] 黄玉盈等,非线性输液管系统的幅频特性分析.力学2000,2000.8,291-292
[18] Duan J, Holmes P, Titi E S. Global existence theory for a generalized Ginzburg-Landau equation. Nonlinearity, 1992, 5, 1303-1314.
[19] Guo Boling. The global solution of the system of equations for complex SchrOdinger field coupled with Boussinesq type self-consistent field. Acta. Math. Sini., 1983, 26, 297-306
[20] Bartuccelli M V, Constantin P, et al. On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation. Phys 1990, D 44, 421-444.
[21] Doering C R, Gibbon J D, Levemore C D. Weak and strong solutions of the complex Ginzburg-Landau equation. Phys., 1994, 71, 285-318.
[22] 蔡燧林,盛骤编.常微分方程组与稳定性理论.高等教育出版社,1986.
[23] 赵建宏,蔡中民.非线性粘弹性圆柱杆在阶跃速度下的迭代解.力学与工程应用,1994.
[24] Holmes C A. Bounds solutions of the nonlinear parabolic amplitude equation for plane poiseuille flow. Pro R London, 1985, A 402, 299-322.
[25] Chidaglia J M, Saut J C. On the initial value problem for the Davey-Stewartson systems. Nonlinearity, 1990, 3, 475-506.
[26] Anker D, Freeman N C. On the solutions of the Davey-Stewartson equation for ling waves. Proc. R Soc London, 1978, A 360, 529-540.
[27] Alain B, Cyrill L, Alexander K. Breathers for a relativistic nonlinear wave equation. Arch. Rational Mech. 2002, 165, 317-345.
[28] Tsutsumi M. Deacy of weak solutions to the Davcy-Stewartson systems. J Math Anal Appl, 1994, 182(3): 680-704.
[29] Hayashi N, Saut J C. Global existence of small solutions to Davcy-Stewartson and the Ishimori systems. Diff and Integ Eqs. 1995, 8, 1675-1687
[30] Guo Boling, Shen Longjun. The global solution of initial value equation for nonlinear Schr?dinger-Boussinesq equation in 3-dimensions. Acta. Math. Sini., 1993, 6, 223-233.
[31] 高经武,蔡中民。一维非线性波动方程的Fourier方法,《固体力学的现代进展》世界图书出版公司万国学术出版社,2000 168-171.
[32] 武际可,周鸥,非线性问题和分叉问题及其数值方法,力学与实践,Vol.16,No.1,1994,1-8.
[33] 张建文,非线性弹性无穷维动力系统的惯性流形与整体吸引子的研究,太原理工大学博士论文,2000.
[34] 郭柏灵,非线性演化方程,上海科技教育出版社,1995.
[35] Guo Boling, Mial Changxing. Asymptotic behavior of coupled Klein-Gordon-Schr?dinger equations. Sci China Ser, 1995, A 25(7), 705-714.
[36] Guo Boling, Gao Hongjun. Finite dimensional behavior of generalized Ginzburg-Landau equation. Progress in Natural Science, 1995, 5(6), 649-610.
[37] Holmes C A. Bounds solutions of the nonlinear parabolic amplitude equation for plane poiseuille flow. Proc R London, 1985, A 402, 299-322.
[38] Yang Linge, Guo Boling. Initial-boundary value problem for the Davey-Stewartson system. Prog Nat Sci, 1997, 7(3), 273-279.
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