非线性项依赖于空间和时间变量的梁方程拟周期解的存在性

Nonlinear beam equations are a class of nonlinear Hamiltonian partial differential equation and are widely used in physics, engineering and materials science. The research on the existence of their quasi-periodic solutions is of great significance for the understanding of their dynamics. The model whose nonlinear term depends on the spatial and time variables has more practical values, but there are few conclusions. To solve these problems, we not only need to estimate the measures of the essentially infinite small divisor conditions, but also need to overcome the complexities of the normal forms and of the measure estimates brought by the time variable. Based on the applicant's researches on nonlinear beam equations, in this project, we apply Birkhoff normal forms and the infinite dimensional KAM theory in proving the existence of beam equations with nonlinear terms depending on the space and time variables. Our research can perfect the KAM theory and can improve the techniques of normal forms and measure estimates. All these jobs have important theoretical values and can provide important theoretical basis for understanding the stability of beam equations.

非线性梁方程是一类非线性Hamilton偏微分方程，在物理、工程、材料学中应用广泛。研究其拟周期解的存在性对于认识方程的动力学性态具有重要意义。非线性项依赖于空间和时间变量的模型更具有应用价值和现实意义，但已有结论尚少。研究此类问题不但需要解决本质上无穷多个小除数条件的测度估计，而且由于强迫项的加入使得化正规型和测度估计变得非常复杂。基于申请人已完成的对非线性梁方程的研究，本项目旨在应用无穷维KAM理论和Birkhoff正规型技术证明非线性项同时依赖于空间和时间变量的高维梁方程拟周期解的存在性。本项目对进一步完善无穷维KAM理论、改进正规型和测度估计技术具有重要理论价值，为深刻认识梁方程模型的稳定性提供重要理论依据。

非线性梁方程是一类非线性Hamilton偏微分方程，在物理、工程、材料学中应用广泛。研究梁方程拟周期解的存在性对于认识方程的动力学性态具有重要意义。非线性项依赖于空间和时间变量的模型具有应用价值和现实意义，但已有结论尚少。本项目应用无穷维KAM理论和Birkhoff正规形技术研究非线性项同时依赖于空间和时间变量的梁方程。此类方程具有拟周期强迫，不满足动量守恒定律。证明结论显示带有正常数势能项的方程在周期边界条件下，对任意的指标集、大多数势能常数和充分小的扰动项，方程具有小振幅、线性稳定的拟周期解，且解的频率接近于未摄系统的频率，方程存在低维不变环面；线性边界条件下完全共振梁方程对于任意的容许指标集而言存在小振幅、线性稳定的拟周期解。另外还得到了一系列关于测度估计和收敛性的技术性引理。本项目对进一步完善无穷维KAM理论、改进正规形和测度估计技术具有重要理论价值，为带有变系数非线性项的偏微分方程拟周期解的研究提供思路，为深刻认识梁方程模型的稳定性提供重要理论依据。