非线性弹性动力学方程组的若干研究

In this project, we will give some research about well-posedness, asymptotoic behavior and controllability of nonlinear elastic wave equations in three space dimensions. Specifically speaking, we will consider the following three topics: 1) the asymptotic behavior for the global classical solutions to the Cauchy problem of nonlinear elastic wave equations with null condition and small data. 2) the global existence to nonlinear elastic wave equations with null condition in exterior domains with Neumann boundary condition for small data. 3) the local exact boundary controllability for nonlinear elastic wave equations with null condition. On one hand, we will employ and develop existing analysis methods, such as vector fields, weighted energy estimates and so on, on the other hand, we will also fully develop the special structure of the equations. We will make our effort to get remarkable progress for the above three topics and promote, enrich and develop the theory of nonlinear elastic wave equations and related quasilinear hyperbolic equations.

本项目计划研究涉及三维非线性弹性动力学方程组的解的渐近性态、适定性以及能控性这三类问题。我们将分别研究满足零条件的三维非线性弹性动力学方程组Cauchy问题小初值经典解的渐近行为；满足零条件的三维非线性弹性动力学方程组具Neumann边界条件的外区域初边值问题小初值经典解的整体存在性；以及非线性弹性动力学方程组的局部精确边界能控性，这些问题的研究在理论及应用上均具有重要性。我们将充分利用并发展已有的分析方法，如向量场方法、加权能量估计等，并充分开发弹性力学方程组所具有的特殊结构。力争在上述几个问题的研究上取得显著进展，推进、丰富和发展非线性弹性动力学方程组以及与其相关的拟线性双曲方程组的理论。

项目负责人在本项目资助期间，主要研究了非线性双曲方程组的经典解，得到了如下几个成果：(1) 对于一类高维的没有相互作用的对角型拟线性双曲组的柯西问题，我们证明了柯西问题小初值经典解的整体存在性，并且给出了其渐近行为及收敛速度；(2)对于三维非线性弹性动力学方程组，我们给出了在满足零条件时小初值经典解整体存在性的一个新的证明。 (3)对于二维非线性弹性动力学方程组，我们给出了经典解的生命跨度估计。