Mathematical Sciences: Geometric Methods and Nonlinear Hyperbolic Equations

新的几何方法将用于分析两个有关非线性双曲型方程的问题。一个推力将集中在建立一个新的衰减估计线性非齐次Klein-Gordon方程在三维空间。当研究与非线性双曲方程相关的线性问题时，就会出现衰减估计。这种方法已被Klainerman和Hormander成功地应用于二次非线性波动方程。一个类似的程序计划用于Klein-Gordon方程，其中将寻求均匀的衰减估计。解决方案是由一个适当的范数的强迫函数-这是不假设有紧密的支持。当这个程序被执行时，应该有可能给出方程整体小解的存在性的直接证明。第二项研究继续先前的工作，即获得光滑的三维可压缩流动寿命的下界。这些理想的、可压缩的流动被认为具有平滑和局部化的小的初始扰动。在随后的扰动的前端附近，最终会检测到一个奇点或不连续点。这些是声波中的冲击，是可以预测的。这项工作的重点是确定奇点是否可以在更早的时间形成，如果可以，则表征其性质。解决方案与其初始配置之间的函数关系预期为涉及初始支持大小的指数关系。这项工作自然会应用于波浪运动和流体动力学的研究。

New geometrical methods will be used in the analysis of two problems concerning nonlinear hyperbolic equations. One thrust will focus on the establishment of a new decay estimate for the linear nonhomogeneous Klein-Gordon equation in three space dimensions. Decay estimates arise when one studies linear problems associated with nonlinear hyperbolic equations. This approach has been suggested by successful applications by Klainerman and Hormander to quadratically nonlinear wave equations. A similar program is planned for the Klein-Gordon equation where uniform decay estimates will be sought. The solution is to be bounded by an appropriate norm of the forcing function - which is not assumed to have compact support. When this program is carried out, it should be possible to give a direct proof of the existence of global small solutions of the equation. A second line of investigation continues earlier work on obtaining lower bounds for the life-span of smooth three- dimensional compressible flows. These ideal, compressible flows are considered with small initial disturbances that are smooth and localized. Near the front of the ensuing disturbance a singularity or discontinuity is eventually detected. These are shocks in the acoustical waves and are predictable. The thrust of this effort is to determine whether or not singularities can form at an earlier time and, if so, to characterize their nature. The functional relationship between the solution and its initial configuration is expected to be an exponential involving the size of the initial support. This work will have natural application to studies of wave motion and fluid dynamics.