Mathematical Sciences: Geometric Methods and Nonlinear Hyperbolic Equations

本文将用新的几何方法分析两个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的线性非齐次Klein-Gordon方程在三维空间。衰减估计出现在研究与非线性双曲方程相关的线性问题时。Klainerman和Hormander在二次非线性波动方程中的成功应用表明了这种方法。一个类似的计划是计划为克莱因-戈登方程的均匀衰减估计将寻求。解是由强迫函数的一个适当的范数所约束的--它不被假设有紧支撑。当这个程序被执行时，应该可以给出方程整体小解的存在性的直接证明。第二条调查线继续早期的工作，获得较低的边界的寿命光滑的三维可压缩流。这些理想的，可压缩流被认为是小的初始扰动是光滑和本地化。在随后的扰动的前沿附近，最终检测到奇异性或不连续性。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，如果可以的话，描述它们的性质。的解决方案和它的初始配置之间的函数关系，预计是一个指数涉及的初始支持的大小。这项工作将有自然的应用研究波动和流体动力学。

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.