Mathematical Sciences: Studies in Nonlinear Analysis

****************************************************************************** 非线性分析研究：摘要 本研究针对与非线性微分方程解的行为相关的两类问题。 第一个问题涉及多孔介质方程（简并非线性抛物线方程）和气体动力学纳维-斯托克斯方程的自相似解的存在性和中间渐近行为。 这里的目标是获得所涉及的流动中各种关键转变的渐近描述，例如多孔介质流动中孔的闭合（聚焦）或气体动力学中冲击波的内爆。 第二个问题涉及约瑟夫森点结负载耦合阵列的动力学研究。这些阵列通过常微分方程的非线性系统进行建模，研究的目的是了解各种锁相解的稳定性和不稳定性的开始。 将特别强调预测锁相稳定性的损失以及这种损失的动态后果。 偏微分方程的自相似解是由方程的基本对称性确定的特别简单结构的解。 它们通常出现在流动中临界转变的渐近描述中，因为这些转变是方程本身的特征，并且不依赖于特定流动的细节。 这种现象的一个例子发生在与非常粘稠流体的涂层流相关的工程实践中。 此类流动中填充孔洞的过程通常由多孔介质方程描述，我们的结果将有助于开发用于研究和控制此类过程的数值方法。 大型负载耦合约瑟夫森结阵列用于许多电子设备，例如微波发生器和参量放大器。 在所有这些装置中，最理想的工作条件是所有结点一致振荡。 不幸的是，对于各种结特性和负载配置，这种锁相状态并不稳定。因此，了解如何根据任何特定阵列的参数值来预测锁相状态的稳定性和不稳定性非常重要。

****************************************************************************** STUDIES IN NONLINEAR ANAYSIS: ABSTRACT The proposed research is directed towards two types of problems concerning the behavior of solutions to nonlinear differential equations. The first of these problems concerns the existence and intermediate asymptotic behavior of self-similar solutions to the porous medium equation (a degenerate nonlinear parabolic equation) and to the Navier-Stokes equations of gas dynamics. The objective here is to obtain the asymptotic description of various critical transitions in the flows involved, such as the closing of holes (focusing) in porous medium flow or the implosion of shock waves in gas dynamics. The second problem concerns the study of the dynamics of of load coupled arrays of Josephson point junctions. These arrays are modeled by nonlinear systems of ordinary differential equations, and the object of the study is to understand the onset of stability and instability of various kinds of phase locked solutions. Particular emphasis will be placed on predicting the loss of stability of phase lock and the dynamical consequences of such a loss. Self-similar solutions to partial differential equations are solutions of a particularly simple structure determined by the underlying symmetries of the equation. They often arise in the asymptotic description of critical transitions in the flow, because these transitions are characteristic of the equations themselves and do not depend on the details of the particular flow. An example of this phenomenon occurs in engineering practice in connection with coating flows of very viscous fluids. The process of filling holes in such flows is often described by a porous medium equation, and our results will be useful in developing numerical methods for the study and control of such processes. Large arrays of load coupled Josephson junctions are used in many electronic devices such as microwave generators and parametric amplifiers. In all these devices the most desirable operating condition is one in which all of the junctions oscillate in unison. Unfortunately, for various junction characteristics and load configurations this phase locked state is not stable. It is therefore important to know how to predict the stability and instability of the phase locked state in terms of the parameter values for any particular array.