Mathematical Sciences: Stability of Solutions of Nonlinear P.D.E.

这项数学研究的主要关注点是 非线性偏微分方程解的渐近稳定性 微分方程 最近开发的几何和 拓扑工具将被利用在分析的 微分算子的谱 中心思想是， 关于一个特殊的解决方案，如稳定的线性化 状态、行波或径向对称解。 等 解决方案在形成定性理论中起着核心作用 因为它们可以被认为是 相关的进化问题。 在这方面， 至关重要的是，要制定评估其 稳定 这项工作基于抽象半群理论，使用 关于操作者的频谱的信息，通过以下方式获得： 线性化问题的解决方案，以预测其 稳定或不稳定。 需要克服的特殊障碍 在任何特定的应用中， 准确估计的位置和多样性的 算子的潜在不稳定特征值。 几个特定系统的稳定性分析将 启动。 需要考虑的应用是 高阶相场模型的行波解，以及 非线性守恒条件下激波剖面稳定性 法律为 后一个问题提出了新的问题，即 由连续光谱发挥，但它主要集中在 而不是理解它。 这方面的工作大部分来自具体的问题， 等熵气体中激波粘性剖面的研究 动力学和非绝热管式反应器。

The primary concern of this mathematical research is the asymptotic stability of solutions of nonlinear partial differential equations. Recently developed geometric and topological tools will be exploited in the analysis of the spectrum of differential operators. The central idea is that of linearizing about a distinguished solution such as a steady state, a travelling wave or a radially symmetric solution. Such solutions play a central role in formulating a qualitative theory in that they can be thought of as asymptotic states for an associated evolution problem. In this regard it is of fundamental importance to develop methods for assessing their stability. This work based on abstract semigroup theory, uses information about the spectrum of an operator, obtained by linearizing about the solution in question, to predict its stability or instability. The particular obstacle to be overcome in any particular application is in accumulating sufficiently accurate estimates for the location and multiplicity of the potentially unstable eigenvalues of the operator. The stability analysis of several particular systems will be initiated. Applications to be considered are the stability of travelling wave solutions of higher order phase field models, and the stability of shock wave profiles on nonlinear conservation laws. The latter problem raises new issues, namely the role played by the continuous spectrum, but it is focused mainly on circumventing, rather than understanding it. Much of this work derives from concrete problems in the study of viscous profiles of shocks arising in isentropic gas dynamics and nonadiabatic tubular reactors.