CAREER: Interdisciplinary Mentoring Program in Analysis, Computation, and Theory (IMPACT)

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。本文主要研究了非线性波和相干结构的稳定性理论，重点研究了连续介质中的锋面传播和可压缩流动的动力学理论。这类问题模拟了现实世界的关键现象，如粘性气体或等离子体中的激波层，反应气体中的爆炸，以及粘性流体中相边界的传播。在这个项目中，我们将探索关于多变量和多维耗散行波的稳定性，特别是可压缩流体流动的稳定性的长期开放问题。我们的技术方法围绕埃文斯函数计算和相关的谱技术、能量估计和渐近常微分方程技术，包括间隙、跟踪和共轭引理、爆破方法和几何奇异摄动理论；还包括在分岔理论，谱理论的线性算子，和非线性偏微分方程的技术。非线性波和相干结构在自然界中无处不在，在生态学、海洋学、流体动力学、燃烧学、等离子体物理学、光学、神经科学和材料科学等许多科学学科中都有出现。这些现象的数学模型非常复杂，需要广泛的新兴分析和计算技术来进行探索。我们的主要目标是了解这些波和结构在干扰存在下的性质。首要的问题是：给定的波或结构在受到干扰时是否会持续存在，还是会分裂成其他东西？我们的目标是开发定量的数学工具来帮助探索这些现象，特别是将在工程、物理、生物、化学和制造业中实际使用的方法。这个项目的大部分直接费用用于资助研究生的研究，作为NSF资助号0639328的一个更大的垂直整合跨学科指导项目的一部分。在研究者的指导下，研究生进行自己的研究，同时也协助指导我们项目的本科生研究人员。这给了本科生更多的帮助和指导的机会，同时也给了研究生一个机会，通过教学来加强他们的理解，也培养了领导能力。这一模式的目标是在数学科学领域培养一支强大而充满活力的年轻学者队伍，他们将有能力应对具有全球竞争力的科学劳动力的挑战。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This work focuses on the stability theory of nonlinear waves and coherent structures, with an emphasis on front propagation arising in the continuum and kinetic theories of compressible flow. This class of problems models key real-world phenomena such as shock layers in a viscous gas or plasma, detonations in a reactive gas, and the propagation of phase boundaries in a viscous fluid. During this project, we will explore long-standing open questions about the stability of many-variable and multi-dimensional dissipative traveling waves, in particular those of compressible fluid flow. Our technical approach centers around Evans function computation and related spectral techniques, energy estimates, and asymptotic ordinary differential equation techniques including the gap, tracking, and conjugation lemmata, blow-up methods, and geometric singular perturbation theory; also included are techniques in bifurcation theory, spectral theory of linear operators, and nonlinear partial differential equations.Nonlinear waves and coherent structures are ubiquitous in nature and occur in many scientific disciplines including ecology, oceanography, fluid dynamics, combustion, plasma physics, optics, neuroscience, and material science. The mathematical models of these phenomena are very complex and will require a wide range of new and emerging analytical and computational techniques for their exploration. Our primary goal is to understand the nature of these waves and structures in the presence of disturbances. The overarching question is: will a given wave or structure persist when it is disturbed, or will it instead bifurcate into something else? Our aim is to develop quantitative mathematical tools to aid in the exploration of these phenomena, particularly methods that will be of practical use in engineering, physics, biology, chemistry, and manufacturing. The majority of the direct costs in this program go to funding graduate student research as part of a larger vertically integrated interdisciplinary mentoring program seeded by NSF Grant No. 0639328. Under the investigator's supervision, graduate students conduct their own research, while also assisting in the mentoring of undergraduate researchers in our program. This gives the undergraduates more opportunities for help and guidance, while also giving the graduate students a chance to reinforce their understanding by teaching, and also to develop leadership skills. The goal of this model is to produce a strong and vibrant pipeline of young scholars in the mathematical sciences who will be well equipped to meet the challenges of a globally competitive scientific workforce.