KUMU PDE Conference Proposal

该奖项将为与会者，特别是来自科学代表性不足群体的研究生、初级研究人员、女性和数学家提供支持，以参加将于2015年4月18日至19日在堪萨斯大学举行的“PDE、动力系统和应用领域的Kumu会议”，该会议由堪萨斯大学(KU)和密苏里大学(MU)的教职员工联合举办。几乎所有重要的物理现象都受基本定律和设计原则的支配，这些基本定律和设计原则将某些量的变化率与其他量的变化率直接联系起来。事实上，考虑到初始条件和运动的物理规律，人们试图预测未来并重建过去。这一重要的观察自然导致了微分方程的想法，从而为理解许多现实世界的问题提供了钥匙。微分方程组在数学物理中有着广泛的应用，在玻色-爱因斯坦凝聚体、流体动力学、图样形成、气体动力学以及光通信网络中的信号建模等领域有着潜在的应用。这次会议将促进来自堪萨斯州和密苏里州附近地区的微分方程及其相关领域的研究人员之间的更大互动。会议计划作为一系列年度会议的第一次，将为区域初级和成熟的研究人员以及研究生提供一个场所，讨论各自领域的最新进展和挑战。此外，青年研究人员将有机会介绍他们自己的工作，并通过与该领域资深专家的互动，对这一重要主题有深入的了解。会议网站：https://www.math.ku.edu/conferences/2015/KUMUPDE/index.htmlComplex非线性系统在科学和工程中大量存在，它们的行为通常用非线性偏微分方程组来建模。在了解其解决方案的行为方面取得的任何进展，对于各种实际应用都是至关重要的，包括流体流动、火焰前沿传播和光纤通信。许多偏微分方程可以方便地描述为无限维动力系统，允许使用动力系统理论的工具和方法来对这些系统的解进行定性和定量的预测。像不变流形这样的对象对理解有限维动力系统的行为有很大的帮助，但非线性偏微分方程和动力系统之间的联系仍然是一个活跃的研究领域。在过去的几十年里，这些领域的研究人员及其应用领域的研究人员之间的合作，为我们理解这种非线性偏微分方程中相干结构的动力学行为、稳定性和稳健性提供了巨大的进步。这次会议的主要主题包括(I)流体动力学、水波和色散偏微分方程组，(Ii)耗散系统中非线性波的存在性、动力学和稳定性，以及(Iii)动力系统和2D-Navier-Stokes方程。在这些广泛的领域中，用于解决许多具有挑战性的问题的技术往往结合了动力系统和偏微分方程组的思想和方法，以及概率论、谱和泛函分析、埃文斯函数和几何奇异摄动理论等。

This award will provide support for participants, especially graduate students, junior researchers, women and mathematicians from under-represented groups in the sciences, to attend the regional conference "KUMU Conference in PDE, Dynamical Systems and Applications" to be held at the University of Kansas from April 18-19, 2015, co-organized by faculty from the University of Kansas (KU) and the University of Missouri (MU). Nearly all important physical phenomena are governed by fundamental laws and design principles that directly relate rates of change of some quantity to that of some other quantity. Indeed, given the initial conditions and the physical laws of motion one seeks to predict the future and reconstruct the past. This important observation naturally leads to the idea of a differential equation, thus providing the key to understanding many real-world problems. Differential equations are widely used as models in mathematical physics and have potential applications to many fields including Bose-Einstein condensates, fluid dynamics, pattern formation, gas dynamics and for modeling signals in optical communication networks. This conference will facilitate greater interaction between researchers in differential equations and its related fields from the area close to Kansas and Missouri. Planned as the first of a series of annual meetings, the conference will provide a venue for regional junior and established researchers, as well as graduate students to discuss the recent advances and challenges in their respective fields. In addition, young researchers will be given the opportunity to present their own work and to gain insights into this important subject through interactions with senior experts in the field. The conference website: https://www.math.ku.edu/conferences/2015/KUMUPDE/index.htmlComplex nonlinear systems abound in science and engineering, and their behavior is often modeled by systems of nonlinear partial differential equations (PDE). Any progress towards understanding the behavior of their solutions is of paramount importance for a variety of practical applications, including fluid flow, flame front propagation and fiber optical communication. Many PDE can be conveniently described as infinite dimensional dynamical systems, allowing for the use of tools and methodologies from dynamical systems theory to make qualitative and quantitative predictions about the solutions of these systems. Objects like invariant manifolds have been a great aid in understanding the behavior of finite-dimensional dynamical systems, but the connections between nonlinear PDE's and dynamical systems is still an area active current research. In the last few decades, collaborations between researchers in these fields, as well as with those working in their applications, have provided tremendous progress in our understanding of the dynamical behavior, stability and robustness of coherent structures in such nonlinear PDE. The main themes of this conference include (i) fluid dynamics, water waves and dispersive PDE's, (ii) existence, dynamics, and stability of nonlinear waves in dissipative systems, and (iii) dynamical systems and 2d-Navier Stokes equations.The techniques used to solve many challenging problems in these broad areas often combine ideas and methodologies from dynamical systems and partial differential equations together with probability theory, spectral and functional analysis, Evans functions, and geometric singular perturbation theory, to name a few.