Topics in gas dynamics and fluid flow

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。研究重点是气体动力学和流体流动方程演化解的数学分析。在气体动力学中，目标是更深入地理解相对论动力学方程及其内部产生的完全洛伦兹不变力。重点讨论这些方程强解的存在性、唯一性、规律性和衰减性质。第二个研究项目研究了轴对称和三维情况下的不可压缩流体动力学。本研究的目的是得到假定爆炸率的下界，并确定它们在扰动下的稳定性。这个项目可以被认为是相关的物理现象，如飓风，龙卷风和漩涡相当接近轴对称。本提案旨在进一步建立这些主题领域之间的联系，并进一步探讨与这两个项目密切相关的其他广泛问题。拟议的研究包含了一个跨学科的方法来解决从不可压缩流体流动到气体动力学等领域的非线性偏微分方程中出现的问题。这些方程所模拟的物理现象在自然界中随处可见。研究人员计划使用的方法涉及基于能量的技术，这些技术已被证明是灵活和鲁棒的，可用于研究各种非线性偏微分方程的规律性。PI的研究愿景包括开发新的工具来研究这些基本模型的基本分析问题；这些工具有望推广到应用非线性PDE中的一系列重要问题。PI计划在纯数学家和应用数学家之间传播通过拟议活动获得的知识。PI将继续探索、开发和实施各种方法，以整合从本科项目到当前研究前沿的各级数学研究。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The research focuses on the mathematical analysis of the evolutionary solutions to the equations from gas dynamics and fluid flow. In gas dynamics the goal is to gain a deeper understanding of relativistic kinetic equations coupled with their internally generated fully Lorentz invariant forces. Attention will be focused on questions of existence, uniqueness, regularity and decay properties of strong solutions to these equations. The second program of research studies incompressible fluid dynamics in both the axisymmetric and three dimensional cases. The goal of this research is to obtain lower bounds on putative blow-up rates and to determine their stability under perturbations. This project can be considered relevant for physical phenomena such as hurricanes, tornadoes, and whirlpools that are fairly close to axisymmetric. This proposal aims to moreover establish connections between these subject areas, and further explore other broad questions closely related to both projects.The proposed research contains an interdisciplinary approach to questions arising in non-linear partial differential equations in areas ranging from incompressible fluid flow to gas dynamics. The physical phenomena that these equations model are encountered all across the natural world. Methods the investigator plans to use involve energy based techniques which have proven to be flexible and robust for studying regularity of a variety of nonlinear partial differential equations. The PI's research vision includes developing new tools to study fundamental analytical problems about these basic models; these tools are expected to be generalizable to a wide array of important problems in applied non-linear PDE. The PI plans to disseminate knowledge obtained through the proposed activity among both pure and applied mathematicians. The PI will continue to explore, develop and implement diverse methods to integrate mathematical research at levels ranging from undergraduate projects to the vanguard of current research.