Nonlinear Partial Differential Equations in Conservation Laws and Applications

这项研究致力于发展新的数学方法和技术来研究一些控制流体流动的非线性偏微分方程组及其相关应用。他们的研究对于理解对STEM至关重要的许多应用的动力学是至关重要的，这些应用包括气体动力学、工程学、材料科学、几何、流体湍流、壳理论、生物学/生物物理学中的活性系统、随机动力学等。虽然一维问题已经被很好地理解了，但多维问题的一般理论在数学上还不够发达。该研究项目将增进对数学和力学以及应用的基本领域的知识；它还将为学生，包括那些来自代表人数不足的群体和妇女的学生提供机会，通过参与积极的应用数学研究，在这些重要领域接受培训。该项目的目标是研究由多维守恒律产生的一些非线性偏微分方程组及其相关应用。主要研究内容如下：(1)气体动力学中跨音速接触间断的存在性和稳定性：这是一个自由边界的混合型问题，本研究将提供新的方法并阐明一般的多维守恒律理论：(2)表面等距浸没的Gauss-Codazzi方程的整体光滑解：Gauss-Codazzi方程的整体光滑解产生曲面的光滑等距浸没，对于一般的表面很难找到这样的整体光滑解；(3)含各种噪声的随机可压缩Navier-Stokes方程的弱解和强解：可压缩流动的随机问题是不发达的和开放的，许多基本问题是困难的；以及(4)生物学中主动流体动力学系统的整体解决方案：主动系统出现在生物学/生物物理学的许多实际应用中，由于其复杂性，其建模和分析具有挑战性，而基本数学问题是广泛开放的。本研究的目的是开发新的分析方法和有效的技术来解决多维无粘性和粘性可压缩流动及其应用中的一些重要问题，并深入了解守恒定律和新兴的现实世界应用中的一般多维问题。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This research project is devoted to developing new mathematical methods and techniques for studying some nonlinear partial differential equations governing the fluid flow and related applications. Their study is crucial for understanding the dynamics of many applications critical for STEM including gas dynamics, engineering, materials science, geometry, fluid turbulence, shell theory, active systems in biology/biophysics, stochastic dynamics, and so on. While the one-dimensional problems are rather well understood, the general theory for the multi-dimensional case is mathematically underdeveloped. The research project will advance knowledge of the fundamental areas of mathematics and mechanics as well as applications; it will also provide opportunities for students, including those from underrepresented groups and women, to receive training in these important areas through participation in the active research in applied mathematics. The goal of the project is to investigate some nonlinear partial differential equations arising from multi-dimensional conservation laws and related applications. The research program focuses on the following topics:(1) the existence and stability of transonic contact discontinuity in gas dynamics: this is a free boundary and mixed-type problem, and this study will provide new methods and shed light on the general multi-dimensional theory of conservation laws;(2) the global smooth solutions to the Gauss-Codazzi equations of isometric immersion of surfaces: a global smooth solution to the Gauss-Codazzi equations yields a smooth isometric immersion of surfaces, and it is challenging to find such a global smooth solution for general surfaces;(3) the weak and strong solutions to the stochastic compressible Navier-Stokes equations with various types of noise: the stochastic problems for the compressible flows are underdeveloped and widely open, and many fundamental problems are difficult; and(4) global solutions to the system of active hydrodynamics in biology: active systems arise in many practical applications in biology/biophysics, and its modeling and analysis are challenging due to its complexity, while fundamental mathematical problems are widely open.The purpose of this research is to develop novel analytic methods and efficient techniques for solving some important problems in multi-dimensional inviscid and viscous compressible flows and applications, and to gain insights into the general multi-dimensional problems of conservation laws and emerging real-world applications.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.