CAREER: PDE approaches to Physical Phenomena driven by Gravity and Diffusion

该项目基于偏微分方程(PDE)方法，研究由重力和扩散驱动的物理上重要的现象。特别是，恒星坍塌或在两种流体之间的交界处产生涡旋等不同现象导致了类似的数学模型。这些模型方程(Euler-Poisson、Navier-Stokes系统、Fokker-Planck方程、动力学输运和Boltzmann方程)被广泛用于描述可压缩流体和气体的运动。它们在数学科学和工程中有着丰富的应用，并对数学提出了严峻的挑战。该项目旨在促进这一数学基础领域的知识，并影响数学的其他领域。这些结果将为其他学科，如天体物理、等离子体物理、空气动力学和计算物理、化学和生物学提供一些证据，并可能导致科学和技术的进步。研究项目将与教育和外联活动相结合，如学生研究项目、一对一辅导活动、课程开发、本科生和研究生暑期学校以及跨学科会议。该项目将通过综合研究、教育和推广活动来加强加州大学河滨分校的应用数学课程。通过暑期班和会议，它将为学生提供极好的学习机会，目标之一是扩大代表不足的群体的参与。特别是以下主题：(I)欧拉-泊松系统的引力坍塌，旋转恒星的动力学，相对论和辐射，(Ii)瑞利-泰勒不稳定性和有或没有表面张力的双流体可压缩Navier-Stokes系统的对应稳定性，(Iii)动力学Fokker-Planck方程的吸收、弹性和非弹性反射边界碰撞，(4)有边界的动力学输运和玻尔兹曼方程的多尺度动力学：理论和计算。我们的目标是找到一个可以捕捉这些物理现象的数学框架，并通过使用PDE方法来开发适当的数学理论。

This project is concerned with the investigation of physically important phenomena driven by gravity and diffusion, based on partial differential equations (PDE) approaches. In particular, such diverse phenomena as collapse of stars or generation of vortices at the interface between two fluids give rise to similar mathematical models. These model equations (Euler-Poisson, Navier-Stokes systems, Fokker-Planck equations, kinetic transport and Boltzmann equations) are widely used to describe the motion of compressible fluids and gases. They have rich applications in mathematical sciences and engineering and pose formidable mathematical challenges. This project aims to advance knowledge in this fundamental area of mathematics and to influence other domains of mathematics. Results will provide some evidence to other disciplines such as astrophysics, plasma physics, aerodynamics, and computational physics, chemistry, and biology, and may lead to scientific and technological advances. The research project will be integrated with educational and outreach activities such as student research projects, one-to-one mentoring activities, course development, summer schools for undergraduate and graduate students and interdisciplinary conferences. The project will strengthen the Applied Mathematics program at University of California-Riverside by means of integrated research, education and outreach activities. Through summer schools and conferences, it will be providing excellent learning opportunities for students with one of the goals being to broaden participation of underrepresented groups.In particular, the following topics will be studied: (i) gravitational collapses of the Euler-Poisson system, the dynamics of rotating stars, relativity and radiation, (ii) Rayleigh-Taylor instability and the counterpart stability of two-fluid compressible Navier-Stokes system with or without surface tension, (iii) absorbing, elastically and inelastically reflecting boundary collisions for the kinetic Fokker-Planck equations, (iv) multi-scale dynamics from the kinetic transport and Boltzmann equations in the presence of boundary: theory and computation. The goal is to find a mathematical framework where these physical phenomena can be captured and to develop an appropriate mathematical theory by using PDE methods.