Singularities and stability in compressible fluids with or without gravity

流体和气体表现出广泛的有趣的物理现象，在科学和工程中具有丰富的应用。它们的动力学是由非线性偏微分方程（PDE）系统，其研究提出了许多数学挑战。这个项目的目的是研究在什么条件下，由可压缩流体动力学（有或没有重力和弹性动力学）产生的偏微分方程的解在有限时间内会产生奇异行为，例如，导致崩溃或爆炸，以及在什么条件下解决方案可以永远存在。这些问题的答案将有助于更好地理解内爆和爆炸波，恒星的稳定性和坍缩，弹性体的长期行为以及其他物理现象。该项目旨在通过新颖的数学方法加深我们对这些高度非平凡现象的理解。该研究项目将通过学生研究项目、指导活动、课程开发和向更广泛的受众传播，与教育和外展活动相结合。该项目的重点是研究几个描述可压缩流动和弹性系统的偏微分方程系统，并确定解决方案是否稳定或在有限时间内发展奇点。主要项目包括：（i）引力Euler-Poisson系统和Einstein-Euler系统的自相似恒星坍缩和稳定性分析，（ii）可压缩Euler方程的强爆炸、收敛激波、空腔和稳定性理论，（iii）均匀旋转多体系统和Euler-Poisson系统的熵旋转解和稳定性理论，以及（iv）弹性体的长时间动力学。这些问题的研究将需要发展新的数学方法，依靠数学分析的偏微分方程，计算机辅助证明，和缩放不变性和自相似性的方程正在研究中。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

Fluids and gases manifest a wide range of interesting physical phenomena with rich applications in science and engineering. Their dynamics is governed by systems of nonlinear partial differential equations (PDEs) whose study presents many mathematical challenges. The objective of this project is to investigate under what conditions solutions to PDEs arising from compressible fluid dynamics with or without gravity and elastodynamics will develop singular behavior in finite time, e.g., leading to collapse or explosion, and under what conditions solutions could exist for all time. Answers to these questions will contribute to a better understanding of implosions and blast waves, stability and collapse of stars, long-time behavior of elastic bodies, and other physical phenomena. The project aims to deepen our understanding of these highly nontrivial phenomena through novel mathematical methods. The research project will be integrated by education and outreach activities through student research projects, mentoring activities, course development and dissemination to a broader audience. The focus of this project is the investigation of several systems of partial differential equations describing compressible flows and elastic systems and determine whether solutions are stable or develop singularities in finite time. The main projects include the study of: (i) self-similar stellar collapse and stability analysis for the gravitational Euler-Poisson system and the Einstein-Euler system, (ii) strong explosion, converging shocks, cavity, and stability theory for compressible Euler equations, (iii) uniformly rotating multi-body system and entropic rotating solutions and stability theory for the Euler-Poisson system, and (iv) long time dynamics for elastic bodies. The study of these problems will require the development of new mathematical approaches relying on mathematical analysis of PDEs, computer-assisted proofs, and the scaling invariance and self-similarity of the equations under study.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.