Singularities and stability in compressible fluids with or without gravity

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.