Dynamics of Compressible Fluids: Gravity, Singularities and Vacuum

Fluids and gases exhibit a broad range of physical phenomena in both laboratory and astrophysical settings, with rich applications across the mathematical sciences and engineering. Their dynamics are governed by systems of nonlinear partial differential equations (PDE), whose analysis presents fundamental mathematical challenges. The objective of this project is to determine the conditions under which solutions of PDE arising in compressible fluid dynamics, with or without self-gravity, develop singular behavior, including shock formation, gravitational collapse, and the emergence of moving free boundaries, as well as the conditions under which global-in-time solutions exist. Addressing these questions will advance our understanding of fundamental physical processes such as supernova explosions near stellar boundaries, stellar formation and end states, the dynamics of vacuum regions, bubble evolution, and related phenomena. By developing novel mathematical techniques, this project seeks to resolve longstanding open problems and deepen our understanding of these highly nonlinear phenomena. The research program will be closely integrated with education and outreach through student research opportunities, mentoring activities, course development, and dissemination of research to the broader scientific community. The project focuses on the formation of singularities and the stability of physically significant structures arising in compressible fluid dynamics through the analysis of initial and initial-boundary value problems for the compressible Euler and Euler-Poisson systems. The project studies (i) nonlinear wave propagation and shock formation near stellar boundaries in the gravitational Euler-Poisson system; (ii) non-radial stellar collapse and the collapse of massive white dwarfs; (iii) the emergence and propagation of physical vacuum free boundaries in compressible gas dynamics; and (iv) the stability theory of hydrostatic bubbles. The goal is to develop rigorous mathematical frameworks that accurately capture these physical phenomena and to establish new analytical theories using modern techniques from nonlinear PDE and applied analysis. 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.