Mathematical Questions in Relativistic Fluids

The general theory of relativity is currently the best description of gravitational phenomena and their interactions with matter. Proposed by Einstein in 1915, it has been overwhelmingly confirmed by experiments and is today an essential part of the toolbox employed by physicists studying astrophysics and cosmology. Mathematically, the theory is rich, and the topic of mathematical general relativity is now an exciting and active field of research among mathematicians.

This project deals with mathematical properties of relativistic fluids, and in particular with the mathematical structure of theories describing the motion of fluids under conditions in which the laws of relativity cannot be neglected. Examples include accretion disks near black holes, the dynamics of the quark-gluon plasma that forms in collisions of heavy-ions in particle accelerators, and the study of fine properties of star evolution.

The mathematical advances and techniques brought about by this project will be important for applications, particularly for the study of viscous effects on mergers of neutron stars. In terms of broader impact, the research has a strong interdisciplinary component and will continue a fruitful interaction between mathematics and physics. This project will also contribute to the education of young scientists, continuing the PI's efforts to disseminate some of the most recent findings in relativistic fluid dynamics to graduate students and post-docs.

Specifically, this project will investigate: (a) local well-posedness of the Einstein-Euler system with a physical vacuum boundary; (b) shock formation for the relativistic Euler equations; (c) causality and local well-posedness of theories of relativistic viscous fluids; and (d) formulations of the equations of relativistic viscous fluids coupled to the Einstein equations that are suitable for the numerical simulations of neutron star mergers. A unifying feature of all the partial differential equations (PDEs) studied in this project is that they form a system with multiple characteristic speeds.

Understanding systems of this type in more than one spatial dimension is currently one of the main challenges in hyperbolic PDEs. All problems will be studied under realistic physical assumptions. This involves, in particular, considering relativistic fluids in three spatial dimensions, with vorticity, and without symmetry assumptions.

Not only is such treatment essential for applications, but it also involves a great deal of rich mathematics.

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.