Geometric Methods for Singular Solutions to Nonlinear Hyperbolic Partial Differential Equations

The principal investigator (PI) will study equations with physical and geometric origins, including Euler’s equations, which describe the motion of fluids such as air and water, and Einstein’s equations of General Relativity, which describe the propagation of gravitational waves – whose recent experimental detection led to the Nobel Prize in Physics. The projects on fluids and related equations will allow one to make rigorous mathematical predictions about the formation and structure of shock waves (e.g. sonic booms).

A key new contribution will be accounting for the presence of swirling motion, a notoriously complex phenomenon that is ubiquitous in nature. The projects in General Relativity will make rigorous predictions about whether a Big Bang occurred in the past, based on assumptions about the present state of the universe. The results will rigorously confirm the dynamic stability of the Big Bang for the full range of situations where it has been expected to occur, thus providing a proof of a conjecture that has its roots in ideas stretching back 50-years.

A common theme unifying the projects in fluids and gravity is that they involve wave-like motion. In previous work, the PI developed new tools for the study of waves, shaped by ideas from geometry. In particular, his recent work has shown that the equations of fluid motion have some unexpected, remarkable commonalities with Einstein’s equations.

These connections allow the PI to blend insights and techniques from the seemingly separate fields of fluids and gravity, which in turn serves as a driving force behind the projects. The research directions are highly interdisciplinary and are ripe with opportunities for training the next generation of researchers across disciplines. Undergraduates, Ph.D. students, and postdoctoral researchers will be involved in the work of the project.

In a first line of research, the PI will study stable shock-forming solutions to the compressible Euler equations in three spatial dimensions, with a focus on giving a complete description of the maximal classical development up to the boundary. A key new feature of the research is that the vorticity and entropy can be non-zero, and the behavior of these quantities must be tracked all the way up to the boundary.

Because the shape of the boundary is unknown in advance, and because elliptic estimates are needed to control the vorticity and entropy, the analysis requires a multitude of new geometric techniques and insights about fluid flow. In a second line of research, the PI will study the shock development problem for various multiple speed quasilinear hyperbolic PDE systems.

This is the problem of describing the transition of initially smooth solutions past their first shock singularity in a manner such that they become unique weak solutions, while simultaneously constructing the shock hypersurface, across which the solution jumps. The shock development problem for multiple speed systems in multiple spatial dimensions is completely open.

This research requires new techniques that account for the distinct singularity strengths exhibited by the different solution variables as they transition across the shock hypersurface. In a third line of research, the PI will study stable Big Bang formation (i.e., stable curvature blowup along an entire spacelike hypersurface) in solutions to Einstein's equations.

The proposed approach is based on a new gauge that will allow one to prove stable Big Bang formation in the entire regime where it has been conjectured to occur. Due to the character of Big Bang singularities and shock singularities, the methods have deep analytical connections to the problems on shocks. Conversely, the techniques relevant for the problems on shocks have their origins in General Relativity. Thus, there is cohesiveness between all the research directions.

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.