CMSA Special Seminar: A physical interpretation of recent singularity theorems

We present a physical interpretation of recent mathematical proofs (Córdoba and Martínez-Zoroa; Buckmaster and Apogee; OpenAI) for singularity formation in nonlinear partial differential equations with smooth forcing. In our formulation, singularity formation is driven by a forward in time process whereby a self-consistent wave–mean-flow interaction couples a similarity profile to a short-wavelength instability of the underlying PDE. This enables a smooth external force to excite short-wavelength instabilities that directly couple to the similarity profile by generating quadratic nonlinearities (Reynolds stresses). These nonlinearities drive the singularity and lead to finite time blowup. The underlying stresses must be computed self-consistently with the similarity profile itself. Our scheme is an explicit construction and allows direct numerical evaluation of the underlying similarity solutions, determined self-consistently. The scheme is not (at present) a proof, but an asymptotic construction. We apply this approach to the 2D incompressible porous media (IPM) equation and the 3D axisymmetric Navier Stokes equations. Finally, we discuss connections of this overall idea to work from the 1990s on droplet break, Euler singularities and instabilities in studies of singularities in PDEs. This is joint work with Kaylie Hausknecht.