Classical Solutions and Singularity Formation in Incompressible Fluids

Summary

[This is a guest post by Diego Córdoba and Luis Martínez-Zoroa . This blog post was initially written in a different file format and converted using AI. — T.] Euler formulated the equations for an inviscid incompressible fluid more than 250 years ago. Published in his 1757 memoir , they belong to the earliest systems of partial differential equations in mathematical physics, following d’Alembert’s work on the wave equation. Understanding the evolution they describe has required the work of many generations of mathematicians. One of the central questions is whether an initially regular fluid flow can develop a singularity in finite time. The recent work on Euler and Navier–Stokes obtained with the aid of large language models has brought renewed attention to this question. These exciting developments build on decades of mathematical research, including works from recent years, when the field has continued to be specially active. In this post we describe some contributions to that body of knowledge, including our own. We will go over some of that recent mathematical literature, and discuss the ideas of constructing blow-ups through a cascade of vortex layers. Partial differential equations provide a common language for problems ranging from wave propagation and fluid motion to geometry and general relativity. For an evolution equation, the Cauchy problem asks us to determine the future from prescribed initial data. A local existence theorem gives a solution for a short time; the next question is whether it continues for all time or develops a singularity at a finite time. This distinction between global existence and finite-time blow-up is already visible in the elementary Riccati equation The solution starts from a finite value and remains smooth until , when it becomes unbounded. The quadratic nonlinearity in three-dimensional Euler suggests a similar possibility, but the fluid problem also involves transport, geometry and a pressure determined nonlocally by the entire flow. The challenge is to understand whether the full equation can sustain the amplification suggested by this simple ODE. The central aim of our research has been to construct classical solutions of incompressible fluid equations that develop singularities in finite time, starting within a regime of local existence and uniqueness. We want the initial data to determine a unique regular evolut…