Here is a 3-paragraph summary of the blog post for mathopen.com:
Recent exciting work by mathematicians Alpöge and Buckmaster, building on earlier results by Córdoba and Martínez-Zoroa, has made significant progress on one of the most notorious open problems in mathematics: the global regularity problem for the incompressible three-dimensional Navier-Stokes equations. This problem sits at the heart of our understanding of fluid dynamics and remains one of the Clay Millennium Prize Problems, making any new developments in the area major news for the mathematical community.
The research focuses on the phenomenon known as "finite time blowup," where solutions to certain fluid equations become singular or infinite in finite time rather than remaining smooth and well-behaved forever. What makes this new work particularly striking is that the blowup occurs even in the presence of a smooth forcing term, meaning the equations are being driven by a perfectly nice external input and still produce catastrophic behavior. The equations studied include the incompressible porous medium equation, the Boussinesq equations, and the incompressible Euler equations.
The broader significance of this work is that it strongly suggests smooth initial data and a smooth forcing term can conspire to produce blowup, which would have profound implications for our theoretical understanding of fluid mechanics. While the Navier-Stokes equations themselves remain unconquered, results like these on closely related systems sharpen our intuition and bring the community closer to resolving one of the greatest unsolved problems in all of mathematics.