Finite time blowup with smooth forcing term for the incompressible porous medium, Boussinesq, and incompressible Euler equations

There’s some exciting very recent work by Alpöge and Buckmaster, building upon prior work by Córdoba and Martínez-Zoroa, in the general topic around the infamous global regularity problem for the incompressible three-dimensional Navier-Stokes equations. It is now widely expected that it should be possible to construct smooth initial data and smooth forcing term that would make these equations develop singularities in finite time; and it should even be possible to do without the forcing term. While these authors do not quite achieve these goals yet, they have made enough of a breakthrough that it looks very feasible to complete these goals in the near future. In particular, they have demonstrated such finite time blowup for three simpler model equations: the incompressible porous medium (IPM) equation, the two-dimensional Boussinesq equation, and the three-dimensional incompressible Euler equations. (The first of these equations was already handled by Córdoba and Martínez-Zoroa, but Alpöge and Buckmaster found a variant of their method that also extended to the other two equations, and has a high likelihood of also extending to Navier-Stokes as well.) As is now remarkably feasible in the modern era of autoformalization agents, their work has also been formalized in Lean.

As one may expect nowadays, the arguments here are heavily AI-assisted, but the authors have been working over the last few weeks to simplify and rewrite the proofs from what they literally call “the worst writeup we had ever seen in the history of mathematics” into something far more readable and of professional quality. This is still a work in progress: unfortunately, they were forced to release their preliminary preprints before they were completely digested and polished, due to external events that are documented on the above link. Nevertheless, the introduction to the Boussinesq paper at least is in pretty good shape, and can serve as an initial starting point. I was also fortunate to have Tristan Buckmaster explain the main ideas of the paper in a half-hour phone conversation, although I still need to work some more (probably with some combination of a blackboard and modern AI tools) to digest things more. For now I will try to write a quick summary of some of the main ideas, the highlighting of which I view as the main value of such work; the actual solving of these problems is only a proxy goal for the primary goal of developing mathematical understanding and insight. Without such understanding, even a problem as infamous as the Navier-Stokes regularity problem of far less intrinsic significance to mathematics than is sometimes promoted in popular media.

The basic strategy, due to Cordoba and Martínez-Zoroa, is to iteratively build up the solution to such equations in stages, repeatedly adding small high frequency corrections to a previous (forced) solution in a manner that makes the solution more singular towards the blowup time while keeping the forcing term well behaved. Rather than work with any specific equation, let’s work with a completely abstract equation

where is the solution, is the nonlinear differential operator representing the equation of motion, and is the forcing term. Of course this is far too general a setting to perform a full analysis, but it should suffice for this brief post.

Suppose that one has already managed to construct a low frequency solution

to this equation, and would like to perturb it to create a new solution

that adds a high frequency correction to the solution that starts emerging near the blowup time, at the cost of some (presumably also) high frequency correction to the forcing term. If one can make the amplitude of the solution correction relatively large while keeping the amplitude of the correction very low, and the frequencies of the corrections increase rapidly with each iteration, then one can hope to iterate this procedure and pass to a limit to obtain a solution

where now exhibits blowup in finite time, while remains smooth.

To make this strategy work, should approximately solve the difference equation

to keep small. If we can somehow neglect nonlinear effects, this basically amounts to solving a linearized equation

The game is then to design the background solution in such a way that the evolution equation exhibits some sort of exploitable instability, in which a solution to such an equation can start off exponentially small at early times, but become large near the blowup time. At this point one may expect nonlinear effects to kick in and make the solution extremely difficult to analyze; but if one can time the emergence of large amplitudes just right, one can hope to arrive at a sweet spot where, by the blowup time, the amplitude has become large enough to disrupt smoothness, but not so large to destabilize the analysis.

In the case of the Boussinesq equation at least, there is an explicit ansatz, described in the introduction to the relevant paper, in which, at times close to blowup and locations close to the origin, behaves linearly in space, and behaves like a high frequency plane wave. Remarkably, this ansatz can be solved exactly (without any nonlinear correction terms), leading to an explicit system of ODE modulation equations that have the required instability property. This is the basic mechanism for blowup; however there are an enormous number of technical complications, for instance relating to spatial cutoffs, that are needed to make the full argument rigorous. The Alpöge–Buckmaster construction has some technical improvements over the older Córdoba–Martínez-Zoroa construction that allow them to treat more general fluid equations; I have not yet digested the precise differences, but the ODEs seem to be more unstable and the high frequency corrections appear to have better spatial localization properties.

Hopefully there will be some better expositions and talks by the authors on this nice result in the future. If I have time and am able to digest the results better, I may also be able to give more details in a followup blog post.

EDIT: there is now also an independent preprint of Ganeshram, Duruisseaux, and Anandkumar that has made a significant advance on the other major approach to finite time blowup, which is to first use numerical or machine learning tools to locate an approximately self-similar blowup profile ansatz, and then demonstrate that it is stable enough to be perturbed to an actual solution. For the Euler equations (with no forcing term or boundary), they have used a physics-informed neural network (PINN) to locate a numerically stable candidate solution; though actually establishing its stability to within the tolerance of the residual error in the solution remains a major challenging task to carry this result all the way through to a full rigorous demonstration of finite time blowup.