Stable blowup for the defocusing Schrödinger equation

The explanation

Start with the simplest wave: a constant, u = c, spread evenly over the space. The equation here is i∂ₜu + Δu = |u|^{p−1}u. For constant data the derivative term Δu is zero. The solution only turns its phase: u(t) = c·e^{−i|c|^{p−1}t}. Its size |u| stays equal to |c| forever. Nothing blows up. The question is whether a wave that is not flat can concentrate into a point of infinite height.

Here u is a complex wave on the torus. The torus is a box whose opposite faces are glued, so waves leave one side and come back on the other. The symbol Δ is the Laplacian, which spreads the wave out. Defocusing means the right side also pushes the wave apart. Blowup means the solution stops being smooth at some finite time T.

The energy of the wave stays fixed and positive. In twelve dimensions, though, it does not control the smoothness needed to continue the solutionSource 1. Each power p has a critical smoothness level, d/2 − 2/(p−1). For d = 12 this is 5 when p = 3 and 5.98 when p = 101, far above 1.

Earlier work built blowup in nearby settings. Tao used a specially designed vector system with forcingSource 2. Merle, Raphaël, Rodnianski and Szeftel found radial blowup for four pairs of dimension and power, under a stated conditionSource 3. Cao-Labora, Gómez-Serrano, Shi and Staffilani did the cubic case on an eight-dimensional torusSource 4. Those two gave only thin families of starting data. One recent result did not claim an open set of dataSource 5.

The paper claims that for some large odd power, an open set of starting data in a Sobolev space H^k blows upSource 6. A Sobolev space measures smoothness through k derivatives. The blowup is self-similar: the spike has width (T−t)^{1/2} and height (T−t)^{−1/(p−1)}Source 7. To show the scaling, suppose p were 5. When T − t = 0.0001, the width is 0.01 and the height grows tenfold. Because the set is open, random Gaussian starting data hit it with positive probabilitySource 8,Source 9. The paper does not claim blowup with probability oneSource 10. The key tool is a limit of very large powers. There, the spike profile has a flat core and a free outer partSource 11.

How sure is this? Lean, a computer proof checker, has a formal statement of the open blowup set and the Gaussian result. So the paper proves them, as far as the Lean statement matches the paper. Lean gives arbitrarily large odd powers, not every large odd power. No specialist review of the paper is published.

What it doesn't show. This does not say which odd power works or give a number for it. It does not claim blowup with probability one, and it treats only dimension twelve.