Stable blowup for a supercritical defocusing Schrödinger equation

Source and verification

This article explains family 371 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

Overview

Positive conserved energy does not always prevent concentration at high regularity. Family 371 claims a Sobolev-open set of finite-time blowup data for a scalar defocusing Schrödinger equation on the twelve-dimensional torus. Openness is essential to its Gaussian-data consequence.

🏷️ Preliminaries: defocusing energy and scaling

Consider

$$i{\partial }_{t}u+\Delta u=|u{|}^{p-1}u\text{on }{\mathbb{T}}^{12},$$

where $p\ge 3$ is odd. The sign is defocusing, with positive conserved energy

$$\begin{array}{c}E(u)\\ =\frac{1}{2}\int |\nabla u{|}^2+\frac{1}{p+1}\int |u{|}^{p+1}.\end{array}$$

The Euclidean equation has the scaling $u_{\lambda }(t,x)={\lambda }^{2/(p-1)}u({\lambda }^{2}t,\lambda x)$. Its critical Sobolev index is

$$s_c=6-\frac{2}{p-1}.$$

For the large powers considered here, $s_c>1$: the equation is energy-supercritical. The conserved energy controls one derivative in $L^2$, while scale-invariant continuation would require stronger control. This calculation is local motivation on the torus, whose fixed period is not preserved by arbitrary Euclidean rescaling.

🏷️ The stable-blowup claim

Family 371, as stated in the manuscript

There exist an odd power $p$, an integer $k>8$, and a nonempty open set $\mathcal{U}\subset H^k({\mathbb{T}}^{12})$ such that every datum in $\mathcal{U}$ has a unique local solution that blows up at some finite time $T$. At a suitable point $x_{\ast }$,

$$\begin{array}{c}(T-t{)}^{1/(p-1)}|u(t,x_{\ast })|\\ ⟶c_0>0.\end{array}$$

In particular, there is no continuous $H^k$ extension through $T$. (OpenAI, 2026)

The power is selected by the construction; this statement should not be read as blowup for every odd power or in every dimension. The formalization documentation makes the selection quantifier especially explicit: admissible odd powers can be chosen arbitrarily large, rather than asserting every sufficiently large odd power.

🏷️ Why an open set implies positive Gaussian probability

Let $g_n$ be independent standard complex Gaussian variables and consider

$$u_0^{\omega }(x)=\sum _{n\in {\mathbb{Z}}^{12}}{g}_n⟨n{⟩}^{-\alpha }{e}^{in\cdot x}.$$

Its expected squared $H^k$ norm is a constant times ${\sum }_n⟨n{⟩}^{2k-2\alpha }$, finite when $\alpha >k+6$. Hence the series belongs to $H^k$ almost surely.

Every nonempty $H^k$ ball has positive probability. To see the mechanism, approximate its center by a finite Fourier polynomial. A finite vector of nondegenerate Gaussians has positive probability of lying near those coefficients. The independent tail has arbitrarily small expected $H^k$ norm once the frequency cutoff is large, hence positive probability of being small. Intersecting these two independent events places the random series in the prescribed ball.

Consequently, a nonempty open blowup set has positive Gaussian probability for each $\alpha >k+6$. A finite-codimension family of blowup data alone would not justify this step. Neither probability-one blowup nor a lower bound uniform in $\alpha$ is claimed.

🏷️ Profile, spectrum, and physical-data openness

The manuscript seeks a self-similar core with spatial scale $(T-t{)}^{1/2}$, amplitude $(T-t{)}^{-1/(p-1)}$, and a logarithmically rotating phase. At large power, the profile approaches a flat-core limit whose matching problem can be analyzed using special functions.

Finite integer and rational certificates control low angular modes and a profile-matching degree. Analytic estimates handle the remaining high modes. The proposed spectral conclusion is that all relevant nondecaying directions arise from symmetries: phase, translations, and the blowup time. Those directions can be adjusted by changing physical parameters rather than imposing arbitrary constraints on the data.

The nonlinear construction initially gives a graph of perturbations decaying toward the profile. Varying the symmetry parameters and applying a degree argument is the further step that turns this graph into an open set of physical initial data. Localization to an expanding torus must preserve global norm control; local convergence alone would allow perturbation mass to escape the region being observed.

Verification boundary

The repository supplies exact-arithmetic certificate checks. These notes have not run them. Even successful finite checks would leave the analytic profile construction, spectral passage, torus localization, and openness argument to be verified separately.

🏷️ Key steps and proof outlines

The profile equations separate amplitude, phase, and pressure

Put $a=1/(p-1)$ and use similarity variables $s=\log (T/(T-t))$, $y=(x-x_{\ast })/\sqrt{T-t}$. The stationary profile equation in dimension twelve is

$$\Delta Q+i(y\cdot \nabla /2+a)Q+bQ=|Q{|}^{1/a}Q.$$

Writing $Q=A{e}^{i\varphi }$ with $A>0$, define $\mu =A^2$ and the radial drift $w=r/2+2{\varphi }^{\prime }$. The imaginary part integrates exactly:

$$w(r)=(6-2a)r^{-11}A(r{)}^{-2}{\int }_0^{r}A(t{)}^2{t}^{11}dt.$$

Regularity at zero fixes the integration constant. The real part becomes

$$-\Delta A+A^{1+1/a}=\left(b+\frac{r^2}{16}-\frac{w^2}{4}\right)A.$$

The construction solves an inner boundary-value problem and an exterior slow-decay problem, then matches them through two parameters. Its finite arithmetic and degree argument certify specific matching and spectral inputs; the nonlinear stability argument additionally requires the resulting nonvanishing profile and all stated tail estimates. In particular $Q=O(r^{-2a})$ and its linearized potential is $O(r^{-2})$ with differentiated versions.

A spectral list is converted into semigroup decay

The source’s spectral theorem identifies fourteen symmetry directions: phase with rate zero, twelve translations with rate $1/2$, and time translation with rate one. It also excludes generalized modes in the relevant half-plane. Merely listing these eigenvalues would not imply stable evolution for an infinite-dimensional operator.

The linear proof works in the real Hilbert space

$$\Vert v{\Vert }_Y^2=\Vert v{\Vert }_{{\dot{H}}^{6-a}}^2+\Vert v{\Vert }_{{\dot{H}}^k}^2,$$

with $k$ sufficiently large. The low order is below six, allowing an $L^{\infty }$ Fourier estimate at low frequency, and above $6-2a$, admitting the profile tail. Free similarity evolution gives squared-norm decay rate $-a$ at the low order. At the high order, choose $k$ after bounding the linearized potential. Its leading multiplication term has a bound independent of $k$; derivatives falling on the potential are smaller tail terms or compact local errors.

This yields an observation inequality

$$\frac{d}{ds}\Vert v{\Vert }_Y^2\le -c\Vert v{\Vert }_Y^2+C\Vert v{\Vert }_{L^2(B_{R_0})}^2.$$

For weakly null bounded initial sequences, local Sobolev compactness makes the observation term vanish on each fixed time interval. The time-one map is consequently a contraction plus a finite-rank operator. Its spectrum outside a smaller disk consists of finite-dimensional eigenspaces. The mode classification now identifies all of them with the fourteen symmetry directions, proving exponential decay on their complement.

The expanding torus has a controlled zero mode and a decaying defect

On the torus of radius $L_s=L{e}^{s/2}$, the low Fourier weight is regularized to $(L_s^{-2}+|n/L_s{|}^2{)}^{6-a}$. Omitting $L_s^{-2}$ would give a zero norm to constants. The same energy argument transfers to these expanding spaces, and compactness on fixed spatial balls transfers stable contraction from the whole-space problem.

Set ${\chi }_s(y)=\chi (y/L_s)$ and cut off the profile by $Q_s={\chi }_{s}Q$. The transport derivative of this cutoff vanishes because $y/L_s$ is the fixed torus coordinate. The remaining defect consists of

$$2\nabla {\chi }_s\cdot \nabla Q+(\Delta {\chi }_s)Q+({\chi }_s-{\chi }_s^p)|Q{|}^{p-1}Q.$$

It is supported at radius comparable to $L_s$ and has norm $O(L_s^{-2-a})$. Since the nonlinearity is polynomial for the selected odd $p$, the nonlinear remainder is quadratic in the perturbation with a uniform local Lipschitz bound.

Forward stable evolution and backward symmetry coordinates

At a fixed similarity-time step, split $v_n=w_n+E_n{u}_n$, where $w_n$ is stable and $u_n\in {\mathbb{R}}^{14}$. Stable evolution contracts forward. To obtain decaying symmetry coordinates, prescribe them instead by

$$u_n=-\sum _{j=n}^{\infty }{D}_{\ast }^{n-j-1}{\mathcal{F}}_j(v_j),$$

where $D_{\ast }$ is the diagonal symmetry step matrix and ${\mathcal{F}}_j$ collects transfer and nonlinear errors. In the norm ${\sup }_n{2}^n(\Vert w_n\Vert +K|u_n|)$, the sum converges even in the neutral phase direction. Choose $K$ to control coupling into the stable component, then the perturbation radius small and $L$ large. A contraction produces a continuous stable graph.

Finally vary physical phase $\theta$, center $\zeta$, and logarithmic blowup time $\beta$. Their leading spectral coordinates are the invertible linear map $(-\theta +b\beta ,\zeta ,\beta )$. On a small parameter sphere the remaining terms are strictly smaller. Brouwer degree therefore places every datum in a physical open $H^k$ ball on the appropriate stable graph. At the resulting center, $(T-t{)}^a|u(t,x_{\ast })|\to |Q(0)|>0$. Openness is obtained by this parameter argument, not by claiming that the graph for fixed similarity parameters is open. (OpenAI, 2026 s “Smooth profiles with a pressurized core,” “Linear decay and transfer to expanding tori,” and “Nonlinear stability, open sets, and Gaussian data”)

See Also

• Family 080 concerns pointwise convergence for the linear Schrödinger equation, a different question from nonlinear blowup.
• Family 367 also combines finite certificates with separate analytic arguments.

🏷️ Manuscripts and verification scope

• Stable self-similar blowup for a supercritical defocusing Schrödinger equation on the torus — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The scope page selects arbitrarily large admissible odd powers and a corresponding Sobolev index, not every sufficiently large odd power. It includes the open blowup set and its positive-probability Gaussian consequence for decay exponent greater than k + 6. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026. Stable self-similar blowup for a supercritical defocusing Schrödinger equation on the torus.