Stable blowup in a defocusing Schrödinger equation

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Family 371 · Partial differential equations
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-26

1. The question

The word “defocusing” suggests that a wave spreads rather than concentrates. For the nonlinear Schrödinger equation, its most immediate mathematical meaning is that the nonlinear part of the conserved energy has a positive sign. Yet positive energy need not control all the derivatives required to keep a solution smooth.

Can a defocusing wave develop a singularity that survives small changes in its initial state? And if it can, does making random initial data arbitrarily regular guarantee global evolution with probability one?

Family 371 claims stable finite-time blowup for a scalar, unforced defocusing Schrödinger equation on the twelve-dimensional torus, for a sufficiently large odd power. The blowup data contain a nonempty open set in a high Sobolev space.

That openness is the decisive feature. The manuscript uses it to show that a prescribed family of Gaussian Fourier laws has positive probability of blowup, even with arbitrarily high finite Sobolev regularity. This is not a claim of probability-one blowup or of instability for every defocusing equation.

2. Background and essential definitions

The equation and its positive energy

On the periodic space

$$\mathbb T^d=(\mathbb R/2\pi\mathbb Z)^d,$$

consider a complex-valued function $u(t,x)$ satisfying

$$i\partial_tu+\Delta u=|u|^{p-1}u,$$

where $p\ge3$ is odd. Opposite faces of a periodic cell are identified. There is no boundary where conditions must be imposed.

For sufficiently regular solutions, the conserved mass and energy are

$$M(u)=\int_{\mathbb T^d}|u|^2\,dx,$$

and

$$E(u)=\int_{\mathbb T^d} \left(\frac12|\nabla u|^2+\frac1{p+1}|u|^{p+1}\right)\,dx.$$

Both energy terms are nonnegative. This distinguishes the defocusing sign from the focusing sign, where the nonlinear energy term is negative.

Nevertheless, these integrals allow a very tall feature concentrated on a very small region. In a supercritical regime, their conservation does not bound the high Sobolev norm needed for classical continuation.

Scaling and supercriticality

The Euclidean version has the scaling

$$u_\lambda(t,x)=\lambda^{2/(p-1)}u(\lambda^2t,\lambda x).$$

Its invariant homogeneous Sobolev order is

$$s_c=\frac d2-\frac2{p-1}.$$

“Energy-supercritical” means $s_c>1$: the scale-invariant regularity lies above the first-derivative level controlled by energy. Although changing scale does not preserve one fixed torus globally, it still describes a concentrating region much smaller than the periodic cell.

The space $H^k(\mathbb T^d)$ controls square-integrable derivatives through order $k$. For large enough $k$, standard local theory gives a unique solution from $H^k$ data. The question is whether that solution can continue for all time.

Blowup, self-similarity and openness

Finite-time blowup here means that the solution cannot be extended continuously in $H^k$ through a finite time $T$. The manuscript proves a stronger visible symptom: its amplitude diverges at a point.

With $a=1/(p-1)$, the claimed core has spatial scale $(T-t)^{1/2}$ and amplitude scale $(T-t)^{-a}$, together with a logarithmically rotating complex phase. This is called self-similar, or type-I, blowup in the manuscript.

An open set of blowup data means that some entire $H^k$ ball consists of data leading to blowup. Nearby data may have different blowup times, locations and phases. Stability permits those parameters to adjust; it does not demand that every perturbed solution blow up at exactly the same spacetime point.

A finite-codimension family is weaker: it can require several exact constraints on the initial data and need not contain any open ball.

3. Development history and previous partial results

Classical Schrödinger Cauchy theory establishes local solutions and many global results in appropriate regimes. In certain whole-space supercritical settings, Killip and Visan showed that a bounded critical Sobolev norm implies scattering. The unresolved issue in such a criterion is obtaining that bound from the initial data or conserved quantities.

Tao constructed blowup for a specially designed vector-valued forced defocusing system, showing that conservation and scaling alone are not enough to settle all regularity questions. That model differs from the unforced scalar equation considered here.

Merle, Raphaël, Rodnianski and Szeftel constructed radial whole-space defocusing blowup on finite-codimension families for selected dimension–power pairs, with a stated profile nondegeneracy condition. Cao-Labora, Gómez-Serrano, Shi and Staffilani developed nonradial periodic blowup, including the cubic equation in dimension eight. Those constructions use hydrodynamic profiles at spatial scales $(T-t)^{1/r}$ with $r>2$.

A more immediate predecessor is Buck, Carballeira, Gómez-Serrano and Shi. They construct asymptotically self-similar scalar defocusing blowup from smooth radial compactly supported data on $\mathbb R^d$ for $(d,p)=(3,27),(4,9),(5,7)$. Their construction is finite-codimensional; it does not assert an open set in the physical Sobolev space.

Open sets of self-similar blowup data have precedents for focusing NLS, including work of Merle–Raphaël–Szeftel and Li. The sign and spectral analysis are different. Family 371 claims to establish the profile and the full stability mechanism needed for an open set in the defocusing periodic problem.

The Gaussian question is motivated by an energy-supercritical branch of an open problem in Deng, Nahmod and Yue. The source explicitly gives its own precise quantifiers rather than identifying its corollary with every possible formulation of that problem. Global-flow results for particular invariant random ensembles also concern different probability measures.

4. What the new manuscript claims

A deterministic open set

There exist an odd power $p$, a finite integer $k>8$, and a nonempty open set

$$\mathcal U\subset H^k(\mathbb T^{12};\mathbb C)$$

such that every initial datum in $\mathcal U$ produces a unique solution on $[0,T)$, for some finite $T$, with

$$u\in C([0,T);H^k)\cap C^1([0,T);H^{k-2}),$$

and no continuous $H^k$ extension through $T$.

More precisely, for some $x_*\in\mathbb T^{12}$ and $c_0>0$,

$$\lim_{t\uparrow T}(T-t)^{1/(p-1)}|u(t,x_*)|=c_0.$$

The power is chosen sufficiently large and odd; $k$ is fixed afterward. The theorem is existential and does not provide an explicit smallest permissible power.

Positive probability under a prescribed Gaussian law

Let $g_n$, indexed by $n\in\mathbb Z^{12}$, be independent standard complex Gaussian random variables. For every $\alpha>k+6$, consider

$$u_0^\omega(x)=\sum_{n\in\mathbb Z^{12}} \frac{g_n(\omega)}{\langle n\rangle^\alpha}e^{in\cdot x}, \qquad \langle n\rangle=(1+|n|^2)^{1/2}.$$

The manuscript states that these data belong to $H^k$ almost surely and have positive probability of finite-time blowup. Increasing $\alpha$ gives arbitrarily high finite Sobolev regularity. A single fixed polynomial decay exponent does not mean that samples are almost surely $C^\infty$.

The probability may depend on $\alpha$. No uniform positive lower bound and no almost-sure blowup assertion are made.

This refutes the universal assertion that every energy-supercritical pair admits a finite Gaussian-decay threshold beyond which all the indicated high-regularity local solutions extend globally almost surely. One fixed pair $(12,p)$ suffices for that logical conclusion.

The source also obtains smooth deterministic blowup examples on product tori of every dimension $d\ge12$ by making the solution independent of the extra coordinates. It does not infer higher-dimensional openness or Gaussian blowup probability from that extension.

5. Why the advance matters

Importance: 5/5. If correct, the family shows robust singularity formation for an unforced scalar defocusing equation and converts it into a failure of a broad proposed random-data global-existence principle.

The deterministic robustness is stronger than producing one special singular solution. It says that blowup occupies a genuine neighborhood in a high-regularity phase space, after allowing the singularity parameters to adjust.

The probabilistic conclusion is equally specific: regularity and full support of the Gaussian law do not remove that neighborhood. Randomness in this setting does not automatically protect the evolution from a stable deterministic singularity.

6. The main idea

Build a profile in the large-power limit

Set $a=1/(p-1)$. For large $p$, this is a small parameter. In the limit $a\to0$, the radial profile develops a nearly flat core and an exterior governed by a simpler equation.

The exterior equation is expressed through confluent hypergeometric functions. Matching its value and derivative to the core becomes a finite-dimensional problem involving the phase frequency and core radius. The manuscript supplies analytic tail estimates and exact rational certificates for the matching and low-angular spectral calculations.

It then constructs genuine profiles at finite, sufficiently large odd powers. These are not merely formal asymptotic series. The finite calculations are reported as part of the manuscript’s argument; no independent certification is asserted by this digest.

Identify all relevant perturbation modes

Use similarity variables

$$y=\frac{x-x_*}{\sqrt{T-t}}, \qquad s=\log\frac{T}{T-t}.$$

Physical blowup at finite time corresponds to $s\to\infty$. Near its center the leading form is

$$u(t,x)\simeq e^{i\theta}(T-t)^{-a} \exp\!\left(ib\log\frac{T-t}{T}\right)Q(y),$$

where $Q$ is the profile and $b$ its phase-rotation frequency. This expression describes the concentrating core; the actual periodic solution uses a localized profile plus a decaying perturbation.

Linearization around $Q$ separates perturbations into spherical-harmonic angular degrees. The claimed spectral classification leaves only the modes corresponding to phase, twelve spatial translations and blowup time in the relevant nondecaying spectral region.

Controlling only finitely many low modes would be insufficient. A uniform outgoing estimate excludes modes escaping to arbitrarily large temporal or angular frequencies. A compact limiting argument preserves algebraic multiplicities in the bounded spectral region.

Transfer stability to the periodic problem

In similarity coordinates, the torus expands as the singularity scale shrinks. Local convergence toward the whole-space profile does not automatically control perturbations far from the core: norm could escape into distant regions of the expanding torus.

The manuscript combines local compact-error estimates with global dissipative control in its similarity-space norms. This transfers decay to expanding tori while retaining the part of the norm that a purely local limiting argument could lose. The dissipation is in the rescaled perturbation estimates, not a claim that the original Schrödinger mass or energy ceases to be conserved.

A nonlinear argument then constructs a graph of perturbations that decay in similarity time.

Let physical parameters absorb the remaining modes

A stable graph by itself could still describe only a finite-codimension set. The source varies the phase, center and blowup time—fourteen real parameters in total—and shows that their variations account for all complementary modes.

A degree argument solves the matching equations for every physical datum in a small $H^k$ ball around a smooth localized reference profile. This is the step that turns a constrained perturbation construction into an open set of actual initial data.

At the center, $Q(0)\ne0$, and the perturbation tends to zero. Undoing the similarity variables gives the stated amplitude limit. Since high $H^k$ regularity controls pointwise values, that divergence prevents continuous $H^k$ continuation.

Use Gaussian support, not random forcing

For $\alpha>k+6$, the expected squared $H^k$ norm of the Fourier series is finite because

$$\sum_{n\in\mathbb Z^{12}}\langle n\rangle^{2k-2\alpha}<\infty.$$

Its law has full support in $H^k$: every nonempty open ball has positive probability. A finite collection of Fourier coefficients has a positive Gaussian density everywhere, while the remaining tail can be small with positive probability.

The deterministic blowup neighborhood is one such open ball. The random solution therefore blows up with positive probability. No stochastic forcing is added to the PDE, and no separate random singularity construction is needed.

7. Scope, limitations, and open questions

The stable theorem concerns dimension twelve and a sufficiently large odd power. It does not show open blowup sets for every energy-supercritical dimension–power pair or settle the optimal dimension and power range.

The Gaussian consequence concerns the specific independent Fourier law above. It does not contradict global existence for other ensembles, especially measures supported on special sets of initial data. It also leaves the blowup probability unquantified.

The result permits changes in blowup time, center and phase under perturbation. It does not assert that arbitrary large data blow up, that the same singularity is the only possible one, or that conservation of mass and energy fails before blowup.

The product construction gives higher-dimensional deterministic examples only. Extending openness, spectral stability or the Gaussian statement to those dimensions requires additional analysis.

8. Sources and further reading

The primary blowup papers linked above locate the contribution relative to whole-space, periodic, type-II and self-similar constructions. The Deng–Nahmod–Yue paper provides the random-data motivation, while the source gives the exact universal assertion it refutes.

For background, nonlinear Schrödinger equations and Sobolev spaces introduce the equation and regularity framework. The manuscript below supplies the deterministic openness and Gaussian consequence reported here.

Source manuscripts

• M1. OpenAI, Stable self-similar blowup for a supercritical defocusing Schrödinger equation on the torus (September 24, 2026).

Sources consulted on 26 October 2026 at repository revision 1581d4c. Download the manuscript citations.