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fac6777 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 | import OAI.MathematicalPhysics.DefocusingNLS.Certificates.RectangleRouche
import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.GaussianCounterexample
namespace OAI
/-! # Stable defocusing Schrödinger blowup
Theorem 1.1 and Corollary 1.2 of OpenAI, *Stable self-similar blowup for a
supercritical defocusing Schrödinger equation on the torus* (September 2026).
-/
open Set Filter Topology MeasureTheory
namespace DefocusingNLS
attribute [local irreducible] HasFiniteTimeSelfSimilarBlowup IsClassicalDefocusingFlow
maximalSobolevInteractionDomain
/-- Theorem 1.1: arbitrarily large odd powers admit a nonempty open family of
classical solutions with finite-time self-similar blowup on the twelve-torus. -/
theorem stable_blowup (p₀ : ℕ) :
∃ p k : ℕ, p₀ ≤ p ∧ Odd p ∧ 3 ≤ p ∧ ∃ hk : 8 < (k : ℝ),
∃ U : Set FourierL2, U.Nonempty ∧ IsOpen U ∧ ∀ f ∈ U,
IsClassicalDefocusingFlow k (by linarith) ((p - 1) / 2) f ∧
HasFiniteTimeSelfSimilarBlowup k (by linarith) ((p - 1) / 2)
(1 / ((p : ℝ) - 1)) f :=
stable_blowup_conditional rectangle_rouche p₀
/-- Corollary 1.2: for every permitted Gaussian Fourier decay exponent,
the same open blowup family has positive Gaussian probability. -/
theorem gaussian_blowup (p₀ : ℕ) :
∃ p k : ℕ, p₀ ≤ p ∧ Odd p ∧ 3 ≤ p ∧ ∃ hk : 8 < (k : ℝ),
∃ U : Set FourierL2, U.Nonempty ∧ IsOpen U ∧
(∀ f ∈ U, IsClassicalDefocusingFlow k (by linarith) ((p - 1) / 2) f ∧
HasFiniteTimeSelfSimilarBlowup k (by linarith) ((p - 1) / 2)
(1 / ((p : ℝ) - 1)) f) ∧
∀ α : ℝ, (k : ℝ) + 6 < α →
(∀ᵐ g ∂fourierGaussianLaw, Memℓp (weightedGaussianCoefficient k α g) 2) ∧
(∀ᵐ g ∂fourierGaussianLaw, ∀ n,
sobolevFourierCoefficient k (weightedGaussianVector k α g) n =
(1 + ‖n‖ ^ 2) ^ (-α / 2) • g n) ∧
0 < weightedGaussianLaw k α U :=
gaussian_blowup_conditional rectangle_rouche p₀
/-- In particular, Gaussian data are not almost surely globally continuable. -/
theorem gaussian_not_almost_sure_global (p₀ : ℕ) :
∃ p k : ℕ, p₀ ≤ p ∧ Odd p ∧ 3 ≤ p ∧ ∃ hk : 8 < (k : ℝ),
∀ α : ℝ, (k : ℝ) + 6 < α →
¬ ∀ᵐ f ∂weightedGaussianLaw k α,
maximalSobolevInteractionDomain k (by linarith) ((p - 1) / 2) f = univ :=
gaussian_not_almost_sure_global_conditional rectangle_rouche p₀
end DefocusingNLS
end OAI
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