File size: 3,382 Bytes
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 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 | import OAI.MathematicalPhysics.DefocusingNLS.Profile.RadialMatchedOpenBlowup
import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.ClassicalMaximalFlow
import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.GaussianFullSupport
namespace OAI
/-! # The paper's main statements conditional on the published rectangle Rouché input
The sole explicit published input is `RectangleRouche`, the rectangle specialization
of the corollary after Ahlfors, *Complex Analysis*, third edition, Chapter 4,
§5.2, Theorem 18, p.153. The conclusions combine the profile, spectral, nonlinear, physical-openness
and probabilistic arguments.
-/
open Set Filter Topology MeasureTheory
namespace DefocusingNLS
attribute [local irreducible] HasFiniteTimeSelfSimilarBlowup IsClassicalDefocusingFlow
maximalSobolevInteractionDomain
/-- Theorem 1.1, with an arbitrary lower bound on the odd power. -/
theorem stable_blowup_conditional (hRou : RectangleRouche) (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 := by
obtain ⟨n, hn, hlarge⟩ := ((radialMatched_exists_open_blowup hRou).and
(eventually_ge_atTop (max 1 p₀))).exists
obtain ⟨N, hk, U, hUn, hUo, hU⟩ := hn
let m := n + radialInnerShootingThreshold
let p := 2 * m + 1
have hm : 1 ≤ m := by dsimp [m]; omega
have hodd : Odd p := ⟨m, rfl⟩
have hp : p₀ ≤ p := by dsimp [p, m]; omega
have hp3 : 3 ≤ p := by dsimp [p]; omega
have hdiv : (p - 1) / 2 = m := by dsimp [p]; omega
have ha : radialShootingA n = 1 / ((p : ℝ) - 1) := by
dsimp [radialShootingA, p, m]
push_cast
congr 1
ring
refine ⟨p, N + 1, hp, hodd, hp3, hk, U, hUn, hUo, ?_⟩
intro f hf
refine ⟨maximalSobolev_classical _ _ _ f, ?_⟩
simpa only [hdiv, ← ha] using hU f hf
/-- Corollary 1.2: the same fixed power and Sobolev order work for every allowed
Gaussian decay exponent. The charged event is an actual open measurable set. -/
theorem gaussian_blowup_conditional (hRou : RectangleRouche) (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 := by
obtain ⟨p, k, hp, hodd, hp3, hk, U, hUn, hUo, hU⟩ := stable_blowup_conditional hRou p₀
refine ⟨p, k, hp, hodd, hp3, hk, U, hUn, hUo, hU, ?_⟩
intro α hα
let : (weightedGaussianLaw k α).IsOpenPosMeasure := weightedGaussianLaw_isOpenPosMeasure k α hα
exact ⟨ae_mem_l2_gaussian_coefficients k α hα, ae_sobolevFourierCoefficient k α hα,
hUo.measure_pos _ hUn⟩
end DefocusingNLS
end OAI
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