File size: 3,058 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 | import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffDiagonalTransfer
import OAI.MathematicalPhysics.DefocusingNLS.Certificates.ContinuousDiagonalLinearStep
namespace OAI
/-! # Continuous normalized linear step data for the actual cutoff profile -/
open scoped SchwartzMap ContDiff NNReal
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
local notation "Radius" => {L : ℝ // 1 ≤ L}
attribute [local irreducible] homogeneousStableCoordinates diagonalRealCoordinates
homogeneousComplexLinearizedStep cutoffProfileEndpoint HasContinuousFiniteDiagonalTorusLinearSteps
theorem cutoffProfile_continuous_diagonal_steps (a b k : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 10 < k) (m : ℕ)
(χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ))
(hχzero : ∀ y : E, 1 ≤ ‖y‖ → χ y = 0)
(hχone : ∀ y : E, ‖y‖ ≤ 1 / 2 → χ y = 1)
(Qp : E → ℂ) (hQp : ContDiff ℝ ∞ Qp) (q : HomogeneousY a k)
(hq : ∀ y, homogeneousPhysicalCLM a k ha ha1 (by linarith) q y = Qp y)
(Q : ℝ) (hQ : 0 ≤ Q)
(hqb : ∀ L : Radius,
‖cutoffProfileCoefficient a k ha1 (by linarith) χ hχ Qp hQp L‖ ≤ Q)
(P : (HomogeneousY a k × HomogeneousY a k) →L[ℂ] (HomogeneousY a k × HomogeneousY a k))
[FiniteDimensional ℂ P.range]
(hcomm : ∀ t, Commute (homogeneousComplexLinearizedStep a b k ha ha1 (by linarith) m q t) P)
(G : P.range →L[ℂ] P.range)
(hG : ∀ t, projectionSemigroupRestriction
(homogeneousComplexLinearizedStep a b k ha ha1 (by linarith) m q) P hcomm t =
NormedSpace.exp ((t : ℝ) • G))
(hspan : (⨆ lam : ℂ, Module.End.eigenspace G.toLinearMap lam) = ⊤)
(hspec : ∀ (lam : ℂ) (v : P.range), v ≠ 0 → G v = lam • v →
lam = 0 ∨ lam = 1 ∨ lam = 1 / 2)
(hkernel : ∀ β : ℝ, 0 < β → ∃ T₀ : ℝ, 0 ≤ T₀ ∧
∀ T : ℝ≥0, T₀ ≤ T → ∃ L₀ : ℝ, ∀ L : Radius, L₀ ≤ L.1 →
∀ f : FourierL2, ‖f‖ ≤ 1 →
homogeneousStableCoordinates a k ha ha1 (by linarith) P
(homogeneousLocalizationCLM a k L.1 ha ha1 (by linarith) L.2 χ f) = 0 →
‖cutoffProfileEndpoint a b k ha ha1 (by linarith) m χ hχ Qp hQp Q hQ hqb T L f‖ < β) :
HasContinuousFiniteDiagonalTorusLinearSteps (homogeneousStableCoordinates a k ha ha1 (by linarith) P)
(cutoffProfileEndpoint a b k ha ha1 (by linarith) m χ hχ Qp hQp Q hQ hqb) := by
have hk8 : 8 < k := by linarith
apply continuous_finiteDiagonalTorusLinearSteps_of_transfer a k ha ha1 hk8 χ (1 / 2) (by norm_num) hχone
(homogeneousStableCoordinates a k ha ha1 hk8 P) G hspan hspec
(cutoffProfileEndpoint a b k ha ha1 hk8 m χ hχ Qp hQp Q hQ hqb) hkernel
· exact fun T => cutoffProfile_diagonal_coordinate_transfer a b k ha ha1 hk m χ hχ hχzero hχone
Qp hQp q hq Q hQ hqb P hcomm G hG hspan hspec T
· exact fun T => cutoffProfileEndpoint_uniform_bound a b k ha ha1 hk8 m χ hχ Qp hQp Q hQ hqb T
end DefocusingNLS
end OAI
|