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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.Linear.HomogeneousDiagonalCoordinates
import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffCoordinateInverse
namespace OAI
/-! # Actual torus coordinate transfer with a contracting backward map -/
open Set
open scoped SchwartzMap ContDiff NNReal
namespace DefocusingNLS
attribute [local irreducible] diagonalRealCoordinates homogeneousStableCoordinates
homogeneousLinearizedStep homogeneousComplexLinearizedStep
local notation "E" => EuclideanSpace ℝ (Fin 12)
local notation "Radius" => {L : ℝ // 1 ≤ L}
theorem cutoffProfile_diagonal_coordinate_transfer (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) (T : ℝ≥0) :
HasContractingTorusCoordinateTransfer a k ha ha1 (by linarith) χ
(diagonalRealCoordinates (homogeneousStableCoordinates a k ha ha1 (by linarith) P)
G hspan hspec) T
(cutoffProfileEndpoint a b k ha ha1 (by linarith) m χ hχ Qp hQp Q hQ hqb T) := by
let : FiniteDimensional ℝ P.range := FiniteDimensional.trans ℝ ℂ P.range
let : FiniteDimensional ℝ (SymmetryCoordinates G) :=
FiniteDimensional.trans ℝ ℂ (SymmetryCoordinates G)
have hk8 : 8 < k := by linarith
apply cutoffProfile_coordinate_transfer_of_inverse a b k ha ha1 hk m χ hχ hχzero hχone
Qp hQp q hq Q hQ hqb _ T
exact homogeneousDiagonalCoordinates_inverse a b k ha ha1 hk8 m q
P hcomm G hG hspan hspec T
end DefocusingNLS
end OAI
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