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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 56 57 58 59 60 61 62 63 64 65 66 | import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingCoordinateSequence
import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffStableTransfer
namespace OAI
/-! # Uniform finite-coordinate transfer for the sampled cutoff profile -/
open Set Filter Topology
open scoped SchwartzMap ContDiff
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
local notation "Radius" => {L : ℝ // 1 ≤ L}
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F]
theorem sampledCutoffProfile_uniform_coordinate_defect (a b k T Q C₀ : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 10 < k) (hT : 0 ≤ T)
(hQ : 0 ≤ Q) (hC₀ : 0 ≤ C₀) (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) (Q₀ : HomogeneousY a k)
(hQ₀ : ∀ y, homogeneousPhysicalCLM a k ha ha1 (by linarith) Q₀ y = Qp y)
(hqb : ∀ L : Radius, ‖cutoffProfileCoefficient a k ha1 (by linarith) χ hχ Qp hQp L‖ ≤ Q)
(hCb : ∀ (L : ℝ) (hL : 1 ≤ L) (f : FourierL2),
‖homogeneousLocalizationCLM a k L ha ha1 (by linarith) hL χ f‖ ≤ C₀ * ‖f‖)
(π : HomogeneousY a k →L[ℝ] F) (A : F →L[ℝ] F)
(hA : ∀ u, π (homogeneousLinearizedTrajectory a b k T ha ha1 (by linarith) hT m Q₀ u
⟨T, hT, le_rfl⟩) = A (π u)) :
∀ ε : ℝ, 0 < ε → ∃ L₀ : ℝ, ∀ L : Radius, L₀ ≤ L.1 → ∀ f : FourierL2, ‖f‖ ≤ 1 →
‖expandingCoordinates a k ha ha1 (by linarith) χ π
(expandingRadiusCurve L.1 T L.2 ⟨T, hT, le_rfl⟩)
(expandingProfileTrajectory a b k L.1 T ha ha1 (by linarith) L.2 hT m Q hQ
(sampledCutoffProfilePath a k L.1 T ha ha1 (by linarith) L.2 χ hχ Qp hQp)
(fun t => hqb (expandingRadiusCurve L.1 T L.2 t)) f ⟨T, hT, le_rfl⟩) -
A (expandingCoordinates a k ha ha1 (by linarith) χ π L f)‖ < ε := by
classical
have hk8 : 8 < k := by linarith
intro ε hε
by_contra h
push Not at h
choose L hL f hf hbad using fun n : ℕ => h (n : ℝ)
have hLinf : Tendsto (fun n => (L n).1) atTop atTop :=
tendsto_atTop_mono hL tendsto_natCast_atTop_atTop
let q := fun n => sampledCutoffProfilePath a k (L n).1 T ha ha1 hk8 (L n).2 χ hχ Qp hQp
have hq (n : ℕ) (t : Icc (0 : ℝ) T) : ‖q n t‖ ≤ Q :=
hqb (expandingRadiusCurve (L n).1 T (L n).2 t)
have hqp (t : Icc (0 : ℝ) T) (y : E) :
Tendsto (fun n => expandingPhysicalContinuous a k (expandingRadius (L n).1 t)
ha ha1 hk8 ((L n).2.trans (expandingRadius_ge (L n).1 t (L n).2 t.2.1)) (q n t) y)
atTop (𝓝 (homogeneousPhysicalCLM a k ha ha1 hk8 Q₀ y)) := by
rw [hQ₀]
exact tendsto_sampledCutoffProfilePath_physical a k T ha ha1 hk8 χ hχ hχzero hχone
Qp hQp (fun n => (L n).1) (fun n => (L n).2) hLinf t y
have hs := expandingProfile_coordinate_sequence a b k T Q 1 C₀ ha ha1 hk hT hQ
zero_le_one hC₀ m χ (1 / 2) (by norm_num) hχone hCb
(fun n => (L n).1) (fun n => (L n).2) hLinf q hq f hf Q₀ π A hA hqp ε hε
obtain ⟨n, hn⟩ := hs.exists
exact (not_lt_of_ge (hbad n)) hn
end DefocusingNLS
end OAI
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