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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 67 68 69 | import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffContinuousStep
import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffStepLinearization
namespace OAI
/-! # Continuous nonlinear cutoff steps with the exact linear endpoint -/
open Set
open scoped SchwartzMap ContDiff NNReal
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
local notation "Radius" => {L : ℝ // 1 ≤ L}
def HasContinuousCutoffNonlinearSteps (a b k : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (m : ℕ)
(χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ))
(Qp : E → ℂ) (hQp : ContDiff ℝ ∞ Qp) (Q : ℝ) (hQ : 0 ≤ Q)
(hqb : ∀ L : Radius, ‖cutoffProfileCoefficient a k ha1 hk χ hχ Qp hQp L‖ ≤ Q)
(T : ℝ≥0) : Prop :=
∃ δ L₀ C : ℝ, 0 < δ ∧ 1 ≤ L₀ ∧ 0 < C ∧
∃ V : {L : ℝ // 1 ≤ L ∧ L₀ ≤ L} × {f : FourierL2 // ‖f‖ ≤ δ} →
C(Icc (0 : ℝ) T, FourierL2),
∃ h : {L : ℝ // 1 ≤ L ∧ L₀ ≤ L} × {f : FourierL2 // ‖f‖ ≤ δ} → FourierL2,
Continuous V ∧ Continuous h ∧
(∀ z, let q := sampledCutoffProfilePath a k z.1.1 T ha ha1 hk z.1.2.1 χ hχ Qp hQp
q + V z = expandingPicard a b k z.1.1 T ha hk z.1.2.1 T.2
(expandingNonlinearReaction a k z.1.1 T ha ha1 hk z.1.2.1 m)
(q ⟨0, le_rfl, T.2⟩ + z.2.1) (q + V z)) ∧
(∀ z, ‖V z‖ ≤ C * (‖z.2.1‖ + z.1.1 ^ (-2 - a))) ∧
(∀ z, V z ⟨T, T.2, le_rfl⟩ =
cutoffProfileEndpoint a b k ha ha1 hk m χ hχ Qp hQp Q hQ hqb T
⟨z.1.1, z.1.2.1⟩ z.2.1 + h z) ∧
(∀ z, z.2.1 = 0 → ‖h z‖ ≤ C * z.1.1 ^ (-2 - a)) ∧
(∀ d : ℝ, 0 < d → d ≤ δ → ∀ L f j, ‖f.1‖ ≤ d → ‖j.1‖ ≤ d →
‖h (L, f) - h (L, j)‖ ≤ C * (d + L.1 ^ (-2 - a)) * ‖f.1 - j.1‖)
theorem exists_cutoffContinuousNonlinear_steps_order (a b k : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k)
(m : ℕ) (ham : 2 * a * (m : ℝ) = 1)
(χ : 𝓢(E, ℝ)) (hχ : HasCompactSupport (χ : E → ℝ))
(hχone : ∀ x : E, ‖x‖ < 1 / 2 → χ x = 1)
(hχzero : ∀ x : E, 1 ≤ ‖x‖ → χ x = 0) (T : ℝ≥0) :
∃ N : ℕ, ∀ (Qp : E → ℂ) (hQp : ContDiff ℝ ∞ Qp) (B : ℝ), 0 ≤ B →
(∀ n ≤ N, ∀ y : E, y ≠ 0 → ‖iteratedFDeriv ℝ n Qp y‖ ≤ B * ‖y‖ ^ (-2 * a - (n : ℝ))) →
(∀ y, stationarySimilarityDefect a b m Qp y = 0) →
∀ (Q : ℝ) (hQ : 0 ≤ Q),
∀ hqb : ∀ L : Radius, ‖cutoffProfileCoefficient a k ha1 hk
(χ.postcompCLM Complex.ofRealCLM) (hasCompactSupport_complexCutoff χ hχ) Qp hQp L‖ ≤ Q,
HasContinuousCutoffNonlinearSteps a b k ha ha1 hk m (χ.postcompCLM Complex.ofRealCLM)
(hasCompactSupport_complexCutoff χ hχ) Qp hQp Q hQ hqb T := by
obtain ⟨N, hN⟩ := exists_cutoffNonlinear_step_continuous a b k T ha ha1 hk T.2 m ham
χ hχ hχone hχzero
refine ⟨N, ?_⟩
intro Qp hQp B hB hsymbol hstationary Q hQ hqb
obtain ⟨δ, L₀, C, hδ, hL₀, hC, V, h, hcV, hch, hsol, hV, hend, hzero, hlip⟩ :=
hN Qp hQp B hB hsymbol hstationary
refine ⟨δ, L₀, C, hδ, hL₀, hC, V, h, hcV, hch, hsol, hV, ?_, hzero, hlip⟩
intro z
obtain ⟨w, hw, he⟩ := hend z
rw [he, cutoffStep_linear_endpoint a b k ha ha1 hk m
(χ.postcompCLM Complex.ofRealCLM) (hasCompactSupport_complexCutoff χ hχ)
Qp hQp Q hQ hqb T ⟨z.1.1, z.1.2.1⟩ z.2.1 w hw]
end DefocusingNLS
end OAI
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