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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 | import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffStableTransfer
import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingProfileOperator
namespace OAI
/-! # Endpoint operators for the actual sampled cutoff profile -/
open scoped SchwartzMap ContDiff NNReal
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
local notation "Radius" => {L : ℝ // 1 ≤ L}
noncomputable def cutoffProfileEndpoint (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) (L : Radius) : FourierL2 →L[ℝ] FourierL2 :=
expandingProfileEndpoint a b k L.1 T ha ha1 hk L.2 T.2 m Q hQ
(sampledCutoffProfilePath a k L.1 T ha ha1 hk L.2 χ hχ Qp hQp)
(fun t => hqb (expandingRadiusCurve L.1 T L.2 t))
@[simp] theorem cutoffProfileEndpoint_apply (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) (L : Radius) (f : FourierL2) :
cutoffProfileEndpoint a b k ha ha1 hk m χ hχ Qp hQp Q hQ hqb T L f =
expandingProfileTrajectory a b k L.1 T ha ha1 hk L.2 T.2 m Q hQ
(sampledCutoffProfilePath a k L.1 T ha ha1 hk L.2 χ hχ Qp hQp)
(fun t => hqb (expandingRadiusCurve L.1 T L.2 t)) f ⟨T, T.2, le_rfl⟩ := rfl
theorem cutoffProfileEndpoint_uniform_bound (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) : ∃ C : ℝ, 0 ≤ C ∧ ∀ L : Radius,
‖cutoffProfileEndpoint a b k ha ha1 hk m χ hχ Qp hQp Q hQ hqb T L‖ ≤ C := by
obtain ⟨K, hK, hb⟩ := exists_expandingProfileTrajectory_bound a b k ha ha1 hk m Q hQ
refine ⟨(K + 1) * Real.exp ((K + 1) * T), by positivity, ?_⟩
intro L
apply ContinuousLinearMap.opNorm_le_bound _ (by positivity)
intro f
rw [cutoffProfileEndpoint_apply]
exact ((expandingProfileTrajectory a b k L.1 T ha ha1 hk L.2 T.2 m Q hQ
(sampledCutoffProfilePath a k L.1 T ha ha1 hk L.2 χ hχ Qp hQp)
(fun t => hqb (expandingRadiusCurve L.1 T L.2 t)) f).norm_coe_le_norm ⟨T, T.2, le_rfl⟩).trans
(hb L.1 T L.2 T.2 _ _ f)
end DefocusingNLS
end OAI
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