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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 | import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffProfileEndpoint
import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffPathContinuity
import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingProfileContinuity
namespace OAI
/-! # Strong continuity of the actual cutoff linear endpoint -/
open scoped SchwartzMap ContDiff NNReal
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
local notation "Radius" => {L : ℝ // 1 ≤ L}
attribute [local irreducible] cutoffProfileEndpoint expandingProfileTrajectory
theorem continuous_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) :
Continuous (fun p : Radius × FourierL2 =>
cutoffProfileEndpoint a b k ha ha1 hk m χ hχ Qp hQp Q hQ hqb T p.1 p.2) := by
have hq := (continuous_sampledCutoffProfilePath a k T ha ha1 hk χ hχ Qp hQp).comp
(continuous_fst : Continuous (fun p : Radius × FourierL2 => p.1))
have hc := continuous_expandingProfileTrajectory_family a b k T Q ha ha1 hk T.2 m hQ
(fun p : Radius × FourierL2 => p.1) continuous_fst
(fun p : Radius × FourierL2 => sampledCutoffProfilePath a k p.1.1 T ha ha1 hk p.1.2 χ hχ Qp hQp)
hq (fun p t => hqb (expandingRadiusCurve p.1.1 T p.1.2 t))
(fun p : Radius × FourierL2 => p.2) continuous_snd
simpa only [cutoffProfileEndpoint_apply, Function.comp_def] using
(continuous_eval_const ⟨T, T.2, le_rfl⟩).comp hc
end DefocusingNLS
end OAI
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