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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 | import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingPerturbationContinuity
import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffStepForcing
namespace OAI
/-! # Scale continuity of the actual sampled cutoff histories
The smooth profile and exact cutoff residual vary continuously in the
finite-time path norm. This supplies the concrete parameter hypotheses
for the nonlinear step continuity theorem.
-/
open Set
open scoped SchwartzMap ContDiff
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
theorem continuous_sampledCutoffProfilePath (a k T : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k)
(χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ))
(Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) :
Continuous (fun L : {L : ℝ // 1 ≤ L} =>
sampledCutoffProfilePath a k L.1 T ha ha1 hk L.2 χ hχ Q hQ) := by
apply ContinuousMap.continuous_of_continuous_uncurry
exact (continuous_cutoffProfile_sample a k ha ha1 hk χ hχ Q hQ).comp
(continuous_expandingRadiusCurve_family T id continuous_id)
theorem continuous_sampledCutoffResidualPath (a k T : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k)
(χ : 𝓢(E, ℝ)) (hχ : HasCompactSupport (χ : E → ℝ))
(hχone : ∀ x : E, ‖x‖ < 1 / 2 → χ x = 1)
(hχzero : ∀ x : E, 2 < ‖x‖ → χ x = 0)
(m : ℕ) (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) :
Continuous (fun L : {L : ℝ // 1 ≤ L} =>
sampledCutoffResidualPath a k L.1 T ha ha1 hk L.2 χ hχ hχone hχzero m Q hQ) := by
apply ContinuousMap.continuous_of_continuous_uncurry
exact (continuous_cutoffResidual_sample a k ha ha1 hk χ hχ hχone hχzero m Q hQ).comp
(continuous_expandingRadiusCurve_family T id continuous_id)
theorem continuous_sampledCutoffDefectPath (a k T : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k)
(χ : 𝓢(E, ℝ)) (hχ : HasCompactSupport (χ : E → ℝ))
(hχone : ∀ x : E, ‖x‖ < 1 / 2 → χ x = 1)
(hχzero : ∀ x : E, 2 < ‖x‖ → χ x = 0)
(m : ℕ) (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) :
Continuous (fun L : {L : ℝ // 1 ≤ L} =>
sampledCutoffDefectPath a k L.1 T ha ha1 hk L.2 χ hχ hχone hχzero m Q hQ) :=
(continuous_sampledCutoffResidualPath a k T ha ha1 hk χ hχ hχone hχzero m Q hQ).const_smul
(-Complex.I : ℂ)
end DefocusingNLS
end OAI
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