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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 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 | import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffResidualDuhamel
import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffApproximateMode
namespace OAI
/-! # Continuous histories for the actual approximate nonlinear solution -/
open Set
open scoped SchwartzMap ContDiff
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
noncomputable def sampledCutoffProfileHistory (a k L T : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) (hT : 0 ≤ T)
(χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ))
(Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) : ℝ → FourierL2 :=
fun t => sampledCutoffProfilePath a k L T ha ha1 hk hL χ hχ Q hQ (projIcc 0 T hT t)
theorem continuous_sampledCutoffProfileHistory (a k L T : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) (hT : 0 ≤ T)
(χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ))
(Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) :
Continuous (sampledCutoffProfileHistory a k L T ha ha1 hk hL hT χ hχ Q hQ) :=
(sampledCutoffProfilePath a k L T ha ha1 hk hL χ hχ Q hQ).continuous.comp continuous_projIcc
theorem sampledCutoffProfileHistory_of_mem (a k L T : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) (hT : 0 ≤ T)
(χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ))
(Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) (t : ℝ) (ht : t ∈ Icc 0 T) :
sampledCutoffProfileHistory a k L T ha ha1 hk hL hT χ hχ Q hQ t =
schwartzTorusSample a k (expandingRadius L t) ha1 hk
(hL.trans (expandingRadius_ge L t hL ht.1))
(radianFourierKernel (cutoffProfileSchwartz (expandingRadius L t)
(mul_pos (lt_of_lt_of_le zero_lt_one hL) (Real.exp_pos _)) χ hχ Q hQ)) := by
unfold sampledCutoffProfileHistory
rw [projIcc_of_mem _ ht]
rfl
noncomputable def sampledCutoffNonlinearPath (a k L T : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L)
(χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ))
(Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) (m : ℕ) : C(Icc (0 : ℝ) T, FourierL2) where
toFun t := expandingOddPower a k (expandingRadiusCurve L T hL t).1
ha ha1 hk (expandingRadiusCurve L T hL t).2 m
(sampledCutoffProfilePath a k L T ha ha1 hk hL χ hχ Q hQ t)
continuous_toFun := (continuous_expandingOddPower_scale a k ha ha1 hk m).comp
((expandingRadiusCurve L T hL).continuous.prodMk
(sampledCutoffProfilePath a k L T ha ha1 hk hL χ hχ Q hQ).continuous)
noncomputable def sampledCutoffForcingPath (a k L T : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L)
(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) : C(Icc (0 : ℝ) T, FourierL2) :=
(-Complex.I) • sampledCutoffNonlinearPath a k L T ha ha1 hk hL
(χ.postcompCLM Complex.ofRealCLM) (hasCompactSupport_complexCutoff χ hχ) Q hQ m +
(-Complex.I) • sampledCutoffResidualPath a k L T ha ha1 hk hL χ hχ hχone hχzero m Q hQ
noncomputable def sampledCutoffForcingHistory (a k L T : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) (hT : 0 ≤ T)
(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) : ℝ → FourierL2 :=
fun t => sampledCutoffForcingPath a k L T ha ha1 hk hL χ hχ hχone hχzero m Q hQ
(projIcc 0 T hT t)
theorem continuous_sampledCutoffForcingHistory (a k L T : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) (hT : 0 ≤ T)
(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 (sampledCutoffForcingHistory a k L T ha ha1 hk hL hT χ hχ hχone hχzero m Q hQ) :=
(sampledCutoffForcingPath a k L T ha ha1 hk hL χ hχ hχone hχzero m Q hQ).continuous.comp
continuous_projIcc
theorem sampledCutoffForcingHistory_of_mem (a k L T : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) (hT : 0 ≤ T)
(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) (t : ℝ) (ht : t ∈ Icc 0 T) :
sampledCutoffForcingHistory a k L T ha ha1 hk hL hT χ hχ hχone hχzero m Q hQ t =
(-Complex.I) • expandingOddPower a k (expandingRadius L t) ha ha1 hk
(hL.trans (expandingRadius_ge L t hL ht.1)) m
(sampledCutoffProfileHistory a k L T ha ha1 hk hL hT
(χ.postcompCLM Complex.ofRealCLM) (hasCompactSupport_complexCutoff χ hχ) Q hQ t) +
(-Complex.I) • sampledCutoffResidualHistory a k L T ha ha1 hk hL hT
χ hχ hχone hχzero m Q hQ t := by
unfold sampledCutoffForcingHistory sampledCutoffProfileHistory sampledCutoffResidualHistory
rw [projIcc_of_mem _ ht]
rfl
theorem sampledCutoffProfileHistory_coefficient (a k L T : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) (hT : 0 ≤ T)
(χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ))
(Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) (t : ℝ) (ht : t ∈ Icc 0 T)
(n : frequencyLattice) :
expandingFourierCoefficient a k (expandingRadius L t)
(sampledCutoffProfileHistory a k L T ha ha1 hk hL hT χ hχ Q hQ t) n =
cutoffSampledMode L χ hχ Q hQ n t := by
rw [sampledCutoffProfileHistory_of_mem a k L T ha ha1 hk hL hT χ hχ Q hQ t ht,
schwartzTorusSample_coefficient]
exact (cutoffSampledMode_eq_physical L (lt_of_lt_of_le zero_lt_one hL)
χ hχ Q hQ n t).symm
theorem sampledCutoffResidualHistory_of_mem (a k L T : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) (hT : 0 ≤ T)
(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) (t : ℝ) (ht : t ∈ Icc 0 T) :
sampledCutoffResidualHistory a k L T ha ha1 hk hL hT χ hχ hχone hχzero m Q hQ t =
schwartzTorusSample a k (expandingRadius L t) ha1 hk
(hL.trans (expandingRadius_ge L t hL ht.1)) (radianFourierKernel
(cutoffResidualSchwartz χ hχ hχone hχzero a (expandingRadius L t)
(mul_pos (lt_of_lt_of_le zero_lt_one hL) (Real.exp_pos _)) m Q hQ)) := by
unfold sampledCutoffResidualHistory
rw [projIcc_of_mem _ ht]
rfl
end DefocusingNLS
end OAI
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