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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 | import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffFourierDefect
namespace OAI
/-! # The actual sampled cutoff profile satisfies the forced mode equation
The smooth stationary-profile premise is explicit. The forcing is the already
constructed three-term cutoff residual, with the same physical normalization.
-/
open scoped SchwartzMap ContDiff
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
theorem cutoffProfileSchwartz_dilation_zero (R : ℝ) (hR : 0 < R)
(χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ))
(Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) :
cutoffProfileSchwartz R hR χ hχ Q hQ =
schwartzPhysicalDilation 0 R hR (compactDilationFamily χ hχ Q hQ R) := by
ext x
simp only [cutoffProfileSchwartz_apply, schwartzPhysicalDilation_apply,
compactDilationFamily_apply, mul_zero, Real.rpow_zero, Complex.ofReal_one,
one_mul, smul_smul, mul_inv_cancel₀ hR.ne', one_smul]
theorem schwartzOddPower_dilation_zero (R : ℝ) (hR : 0 < R)
(ψ : 𝓢(E, ℂ)) (m : ℕ) :
schwartzOddPower m (schwartzPhysicalDilation 0 R hR ψ) =
schwartzPhysicalDilation 0 R hR (schwartzOddPower m ψ) := by
ext x
simp only [schwartzOddPower_apply, schwartzPhysicalDilation_apply, mul_zero,
Real.rpow_zero, Complex.ofReal_one, one_mul]
theorem cutoffProfileSchwartz_zero_outside (R : ℝ) (hR : 0 < R)
(χ : 𝓢(E, ℝ)) (hχ : HasCompactSupport (χ : E → ℝ))
(hχzero : ∀ x : E, 2 < ‖x‖ → χ x = 0)
(Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) (x : E) (hx : 2 * R < ‖x‖) :
cutoffProfileSchwartz R hR (χ.postcompCLM Complex.ofRealCLM)
(hasCompactSupport_complexCutoff χ hχ) Q hQ x = 0 := by
have hr : 2 < ‖R⁻¹ • x‖ := by
rw [norm_smul, Real.norm_eq_abs, abs_inv, abs_of_pos hR]
exact (lt_inv_mul_iff₀' hR).mpr hx
simp only [cutoffProfileSchwartz_apply, SchwartzMap.postcompCLM_apply,
Complex.ofRealCLM_apply, hχzero _ hr, Complex.ofReal_zero, zero_mul]
theorem cutoff_normalized_nonlinearity_coefficient (a k R : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hR : 1 ≤ R) (m : ℕ)
(χ : 𝓢(E, ℝ)) (hχ : HasCompactSupport (χ : E → ℝ))
(hχzero : ∀ x : E, 2 < ‖x‖ → χ x = 0)
(Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) (n : frequencyLattice) :
physicalSchwartzCoefficient 1 n (schwartzOddPower m
(compactDilationFamily (χ.postcompCLM Complex.ofRealCLM)
(hasCompactSupport_complexCutoff χ hχ) Q hQ R)) =
expandingFourierCoefficient a k R
(expandingOddPower a k R ha ha1 hk hR m
(schwartzTorusSample a k R ha1 hk hR (radianFourierKernel
(cutoffProfileSchwartz R (lt_of_lt_of_le zero_lt_one hR)
(χ.postcompCLM Complex.ofRealCLM)
(hasCompactSupport_complexCutoff χ hχ) Q hQ)))) n := by
rw [sampled_oddPower_coefficient a k R ha ha1 hk hR m _
(cutoffProfileSchwartz_zero_outside R (lt_of_lt_of_le zero_lt_one hR)
χ hχ hχzero Q hQ)]
change _ = physicalSchwartzCoefficient R n _
rw [cutoffProfileSchwartz_dilation_zero, schwartzOddPower_dilation_zero,
physicalSchwartzCoefficient_dilation]
theorem cutoff_normalized_defect_coefficient (a b R : ℝ) (hR : 0 < R)
(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, 2 < ‖x‖ → χ x = 0)
(Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q)
(hstationary : ∀ y, stationarySimilarityDefect a b m Q y = 0) (n : frequencyLattice) :
physicalSchwartzCoefficient 1 n (normalizedCutoffDefectSchwartz a b R m
(χ.postcompCLM Complex.ofRealCLM) (hasCompactSupport_complexCutoff χ hχ) Q hQ) =
physicalSchwartzCoefficient R n
(cutoffResidualSchwartz χ hχ hχone hχzero a R hR m Q hQ) := by
have he : schwartzPhysicalDilation 0 R hR
(normalizedCutoffDefectSchwartz a b R m
(χ.postcompCLM Complex.ofRealCLM) (hasCompactSupport_complexCutoff χ hχ) Q hQ) =
cutoffResidualSchwartz χ hχ hχone hχzero a R hR m Q hQ := by
ext y
rw [schwartzPhysicalDilation_apply,
normalizedCutoffDefectSchwartz_apply a b R hR m χ hχ Q hQ,
hstationary,
cutoffResidualSchwartz_apply χ hχ hχone hχzero a R hR m ham Q hQ]
simp only [mul_zero, zero_add, Real.rpow_zero, Complex.ofReal_one, one_mul,
smul_smul, mul_inv_cancel₀ hR.ne', one_smul]
rw [← he, physicalSchwartzCoefficient_dilation]
noncomputable def cutoffSampledMode (L : ℝ) (χ : 𝓢(E, ℂ))
(hχ : HasCompactSupport (χ : E → ℂ)) (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q)
(n : frequencyLattice) (t : ℝ) : ℂ :=
physicalSchwartzCoefficient 1 n (compactDilationFamily χ hχ Q hQ (expandingRadius L t))
theorem cutoffSampledMode_eq_physical (L : ℝ) (hL : 0 < L)
(χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ))
(Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) (n : frequencyLattice) (t : ℝ) :
cutoffSampledMode L χ hχ Q hQ n t =
physicalSchwartzCoefficient (expandingRadius L t) n
(cutoffProfileSchwartz (expandingRadius L t)
(mul_pos hL (Real.exp_pos _)) χ hχ Q hQ) := by
exact (cutoffProfile_coefficient_fixed_scale _ (mul_pos hL (Real.exp_pos _))
χ hχ Q hQ n).symm
theorem hasDerivAt_cutoffSampledMode (L s : ℝ) (hL : 0 < L)
(χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ))
(Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) (n : frequencyLattice) :
HasDerivAt (cutoffSampledMode L χ hχ Q hQ n)
(physicalSchwartzCoefficient 1 n (cutoffVelocitySchwartz χ hχ Q hQ (expandingRadius L s))) s := by
have hd := hasDerivAt_cutoffProfile_coefficient L s hL χ hχ Q hQ n
have he : (fun t => schwartzLatticeCoefficient (expandingRadius L t)
(radianFourierKernel (cutoffProfileSchwartz (expandingRadius L t)
(mul_pos hL (Real.exp_pos _)) χ hχ Q hQ)) n) =
cutoffSampledMode L χ hχ Q hQ n := by
funext t
exact (cutoffSampledMode_eq_physical L hL χ hχ Q hQ n t).symm
rw [he] at hd
convert hd using 1
simp only [physicalSchwartzCoefficient_apply, schwartzLatticeCoefficient,
mul_one, inv_one, one_smul, radianFourierKernel_apply]
rfl
/-- The sampled profile has the actual Fourier mode derivative, including the
same sampled residual used in the decaying Duhamel estimate. -/
theorem hasDerivAt_cutoffSampledMode_equation (a b k L s : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 0 < L)
(hRs : 1 ≤ expandingRadius L s) (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, 2 < ‖x‖ → χ x = 0)
(Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q)
(hstationary : ∀ y, stationarySimilarityDefect a b m Q y = 0) (n : frequencyLattice) :
let R := expandingRadius L s
let ψ := cutoffProfileSchwartz R (lt_of_lt_of_le zero_lt_one hRs)
(χ.postcompCLM Complex.ofRealCLM) (hasCompactSupport_complexCutoff χ hχ) Q hQ
let U := schwartzTorusSample a k R ha1 hk hRs (radianFourierKernel ψ)
let residual := cutoffResidualSchwartz χ hχ hχone hχzero a R
(lt_of_lt_of_le zero_lt_one hRs) m Q hQ
HasDerivAt
(cutoffSampledMode L (χ.postcompCLM Complex.ofRealCLM)
(hasCompactSupport_complexCutoff χ hχ) Q hQ n)
(((-(a : ℂ)) + Complex.I * ((b : ℂ) -
(((R ^ (2 : ℕ))⁻¹ : ℝ) : ℂ) * ((‖n‖ ^ 2 : ℝ) : ℂ))) *
cutoffSampledMode L (χ.postcompCLM Complex.ofRealCLM)
(hasCompactSupport_complexCutoff χ hχ) Q hQ n s -
Complex.I * expandingFourierCoefficient a k R
(expandingOddPower a k R ha ha1 hk hRs m U) n -
Complex.I * physicalSchwartzCoefficient R n residual) s := by
intro R ψ U residual
have hd := hasDerivAt_cutoffSampledMode L s hL
(χ.postcompCLM Complex.ofRealCLM) (hasCompactSupport_complexCutoff χ hχ) Q hQ n
have he := physicalSchwartzCoefficient_cutoffDefect a b R m
(χ.postcompCLM Complex.ofRealCLM) (hasCompactSupport_complexCutoff χ hχ) Q hQ n
rw [cutoff_normalized_defect_coefficient a b R
(lt_of_lt_of_le zero_lt_one hRs) m ham χ hχ hχone hχzero Q hQ hstationary n,
cutoff_normalized_nonlinearity_coefficient a k R ha ha1 hk hRs m
χ hχ hχzero Q hQ n] at he
apply hd.congr_deriv
change physicalSchwartzCoefficient 1 n
(cutoffVelocitySchwartz (χ.postcompCLM Complex.ofRealCLM)
(hasCompactSupport_complexCutoff χ hχ) Q hQ R) = _
change physicalSchwartzCoefficient R n residual =
Complex.I * physicalSchwartzCoefficient 1 n
(cutoffVelocitySchwartz (χ.postcompCLM Complex.ofRealCLM)
(hasCompactSupport_complexCutoff χ hχ) Q hQ R) +
(-(((R ^ (2 : ℕ))⁻¹ : ℝ) : ℂ) * ((‖n‖ ^ 2 : ℝ) : ℂ) +
Complex.I * (a : ℂ) + (b : ℂ)) *
cutoffSampledMode L (χ.postcompCLM Complex.ofRealCLM)
(hasCompactSupport_complexCutoff χ hχ) Q hQ n s -
expandingFourierCoefficient a k R (expandingOddPower a k R ha ha1 hk hRs m U) n at he
linear_combination Complex.I * he +
(physicalSchwartzCoefficient 1 n
(cutoffVelocitySchwartz (χ.postcompCLM Complex.ofRealCLM)
(hasCompactSupport_complexCutoff χ hχ) Q hQ R) +
(a : ℂ) * cutoffSampledMode L (χ.postcompCLM Complex.ofRealCLM)
(hasCompactSupport_complexCutoff χ hχ) Q hQ n s) * Complex.I_sq
end DefocusingNLS
end OAI
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