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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 | import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffTimeLipschitz
import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.ExponentialModulationTaylor
namespace OAI
/-! # Uniform phase, time and translation Taylor estimate for the cutoff profile -/
open scoped SchwartzMap ContDiff
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
local notation "Radius" => {L : ℝ // 1 ≤ L}
theorem cutoffProfile_modulation_taylor (a k : ℝ) (ha : 0 < a) (ha1 : a < 1)
(hk : 8 < k) (χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ))
(hχzero : ∀ y : E, 1 ≤ ‖y‖ → χ y = 0)
(Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q)
(hsymbol : ∀ n : ℕ, ∃ D : ℝ, 0 ≤ D ∧ ∀ y : E, y ≠ 0 →
‖iteratedFDeriv ℝ n Q y‖ ≤ D * ‖y‖ ^ (-2 * a - (n : ℝ))) :
∃ C : ℝ, 0 ≤ C ∧ ∀ (L : Radius), 2 ≤ L.1 → ∀ (t : ℝ) (v : E) (z : ℂ),
|t| ≤ 2 * Real.log 2 → |t| ≤ 1 → ‖z‖ ≤ 1 →
‖Complex.exp z • sobolevTranslation (euclideanToTorus ((1 / L.1) • v))
(fixedCutoffProfile a k ha ha1 hk L χ hχ Q hQ (expandingRadius L.1 t)) -
fixedCutoffProfile a k ha ha1 hk L χ hχ Q hQ L.1 -
(z • fixedCutoffProfile a k ha ha1 hk L χ hχ Q hQ L.1 +
t • fixedCutoffProfile a k ha ha1 hk L χ hχ (cartesianTransport Q)
(cartesianTransport_contDiff Q hQ) L.1 +
fixedCutoffProfile a k ha ha1 hk L χ hχ (cartesianDerivative v Q)
(cartesianDerivative_contDiff Q hQ v) L.1)‖ ≤
C * (‖z‖ ^ 2 + |t| ^ 2 + ‖v‖ ^ 2 + ‖v‖ / L.1) := by
obtain ⟨A, hA, htaylor⟩ := cutoffProfile_mixed_taylor a k ha ha1 hk χ hχ hχzero Q hQ hsymbol
obtain ⟨B, hB, hbase⟩ := cutoffProfile_sampling_of_global_symbol a k ha ha1 hk χ hχ hχzero Q hQ hsymbol
obtain ⟨D, hD, hfirst⟩ := cutoffProfile_mixed_first_bound a k ha ha1 hk χ hχ hχzero Q hQ hsymbol
refine ⟨A + B + 2 * D, by positivity, ?_⟩
intro L hL t v z ht ht1 hz
let u := sobolevTranslation (euclideanToTorus ((1 / L.1) • v))
(fixedCutoffProfile a k ha ha1 hk L χ hχ Q hQ (expandingRadius L.1 t))
let q := fixedCutoffProfile a k ha ha1 hk L χ hχ Q hQ L.1
let d := t • fixedCutoffProfile a k ha ha1 hk L χ hχ (cartesianTransport Q)
(cartesianTransport_contDiff Q hQ) L.1 +
fixedCutoffProfile a k ha ha1 hk L χ hχ (cartesianDerivative v Q)
(cartesianDerivative_contDiff Q hQ v) L.1
have hrem : ‖u - q - d‖ ≤ A * (|t| ^ 2 + ‖v‖ ^ 2 + ‖v‖ / L.1) := by
simpa only [d, sub_add_eq_sub_sub] using htaylor L hL t v ht
have hq : ‖q‖ ≤ B := by
simpa only [q, fixedCutoffProfile_self, physicalSchwartzTorusSamplingCLM_apply] using hbase L.1 L.2
have hu : ‖u - q‖ ≤ D * (|t| + ‖v‖) := hfirst L hL t v ht ht1
have heq : z • q + t • fixedCutoffProfile a k ha ha1 hk L χ hχ (cartesianTransport Q)
(cartesianTransport_contDiff Q hQ) L.1 +
fixedCutoffProfile a k ha ha1 hk L χ hχ (cartesianDerivative v Q)
(cartesianDerivative_contDiff Q hQ v) L.1 = z • q + d := by
dsimp only [d]
abel
change ‖Complex.exp z • u - q - _‖ ≤ _
rw [heq]
calc
_ ≤ ‖u - q - d‖ + ‖z‖ ^ 2 * ‖q‖ + 2 * ‖z‖ * ‖u - q‖ :=
exponential_modulation_taylor z hz u q d
_ ≤ A * (|t| ^ 2 + ‖v‖ ^ 2 + ‖v‖ / L.1) + ‖z‖ ^ 2 * B +
2 * ‖z‖ * (D * (|t| + ‖v‖)) :=
add_le_add (add_le_add hrem (mul_le_mul_of_nonneg_left hq (sq_nonneg _)))
(mul_le_mul_of_nonneg_left hu (by positivity))
_ ≤ _ := by
let R := ‖z‖ ^ 2 + |t| ^ 2 + ‖v‖ ^ 2 + ‖v‖ / L.1
have hvL : 0 ≤ ‖v‖ / L.1 := by positivity
have hc : ‖z‖ * (|t| + ‖v‖) ≤ R := by
dsimp only [R]
nlinarith [sq_nonneg (‖z‖ - |t|), sq_nonneg (‖z‖ - ‖v‖)]
have h1 : A * (|t| ^ 2 + ‖v‖ ^ 2 + ‖v‖ / L.1) ≤ A * R :=
mul_le_mul_of_nonneg_left (by dsimp [R]; nlinarith [sq_nonneg ‖z‖]) hA
have h2 : ‖z‖ ^ 2 * B ≤ B * R := by
have := mul_le_mul_of_nonneg_left
(show ‖z‖ ^ 2 ≤ R by dsimp [R]; nlinarith [sq_nonneg |t|, sq_nonneg ‖v‖]) hB
nlinarith only [this]
have h3 := mul_le_mul_of_nonneg_left hc (show 0 ≤ 2 * D by positivity)
dsimp only [R] at h1 h2 h3
nlinarith only [h1, h2, h3]
end DefocusingNLS
end OAI
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