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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 | import OAI.MathematicalPhysics.DefocusingNLS.Linear.SchwartzCutoffDerivative
namespace OAI
/-! # Uniform bound for the cutoff's directional derivative term -/
open scoped SchwartzMap LineDeriv ContDiff
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
private theorem cutoff_directional_sum (L : ℝ) (hL : 0 < L)
(χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ))
(Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) (v : E) :
cutoffProfileSchwartz L hL (∂_{v} χ) (schwartzCutoff_lineDeriv_compact χ hχ v) Q hQ =
∑ j : Fin 12, (v j) • cutoffProfileSchwartz L hL
(∂_{(EuclideanSpace.basisFun (Fin 12) ℝ) j} χ)
(schwartzCutoff_lineDeriv_compact χ hχ _) Q hQ := by
ext y
have hrep : (∑ j : Fin 12, v j • (EuclideanSpace.basisFun (Fin 12) ℝ) j) = v := by
simpa only [EuclideanSpace.basisFun_repr] using (EuclideanSpace.basisFun (Fin 12) ℝ).sum_repr v
have he := congrArg (fderiv ℝ (χ : E → ℂ) (L⁻¹ • y)) hrep
simp only [map_sum, map_smul] at he
simp only [cutoffProfileSchwartz_apply, sum_apply, smul_apply,
SchwartzMap.lineDerivOp_apply_eq_fderiv, Complex.real_smul]
rw [← he]
simp only [Finset.sum_mul, Complex.real_smul, mul_assoc]
theorem cutoff_directional_sample_bound (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 : ℝ) (hL : 1 ≤ L) (v : E),
‖physicalSchwartzTorusSamplingCLM a k L ha1 hk hL
(cutoffProfileSchwartz L (by linarith) (∂_{v} χ)
(schwartzCutoff_lineDeriv_compact χ hχ v) Q hQ)‖ ≤ C * ‖v‖ := by
classical
have hb (j : Fin 12) := cutoffProfile_sampling_of_global_symbol a k ha ha1 hk
(∂_{(EuclideanSpace.basisFun (Fin 12) ℝ) j} χ)
(schwartzCutoff_lineDeriv_compact χ hχ _)
(schwartzCutoff_lineDeriv_zero χ hχzero _) Q hQ hsymbol
choose B hB hbound using hb
refine ⟨∑ j, B j, Finset.sum_nonneg (fun j _ => hB j), ?_⟩
intro L hL v
rw [cutoff_directional_sum L (by linarith) χ hχ Q hQ v, map_sum]
calc
_ ≤ ∑ j : Fin 12, ‖physicalSchwartzTorusSamplingCLM a k L ha1 hk hL
((v j) • cutoffProfileSchwartz L (by linarith)
(∂_{(EuclideanSpace.basisFun (Fin 12) ℝ) j} χ)
(schwartzCutoff_lineDeriv_compact χ hχ _) Q hQ)‖ := norm_sum_le _ _
_ ≤ ∑ j : Fin 12, ‖v‖ * B j := by
apply Finset.sum_le_sum
intro j _
rw [ContinuousLinearMap.map_smul_of_tower, norm_smul]
exact mul_le_mul (PiLp.norm_apply_le v j) (hbound j L hL)
(norm_nonneg _) (norm_nonneg _)
_ = _ := by rw [← Finset.mul_sum]; ring
theorem cutoff_directional_error_bound (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 : ℝ) (hL : 1 ≤ L) (v : E),
‖physicalSchwartzTorusSamplingCLM a k L ha1 hk hL
(∂_{v} (cutoffProfileSchwartz L (by linarith) χ hχ Q hQ)) -
physicalSchwartzTorusSamplingCLM a k L ha1 hk hL
(cutoffProfileSchwartz L (by linarith) χ hχ (cartesianDerivative v Q)
(cartesianDerivative_contDiff Q hQ v))‖ ≤ C / L * ‖v‖ := by
obtain ⟨C, hC, hb⟩ := cutoff_directional_sample_bound a k ha ha1 hk χ hχ hχzero Q hQ hsymbol
refine ⟨C, hC, ?_⟩
intro L hL v
rw [cutoffProfileSchwartz_lineDeriv L (by linarith) χ hχ Q hQ v, map_add,
add_sub_cancel_left, ContinuousLinearMap.map_smul_of_tower, norm_smul, Real.norm_eq_abs,
abs_of_pos (inv_pos.mpr (by linarith : 0 < L))]
calc
_ ≤ L⁻¹ * (C * ‖v‖) := mul_le_mul_of_nonneg_left (hb L hL v) (by positivity)
_ = _ := by ring
end DefocusingNLS
end OAI
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