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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 | import OAI.MathematicalPhysics.DefocusingNLS.Linear.HomogeneousProfileDerivative
import OAI.MathematicalPhysics.DefocusingNLS.Profile.CartesianTransportCalculus
namespace OAI
/-! # Dilation transport preserves the finite-order symbol bounds -/
open scoped ContDiff
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
private theorem halfIdentity_fderiv :
fderiv ℝ (fun y : E => (1 / 2 : ℝ) • y) =
fun _ => (1 / 2 : ℝ) • ContinuousLinearMap.id ℝ E := by
funext x
exact ((hasFDerivAt_id x).const_smul (1 / 2 : ℝ)).fderiv
private theorem halfIdentity_higher_norm (j : ℕ) (hj : 2 ≤ j) (x : E) :
‖iteratedFDeriv ℝ j (fun y : E => (1 / 2 : ℝ) • y) x‖ = 0 := by
obtain ⟨n, rfl⟩ := Nat.exists_eq_add_of_le hj
rw [show 2 + n = (n + 1) + 1 by omega, ← norm_iteratedFDeriv_fderiv,
halfIdentity_fderiv, iteratedFDeriv_succ_const]
simp only [Pi.zero_apply, norm_zero]
private theorem halfIdentity_first_norm (x : E) :
‖iteratedFDeriv ℝ 1 (fun y : E => (1 / 2 : ℝ) • y) x‖ ≤ 1 := by
rw [norm_iteratedFDeriv_one, halfIdentity_fderiv, norm_smul]
norm_num
private theorem halfIdentity_zero_norm (x : E) :
‖iteratedFDeriv ℝ 0 (fun y : E => (1 / 2 : ℝ) • y) x‖ ≤ ‖x‖ := by
rw [norm_iteratedFDeriv_zero, norm_smul]
norm_num
linarith [norm_nonneg x]
theorem cartesianTransport_global_symbol (a : ℝ) (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))) :
∀ n : ℕ, ∃ D : ℝ, 0 ≤ D ∧ ∀ y : E, y ≠ 0 →
‖iteratedFDeriv ℝ n (cartesianTransport Q) y‖ ≤ D * ‖y‖ ^ (-2 * a - (n : ℝ)) := by
intro n
obtain ⟨D₀, hD₀, hb₀⟩ := hsymbol n
obtain ⟨D₁, hD₁, hb₁⟩ := hsymbol (n + 1)
let D := max D₀ D₁
let C := ∑ i ∈ Finset.range (n + 1), (n.choose i : ℝ)
refine ⟨C * D, mul_nonneg (Finset.sum_nonneg (fun _ _ => Nat.cast_nonneg _))
(hD₀.trans (le_max_left _ _)), ?_⟩
intro y hy
have hyr : 0 < ‖y‖ := norm_pos_iff.mpr hy
have hprod := norm_iteratedFDeriv_clm_apply (contDiff_infty_iff_fderiv.mp hQ).2
(contDiff_id.const_smul (1 / 2 : ℝ)) y (by simp : (n : ℕ∞ω) ≤ ∞)
simp only [id_eq] at hprod
change ‖iteratedFDeriv ℝ n (cartesianTransport Q) y‖ ≤ _ at hprod
apply hprod.trans
calc
_ ≤ ∑ i ∈ Finset.range (n + 1), (n.choose i : ℝ) * (D * ‖y‖ ^ (-2 * a - (n : ℝ))) := by
apply Finset.sum_le_sum
intro i hi
have hin : i ≤ n := Nat.le_of_lt_succ (Finset.mem_range.mp hi)
by_cases h0 : n - i = 0
· have hi_eq : i = n := by omega
subst i
rw [Nat.sub_self]
have hb : ‖iteratedFDeriv ℝ n (fderiv ℝ Q) y‖ ≤ D * ‖y‖ ^ (-2 * a - ((n + 1 : ℕ) : ℝ)) := by
rw [norm_iteratedFDeriv_fderiv]
exact (hb₁ y hy).trans (mul_le_mul_of_nonneg_right (le_max_right _ _) (by positivity))
calc
_ ≤ (n.choose n : ℝ) * (D * ‖y‖ ^ (-2 * a - ((n + 1 : ℕ) : ℝ))) * ‖y‖ :=
mul_le_mul (mul_le_mul_of_nonneg_left hb (Nat.cast_nonneg _))
(halfIdentity_zero_norm y) (norm_nonneg _) (by positivity)
_ = _ := by
have hp : ‖y‖ ^ (-2 * a - ((n + 1 : ℕ) : ℝ)) * ‖y‖ = ‖y‖ ^ (-2 * a - (n : ℝ)) := by
calc
_ = ‖y‖ ^ (-2 * a - ((n + 1 : ℕ) : ℝ)) * ‖y‖ ^ (1 : ℝ) := by
rw [Real.rpow_one]
_ = ‖y‖ ^ (-2 * a - ((n + 1 : ℕ) : ℝ) + 1) := (Real.rpow_add hyr _ _).symm
_ = _ := by congr 1; push_cast; ring
rw [mul_assoc, mul_assoc, hp]
· by_cases h1 : n - i = 1
· have hi_eq : i + 1 = n := by omega
rw [h1]
have hb : ‖iteratedFDeriv ℝ i (fderiv ℝ Q) y‖ ≤ D * ‖y‖ ^ (-2 * a - (n : ℝ)) := by
rw [norm_iteratedFDeriv_fderiv, hi_eq]
exact (hb₀ y hy).trans (mul_le_mul_of_nonneg_right (le_max_left _ _) (by positivity))
calc
_ ≤ (n.choose i : ℝ) * (D * ‖y‖ ^ (-2 * a - (n : ℝ))) * 1 :=
mul_le_mul (mul_le_mul_of_nonneg_left hb (Nat.cast_nonneg _))
(halfIdentity_first_norm y) (norm_nonneg _) (by positivity)
_ = _ := mul_one _
· rw [halfIdentity_higher_norm (n - i) (by omega) y, mul_zero]
positivity
_ = C * D * ‖y‖ ^ (-2 * a - (n : ℝ)) := by
rw [← Finset.sum_mul]
dsimp only [C]
ring
end DefocusingNLS
end OAI
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