File size: 3,428 Bytes
d5b562f | 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 | import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingDuhamelDuality
namespace OAI
/-! # Uniform domination for the actual dual Duhamel integrand -/
open Set MeasureTheory
open scoped SchwartzMap
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
theorem expandingDualDuhamelIntegrand_eq (a b k L t : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) (ht : 0 ≤ t)
(φ : 𝓢(E, ℂ)) (r : ℝ → FourierL2) (s : ℝ) :
expandingDualDuhamelIntegrand a b k L t ha ha1 hk hL φ r s =
expandingFourierTest a k (expandingRadius L t) ha ha1 hk
(hL.trans (expandingRadius_ge L t hL ht)) φ
(expandingDuhamelIntegrand a b k L ha hk hL t r s) := by
by_cases hs : s ∈ Icc 0 t
· simp only [expandingDualDuhamelIntegrand, dite_eq_left hs,
expandingDuhamelIntegrand_of_mem _ _ _ _ _ _ _ _ _ _ hs]
have h := expandingFreeStep_fourier_duality a b k (expandingRadius L s) (t - s)
ha ha1 hk (hL.trans (expandingRadius_ge L s hL hs.1)) (sub_nonneg.mpr hs.2) φ (r s)
simpa only [expandingDuhamel_destination] using h.symm
· simp only [expandingDualDuhamelIntegrand, expandingDuhamelIntegrand,
dite_eq_right hs, map_zero]
theorem integrableOn_expandingDualDuhamelIntegrand (a b k L t : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) (ht : 0 ≤ t)
(φ : 𝓢(E, ℂ)) (r : ℝ → FourierL2) (hr : ContinuousOn r (Icc 0 t)) :
IntegrableOn (expandingDualDuhamelIntegrand a b k L t ha ha1 hk hL φ r) (Icc 0 t) := by
have he : expandingDualDuhamelIntegrand a b k L t ha ha1 hk hL φ r =
fun s => expandingFourierTest a k (expandingRadius L t) ha ha1 hk
(hL.trans (expandingRadius_ge L t hL ht)) φ
(expandingDuhamelIntegrand a b k L ha hk hL t r s) :=
funext (expandingDualDuhamelIntegrand_eq a b k L t ha ha1 hk hL ht φ r)
rw [he]
exact (expandingFourierTest a k (expandingRadius L t) ha ha1 hk
(hL.trans (expandingRadius_ge L t hL ht)) φ).integrable_comp
(integrableOn_expandingDuhamelIntegrand a b k L ha hk hL t r hr)
theorem expandingDualDuhamelIntegrand_bound (a b k L t M : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) (ht : 0 ≤ t)
(φ : 𝓢(E, ℂ)) (r : ℝ → FourierL2) (s : ℝ) (hs : s ∈ Icc 0 t)
(hr : ‖r s‖ ≤ M) :
‖expandingDualDuhamelIntegrand a b k L t ha ha1 hk hL φ r s‖ ≤
(expandingEmbeddingBound a k * SchwartzMap.seminorm ℂ 0 0 φ) * M := by
rw [expandingDualDuhamelIntegrand_eq a b k L t ha ha1 hk hL ht,
expandingDuhamelIntegrand_of_mem _ _ _ _ _ _ _ _ _ _ hs]
let F := expandingFourierTest a k (expandingRadius L t) ha ha1 hk
(hL.trans (expandingRadius_ge L t hL ht)) φ
have hf := expandingFreeStep_norm_bound a b k (expandingRadius L s) (t - s) ha hk
(hL.trans (expandingRadius_ge L s hL hs.1)) (sub_nonneg.mpr hs.2) (r s)
have he : Real.exp (-a * (t - s) / 2) ≤ 1 := Real.exp_le_one_iff.mpr (by nlinarith [hs.2])
have hstep : ‖expandingFreeStep a b k (expandingRadius L s) (t - s) ha hk
(hL.trans (expandingRadius_ge L s hL hs.1)) (sub_nonneg.mpr hs.2) (r s)‖ ≤ M :=
hf.trans ((mul_le_of_le_one_left (norm_nonneg _) he).trans hr)
exact (F.le_opNorm _).trans
(mul_le_mul (expandingFourierTest_norm_le a k _ ha ha1 hk _ φ) hstep
(norm_nonneg _) (by unfold expandingEmbeddingBound; positivity))
end DefocusingNLS
end OAI
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