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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 | import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffPicardResidual
import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingApproximateExistence
import OAI.MathematicalPhysics.DefocusingNLS.Linear.SchwartzProfileSampling
namespace OAI
/-! # Exact local evolution near the sampled stationary cutoff
Finite symbol jets control the whole cutoff path and its true mild residual.
The starting-scale smallness gives a genuine solution of the nonlinear
equation, with no globally Lipschitz extension of that equation.
-/
open Set Metric
open scoped SchwartzMap ContDiff
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
theorem exists_cutoffProfile_localSolution (a b k : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k)
(χ : 𝓢(E, ℝ)) (hχ : HasCompactSupport (χ : E → ℝ))
(hχone : ∀ x : E, ‖x‖ < 1 / 2 → χ x = 1)
(hχzero : ∀ x : E, 1 ≤ ‖x‖ → χ x = 0)
(m : ℕ) (ham : 2 * a * (m : ℝ) = 1) :
∃ N : ℕ, ∀ (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) (D : ℝ), 0 ≤ D →
(∀ n ≤ N, ∀ y : E, y ≠ 0 →
‖iteratedFDeriv ℝ n Q y‖ ≤ D * ‖y‖ ^ (-2 * a - (n : ℝ))) →
(∀ y, stationarySimilarityDefect a b m Q y = 0) →
∃ C K : ℝ, 0 ≤ C ∧ 0 ≤ K ∧ ∀ (L T : ℝ) (hL : 1 ≤ L) (hT : 0 ≤ T),
T * K < 1 →
let q := sampledCutoffProfilePath a k L T ha ha1 hk hL
(χ.postcompCLM Complex.ofRealCLM) (hasCompactSupport_complexCutoff χ hχ) Q hQ
let q₀ := sampledCutoffProfileHistory a k L T ha ha1 hk hL hT
(χ.postcompCLM Complex.ofRealCLM) (hasCompactSupport_complexCutoff χ hχ) Q hQ 0
∀ u₀ : FourierL2, dist u₀ q₀ + C * L ^ (-2 - a) ≤ 1 - T * K →
∃ u : C(Icc (0 : ℝ) T, FourierL2),
u = expandingPicard a b k L T ha hk hL hT
(expandingNonlinearReaction a k L T ha ha1 hk hL m) u₀ u ∧
dist u q ≤ 1 ∧
dist u q ≤ (dist u₀ q₀ + C * L ^ (-2 - a)) / (1 - T * K) := by
have hzero₂ : ∀ x : E, 2 < ‖x‖ → χ x = 0 := fun x hx => hχzero x (by linarith)
have hzeroC : ∀ x : E, 1 ≤ ‖x‖ → (χ.postcompCLM Complex.ofRealCLM) x = 0 := by
intro x hx
simp only [SchwartzMap.postcompCLM_apply, hχzero x hx, map_zero]
obtain ⟨N₁, hsamp⟩ := exists_cutoffProfile_sampling_bound a k ha ha1 hk
(χ.postcompCLM Complex.ofRealCLM) (hasCompactSupport_complexCutoff χ hχ) hzeroC
obtain ⟨N₂, hres⟩ := exists_cutoffPicard_residual_bound a b k ha ha1 hk
χ hχ hχone hzero₂ m ham
refine ⟨max N₁ N₂, ?_⟩
intro Q hQ D hD hsymbol hstationary
obtain ⟨R, hR, hbR⟩ := hsamp Q hQ D hD
(fun n hn => hsymbol n (hn.trans (le_max_left _ _)))
obtain ⟨C, hC, hbC⟩ := hres Q hQ D hD
(fun n hn => hsymbol n (hn.trans (le_max_right _ _))) hstationary
obtain ⟨K, hK, hex⟩ := exists_expandingNonlinear_near_approximation a b k ha ha1 hk
m R 1 hR (by norm_num)
refine ⟨C, K, hC, hK, ?_⟩
intro L T hL hT hTK q q₀ u₀ hsmall
have hqR : ∀ t, ‖q t‖ ≤ R := by
intro t
exact hbR (expandingRadius L t) (expandingRadiusCurve L T hL t).2
have hdef : dist q (expandingPicard a b k L T ha hk hL hT
(expandingNonlinearReaction a k L T ha ha1 hk hL m) u₀ q) ≤
dist u₀ q₀ + C * L ^ (-2 - a) := by
calc
_ ≤ dist q (expandingPicard a b k L T ha hk hL hT
(expandingNonlinearReaction a k L T ha ha1 hk hL m) q₀ q) +
dist (expandingPicard a b k L T ha hk hL hT
(expandingNonlinearReaction a k L T ha ha1 hk hL m) q₀ q)
(expandingPicard a b k L T ha hk hL hT
(expandingNonlinearReaction a k L T ha ha1 hk hL m) u₀ q) := dist_triangle _ _ _
_ ≤ C * L ^ (-2 - a) + dist q₀ u₀ := add_le_add (hbC L T hL hT)
(expandingPicard_initial_dist_le a b k L T ha hk hL hT _ q₀ u₀ q)
_ = _ := by rw [dist_comm q₀ u₀, add_comm]
exact hex L T hL hT hTK q hqR u₀ (dist u₀ q₀ + C * L ^ (-2 - a)) hdef
(by simpa only [mul_one] using hsmall)
end DefocusingNLS
end OAI
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