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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 | import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingStepMap
import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffStepForcing
namespace OAI
/-! # The nonlinear step for the actual sampled cutoff profile
The fixed-step bounds follow from a smooth stationary profile with finitely
many symbol estimates.
-/
open Set
open scoped SchwartzMap ContDiff
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
theorem exists_cutoffNonlinear_step (a b k M : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hM : 0 ≤ M)
(m : ℕ) (ham : 2 * a * (m : ℝ) = 1)
(χ : 𝓢(E, ℝ)) (hχ : HasCompactSupport (χ : E → ℝ))
(hχone : ∀ x : E, ‖x‖ < 1 / 2 → χ x = 1)
(hχsupport : ∀ x : E, 1 ≤ ‖x‖ → χ x = 0) :
∃ 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) →
∃ δ L₀ C : ℝ, 0 < δ ∧ 1 ≤ L₀ ∧ 0 < C ∧ ∀ (L : ℝ) (hL : 1 ≤ L), L₀ ≤ L →
let q := sampledCutoffProfilePath a k L M ha ha1 hk hL
(χ.postcompCLM Complex.ofRealCLM) (hasCompactSupport_complexCutoff χ hχ) Q hQ
∃ (V : {f : FourierL2 // ‖f‖ ≤ δ} → C(Icc (0 : ℝ) M, FourierL2))
(h : {f : FourierL2 // ‖f‖ ≤ δ} → FourierL2),
(∀ f, q + V f = expandingPicard a b k L M ha hk hL hM
(expandingNonlinearReaction a k L M ha ha1 hk hL m)
(q ⟨0, le_rfl, hM⟩ + f.1) (q + V f)) ∧
(∀ f, ‖V f‖ ≤ C * (‖f.1‖ + L ^ (-2 - a))) ∧
(∀ f, ∃ w : C(Icc (0 : ℝ) M, FourierL2),
(∀ t : Icc (0 : ℝ) M, w t = expandingFreeStep a b k L t ha hk hL t.2.1 f.1 +
expandingDuhamel a b k L ha hk hL t
(expandingReactionHistory M hM
(expandingProfileReaction a k M ha ha1 hk m (expandingRadiusCurve L M hL) q 0) w)) ∧
V f ⟨M, hM, le_rfl⟩ = w ⟨M, hM, le_rfl⟩ + h f) ∧
(∀ z, z.1 = 0 → ‖h z‖ ≤ C * L ^ (-2 - a)) ∧
(∀ d : ℝ, 0 < d → d ≤ δ → ∀ f j, ‖f.1‖ ≤ d → ‖j.1‖ ≤ d →
‖h f - h j‖ ≤ C * (d + L ^ (-2 - a)) * ‖f.1 - j.1‖) := by
let hχzero := fun x : E => fun hx : 2 < ‖x‖ => hχsupport x (by linarith)
obtain ⟨N, hN⟩ := exists_cutoffStep_profile_forcing_bounds a k ha ha1 hk m χ hχ hχone hχsupport
refine ⟨N, ?_⟩
intro Q hQ D hD hsymbol hstationary
obtain ⟨R, B, hR, hB, hbounds⟩ := hN Q hQ D hD hsymbol
obtain ⟨δ, A, hδ, hA, hstep⟩ := exists_expandingNonlinear_stepMap a b k M ha ha1 hk hM m R hR
obtain ⟨L₀, hL₀, hsmall⟩ := exists_cutoffResidual_small_scale a B δ ha hδ
let C := A * (1 + B)
have hC : 0 < C := by dsimp [C]; positivity
have hAC : A ≤ C := by dsimp [C]; nlinarith [mul_nonneg hA.le hB]
have hABC : A * B ≤ C := by dsimp [C]; nlinarith
refine ⟨δ, L₀, C, hδ, hL₀, hC, ?_⟩
intro L hL hLlarge q
let g := sampledCutoffDefectPath a k L M ha ha1 hk hL χ hχ hχone hχzero m Q hQ
let G := B * L ^ (-2 - a)
have hpow : 0 ≤ L ^ (-2 - a) := Real.rpow_nonneg (by linarith) _
have hG : 0 ≤ G := mul_nonneg hB hpow
have hGδ : G ≤ δ := hsmall L hLlarge
have hq : ∀ t, ‖q t‖ ≤ R := fun t => (hbounds L M hL t).1
have hg : ∀ t, ‖g t‖ ≤ G := fun t => (hbounds L M hL t).2
obtain ⟨V, h, hV, hV_bound, hEnd, hZero, hLip⟩ := hstep L hL q g G hG hGδ hq hg
have hcoeff (x : ℝ) (hx : 0 ≤ x) : A * (x + G) ≤ C * (x + L ^ (-2 - a)) := by
calc
A * (x + G) = A * x + (A * B) * L ^ (-2 - a) := by dsimp [G]; ring
_ ≤ C * x + C * L ^ (-2 - a) := add_le_add
(mul_le_mul_of_nonneg_right hAC hx) (mul_le_mul_of_nonneg_right hABC hpow)
_ = C * (x + L ^ (-2 - a)) := by ring
refine ⟨V, h, ?_, ?_, ?_, ?_, ?_⟩
· intro f
exact expandingPerturbation_add_mild a b k L M ha ha1 hk hL hM m q g (V f)
(q ⟨0, le_rfl, hM⟩) f.1
(sampledCutoffProfile_forced_mild a b k L M ha ha1 hk hL hM m ham
χ hχ hχone hχzero Q hQ hstationary) (hV f)
· intro f
exact (hV_bound f).trans (hcoeff _ (norm_nonneg _))
· intro f
refine ⟨expandingProfileTrajectory a b k L M ha ha1 hk hL hM m R hR q hq f.1,
expandingProfileTrajectory_eq a b k L M ha ha1 hk hL hM m R hR q hq f.1, hEnd f⟩
· intro z hz
exact (hZero z hz).trans (by
change A * (B * L ^ (-2 - a)) ≤ C * L ^ (-2 - a)
rw [← mul_assoc]
exact mul_le_mul_of_nonneg_right hABC hpow)
· intro d hd hdδ f j hf hj
exact (hLip d hd hdδ f j hf hj).trans
(mul_le_mul_of_nonneg_right (hcoeff d hd.le) (norm_nonneg _))
end DefocusingNLS
end OAI
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