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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 | import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingStepMapContinuity
import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffPathContinuity
namespace OAI
/-! # The continuous nonlinear step for the actual cutoff profile
The same family has the exact unforced mild equation, a continuous endpoint
remainder, and the scale-dependent remainder estimates. The stationary
profile and its finite symbol bounds are the hypotheses for the step
estimates.
-/
open Set
open scoped SchwartzMap ContDiff
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
attribute [local irreducible] expandingPicard expandingPerturbationReaction
expandingProfileTrajectory sampledCutoffProfilePath sampledCutoffDefectPath
theorem exists_cutoffNonlinear_step_continuous (a b k T : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hT : 0 ≤ T)
(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 ∧
let q := fun L : {L : ℝ // 1 ≤ L ∧ L₀ ≤ L} =>
sampledCutoffProfilePath a k L.1 T ha ha1 hk L.2.1
(χ.postcompCLM Complex.ofRealCLM) (hasCompactSupport_complexCutoff χ hχ) Q hQ
∃ (V : {L : ℝ // 1 ≤ L ∧ L₀ ≤ L} × {f : FourierL2 // ‖f‖ ≤ δ} →
C(Icc (0 : ℝ) T, FourierL2))
(h : {L : ℝ // 1 ≤ L ∧ L₀ ≤ L} × {f : FourierL2 // ‖f‖ ≤ δ} → FourierL2),
Continuous V ∧ Continuous h ∧
(∀ z, q z.1 + V z = expandingPicard a b k z.1.1 T ha hk z.1.2.1 hT
(expandingNonlinearReaction a k z.1.1 T ha ha1 hk z.1.2.1 m)
(q z.1 ⟨0, le_rfl, hT⟩ + z.2.1) (q z.1 + V z)) ∧
(∀ z, ‖V z‖ ≤ C * (‖z.2.1‖ + z.1.1 ^ (-2 - a))) ∧
(∀ z, ∃ w : C(Icc (0 : ℝ) T, FourierL2),
(∀ t : Icc (0 : ℝ) T, w t =
expandingFreeStep a b k z.1.1 t ha hk z.1.2.1 t.2.1 z.2.1 +
expandingDuhamel a b k z.1.1 ha hk z.1.2.1 t
(expandingReactionHistory T hT
(expandingProfileReaction a k T ha ha1 hk m
(expandingRadiusCurve z.1.1 T z.1.2.1) (q z.1) 0) w)) ∧
V z ⟨T, hT, le_rfl⟩ = w ⟨T, hT, le_rfl⟩ + h z) ∧
(∀ z, z.2.1 = 0 → ‖h z‖ ≤ C * z.1.1 ^ (-2 - a)) ∧
(∀ d : ℝ, 0 < d → d ≤ δ → ∀ L f j, ‖f.1‖ ≤ d → ‖j.1‖ ≤ d →
‖h (L, f) - h (L, j)‖ ≤ C * (d + L.1 ^ (-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_continuous a b k T ha ha1 hk hT m R hR
obtain ⟨L₀, hL₀, hsmall⟩ := exists_cutoffResidual_small_scale a B δ ha hδ
let P := {L : ℝ // 1 ≤ L ∧ L₀ ≤ L}
let l : P → {L : ℝ // 1 ≤ L} := fun L => ⟨L.1, L.2.1⟩
have hl : Continuous l := continuous_subtype_val.subtype_mk _
let q : P → C(Icc (0 : ℝ) T, FourierL2) := fun L =>
sampledCutoffProfilePath a k L.1 T ha ha1 hk L.2.1
(χ.postcompCLM Complex.ofRealCLM) (hasCompactSupport_complexCutoff χ hχ) Q hQ
let g : P → C(Icc (0 : ℝ) T, FourierL2) := fun L =>
sampledCutoffDefectPath a k L.1 T ha ha1 hk L.2.1 χ hχ hχone hχzero m Q hQ
let G : P → ℝ := fun L => B * L.1 ^ (-2 - a)
have hq : Continuous q := (continuous_sampledCutoffProfilePath a k T ha ha1 hk
(χ.postcompCLM Complex.ofRealCLM) (hasCompactSupport_complexCutoff χ hχ) Q hQ).comp hl
have hg : Continuous g := (continuous_sampledCutoffDefectPath a k T ha ha1 hk
χ hχ hχone hχzero m Q hQ).comp hl
have hqb : ∀ L t, ‖q L t‖ ≤ R := fun L t => (hbounds L.1 T L.2.1 t).1
have hgb : ∀ L t, ‖g L t‖ ≤ G L := fun L t => (hbounds L.1 T L.2.1 t).2
have hG : ∀ L : P, 0 ≤ G L :=
fun L => mul_nonneg hB (Real.rpow_nonneg (by linarith [L.2.1]) _)
have hGδ : ∀ L : P, G L ≤ δ := fun L => hsmall L.1 L.2.2
specialize hstep P l q g G hl hq hg hG hGδ hqb hgb
obtain ⟨V, h, hcV, hch, hV, hVbound, hEnd, hzero, hlip⟩ := hstep
let C := A * (1 + B)
have hC : 0 < C := by dsimp [C]; positivity
have hcoeff (L : P) (x : ℝ) (hx : 0 ≤ x) :
A * (x + G L) ≤ C * (x + L.1 ^ (-2 - a)) := by
have hp : 0 ≤ L.1 ^ (-2 - a) := Real.rpow_nonneg (by linarith [L.2.1]) _
have hAA : A ≤ C := by dsimp [C]; nlinarith [mul_nonneg hA.le hB]
have hAB : A * B ≤ C := by dsimp [C]; nlinarith
calc
A * (x + G L) = A * x + (A * B) * L.1 ^ (-2 - a) := by dsimp [G]; ring
_ ≤ C * x + C * L.1 ^ (-2 - a) := add_le_add
(mul_le_mul_of_nonneg_right hAA hx) (mul_le_mul_of_nonneg_right hAB hp)
_ = C * (x + L.1 ^ (-2 - a)) := by ring
refine ⟨δ, L₀, C, hδ, hL₀, hC, V, h, hcV, hch, ?_, ?_, ?_, ?_, ?_⟩
· intro z
exact expandingPerturbation_add_mild a b k z.1.1 T ha ha1 hk z.1.2.1 hT m
(q z.1) (g z.1) (V z) (q z.1 ⟨0, le_rfl, hT⟩) z.2.1
(sampledCutoffProfile_forced_mild a b k z.1.1 T ha ha1 hk z.1.2.1 hT m ham
χ hχ hχone hχzero Q hQ hstationary) (hV z)
· intro z
exact (hVbound z).trans (hcoeff z.1 _ (norm_nonneg _))
· intro z
refine ⟨expandingProfileTrajectory a b k z.1.1 T ha ha1 hk z.1.2.1 hT
m R hR (q z.1) (hqb z.1) z.2.1, ?_, hEnd z⟩
exact expandingProfileTrajectory_eq a b k z.1.1 T ha ha1 hk z.1.2.1 hT
m R hR (q z.1) (hqb z.1) z.2.1
· intro z hz
have hb := hcoeff z.1 0 (by norm_num)
simp only [zero_add] at hb
exact (hzero z hz).trans hb
· intro d hd hdδ L f j hf hj
exact (hlip d hd hdδ L f j hf hj).trans
(mul_le_mul_of_nonneg_right (hcoeff L d hd.le) (norm_nonneg _))
end DefocusingNLS
end OAI
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