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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 | import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.ContinuousTorusSequence
import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffContinuousLinearization
import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffEndpointContinuity
import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.StableGraphStepDecay
namespace OAI
/-! # A continuous stable graph of actual global cutoff trajectories -/
open Set Filter Topology
open scoped SchwartzMap ContDiff NNReal
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
local notation "Radius" => {L : ℝ // 1 ≤ L}
abbrev CutoffStableParameter {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
(ρ L₀ : ℝ) (π : Radius → FourierL2 →L[ℝ] F) :=
{p : Radius × FourierL2 // L₀ ≤ p.1.1 ∧ ‖p.2‖ ≤ ρ / 4 ∧ π p.1 p.2 = 0}
def HasContinuousCutoffStableOrbits {F : Type*}
[NormedAddCommGroup F] [NormedSpace ℝ F]
(a b k : ℝ) (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (m : ℕ)
(χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ))
(Qp : E → ℂ) (hQp : ContDiff ℝ ∞ Qp) : Prop :=
∃ ρ : ℝ, 0 < ρ ∧ ∃ C : ℝ, 0 < C ∧ ∃ L₀ : ℝ,
∃ ζ : Radius → F →L[ℝ] FourierL2, ∃ π : Radius → FourierL2 →L[ℝ] F,
ContinuousOn ζ {L : Radius | L₀ ≤ L.1} ∧
Continuous (fun q : Radius × FourierL2 => π q.1 q.2) ∧
(∀ L : Radius, L₀ ≤ L.1 → (∀ v, π L (ζ L v) = v) ∧ ‖ζ L‖ ≤ 1 ∧ ‖π L‖ ≤ C) ∧
∃ G : CutoffStableParameter ρ L₀ π → FourierL2, Continuous G ∧
∀ p, ‖G p‖ ≤ 2 * ρ ∧ stableFrameProjection (ζ p.1.1) (π p.1.1) (G p) = p.1.2 ∧
∃ u : ℝ → FourierL2, ∃ hu : ContinuousOn u (Ici 0),
u 0 = sampledCutoffProfileOrbit a k p.1.1.1 ha1 hk p.1.1.2 χ hχ Qp hQp 0 + G p ∧
(∀ (S : ℝ) (hS : 0 < S), expandingSlabRestriction u hu S =
expandingPicard a b k p.1.1.1 S ha hk p.1.1.2 hS.le
(expandingNonlinearReaction a k p.1.1.1 S ha ha1 hk p.1.1.2 m) (u 0)
(expandingSlabRestriction u hu S)) ∧
Tendsto (fun s => ‖u s - sampledCutoffProfileOrbit a k p.1.1.1 ha1 hk p.1.1.2 χ hχ Qp hQp s‖)
atTop (𝓝 0)
theorem exists_continuous_cutoffStable_orbits {F : Type*}
[NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F]
(a b k : ℝ) (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (m : ℕ)
(χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ))
(Qp : E → ℂ) (hQp : ContDiff ℝ ∞ Qp) (Q : ℝ) (hQ : 0 ≤ Q)
(hqb : ∀ L : Radius, ‖cutoffProfileCoefficient a k ha1 hk χ hχ Qp hQp L‖ ≤ Q)
(T : ℝ≥0) (hT : 0 < (T : ℝ))
(hratio : 2 * Real.exp (-(2 + a) * T / 2) ≤ 1)
(hlinear : HasContinuousTorusLinearStep (F := F) T
(cutoffProfileEndpoint a b k ha ha1 hk m χ hχ Qp hQp Q hQ hqb T))
(hnonlinear : HasContinuousCutoffNonlinearSteps a b k ha ha1 hk m χ hχ Qp hQp Q hQ hqb T) :
HasContinuousCutoffStableOrbits (F := F) a b k ha ha1 hk m χ hχ Qp hQp := by
classical
obtain ⟨δ, Lnl, B, hδ, _, hB, V, h, hcV, hch, hsol, hV, hend, hzero, hlip⟩ := hnonlinear
obtain ⟨C, hC, ρ, hρ, hρδ, Lg, ζ, π, hζc, hπc, hbound, hπζ, hseq⟩ :=
exists_continuous_cutoffStable_sequence_threshold a δ B ha hδ hB.le T hratio
(cutoffProfileEndpoint a b k ha ha1 hk m χ hχ Qp hQp Q hQ hqb T)
(continuous_cutoffProfileEndpoint a b k ha ha1 hk m χ hχ Qp hQp Q hQ hqb T) hlinear
let L₀ := max Lg Lnl
let ζ₀ := fun L => ζ L 0
let π₀ := fun L => π L 0
let P := CutoffStableParameter ρ L₀ π₀
let l : P → Radius := fun p => p.1.1
let w : P → FourierL2 := fun p => p.1.2
have hlc : Continuous l := continuous_fst.comp continuous_subtype_val
have hwc : Continuous w := continuous_snd.comp continuous_subtype_val
have hl (p : P) : Lg ≤ (l p).1 := (le_max_left _ _).trans p.2.1
let ln := fun n (p : P) => expandingDiscreteRadius (l p) T n
have hln (n : ℕ) (p : P) : Lnl ≤ (ln n p).1 :=
((le_max_right _ _).trans p.2.1).trans (expandingDiscreteRadius_ge (l p) T n)
let r := fun n (p : P) => (⟨(ln n p).1, (ln n p).2, hln n p⟩ : {R : ℝ // 1 ≤ R ∧ Lnl ≤ R})
have hrc (n : ℕ) : Continuous (r n) :=
(continuous_subtype_val.comp ((continuous_expandingDiscreteRadius T n).comp hlc)).subtype_mk _
let H := fun p n v => h (r n p, v)
have hHc (n : ℕ) : Continuous (fun q : P × {v : FourierL2 // ‖v‖ ≤ δ} => H q.1 n q.2) :=
hch.comp (((hrc n).comp continuous_fst).prodMk continuous_snd)
obtain ⟨z, hzc, hz0, hzstep, hzn⟩ := hseq P l hlc hl H hHc
(fun p n d hd hdδ v u hv hu => hlip d hd hdδ (r n p) v u hv hu)
(fun p n v hv => hzero (r n p, v) hv)
w hwc (fun p => p.2.2.1) (fun p => p.2.2.2)
refine ⟨ρ, hρ, C, hC, L₀, ζ₀, π₀, ?_, hπc 0, ?_, (fun p => z p 0), hzc 0, ?_⟩
· apply (hζc 0).mono
intro L hL
exact (le_max_left Lg Lnl).trans hL
· intro L hL
have hLg : Lg ≤ L.1 := (le_max_left _ _).trans hL
exact ⟨hπζ L hLg 0, (hbound L hLg 0).1, (hbound L hLg 0).2⟩
· intro p
refine ⟨by simpa using hzn p 0, hz0 p, ?_⟩
have hzδ (n : ℕ) : ‖z p n‖ ≤ δ := (hzn p n).trans ((mul_le_of_le_one_right
(by positivity : 0 ≤ 2 * ρ) (pow_le_one₀ (by norm_num : (0 : ℝ) ≤ 1 / 2) (by norm_num))).trans hρδ)
let Z := fun n => (⟨z p n, hzδ n⟩ : {f : FourierL2 // ‖f‖ ≤ δ})
let W := fun n => V (r n p, Z n)
have hWM (n : ℕ) : W n ⟨T, T.2, le_rfl⟩ = z p (n + 1) := by
calc
_ = cutoffProfileEndpoint a b k ha ha1 hk m χ hχ Qp hQp Q hQ hqb T (ln n p) (z p n) +
H p n (Z n) := hend (r n p, Z n)
_ = cutoffProfileEndpoint a b k ha ha1 hk m χ hχ Qp hQp Q hQ hqb T (ln n p) (z p n) +
cutoffRemainderExtension δ (H p) n (z p n) := by
rw [cutoffRemainderExtension, dite_eq_left (hzδ n)]
_ = _ := (hzstep p n).symm
have hWbound (n : ℕ) : ‖W n‖ ≤ B * (‖z p n‖ +
(expandingRadius (l p).1 ((n : ℝ) * T)) ^ (-2 - a)) := hV (r n p, Z n)
have hr : Real.exp (-(2 + a) * T / 2) ≤ 1 / 2 := by linarith
have hWgeo := cutoffStep_path_geometric_bound a (l p).1 T B (2 * ρ)
(lt_of_lt_of_le zero_lt_one (l p).2) hB.le hr (z p) W (hzn p) hWbound
exact exists_cutoffGlobal_orbit_of_steps a b k (l p).1 T ha ha1 hk (l p).2 hT m χ hχ Qp hQp
(z p) W hWM (fun n => hsol (r n p, Z n))
(B * (2 * ρ + (l p).1 ^ (-2 - a))) (1 / 2) (by norm_num) (by norm_num) hWgeo
end DefocusingNLS
end OAI
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