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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 68 69 | import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingProfileContinuity
import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingStepExistence
namespace OAI
/-! # A continuous actual nonlinear step for varying scale and initial data
The already proved uniform small-data construction now yields a jointly
continuous selected trajectory and endpoint remainder. Uniqueness of the
small trajectory makes this independent of the existence proof's choices.
-/
open Set
namespace DefocusingNLS
attribute [local irreducible] expandingPicard expandingPerturbationReaction
expandingProfileTrajectory
theorem exists_expandingContinuous_step
{P : Type*} [TopologicalSpace P] (a b k T : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hT : 0 ≤ T)
(m : ℕ) (R : ℝ) (hR : 0 ≤ R) :
∃ δ A : ℝ, 0 < δ ∧ 0 < A ∧ ∀
(L : P → {L : ℝ // 1 ≤ L}) (q g : P → C(Icc (0 : ℝ) T, FourierL2)) (G : ℝ),
Continuous L → Continuous q → Continuous g → 0 ≤ G → G ≤ δ →
(hq : ∀ p t, ‖q p t‖ ≤ R) → (∀ p t, ‖g p t‖ ≤ G) →
∃ (V : P × {f : FourierL2 // ‖f‖ ≤ δ} → C(Icc (0 : ℝ) T, FourierL2))
(h : P × {f : FourierL2 // ‖f‖ ≤ δ} → FourierL2),
Continuous V ∧ Continuous h ∧
(∀ z, V z = expandingPicard a b k (L z.1).1 T ha hk (L z.1).2 hT
(expandingPerturbationReaction a k (L z.1).1 T ha ha1 hk (L z.1).2 m
(q z.1) (g z.1)) z.2.1 (V z)) ∧
(∀ z, ‖V z‖ ≤ A * (‖z.2.1‖ + G)) ∧ (∀ z, ‖V z‖ ≤ 1 / 2) ∧
(∀ z, V z ⟨T, hT, le_rfl⟩ =
expandingProfileTrajectory a b k (L z.1).1 T ha ha1 hk (L z.1).2 hT
m R hR (q z.1) (hq z.1) z.2.1 ⟨T, hT, le_rfl⟩ + h z) := by
obtain ⟨δ, A, hδ, hA, _, hexists⟩ :=
exists_expandingPerturbation_step_radius a b k T ha ha1 hk hT m R hR
refine ⟨δ, A, hδ, hA, ?_⟩
intro L q g G hL hq hg hG hGδ hqb hgb
choose V hV hVbound hVhalf using
(fun z : P × {f : FourierL2 // ‖f‖ ≤ δ} =>
hexists (L z.1).1 (L z.1).2 (q z.1) (g z.1) G hG hGδ (hqb z.1) (hgb z.1)
z.2.1 z.2.2)
let h : P × {f : FourierL2 // ‖f‖ ≤ δ} → FourierL2 := fun z =>
V z ⟨T, hT, le_rfl⟩ - expandingProfileTrajectory a b k (L z.1).1 T ha ha1 hk
(L z.1).2 hT m R hR (q z.1) (hqb z.1) z.2.1 ⟨T, hT, le_rfl⟩
have hu₀ : Continuous (fun z : P × {f : FourierL2 // ‖f‖ ≤ δ} => z.2.1) :=
continuous_subtype_val.comp continuous_snd
have hcV : Continuous V := continuous_expandingPerturbation_family
(P := P × {f : FourierL2 // ‖f‖ ≤ δ}) a b k T R
ha ha1 hk hT m hR (fun z => L z.1) (hL.comp continuous_fst)
(fun z => q z.1) (fun z => g z.1) (hq.comp continuous_fst) (hg.comp continuous_fst)
(fun z => hqb z.1) (fun z => z.2.1) hu₀ V hV (fun z => (hVhalf z).trans (by norm_num))
have hch : Continuous h := continuous_expandingStepRemainder_family
(P := P × {f : FourierL2 // ‖f‖ ≤ δ}) a b k T R
ha ha1 hk hT m hR (fun z => L z.1) (hL.comp continuous_fst)
(fun z => q z.1) (fun z => g z.1) (hq.comp continuous_fst) (hg.comp continuous_fst)
(fun z => hqb z.1) (fun z => z.2.1) hu₀ V hV (fun z => (hVhalf z).trans (by norm_num))
refine ⟨V, h, hcV, hch, hV, hVbound, hVhalf, ?_⟩
intro z
dsimp [h]
abel
end DefocusingNLS
end OAI
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