File size: 3,472 Bytes
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 | import OAI.MathematicalPhysics.DefocusingNLS.Profile.RadialMatchedProfileBounds
import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffMovingPath
import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingProfilePropagator
namespace OAI
/-! # Uniform finite-slab evolution for the actual cutoff profile
The matched profile's proved symbol bounds give a single norm bound on
all of its expanding-torus samples. The actual linearized potential then
has bounded real-linear propagators uniformly in the starting radius.
-/
open Set
open scoped SchwartzMap ContDiff
namespace DefocusingNLS
open ProfileCertificate
local notation "E" => EuclideanSpace ℝ (Fin 12)
theorem exists_matchedCutoffProfile_uniform_bound (n : ℕ) (z : ProfileMatchingBall)
(hX : HasRadialExterior (radialShootingNu (n + radialInnerShootingThreshold) z)
(n + radialInnerShootingThreshold) (radialShootingM z) (Real.log innerBoundaryRadius))
(hz : radialMatchingMap n z = 0) (k : ℝ) (hk : 8 < k)
(χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ))
(hχzero : ∀ y : E, 1 ≤ ‖y‖ → χ y = 0) :
let ha := (radialShootingA_bounds n (profileMatchingParameter z)).1
let ha1 := (radialShootingA_bounds n (profileMatchingParameter z)).2
let hQ := radialMatchedCartesian_contDiff n z hX hz
∃ R : ℝ, 0 ≤ R ∧ ∀ (L T : ℝ) (hL : 1 ≤ L) (t : Icc (0 : ℝ) T),
‖sampledCutoffProfilePath (radialShootingA n) k L T ha ha1 hk hL
χ hχ (radialMatchedCartesian n z) hQ t‖ ≤ R := by
intro ha ha1 hQ
obtain ⟨R, hR, hbound⟩ := radialMatchedCutoff_sampling_bound n z hX hz k hk χ hχ hχzero
refine ⟨R, hR, ?_⟩
intro L T hL t
exact hbound (expandingRadiusCurve L T hL t).1 (expandingRadiusCurve L T hL t).2
theorem exists_matchedCutoffLinearized_propagator (n : ℕ) (z : ProfileMatchingBall)
(hX : HasRadialExterior (radialShootingNu (n + radialInnerShootingThreshold) z)
(n + radialInnerShootingThreshold) (radialShootingM z) (Real.log innerBoundaryRadius))
(hz : radialMatchingMap n z = 0) (k : ℝ) (hk : 8 < k)
(χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ))
(hχzero : ∀ y : E, 1 ≤ ‖y‖ → χ y = 0) :
let a := radialShootingA n
let b := radialShootingB (profileMatchingParameter z)
let m := n + radialInnerShootingThreshold
let ha := (radialShootingA_bounds n (profileMatchingParameter z)).1
let ha1 := (radialShootingA_bounds n (profileMatchingParameter z)).2
let hQ := radialMatchedCartesian_contDiff n z hX hz
∃ K : ℝ, 0 ≤ K ∧ ∀ (L T : ℝ) (hL : 1 ≤ L) (hT : 0 ≤ T),
let q := sampledCutoffProfilePath a k L T ha ha1 hk hL χ hχ (radialMatchedCartesian n z) hQ
∃ U : FourierL2 →L[ℝ] C(Icc (0 : ℝ) T, FourierL2),
(∀ u₀ t, U u₀ t = expandingFreeStep a b k L t ha hk hL t.2.1 u₀ +
expandingDuhamel a b k L ha hk hL t (expandingReactionHistory T hT
(expandingProfileReaction a k T ha ha1 hk m (expandingRadiusCurve L T hL) q 0)
(U u₀))) ∧
‖U‖ ≤ (K + 1) * Real.exp ((K + 1) * T) := by
intro a b m ha ha1 hQ
obtain ⟨R, hR, hbound⟩ := exists_matchedCutoffProfile_uniform_bound n z hX hz k hk χ hχ hχzero
obtain ⟨K, hK, hprop⟩ := exists_expandingProfile_propagator a b k ha ha1 hk m R hR
refine ⟨K, hK, ?_⟩
intro L T hL hT
exact hprop L T hL hT _ (hbound L T hL)
end DefocusingNLS
end OAI
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