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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 | import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.MaximalStrongSolution
import Mathlib.Analysis.Calculus.ContDiff.Deriv
namespace OAI
/-! # The maximal Sobolev flow is a classical solution of the defocusing equation -/
open Set
open scoped ContDiff
namespace DefocusingNLS
def IsClassicalDefocusingFlow (k : ℝ) (hk : 6 < k) (m : ℕ) (f : FourierL2) : Prop :=
let u := maximalSobolevSchrodingerFlow k hk m f
let D := maximalSobolevInteractionDomain k hk m f
u 0 = f ∧ ContinuousOn u D ∧
ContDiffOn ℝ 1 (fun t => lowerSobolevInclusion (u t)) D ∧
∀ t ∈ D, HasDerivAt (fun s => lowerSobolevInclusion (u s))
(lowerSobolevGenerator (u t) - Complex.I • lowerSobolevInclusion
(sobolevOddPower k hk m (u t))) t
theorem maximalSobolev_classical (k : ℝ) (hk : 6 < k) (m : ℕ) (f : FourierL2) :
IsClassicalDefocusingFlow k hk m f := by
let u := maximalSobolevSchrodingerFlow k hk m f
let D := maximalSobolevInteractionDomain k hk m f
let v := fun t => lowerSobolevGenerator (u t) - Complex.I • lowerSobolevInclusion
(sobolevOddPower k hk m (u t))
have hu : ContinuousOn u D := continuousOn_maximalSobolevSchrodingerFlow k hk m f
have hv : ContinuousOn v D :=
(lowerSobolevGenerator.continuous.comp_continuousOn hu).sub
((lowerSobolevInclusion.continuous.comp_continuousOn
((contDiff_sobolevOddPower k hk m).continuous.comp_continuousOn hu)).const_smul Complex.I)
have hd (t : ℝ) (ht : t ∈ D) :
HasDerivAt (fun s => lowerSobolevInclusion (u s)) (v t) t :=
hasDerivAt_maximalSobolevSchrodingerFlow k hk m f ht
have hopen : IsOpen D := isOpen_maximalSobolevInteractionDomain k hk m f
have hdiff : ContDiffOn ℝ 1 (fun t => lowerSobolevInclusion (u t)) D := by
rw [show (1 : ℕ∞ω) = 0 + 1 from rfl, contDiffOn_succ_iff_deriv_of_isOpen hopen]
refine ⟨fun t ht => (hd t ht).differentiableAt.differentiableWithinAt, ?_, ?_⟩
· norm_num
· apply contDiffOn_zero.mpr
apply hv.congr
intro t ht
exact (hd t ht).deriv
exact ⟨maximalSobolevSchrodingerFlow_initial k hk m f, hu, hdiff, hd⟩
end DefocusingNLS
end OAI
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