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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 | import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingEnergyScaling
import Mathlib.Analysis.InnerProductSpace.Calculus
namespace OAI
/-! # Polarization of the exact expanding-torus energy identities -/
namespace DefocusingNLS
theorem expandingEnergy_inner (a k L : ℝ) (hL : 1 ≤ L) (f g : FourierL2) :
inner ℝ f g = inner ℝ (expandingLowEnergy a k L hL f) (expandingLowEnergy a k L hL g) +
inner ℝ (expandingHighEnergy a k L hL f) (expandingHighEnergy a k L hL g) := by
have h := expandingEnergy_norm_sq a k L hL (f + g)
simp only [map_add] at h
rw [norm_add_sq_real, norm_add_sq_real, norm_add_sq_real,
expandingEnergy_norm_sq a k L hL f, expandingEnergy_norm_sq a k L hL g] at h
linarith
theorem expandingLowEnergy_free_inner (a b k L s : ℝ)
(ha : 0 < a) (hk : 8 < k) (hL : 1 ≤ L) (hs : 0 ≤ s) (f g : FourierL2) :
inner ℝ (expandingLowEnergy a k (expandingRadius L s)
(hL.trans (expandingRadius_ge L s hL hs)) (expandingFreeStep a b k L s ha hk hL hs f))
(expandingLowEnergy a k (expandingRadius L s)
(hL.trans (expandingRadius_ge L s hL hs)) (expandingFreeStep a b k L s ha hk hL hs g)) =
Real.exp (-a * s) * inner ℝ (expandingLowEnergy a k L hL f) (expandingLowEnergy a k L hL g) := by
have h := expandingLowEnergy_free a b k L s ha hk hL hs (f + g)
simp only [map_add] at h
rw [norm_add_sq_real, norm_add_sq_real,
expandingLowEnergy_free a b k L s ha hk hL hs f,
expandingLowEnergy_free a b k L s ha hk hL hs g] at h
nlinarith [h]
theorem expandingHighEnergy_free_inner (a b k L s : ℝ)
(ha : 0 < a) (hk : 8 < k) (hL : 1 ≤ L) (hs : 0 ≤ s) (f g : FourierL2) :
inner ℝ (expandingHighEnergy a k (expandingRadius L s)
(hL.trans (expandingRadius_ge L s hL hs)) (expandingFreeStep a b k L s ha hk hL hs f))
(expandingHighEnergy a k (expandingRadius L s)
(hL.trans (expandingRadius_ge L s hL hs)) (expandingFreeStep a b k L s ha hk hL hs g)) =
Real.exp ((6 - 2 * a - k) * s) *
inner ℝ (expandingHighEnergy a k L hL f) (expandingHighEnergy a k L hL g) := by
have h := expandingHighEnergy_free a b k L s ha hk hL hs (f + g)
simp only [map_add] at h
rw [norm_add_sq_real, norm_add_sq_real,
expandingHighEnergy_free a b k L s ha hk hL hs f,
expandingHighEnergy_free a b k L s ha hk hL hs g] at h
nlinarith [h]
end DefocusingNLS
end OAI
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