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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 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 | import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingResidualStability
namespace OAI
/-! # Exact trajectories in a neighborhood of an approximate mild solution
The contraction is restricted to a closed ball of paths, so only the actual
local Lipschitz estimate for the polynomial reaction is used.
-/
open Set Metric
namespace DefocusingNLS
attribute [local irreducible] expandingFreeStep
theorem expandingMildSolution_initial (a b k L T : ℝ)
(ha : 0 < a) (hk : 8 < k) (hL : 1 ≤ L) (hT : 0 ≤ T)
(F : C((Icc (0 : ℝ) T) × FourierL2, FourierL2)) (u₀ : FourierL2)
(u : C(Icc (0 : ℝ) T, FourierL2))
(hu : u = expandingPicard a b k L T ha hk hL hT F u₀ u) :
u ⟨0, le_rfl, hT⟩ = u₀ := by
have h := congrArg (fun v : C(Icc (0 : ℝ) T, FourierL2) => v ⟨0, le_rfl, hT⟩) hu
simpa [expandingPicard, expandingDuhamel] using h
theorem expandingMild_neighborhood_unique (a b k L T : ℝ)
(ha : 0 < a) (hk : 8 < k) (hL : 1 ≤ L) (hT : 0 ≤ T)
(F : C((Icc (0 : ℝ) T) × FourierL2, FourierL2)) (u₀ : FourierL2)
(u v : C(Icc (0 : ℝ) T, FourierL2)) (K : ℝ) (hK : 0 ≤ K) (hTK : T * K < 1)
(hF : ∀ t, ‖F (t, u t) - F (t, v t)‖ ≤ K * ‖u t - v t‖)
(hu : u = expandingPicard a b k L T ha hk hL hT F u₀ u)
(hv : v = expandingPicard a b k L T ha hk hL hT F u₀ v) : u = v := by
have h := expandingPicard_pair_dist_le a b k L T ha hk hL hT F u₀ K hK u v hF
rw [← hu, ← hv] at h
apply dist_eq_zero.mp
nlinarith [dist_nonneg (x := u) (y := v)]
theorem exists_expandingMild_near_approximation (a b k L T : ℝ)
(ha : 0 < a) (hk : 8 < k) (hL : 1 ≤ L) (hT : 0 ≤ T)
(F : C((Icc (0 : ℝ) T) × FourierL2, FourierL2)) (u₀ : FourierL2)
(v : C(Icc (0 : ℝ) T, FourierL2)) (ρ K ε : ℝ)
(hρ : 0 ≤ ρ) (hK : 0 ≤ K) (hTK : T * K < 1)
(hF : ∀ u ∈ closedBall v ρ, ∀ w ∈ closedBall v ρ, ∀ t,
‖F (t, u t) - F (t, w t)‖ ≤ K * ‖u t - w t‖)
(hdef : dist v (expandingPicard a b k L T ha hk hL hT F u₀ v) ≤ ε)
(hsmall : ε ≤ (1 - T * K) * ρ) :
∃ u : C(Icc (0 : ℝ) T, FourierL2),
u = expandingPicard a b k L T ha hk hL hT F u₀ u ∧
dist u v ≤ ρ ∧ dist u v ≤ ε / (1 - T * K) := by
let P := expandingPicard a b k L T ha hk hL hT F u₀
have hv : v ∈ closedBall v ρ := by simpa using hρ
have hmap : MapsTo P (closedBall v ρ) (closedBall v ρ) := by
intro u hu
have hp := expandingPicard_pair_dist_le a b k L T ha hk hL hT F u₀ K hK u v
(hF u hu v hv)
have hd : dist u v ≤ ρ := hu
change dist (P u) v ≤ ρ
calc
dist (P u) v ≤ dist (P u) (P v) + dist (P v) v := dist_triangle _ _ _
_ ≤ T * K * dist u v + ε := add_le_add hp (by simpa only [dist_comm] using hdef)
_ ≤ ρ := by nlinarith [mul_le_mul_of_nonneg_left hd (mul_nonneg hT hK)]
have hcontract : ContractingWith ⟨T * K, mul_nonneg hT hK⟩
(hmap.restrict P (closedBall v ρ) (closedBall v ρ)) := by
refine ⟨hTK, LipschitzWith.of_dist_le_mul ?_⟩
intro u w
exact expandingPicard_pair_dist_le a b k L T ha hk hL hT F u₀ K hK u.1 w.1
(hF u.1 u.2 w.1 w.2)
obtain ⟨u, hu, hfix, _⟩ := hcontract.exists_fixedPoint'
isClosed_closedBall.isComplete hmap hv (edist_ne_top _ _)
have heq : u = P u := hfix.symm
refine ⟨u, heq, hu, ?_⟩
have hb := expandingMild_residual_stability a b k L T ha hk hL hT F u₀ u₀
K hK hTK u v (hF u hu v hv) heq ε hdef
simpa only [dist_self, zero_add] using hb
/-- The local existence radius and Lipschitz constant are independent of the
starting torus scale. The reaction is the actual odd-power nonlinearity. -/
theorem exists_expandingNonlinear_near_approximation (a b k : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (m : ℕ)
(R ρ : ℝ) (hR : 0 ≤ R) (hρ : 0 ≤ ρ) :
∃ K : ℝ, 0 ≤ K ∧ ∀ (L T : ℝ) (hL : 1 ≤ L) (hT : 0 ≤ T), T * K < 1 →
∀ (v : C(Icc (0 : ℝ) T, FourierL2)), (∀ t, ‖v t‖ ≤ R) →
∀ (u₀ : FourierL2) (ε : ℝ),
dist v (expandingPicard a b k L T ha hk hL hT
(expandingNonlinearReaction a k L T ha ha1 hk hL m) u₀ v) ≤ ε →
ε ≤ (1 - T * K) * ρ →
∃ u : C(Icc (0 : ℝ) T, FourierL2),
u = expandingPicard a b k L T ha hk hL hT
(expandingNonlinearReaction a k L T ha ha1 hk hL m) u₀ u ∧
dist u v ≤ ρ ∧ dist u v ≤ ε / (1 - T * K) := by
obtain ⟨K, hK, hb⟩ := exists_expandingOddPower_lipschitz_on_ball a k ha ha1 hk m
(R + ρ) (add_nonneg hR hρ)
refine ⟨K, hK, ?_⟩
intro L T hL hT hTK v hv u₀ ε hdef hsmall
apply exists_expandingMild_near_approximation a b k L T ha hk hL hT _ u₀ v
ρ K ε hρ hK hTK _ hdef hsmall
intro u hu w hw t
have hnorm (z : C(Icc (0 : ℝ) T, FourierL2)) (hz : z ∈ closedBall v ρ) :
‖z t‖ ≤ R + ρ := by
have hd := ContinuousMap.dist_apply_le_dist (f := z) (g := v) t
have hz' : dist z v ≤ ρ := hz
calc
‖z t‖ ≤ ‖v t‖ + ‖z t - v t‖ := by
have he : v t + (z t - v t) = z t := by abel
simpa only [he] using norm_add_le (v t) (z t - v t)
_ ≤ R + ρ := add_le_add (hv t) (by simpa only [dist_eq_norm] using hd.trans hz')
change ‖(-Complex.I) • expandingOddPower a k (expandingRadiusCurve L T hL t).1
ha ha1 hk (expandingRadiusCurve L T hL t).2 m (u t) -
(-Complex.I) • expandingOddPower a k (expandingRadiusCurve L T hL t).1
ha ha1 hk (expandingRadiusCurve L T hL t).2 m (w t)‖ ≤ _
rw [← smul_sub, norm_smul]
simp only [norm_neg, Complex.norm_I, one_mul]
exact hb _ (expandingRadiusCurve L T hL t).2 _ _ (hnorm u hu) (hnorm w hw)
end DefocusingNLS
end OAI
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