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37bcf52 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 | import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.LogarithmicSpatialForce
import OAI.MathematicalPhysics.NavierStokes.ShearFlows.ForceEvaluation
namespace OAI
/-! The logarithmic estimate applies to an initialized field as soon as all
its original derivatives are bounded. Periodicity in time is unnecessary. -/
noncomputable section
open scoped ContDiff BigOperators
open ShearFlows
namespace ForcedComputation
theorem boundedMixed_derivative {V : Velocity} (hb : BoundedMixedDerivatives V)
(β : List (Fin 4)) : BoundedMixedDerivatives (mixedDerivative V β) := by
intro α
simpa only [mixedDerivative_append] using hb (α ++ β)
theorem boundedMixed_sub {V W : Velocity} (hV : ContDiff ℝ ∞ V)
(hW : ContDiff ℝ ∞ W) (hVB : BoundedMixedDerivatives V)
(hWB : BoundedMixedDerivatives W) : BoundedMixedDerivatives (V - W) := by
intro α
obtain ⟨A, hA, hbA⟩ := hVB α
obtain ⟨B, hB, hbB⟩ := hWB α
refine ⟨A + B, add_nonneg hA hB, fun y => ?_⟩
rw [mixedDerivative_sub hV hW]
exact (norm_sub_le _ _).trans (add_le_add (hbA y) (hbB y))
theorem boundedMixed_sum {ι : Type*} [Fintype ι] (V : ι → Velocity)
(hs : ∀ i, ContDiff ℝ ∞ (V i)) (hb : ∀ i, BoundedMixedDerivatives (V i)) :
BoundedMixedDerivatives (∑ i, V i) := by
intro α
choose C hC hbC using fun i => hb i α
refine ⟨∑ i, C i, Finset.sum_nonneg (fun i _ => hC i), fun y => ?_⟩
rw [mixedDerivative_sum V hs]
simpa only [Finset.sum_apply] using
(norm_sum_le (s := Finset.univ) (f := fun i => mixedDerivative (V i) α y)).trans
(Finset.sum_le_sum fun i _ => hbC i y)
theorem bounded_uniformTimeBounds {V : Velocity} (hV : ContDiff ℝ ∞ V)
(hb : BoundedMixedDerivatives V) : UniformTimeBounds (fun x s => V (s, x)) := by
intro n
obtain ⟨C, hC, hbound⟩ := hb (List.replicate n 0)
exact ⟨C, hC, fun x t => by rw [iteratedDeriv_eq_mixedTime hV]; exact hbound (t, x)⟩
theorem bounded_spatialPhase {V : Velocity} (hV : ContDiff ℝ ∞ V)
(hb : BoundedMixedDerivatives V) (α : List (Fin 3)) :
BoundedMixedDerivatives (spatialPhase V α) := by
rw [spatialPhase_eq_mixed hV]
exact boundedMixed_derivative hb _
theorem bounded_quadraticPhase {V : Velocity} (hV : ContDiff ℝ ∞ V)
(hb : BoundedMixedDerivatives V) (ha : ZeroAdvection V) :
BoundedMixedDerivatives (fun y : SpaceTime => quadraticPhase V y.2 y.1) := by
have he : (fun y : SpaceTime => quadraticPhase V y.2 y.1) = mixedDerivative V [0] - V := by
funext y
simp only [quadraticPhase, ha y.1 y.2, add_zero, timeDerivative_eq hV]
rfl
rw [he]
exact boundedMixed_sub (mixedDerivative_smooth hV _) hV (boundedMixed_derivative hb _) hb
theorem bounded_viscousPhase {V : Velocity} (hV : ContDiff ℝ ∞ V)
(hb : BoundedMixedDerivatives V) :
BoundedMixedDerivatives (fun y : SpaceTime => viscousPhase V y.2 y.1) := by
have he : (fun y : SpaceTime => viscousPhase V y.2 y.1) =
∑ j : Fin 3, mixedDerivative V [j.succ,j.succ] := by
funext y
simpa only [viscousPhase, Finset.sum_apply] using laplacian_eq_mixed hV y.1 y.2
rw [he]
exact boundedMixed_sum _ (fun _ => mixedDerivative_smooth hV _)
(fun _ => boundedMixed_derivative hb _)
theorem bounded_profile_mixed_decay {V : Velocity} (hV : ContDiff ℝ ∞ V)
(hb : BoundedMixedDerivatives V) (r n : ℕ) (α : List (Fin 3)) :
∃ C : ℝ, 0 ≤ C ∧ ∀ x t, 0 ≤ t →
‖iteratedDeriv n
(fun s => spatialWord α (fun y => logarithmicProfile r (fun q => V (q, y)) s) x) t‖ ≤
C * (1 + t)⁻¹ ^ (r + n) := by
have hphase (x : Space) : ContDiff ℝ ∞ (fun s => spatialPhase V α (s, x)) :=
(spatialPhase_smooth hV α).comp (contDiff_id.prodMk contDiff_const)
obtain ⟨C, hC, hbound⟩ := logarithmicProfile_uniform_decay r n hphase
(bounded_uniformTimeBounds (spatialPhase_smooth hV α) (bounded_spatialPhase hV hb α))
refine ⟨C, hC, fun x t ht => ?_⟩
have he : (fun s => spatialWord α (fun y => logarithmicProfile r (fun q => V (q, y)) s) x) =
logarithmicProfile r (fun s => spatialPhase V α (s, x)) :=
funext (fun s => spatialWord_logarithmicProfile r V α s x)
rw [he]
exact hbound x t ht
theorem bounded_slowForce_spatial_time_decay (ν : ℝ) {V : Velocity}
(hV : ContDiff ℝ ∞ V) (hb : BoundedMixedDerivatives V) (ha : ZeroAdvection V)
(α : List (Fin 3)) (n : ℕ) :
∃ C : ℝ, 0 ≤ C ∧ ∀ x t, 0 ≤ t →
‖iteratedDeriv n (fun s => spatialWord α (fun y => slowForce ν V (s, y)) x) t‖ ≤
C * (1 + t)⁻¹ ^ (1 + n) := by
let Q : Velocity := fun y => quadraticPhase V y.2 y.1
let D : Velocity := fun y => viscousPhase V y.2 y.1
have hQ : ContDiff ℝ ∞ Q := quadraticPhase_smooth hV
have hD : ContDiff ℝ ∞ D := viscousPhase_smooth hV
obtain ⟨A, hA, hQA⟩ := bounded_profile_mixed_decay hQ (bounded_quadraticPhase hV hb ha) 2 n α
obtain ⟨B, hB, hDB⟩ := bounded_profile_mixed_decay hD (bounded_viscousPhase hV hb) 1 n α
refine ⟨A + ‖ν‖ * B, by positivity, fun x t ht0 => ?_⟩
have hQx : ContDiff ℝ ∞ (fun s => spatialPhase Q α (s, x)) :=
(spatialPhase_smooth hQ α).comp (contDiff_id.prodMk contDiff_const)
have hDx : ContDiff ℝ ∞ (fun s => spatialPhase D α (s, x)) :=
(spatialPhase_smooth hD α).comp (contDiff_id.prodMk contDiff_const)
have hqt := logarithmicProfile_contDiffAt 2 hQx (show -1 < t by linarith)
have hdt := logarithmicProfile_contDiffAt 1 hDx (show -1 < t by linarith)
have he : (fun s => spatialWord α (fun y => slowForce ν V (s, y)) x) =
fun s => logarithmicProfile 2 (fun q => spatialPhase Q α (q, x)) s -
ν • logarithmicProfile 1 (fun q => spatialPhase D α (q, x)) s :=
funext (fun s => spatialWord_slowForce hV ν α s x)
rw [he, iteratedDeriv_fun_sub (hqt.of_le (by simp))
((hdt.const_smul ν).of_le (by simp)), iteratedDeriv_fun_const_smul_field]
have hQA' := hQA x t ht0
have hDB' := hDB x t ht0
simp_rw [spatialWord_logarithmicProfile] at hQA' hDB'
have hinv : 0 ≤ (1 + t)⁻¹ := by positivity
have hinvle : (1 + t)⁻¹ ≤ 1 :=
(inv_le_one₀ (by linarith : 0 < 1 + t)).mpr (by linarith)
have hpow : (1 + t)⁻¹ ^ (2 + n) ≤ (1 + t)⁻¹ ^ (1 + n) := by
calc
_ = (1 + t)⁻¹ ^ (1 + n) * (1 + t)⁻¹ := by rw [← pow_succ]; congr 1; omega
_ ≤ (1 + t)⁻¹ ^ (1 + n) * 1 :=
mul_le_mul_of_nonneg_left hinvle (pow_nonneg hinv _)
_ = _ := mul_one _
calc
_ ≤ ‖iteratedDeriv n (logarithmicProfile 2 (fun q => spatialPhase Q α (q, x))) t‖ +
‖ν • iteratedDeriv n (logarithmicProfile 1 (fun q => spatialPhase D α (q, x))) t‖ :=
norm_sub_le _ _
_ ≤ A * (1 + t)⁻¹ ^ (2 + n) + ‖ν‖ * (B * (1 + t)⁻¹ ^ (1 + n)) := by
rw [norm_smul]
exact add_le_add hQA' (mul_le_mul_of_nonneg_left hDB' (norm_nonneg ν))
_ ≤ A * (1 + t)⁻¹ ^ (1 + n) + ‖ν‖ * (B * (1 + t)⁻¹ ^ (1 + n)) :=
add_le_add (mul_le_mul_of_nonneg_left hpow hA) le_rfl
_ = _ := by ring
end ForcedComputation
end
end OAI
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