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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 | import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Flow.CompactPeriodization
namespace OAI
/-! On the support chart, differentiation commutes exactly with periodization.
Consequently every mixed rapid estimate transfers with the same constant. -/
noncomputable section
open Set Filter
open scoped Topology
namespace RapidForcing.CompactEmbedding
theorem chart_closed : IsClosed chart := by
have he : chart = (⋂ i : Fin 3, {x : Space | 1 / 4 ≤ x i}) ∩
(⋂ i : Fin 3, {x : Space | x i ≤ 3 / 4}) := by
ext x
simp [chart]
rw [he]
exact (isClosed_iInter fun i => isClosed_le continuous_const (EuclideanSpace.proj i).continuous).inter
(isClosed_iInter fun i => isClosed_le (EuclideanSpace.proj i).continuous continuous_const)
def ChartSupported {E : Type} [Zero E] (f : Field E) : Prop :=
∀ t x, x ∉ chart → f t x = 0
namespace ChartSupported
variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : Field E}
omit [NormedSpace ℝ E] in
theorem tsupport (hf : ChartSupported f) (t : ℝ) : tsupport (f t) ⊆ chart := by
apply closure_minimal _ chart_closed
intro x hx
by_contra hn
exact hx (hf t x hn)
theorem spatialD (hf : ChartSupported f) (i : Fin 3) :
ChartSupported (RapidForcing.spatialD i f) := by
intro t x hx
apply image_eq_zero_of_notMem_tsupport
intro hm
exact hx (hf.tsupport t (tsupport_fderiv_apply_subset ℝ (basis i) hm))
theorem timeD (hf : ChartSupported f) : ChartSupported (RapidForcing.timeD f) := by
intro t x hx
change derivWithin (fun s => f s x) (Ici 0) t = 0
rw [show (fun s => f s x) = (fun _ => (0 : E)) from funext (fun s => hf s x hx)]
simp
theorem spatialIter (hf : ChartSupported f) (i : Fin 3) (n : ℕ) :
ChartSupported ((RapidForcing.spatialD i)^[n] f) := by
induction n with
| zero => exact hf
| succ n ih => simpa only [Function.iterate_succ_apply'] using ih.spatialD i
theorem timeIter (hf : ChartSupported f) (n : ℕ) :
ChartSupported (RapidForcing.timeD^[n] f) := by
induction n with
| zero => exact hf
| succ n ih => simpa only [Function.iterate_succ_apply'] using ih.timeD
theorem mixedD (hf : ChartSupported f) (l : ℕ) (α : MultiIndex) :
ChartSupported (RapidForcing.mixedD l α f) :=
(((hf.spatialIter 2 (α 2)).spatialIter 1 (α 1)).spatialIter 0 (α 0)).timeIter l
end ChartSupported
section Derivatives
variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : Field E}
theorem periodic_spatialD (hf : ChartSupported f) (i : Fin 3) :
spatialD i (periodic f) = periodic (spatialD i f) := by
funext t x
by_cases hx : ∀ j, x j ≠ (⌊x j⌋ : ℤ)
· have hr := representative_local_translation hx
have he : periodic f t =ᶠ[𝓝 x]
(fun y => f t (y - latticeVector (fun j => ⌊x j⌋))) :=
hr.mono (fun y hy => congrArg (f t) hy)
change fderiv ℝ (periodic f t) x (basis i) =
fderiv ℝ (f t) (representative x) (basis i)
rw [he.fderiv_eq, fderiv_comp_sub, hr.eq_of_nhds]
· push Not at hx
obtain ⟨j, hj⟩ := hx
have hr := representative_local_outside hj
have he : periodic f t =ᶠ[𝓝 x] (fun _ => (0 : E)) :=
hr.mono (fun y hy => hf t (representative y) hy)
have hd := (hf.spatialD i) t (representative x) hr.self_of_nhds
change fderiv ℝ (periodic f t) x (basis i) = spatialD i f t (representative x)
rw [he.fderiv_eq, hd]
simp
theorem periodic_timeD : timeD (periodic f) = periodic (timeD f) := rfl
theorem periodic_spatialIter (hf : ChartSupported f) (i : Fin 3) (n : ℕ) :
(spatialD i)^[n] (periodic f) = periodic ((spatialD i)^[n] f) := by
induction n with
| zero => rfl
| succ n ih =>
rw [Function.iterate_succ_apply', ih,
periodic_spatialD (hf.spatialIter i n), Function.iterate_succ_apply']
theorem periodic_timeIter (n : ℕ) : timeD^[n] (periodic f) = periodic (timeD^[n] f) := by
induction n with
| zero => rfl
| succ n ih =>
rw [Function.iterate_succ_apply', ih, periodic_timeD, Function.iterate_succ_apply']
theorem periodic_mixedD (hf : ChartSupported f) (l : ℕ) (α : MultiIndex) :
mixedD l α (periodic f) = periodic (mixedD l α f) := by
unfold mixedD spatialMulti
rw [periodic_spatialIter hf,
periodic_spatialIter (hf.spatialIter 2 (α 2)),
periodic_spatialIter ((hf.spatialIter 2 (α 2)).spatialIter 1 (α 1)), periodic_timeIter]
theorem periodic_mixed_bound (hf : ChartSupported f) {J l : ℕ} {α : MultiIndex} {C : ℝ}
(hb : ∀ t, 0 ≤ t → ∀ x, (1 + t) ^ J * ‖mixedD l α f t x‖ ≤ C) :
∀ t, 0 ≤ t → ∀ x, (1 + t) ^ J * ‖mixedD l α (periodic f) t x‖ ≤ C := by
intro t ht x
rw [periodic_mixedD hf]
exact hb t ht (representative x)
theorem periodic_rapid (hf : ChartSupported f) (hr : Rapid f) : Rapid (periodic f) := by
intro J l α
obtain ⟨C, hC, hb⟩ := hr J l α
exact ⟨C, hC, periodic_mixed_bound hf hb⟩
end Derivatives
end RapidForcing.CompactEmbedding
end
end OAI
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