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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 | import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.CompactRapidEmbedding
namespace OAI
/-! Periodization in a chart strictly separated from every integer face. The
fractional representative is only a definition: smoothness follows from local
translation or local zero, so no differentiability of the fractional part at
an integer is asserted. -/
noncomputable section
open Set Filter
open scoped Topology ContDiff
namespace RapidForcing.CompactEmbedding
def latticeVector (k : Fin 3 → ℤ) : Space := WithLp.toLp 2 (fun i => (k i : ℝ))
def representative (x : Space) : Space := WithLp.toLp 2 (fun i => Int.fract (x i))
@[simp] theorem representative_apply (x : Space) (i : Fin 3) :
representative x i = Int.fract (x i) := rfl
@[simp] theorem latticeVector_apply (k : Fin 3 → ℤ) (i : Fin 3) :
latticeVector k i = (k i : ℝ) := rfl
theorem representative_add_lattice (x : Space) (k : Fin 3 → ℤ) :
representative (x + latticeVector k) = representative x := by
ext i
exact Int.fract_add_intCast (x i) (k i)
theorem representative_eq {x : Space} (hx : ∀ i, x i ∈ Ico (0 : ℝ) 1) :
representative x = x := by
ext i
exact Int.fract_eq_self.mpr (hx i)
def periodic {E : Type} (f : Field E) : Field E := fun t x => f t (representative x)
theorem periodic_add_lattice {E : Type} (f : Field E) (t : ℝ)
(x : Space) (k : Fin 3 → ℤ) :
periodic f t (x + latticeVector k) = periodic f t x := by
simp only [periodic, representative_add_lattice]
theorem periodic_eq_on_chart {E : Type} (f : Field E) (t : ℝ) {x : Space}
(hx : x ∈ chart) : periodic f t x = f t x := by
rw [periodic, representative_eq]
intro i
exact ⟨(chart_inside_unit hx i).1.le, (chart_inside_unit hx i).2⟩
theorem representative_local_translation {x : Space}
(hx : ∀ i, x i ≠ (⌊x i⌋ : ℤ)) :
representative =ᶠ[𝓝 x] (fun y => y - latticeVector (fun i => ⌊x i⌋)) := by
have hlocal : ∀ i : Fin 3, ∀ᶠ y in 𝓝 x,
(⌊x i⌋ : ℝ) < y i ∧ y i < (⌊x i⌋ : ℝ) + 1 := by
intro i
exact ((EuclideanSpace.proj i).continuous.continuousAt).eventually
(Ioo_mem_nhds ((Int.floor_le (x i)).lt_of_ne (Ne.symm (hx i)))
(Int.lt_floor_add_one (x i)))
filter_upwards [Filter.eventually_all.mpr hlocal] with y hy
ext i
have hk : ⌊y i⌋ = ⌊x i⌋ := Int.floor_eq_iff.mpr ⟨(hy i).1.le, (hy i).2⟩
simp only [representative_apply, PiLp.sub_apply, latticeVector_apply, Int.fract, hk]
theorem fract_near_integer {x : ℝ} {k : ℤ}
(hx : x ∈ Ioo ((k : ℝ) - 1 / 8) ((k : ℝ) + 1 / 8)) :
Int.fract x < 1 / 4 ∨ 3 / 4 < Int.fract x := by
have hfract : Int.fract (x - (k : ℝ)) = Int.fract x := Int.fract_sub_intCast x k
rw [← hfract]
by_cases hp : 0 ≤ x - (k : ℝ)
· left
rw [Int.fract_eq_self.mpr ⟨hp, by linarith [hx.2]⟩]
linarith [hx.2]
· right
have hk : ⌊x - (k : ℝ)⌋ = (-1 : ℤ) :=
Int.floor_eq_iff.mpr ⟨by norm_num; linarith [hx.1], by norm_num; linarith⟩
rw [Int.fract, hk]
norm_num
linarith [hx.1]
theorem representative_local_outside {x : Space} {i : Fin 3}
(hx : x i = (⌊x i⌋ : ℤ)) : ∀ᶠ y in 𝓝 x, representative y ∉ chart := by
have hl : (⌊x i⌋ : ℝ) - 1 / 8 < x i := by linarith [hx]
have hr : x i < (⌊x i⌋ : ℝ) + 1 / 8 := by linarith [hx]
filter_upwards [((EuclideanSpace.proj i).continuous.continuousAt).eventually
(Ioo_mem_nhds hl hr)] with y hy
intro hc
rcases fract_near_integer hy with h | h
· exact (not_lt_of_ge (hc.1 i)) h
· exact (not_lt_of_ge (hc.2 i)) h
theorem periodic_local {E : Type} [Zero E] (f : Field E)
(hf : ∀ t x, x ∉ chart → f t x = 0) (x : Space) :
(∃ k : Fin 3 → ℤ, ∀ᶠ y in 𝓝 x, ∀ t, periodic f t y = f t (y - latticeVector k)) ∨
(∀ᶠ y in 𝓝 x, ∀ t, periodic f t y = 0) := by
by_cases hx : ∀ i, x i ≠ (⌊x i⌋ : ℤ)
· left
refine ⟨fun i => ⌊x i⌋, ?_⟩
filter_upwards [representative_local_translation hx] with y hy t
simp only [periodic, hy]
· right
push Not at hx
obtain ⟨i, hi⟩ := hx
filter_upwards [representative_local_outside hi] with y hy t
exact hf t (representative y) hy
theorem periodic_smooth {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E]
{f : Field E} (hs : ContDiff ℝ ∞ (Function.uncurry f))
(hf : ∀ t x, x ∉ chart → f t x = 0) :
ContDiff ℝ ∞ (Function.uncurry (periodic f)) := by
apply contDiff_iff_contDiffAt.mpr
rintro ⟨t, x⟩
rcases periodic_local f hf x with ⟨k, hk⟩ | hz
· have he : Function.uncurry (periodic f) =ᶠ[𝓝 (t, x)]
(fun p : ℝ × Space => f p.1 (p.2 - latticeVector k)) := by
filter_upwards [continuous_snd.continuousAt.eventually hk] with p hp
exact hp p.1
exact (hs.comp (contDiff_fst.prodMk (contDiff_snd.sub contDiff_const))).contDiffAt.congr_of_eventuallyEq he
· have he : Function.uncurry (periodic f) =ᶠ[𝓝 (t, x)] (fun _ => (0 : E)) := by
filter_upwards [continuous_snd.continuousAt.eventually hz] with p hp
exact hp p.1
exact contDiffAt_const.congr_of_eventuallyEq he
end RapidForcing.CompactEmbedding
end
end OAI
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