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caec230 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 | import OAI.MathematicalPhysics.RapidForcing.Support
import OAI.MathematicalPhysics.RapidForcing.Profiles
namespace OAI
open scoped BigOperators ENNReal Topology
open Set MeasureTheory
namespace RapidForcing
lemma supportBox_subset_of_center {c c' : Space} {r R : ℝ}
(h : ∀ i : Fin 3, |c' i - c i| + r ≤ R) :
supportBox c' r ⊆ supportBox c R := by
intro x hx i
calc
|x i - c i| = |(x i - c' i) + (c' i - c i)| := by congr 1; ring
_ ≤ |x i - c' i| + |c' i - c i| := abs_add_le _ _
_ ≤ R := by have := hx i; have := h i; linarith
lemma supportBox_subset_K {c : Space} {r : ℝ} (hr : r ≤ 1)
(hx : -1 ≤ c 0 ∧ c 0 ≤ 0) (hy : 0 ≤ c 1 ∧ c 1 ≤ 1) (hz : c 2 = 0) :
supportBox c r ⊆ K := by
intro x hxbox
have h0 := abs_le.mp (hxbox 0)
have h1 := abs_le.mp (hxbox 1)
have h2 := abs_le.mp (hxbox 2)
change -2 ≤ x 0 ∧ x 0 ≤ 1 ∧ -1 ≤ x 1 ∧ x 1 ≤ 2 ∧
-1 ≤ x 2 ∧ x 2 ≤ 1
constructor
· linarith [hx.1]
constructor
· linarith [hx.2]
constructor
· linarith [hy.1]
constructor
· linarith [hy.2]
constructor <;> linarith
lemma grid_coordinate_mem {d : ScaleData} {n m : ℕ} (hm : m ≤ d.D n) :
0 ≤ (m : ℝ) * d.δ n ∧ (m : ℝ) * d.δ n ≤ 1 := by
refine ⟨mul_nonneg (Nat.cast_nonneg _) (d.delta_nonneg _), ?_⟩
rw [d.delta_eq_inv_D, ← div_eq_mul_inv]
exact (div_le_one (Nat.cast_pos.mpr (d.D_pos _))).mpr (Nat.cast_le.mpr hm)
lemma addressDisplacement_bound (M : Machine) (w : M.Input) (n m : ℕ) (i : Fin 3) :
|addressDisplacement M (M.scaleData w) n m i| ≤ (M.scaleData w).b n := by
let d := M.scaleData w
have hb := (d.b_pos n).le
have hδ := d.delta_nonneg (n + 1)
have hd : (M.terminalDigit d (m % d.capacity n) : ℝ) ≤ 1 :=
Nat.cast_le_one.mpr (M.terminalDigit_le_one _ _)
have hd0 : (0 : ℝ) ≤ (M.terminalDigit d (m % d.capacity n) : ℝ) := Nat.cast_nonneg _
have hn : (M.nextDigit d n (m % d.capacity n) : ℝ) ≤ (d.capacity (n + 1) : ℝ) :=
Nat.cast_le.mpr (M.nextDigit_lt w (Nat.mod_lt _ (d.capacity_pos n))).le
fin_cases i
· change |-d.b n * (M.terminalDigit d (m % d.capacity n) : ℝ)| ≤ d.b n
rw [abs_of_nonpos (mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr hb) hd0)]
nlinarith
· change |d.δ (n + 1) * (M.nextDigit d n (m % d.capacity n) : ℝ)| ≤ d.b n
rw [abs_of_nonneg (mul_nonneg hδ (Nat.cast_nonneg _))]
exact mul_le_mul_of_nonneg_left hn hδ
· change |(0 : ℝ)| ≤ d.b n
simpa using hb
lemma addressPath_bound (M : Machine) (w : M.Input) (n m : ℕ) (σ : ℝ) (i : Fin 3) :
|addressPath M (M.scaleData w) n m σ i - addressCenter (M.scaleData w) n m i| ≤
(M.scaleData w).b n := by
have h := addressDisplacement_bound M w n m i
change |(addressCenter (M.scaleData w) n m i +
θ σ * addressDisplacement M (M.scaleData w) n m i) -
addressCenter (M.scaleData w) n m i| ≤ _
rw [add_sub_cancel_left, abs_mul, abs_of_nonneg (theta_nonneg σ)]
exact (mul_le_mul_of_nonneg_left h (theta_nonneg σ)).trans
(mul_le_of_le_one_left (M.scaleData w |>.b_pos n).le (theta_le_one σ))
lemma addressCurl_tsupport_subset_box (M : Machine) (w : M.Input)
(n m : ℕ) (σ : ℝ) :
tsupport (movingCurl (addressPath M (M.scaleData w) n m)
((M.scaleData w).δ n) σ) ⊆
supportBox (addressCenter (M.scaleData w) n m) ((M.scaleData w).δ n / 4) := by
apply (movingCurl_tsupport_subset _ ((M.scaleData w).delta_pos n) _).trans
apply supportBox_subset_of_center
intro i
have := addressPath_bound M w n m σ i
have := (M.scaleData w).b_lt_delta_div_eight n
linarith
lemma addressCurl_tsupport_subset_K (M : Machine) (w : M.Input)
(n m : ℕ) (hm : m ≤ (M.scaleData w).D n) (σ : ℝ) :
tsupport (movingCurl (addressPath M (M.scaleData w) n m)
((M.scaleData w).δ n) σ) ⊆ K := by
apply (addressCurl_tsupport_subset_box M w n m σ).trans
apply supportBox_subset_K
· have := (M.scaleData w).delta_le_one n; linarith
· exact ⟨le_refl _, by norm_num [addressCenter]⟩
· exact grid_coordinate_mem hm
· rfl
lemma initial_coordinate_mem (M : Machine) (w : M.Input) :
0 ≤ (M.scaleData w).δ 0 * (M.initialDigit w : ℝ) ∧
(M.scaleData w).δ 0 * (M.initialDigit w : ℝ) < 1 := by
refine ⟨mul_nonneg ((M.scaleData w).delta_nonneg _) (Nat.cast_nonneg _), ?_⟩
calc
(M.scaleData w).δ 0 * (M.initialDigit w : ℝ) <
(M.scaleData w).δ 0 * ((M.scaleData w).capacity 0 : ℝ) :=
mul_lt_mul_of_pos_left (Nat.cast_lt.mpr (M.initialDigit_lt w))
((M.scaleData w).delta_pos _)
_ ≤ (1 / 2 : ℝ) ^ (15 * (0 + 1)) := (M.scaleData w).delta_capacity_le 0
_ < 1 := by norm_num
lemma loadingCurl_tsupport_subset_K (M : Machine) (w : M.Input) (t : ℝ) :
tsupport (movingCurl (fun s => θ s •
vec (-1) ((M.scaleData w).δ 0 * (M.initialDigit w : ℝ)) 0) 1 t) ⊆ K := by
apply (movingCurl_tsupport_subset _ (by norm_num : (0 : ℝ) < 1) _).trans
apply supportBox_subset_K
· norm_num
· change -1 ≤ θ t * (-1) ∧ θ t * (-1) ≤ 0
constructor <;> linarith [theta_nonneg t, theta_le_one t]
· change 0 ≤ θ t * ((M.scaleData w).δ 0 * (M.initialDigit w : ℝ)) ∧
θ t * ((M.scaleData w).δ 0 * (M.initialDigit w : ℝ)) ≤ 1
obtain ⟨hy, hy'⟩ := initial_coordinate_mem M w
exact ⟨mul_nonneg (theta_nonneg t) hy,
(mul_le_of_le_one_left hy (theta_le_one t)).trans hy'.le⟩
· simp
theorem addressedVelocity_supported (M : Machine) (w : M.Input) :
Supported (addressedVelocity M w) := by
apply supported_iff_zero_off.mpr
intro t _ x hx
unfold addressedVelocity
split_ifs with ht
· apply image_eq_zero_of_notMem_tsupport
intro h
exact hx (loadingCurl_tsupport_subset_K M w t h)
· apply Finset.sum_eq_zero
intro m hm
apply image_eq_zero_of_notMem_tsupport
intro h
apply hx
exact addressCurl_tsupport_subset_K M w _ m
(Nat.le_of_lt_succ (Finset.mem_range.mp hm)) _ h
lemma Supported.residual {u : Field Space} (hu : Supported u) (ν : ℝ) :
Supported (residual ν u) := by
apply supported_iff_zero_off.mpr
intro t ht x hx
have htD := hu.timeD.zero_off ht hx
have hL : laplace u t x = 0 := by
apply Finset.sum_eq_zero
intro i _
exact ((hu.spatialD i).spatialD i).zero_off ht hx
have hA : advection u t x = 0 := by
apply Finset.sum_eq_zero
intro i _
rw [(hu.spatialD i).zero_off ht hx, smul_zero]
change RapidForcing.timeD u t x + advection u t x - ν • laplace u t x = 0
rw [htD, hA, hL]
simp
theorem addressedForce_supported (ν : ℝ) (M : Machine) (w : M.Input) :
Supported (addressedForce ν M w) :=
(addressedVelocity_supported M w).residual ν
end RapidForcing
end OAI
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