File size: 3,137 Bytes
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 | import OAI.MathematicalPhysics.RapidForcing.CurlSupport
namespace OAI
open scoped BigOperators ENNReal Topology
open Set MeasureTheory
namespace RapidForcing
lemma addressCurl_tsupport_disjoint (M : Machine) (w : M.Input)
(n m m' : ℕ) (hmm' : m ≠ m') (σ : ℝ) :
Disjoint (tsupport (movingCurl (addressPath M (M.scaleData w) n m)
((M.scaleData w).δ n) σ))
(tsupport (movingCurl (addressPath M (M.scaleData w) n m')
((M.scaleData w).δ n) σ)) := by
apply Set.disjoint_left.mpr
intro x hx hx'
have h := abs_le.mp (addressCurl_tsupport_subset_box M w n m σ hx 1)
have h' := abs_le.mp (addressCurl_tsupport_subset_box M w n m' σ hx' 1)
simp only [addressCenter, vec_one_coord] at h h'
rcases lt_or_gt_of_ne hmm' with hm | hm
· have hg : (m : ℝ) + 1 ≤ (m' : ℝ) := by exact_mod_cast hm
have hδ := (M.scaleData w).delta_pos n
nlinarith
· have hg : (m' : ℝ) + 1 ≤ (m : ℝ) := by exact_mod_cast hm
have hδ := (M.scaleData w).delta_pos n
nlinarith
@[simp] lemma cross_zero_left (v : Space) : cross 0 v = 0 := by
simp [cross]
@[simp] lemma curl_zero : curl (fun _ : Space => 0) = fun _ => 0 := by
funext x
simp [curl]
lemma movingCurl_eq_zero_of_deriv_eq_zero (c : ℝ → Space) (δ σ : ℝ)
(h : deriv c σ = 0) : movingCurl c δ σ = fun _ => 0 := by
funext x
simp [movingCurl, h]
lemma movingCurl_left_collar (a v : Space) (δ : ℝ) {σ : ℝ} (hσ : σ < 1 / 4) :
movingCurl (fun s => a + θ s • v) δ σ = fun _ => 0 := by
apply movingCurl_eq_zero_of_deriv_eq_zero
have he : (fun s => a + θ s • v) =ᶠ[𝓝 σ] (fun _ => a) := by
filter_upwards [Iio_mem_nhds hσ] with s hs
rw [theta_zero (le_of_lt hs), zero_smul, add_zero]
rw [he.deriv_eq, deriv_const]
lemma movingCurl_right_collar (a v : Space) (δ : ℝ) {σ : ℝ} (hσ : 3 / 4 < σ) :
movingCurl (fun s => a + θ s • v) δ σ = fun _ => 0 := by
apply movingCurl_eq_zero_of_deriv_eq_zero
have he : (fun s => a + θ s • v) =ᶠ[𝓝 σ] (fun _ => a + v) := by
filter_upwards [Ioi_mem_nhds hσ] with s hs
rw [theta_one (le_of_lt hs), one_smul]
rw [he.deriv_eq, deriv_const]
theorem addressedVelocity_initial_collar (M : Machine) (w : M.Input)
{t : ℝ} (ht : t < 1 / 4) : addressedVelocity M w t = fun _ => 0 := by
funext x
unfold addressedVelocity
rw [ite_eq_left (by linarith : t ≤ 1)]
have h := movingCurl_left_collar 0
(vec (-1) ((M.scaleData w).δ 0 * (M.initialDigit w : ℝ)) 0) 1 ht
simp only [zero_add] at h
exact congrFun h x
theorem addressedVelocity_initial (M : Machine) (w : M.Input) (x : Space) :
addressedVelocity M w 0 x = 0 := by
exact congrFun (addressedVelocity_initial_collar M w (by norm_num)) x
theorem addressed_fields_mixed_support (ν : ℝ) (M : Machine) (w : M.Input)
(l : ℕ) (α : MultiIndex) :
IsCompact K ∧ Supported (mixedD l α (addressedVelocity M w)) ∧
Supported (mixedD l α (addressedForce ν M w)) :=
⟨K_compact, (addressedVelocity_supported M w).mixedD l α,
(addressedForce_supported ν M w).mixedD l α⟩
end RapidForcing
end OAI
|