| import OAI.MathematicalPhysics.RapidForcing.CurlSupport |
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| namespace OAI |
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| open scoped BigOperators ENNReal Topology |
| open Set MeasureTheory |
| namespace RapidForcing |
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| 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 |
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| @[simp] lemma cross_zero_left (v : Space) : cross 0 v = 0 := by |
| simp [cross] |
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| @[simp] lemma curl_zero : curl (fun _ : Space => 0) = fun _ => 0 := by |
| funext x |
| simp [curl] |
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| 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] |
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| 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] |
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| 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] |
|
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| 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 |
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| 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 |
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| 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⟩ |
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| end RapidForcing |
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| end OAI |
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