| 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 |
|
|
