| import Mathlib |
| import OAI.MathematicalPhysics.SheetFlows.ScheduledForce |
|
|
| namespace OAI |
|
|
| /-! SheetFlows classical regularity. -/ |
|
|
| noncomputable section |
| open Set MeasureTheory |
| open scoped BigOperators |
|
|
| open Set MeasureTheory Filter |
| open scoped BigOperators Topology |
| namespace Solenoidal |
|
|
| structure TorusC1 {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (g : Space → E) : Prop where |
| differentiable : Differentiable ℝ g |
| continuous_partial : ∀ j, Continuous (fun x => fderiv ℝ g x (basis j)) |
| periodic : ∀ x k, g (x + deck k) = g x |
|
|
| theorem TorusC1.of_smooth {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {g : Space → E} (hg : ContDiff ℝ (⊤ : ℕ∞) g) |
| (hp : ∀ x k, g (x + deck k) = g x) : TorusC1 g := |
| ⟨hg.differentiable (by simp), fun j => (directional_contDiff hg (basis j)).continuous, hp⟩ |
|
|
| theorem TorusC1.const {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (c : E) : TorusC1 (fun _ : Space => c) := by |
| refine ⟨differentiable_const c, ?_, fun _ _ => rfl⟩ |
| intro j |
| simpa using (continuous_const : Continuous (fun _ : Space => (0 : E))) |
|
|
| theorem TorusC1.add {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {g h : Space → E} (hg : TorusC1 g) (hh : TorusC1 h) : TorusC1 (g + h) := by |
| refine ⟨hg.differentiable.add hh.differentiable, ?_, ?_⟩ |
| · intro j |
| have he (x : Space) : fderiv ℝ (g + h) x (basis j) = |
| fderiv ℝ g x (basis j) + fderiv ℝ h x (basis j) := by |
| rw [fderiv_add (hg.differentiable x) (hh.differentiable x)] |
| rfl |
| simpa only [he] using (hg.continuous_partial j).fun_add (hh.continuous_partial j) |
| · intro x k |
| simp only [Pi.add_apply, hg.periodic, hh.periodic] |
|
|
| theorem TorusC1.sub {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {g h : Space → E} (hg : TorusC1 g) (hh : TorusC1 h) : TorusC1 (g - h) := by |
| refine ⟨hg.differentiable.sub hh.differentiable, ?_, ?_⟩ |
| · intro j |
| have he (x : Space) : fderiv ℝ (g - h) x (basis j) = |
| fderiv ℝ g x (basis j) - fderiv ℝ h x (basis j) := by |
| rw [fderiv_sub (hg.differentiable x) (hh.differentiable x)] |
| rfl |
| simpa only [he] using (hg.continuous_partial j).fun_sub (hh.continuous_partial j) |
| · intro x k |
| simp only [Pi.sub_apply, hg.periodic, hh.periodic] |
|
|
| theorem scalarPartial_mul {g h : Space → ℝ} |
| (hg : Differentiable ℝ g) (hh : Differentiable ℝ h) (x : Space) (j : Fin 3) : |
| fderiv ℝ (g * h) x (basis j) = |
| g x * fderiv ℝ h x (basis j) + h x * fderiv ℝ g x (basis j) := by |
| rw [fderiv_mul (hg x) (hh x)] |
| rfl |
|
|
| theorem TorusC1.mul {g h : Space → ℝ} (hg : TorusC1 g) (hh : TorusC1 h) : |
| TorusC1 (g * h) := by |
| refine ⟨hg.differentiable.mul hh.differentiable, ?_, ?_⟩ |
| · intro j |
| simp only [scalarPartial_mul hg.differentiable hh.differentiable] |
| exact (hg.differentiable.continuous.fun_mul (hh.continuous_partial j)).add |
| (hh.differentiable.continuous.fun_mul (hg.continuous_partial j)) |
| · intro x k |
| simp only [Pi.mul_apply, hg.periodic, hh.periodic] |
|
|
| theorem TorusC1.sum {ι : Type*} [Fintype ι] {E : Type*} |
| [NormedAddCommGroup E] [NormedSpace ℝ E] (g : ι → Space → E) |
| (hg : ∀ i, TorusC1 (g i)) : TorusC1 (∑ i, g i) := by |
| classical |
| refine ⟨Differentiable.sum (fun i _ => (hg i).differentiable), ?_, ?_⟩ |
| · intro j |
| have he (x : Space) : fderiv ℝ (∑ i, g i) x (basis j) = |
| ∑ i, fderiv ℝ (g i) x (basis j) := by |
| rw [fderiv_sum (fun i _ => (hg i).differentiable x), sum_apply] |
| simpa only [he] using continuous_finsetSum _ (fun i _ => (hg i).continuous_partial j) |
| · intro x k |
| simp only [Finset.sum_apply, fun i => (hg i).periodic] |
|
|
| theorem fderiv_component {g : Space → Space} (hg : Differentiable ℝ g) |
| (k : Fin 3) (x v : Space) : |
| fderiv ℝ (fun y => g y k) x v = fderiv ℝ g x v k := by |
| rw [fderiv_pi (φ := fun j y => g y j) (fun j => |
| (differentiable_apply j).fun_comp hg |>.differentiableAt)] |
| rfl |
|
|
| theorem TorusC1.component {g : Space → Space} (hg : TorusC1 g) (k : Fin 3) : |
| TorusC1 (fun x => g x k) := by |
| refine ⟨(differentiable_apply k).fun_comp hg.differentiable, ?_, ?_⟩ |
| · intro j |
| simp only [fderiv_component hg.differentiable] |
| exact (continuous_apply k).comp (hg.continuous_partial j) |
| · intro x l |
| rw [hg.periodic] |
|
|
| theorem periodic_integral_divergence_zero_C1 {E : Type*} [NormedAddCommGroup E] |
| [NormedSpace ℝ E] (g : Fin 3 → Space → E) (hg : ∀ j, TorusC1 (g j)) : |
| (∫ x in fundamentalCell, ∑ j : Fin 3, fderiv ℝ (g j) x (basis j)) = 0 := by |
| rw [setIntegral_congr_set cell_ae_eq_Icc] |
| have hc := continuous_finsetSum Finset.univ (fun j _ => (hg j).continuous_partial j) |
| simp only [basis] at * |
| rw [integral_divergence_of_hasFDerivAt_off_countable' (0 : Space) (fun _ => 10) |
| (fun _ => by norm_num) g (fun j => fderiv ℝ (g j)) ∅ Set.countable_empty |
| (fun j => (hg j).differentiable.continuous.continuousOn) |
| (fun x _ j => (hg j).differentiable x |>.hasFDerivAt) |
| hc.integrableOn_Icc] |
| apply Finset.sum_eq_zero |
| intro j _ |
| have he : (fun y : Fin 2 → ℝ => g j (j.insertNth 10 y)) = |
| (fun y : Fin 2 → ℝ => g j (j.insertNth 0 y)) := by |
| funext y |
| rw [face_front_eq_back_deck] |
| exact (hg j).periodic _ _ |
| change (∫ y in Set.Icc _ _, g j (j.insertNth 10 y)) - |
| (∫ y in Set.Icc _ _, g j (j.insertNth 0 y)) = 0 |
| rw [he, sub_self] |
| |
| theorem periodic_integral_partial_zero_C1 {E : Type*} [NormedAddCommGroup E] |
| [NormedSpace ℝ E] {g : Space → E} (hg : TorusC1 g) (j : Fin 3) : |
| (∫ x in fundamentalCell, fderiv ℝ g x (basis j)) = 0 := by |
| classical |
| have h := periodic_integral_divergence_zero_C1 (fun k => if k = j then g else 0) |
| (fun k => by split_ifs <;> first | exact hg | exact TorusC1.const 0) |
| have he (k : Fin 3) (x : Space) : |
| fderiv ℝ (if k = j then g else 0) x (basis k) = |
| if k = j then fderiv ℝ g x (basis j) else 0 := by |
| by_cases hk : k = j |
| · subst k |
| simp |
| · simp [hk] |
| simpa only [he, Finset.sum_ite_eq', Finset.mem_univ, ite_true] using h |
|
|
| theorem periodic_integration_by_parts {g h : Space → ℝ} |
| (hg : TorusC1 g) (hh : TorusC1 h) (j : Fin 3) : |
| (∫ x in fundamentalCell, g x * fderiv ℝ h x (basis j)) = |
| -(∫ x in fundamentalCell, h x * fderiv ℝ g x (basis j)) := by |
| have hz := periodic_integral_partial_zero_C1 (hg.mul hh) j |
| simp only [scalarPartial_mul hg.differentiable hh.differentiable] at hz |
| rw [integral_add (f := fun x => g x * fderiv ℝ h x (basis j)) |
| (g := fun x => h x * fderiv ℝ g x (basis j)) |
| (continuous_integrableOn_cell (hg.differentiable.continuous.fun_mul (hh.continuous_partial j))) |
| (continuous_integrableOn_cell (hh.differentiable.continuous.fun_mul (hg.continuous_partial j)))] at hz |
| linarith |
|
|
| theorem cylinder_spatial_slice {E : Type*} [TopologicalSpace E] |
| {g : ℝ → Space → E} {T t : ℝ} |
| (hg : ContinuousOn (fun z : SpaceTime => g z.1 z.2) (cylinder T)) |
| (ht : t ∈ Set.Icc 0 T) : Continuous (g t) := by |
| simpa only [Function.comp_def, id_eq] using hg.comp_continuous |
| (continuous_const.prodMk continuous_id) (fun x => ⟨ht, Set.mem_univ x⟩) |
|
|
| theorem spatialPartial_periodic_of_differentiable {u : Field} {t : ℝ} |
| (hu : Differentiable ℝ (u t)) (hp : SpatiallyPeriodic u) (j : Fin 3) (x : Space) (k : Fin 3 → ℤ) : |
| spatialPartial u j t (x + deck k) = spatialPartial u j t x := |
| directional_translation hu (fun y => hp t y k) (basis j) x |
|
|
| theorem ClassicalSolution.spatialC1 {ν : ℝ} {f u : Field} {p : Pressure} |
| (h : ClassicalSolution ν f u p) {t : ℝ} (ht : 0 ≤ t) : TorusC1 (u t) := by |
| refine ⟨h.differentiable_x t ht, ?_, h.spatial_periodic_u t⟩ |
| intro j |
| exact cylinder_spatial_slice (h.continuous_x t ht j) ⟨ht, le_refl t⟩ |
|
|
| theorem ClassicalSolution.spatialPartialC1 {ν : ℝ} {f u : Field} {p : Pressure} |
| (h : ClassicalSolution ν f u p) {t : ℝ} (ht : 0 ≤ t) (j : Fin 3) : |
| TorusC1 (spatialPartial u j t) := by |
| refine ⟨h.differentiable_xx t ht j, ?_, ?_⟩ |
| · intro k |
| exact cylinder_spatial_slice (h.continuous_xx t ht j k) ⟨ht, le_refl t⟩ |
| · exact spatialPartial_periodic_of_differentiable (h.differentiable_x t ht) |
| h.spatial_periodic_u j |
|
|
| theorem ClassicalSolution.pressureC1 {ν : ℝ} {f u : Field} {p : Pressure} |
| (h : ClassicalSolution ν f u p) {t : ℝ} (ht : 0 ≤ t) : TorusC1 (p t) := by |
| refine ⟨h.differentiable_p t ht, ?_, h.spatial_periodic_p t⟩ |
| intro j |
| exact (continuous_apply j).comp |
| (cylinder_spatial_slice (h.continuous_px t ht) ⟨ht, le_refl t⟩) |
|
|
| theorem fderiv_apply_eq_sum {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (g : Space → E) (x d : Space) : |
| fderiv ℝ g x d = ∑ j : Fin 3, d j • fderiv ℝ g x (basis j) := by |
| have hd : d = ∑ j : Fin 3, d j • basis j := by |
| ext k |
| fin_cases k <;> simp [basis, Fin.sum_univ_succ] |
| conv_lhs => rw [hd] |
| simp only [map_sum, map_smul] |
|
|
| theorem periodic_transport_zero {q : Space → ℝ} {v : Space → Space} |
| (hq : TorusC1 q) (hv : TorusC1 v) |
| (hdiv : ∀ x, ∑ j : Fin 3, fderiv ℝ v x (basis j) j = 0) : |
| (∫ x in fundamentalCell, fderiv ℝ q x (v x)) = 0 := by |
| have hz := periodic_integral_divergence_zero_C1 |
| (fun j x => q x * v x j) (fun j => hq.mul (hv.component j)) |
| have he (x : Space) : |
| (∑ j, fderiv ℝ (fun y => q y * v y j) x (basis j)) = |
| fderiv ℝ q x (v x) := by |
| have hmul (j : Fin 3) : fderiv ℝ (fun y => q y * v y j) x (basis j) = |
| q x * fderiv ℝ (fun y => v y j) x (basis j) + |
| v x j * fderiv ℝ q x (basis j) := |
| scalarPartial_mul hq.differentiable (hv.component j).differentiable x j |
| simp only [hmul, fderiv_component hv.differentiable, Finset.sum_add_distrib, |
| ← Finset.mul_sum, hdiv, mul_zero, zero_add] |
| rw [fderiv_apply_eq_sum] |
| rfl |
| simpa only [he] using hz |
|
|
| def kineticDensity (a : Space) : ℝ := ∑ i : Fin 3, a i * a i / 2 |
|
|
| theorem kineticDensity_nonneg (a : Space) : 0 ≤ kineticDensity a := by |
| unfold kineticDensity |
| exact Finset.sum_nonneg (fun i _ => div_nonneg (mul_self_nonneg _) (by norm_num)) |
|
|
| theorem kineticDensity_continuous : Continuous kineticDensity := by |
| unfold kineticDensity |
| exact continuous_finsetSum Finset.univ (fun i _ => |
| ((continuous_apply i).fun_mul (continuous_apply i)).div_const 2) |
|
|
| theorem fderiv_kineticDensity {w : Space → Space} (hw : Differentiable ℝ w) |
| (x d : Space) : |
| fderiv ℝ (fun y => kineticDensity (w y)) x d = |
| ∑ i : Fin 3, w x i * (fderiv ℝ w x d) i := by |
| have hg (i : Fin 3) : Differentiable ℝ (fun y => w y i) := |
| (differentiable_apply i).fun_comp hw |
| have he : (fun y => kineticDensity (w y)) = ∑ i : Fin 3, (fun y => w y i * w y i * (1/2 : ℝ)) := by |
| funext y |
| simp [kineticDensity, div_eq_mul_inv] |
| rw [he, fderiv_sum (fun i _ => (((hg i).fun_mul (hg i)).mul_const (1/2 : ℝ)) x), sum_apply] |
| apply Finset.sum_congr rfl |
| intro i _ |
| rw [fderiv_mul_const (((hg i).fun_mul (hg i)) x), fderiv_fun_mul ((hg i) x) ((hg i) x)] |
| simp only [smul_apply, add_apply, |
| smul_eq_mul, fderiv_component hw] |
| ring |
|
|
| theorem TorusC1.kineticDensity {w : Space → Space} (hw : TorusC1 w) : |
| TorusC1 (fun y => kineticDensity (w y)) := by |
| have hc (i : Fin 3) := ((hw.component i).mul (hw.component i)).mul (TorusC1.const (1/2 : ℝ)) |
| convert TorusC1.sum (fun i : Fin 3 => fun y => w y i * w y i * (1/2 : ℝ)) hc using 1 |
| ext y |
| simp [Solenoidal.kineticDensity, div_eq_mul_inv] |
|
|
| theorem periodic_transport_energy_zero {w v : Space → Space} |
| (hw : TorusC1 w) (hv : TorusC1 v) |
| (hdiv : ∀ x, ∑ j : Fin 3, fderiv ℝ v x (basis j) j = 0) : |
| (∫ x in fundamentalCell, ∑ i : Fin 3, w x i * (fderiv ℝ w x (v x)) i) = 0 := by |
| simpa only [fderiv_kineticDensity hw.differentiable] using |
| periodic_transport_zero hw.kineticDensity hv hdiv |
|
|
| end Solenoidal |
| end |
|
|
| end OAI |
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|
