| import Mathlib |
| import OAI.Analysis.Conductivity.Flux.TwoFluxColumns |
|
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| namespace OAI |
|
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| noncomputable section |
|
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| open MeasureTheory |
| open scoped ENNReal |
| open Matrix Filter Topology |
| open Set MeasureTheory Filter Topology |
| open scoped BigOperators |
| open Set MeasureTheory Filter Topology |
| open scoped Manifold |
| open Set Filter |
| open scoped Topology |
| open Set Filter MeasureTheory |
| open scoped Topology Manifold ENNReal |
| open Set |
| namespace ScalarConductivity |
|
|
| section |
|
|
| theorem exists_adapted_spatial_basis |
| {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] (hdim : Module.finrank ℝ E = 3) |
| (D : E →L[ℝ] (Fin 2 → ℝ)) (hD : Function.Surjective D) : |
| ∃ v : Module.Basis (Fin 3) ℝ E, |
| D (v 0) = Pi.single 0 1 ∧ D (v 1) = Pi.single 1 1 ∧ D (v 2) = 0 := by |
| classical |
| obtain ⟨Q, hQ⟩ := ContinuousLinearMap.HasRightInverse.of_surjective_of_finiteDimensional hD |
| have hk := D.toLinearMap.finrank_range_add_finrank_ker |
| rw [LinearMap.range_eq_top.mpr hD, finrank_top, |
| Module.finrank_pi, Fintype.card_fin, hdim] at hk |
| have hkp : 0 < Module.finrank ℝ D.ker := by omega |
| obtain ⟨w, hw⟩ := Module.finrank_pos_iff_exists_ne_zero.mp hkp |
| have hw0 : (w : E) ≠ 0 := fun h => hw (Subtype.ext h) |
| have hDw : D (w : E) = 0 := w.property |
| let v : Fin 3 → E := ![Q (Pi.single 0 1), Q (Pi.single 1 1), w] |
| have hv : LinearIndependent ℝ v := by |
| rw [Fintype.linearIndependent_iff] |
| intro c hc |
| have hs : c 0 • Q (Pi.single 0 1) + c 1 • Q (Pi.single 1 1) + c 2 • (w : E) = 0 := by |
| simpa [v, Fin.sum_univ_succ, add_assoc] using hc |
| have hh := congrArg D hs |
| simp only [map_add, map_smul, hQ (Pi.single 0 1), hQ (Pi.single 1 1), hDw, |
| smul_zero, add_zero, map_zero] at hh |
| have h0 : c 0 = 0 := by simpa using congr_fun hh 0 |
| have h1 : c 1 = 0 := by simpa using congr_fun hh 1 |
| have h2 : c 2 = 0 := by |
| simp only [h0, h1, zero_smul, zero_add] at hs |
| exact (smul_eq_zero.mp hs).resolve_right hw0 |
| intro i |
| fin_cases i <;> assumption |
| let b := basisOfLinearIndependentOfCardEqFinrank hv (by simpa using hdim.symm) |
| have hb : (b : Fin 3 → E) = v := coe_basisOfLinearIndependentOfCardEqFinrank hv _ |
| refine ⟨b, ?_, ?_, ?_⟩ |
| · rw [hb] |
| exact hQ _ |
| · rw [hb] |
| exact hQ _ |
| · rw [hb] |
| exact hDw |
|
|
| open Filter Topology |
|
|
| def rawTwoSpatial {E : Type*} [AddCommGroup E] [Module ℝ E] |
| (v : Fin 3 → E) (r : Fin 5 → ℝ) : Fin 2 → E := |
| ![r 0 • v 0 + r 1 • v 1 + r 3 • v 2, |
| r 1 • v 0 + r 2 • v 1 + r 4 • v 2] |
|
|
| lemma rawTwoSpatial_symmetry |
| {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (v : Fin 3 → E) (D : E →L[ℝ] (Fin 2 → ℝ)) |
| (h0 : D (v 0) = Pi.single 0 1) (h1 : D (v 1) = Pi.single 1 1) |
| (h2 : D (v 2) = 0) (r : Fin 5 → ℝ) : |
| D (rawTwoSpatial v r 0) 1 = D (rawTwoSpatial v r 1) 0 := by |
| simp [rawTwoSpatial, map_add, map_smul, h0, h1, h2] |
|
|
| lemma adapted_basis_coordinate |
| {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (v : Module.Basis (Fin 3) ℝ E) (D : E →L[ℝ] (Fin 2 → ℝ)) |
| (h0 : D (v 0) = Pi.single 0 1) (h1 : D (v 1) = Pi.single 1 1) |
| (h2 : D (v 2) = 0) (x : E) (i : Fin 2) : |
| D x i = v.repr x i.castSucc := by |
| have hx := congrArg D (v.sum_repr x) |
| rw [Fin.sum_univ_three] at hx |
| simp only [map_add, map_smul, h0, h1, h2, smul_zero, add_zero] at hx |
| fin_cases i |
| · simpa using (congr_fun hx 0).symm |
| · simpa using (congr_fun hx 1).symm |
|
|
| lemma exists_rawTwoSpatial |
| {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (v : Module.Basis (Fin 3) ℝ E) (D : E →L[ℝ] (Fin 2 → ℝ)) |
| (h0 : D (v 0) = Pi.single 0 1) (h1 : D (v 1) = Pi.single 1 1) |
| (h2 : D (v 2) = 0) (R : Fin 2 → E) |
| (hR : D (R 0) 1 = D (R 1) 0) : |
| ∃ r : Fin 5 → ℝ, rawTwoSpatial v r = R := by |
| have hs : v.repr (R 0) 1 = v.repr (R 1) 0 := by |
| simpa [adapted_basis_coordinate v D h0 h1 h2] using hR |
| refine ⟨![v.repr (R 0) 0, v.repr (R 0) 1, v.repr (R 1) 1, |
| v.repr (R 0) 2, v.repr (R 1) 2], ?_⟩ |
| funext i |
| fin_cases i |
| · change v.repr (R 0) 0 • v 0 + v.repr (R 0) 1 • v 1 + v.repr (R 0) 2 • v 2 = R 0 |
| simpa only [Fin.sum_univ_three] using v.sum_repr (R 0) |
| · change v.repr (R 0) 1 • v 0 + v.repr (R 1) 1 • v 1 + v.repr (R 1) 2 • v 2 = R 1 |
| rw [hs] |
| simpa only [Fin.sum_univ_three] using v.sum_repr (R 1) |
|
|
| lemma rawTwoSpatial_symbol |
| {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (v : Fin 3 → E) (L : E →L[ℝ] ℝ) (r : Fin 5 → ℝ) : |
| divSymbolTwo (fun i => L (v i)) *ᵥ r = fun j => L (rawTwoSpatial v r j) := by |
| ext j |
| fin_cases j <;> simp [rawTwoSpatial, divSymbolTwo, Matrix.mulVec, |
| dotProduct, Fin.sum_univ_succ, map_add, map_smul] <;> ring |
|
|
| lemma smooth_rawTwoSpatial |
| {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (v : Fin 3 → E) (G : Fin 5 → SmoothScalar E) (j : Fin 2) : |
| ContDiff ℝ (↑(⊤ : ℕ∞)) (fun x => rawTwoSpatial v (fun i => (G i).val x) j) := by |
| fin_cases j <;> |
| exact (((smoothScalar_contDiff (G _)).smul contDiff_const).add |
| ((smoothScalar_contDiff (G _)).smul contDiff_const)).add |
| ((smoothScalar_contDiff (G _)).smul contDiff_const) |
|
|
| lemma support_rawTwoSpatial |
| {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (v : Fin 3 → E) (G : Fin 5 → SmoothScalar E) {K : Set E} |
| (hK : IsClosed K) (hG : ∀ i, tsupport (G i).val ⊆ K) (j : Fin 2) : |
| tsupport (fun x => rawTwoSpatial v (fun i => (G i).val x) j) ⊆ K := by |
| apply closure_minimal _ hK |
| intro x hx |
| by_contra hn |
| have hz (i : Fin 5) : (G i).val x = 0 := |
| image_eq_zero_of_notMem_tsupport (fun ht => hn (hG i ht)) |
| apply hx |
| fin_cases j <;> simp [rawTwoSpatial, hz] |
|
|
| lemma rawTwoSpatial_pairing |
| {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (v : Fin 3 → E) (G : Fin 5 → SmoothScalar E) (ψ : SmoothScalar E) |
| (j : Fin 2) (x : E) : |
| fderiv ℝ ψ.val x (rawTwoSpatial v (fun i => (G i).val x) j) = |
| ∑ i, (twoFluxColumns G j i).val x * (smoothDirection (v i) ψ).val x := by |
| fin_cases j <;> simp [rawTwoSpatial, twoFluxColumns, smoothDirection, |
| Fin.sum_univ_succ, map_add, map_smul, add_assoc] |
|
|
| end |
|
|
| open MeasureTheory Filter Topology |
|
|
| lemma uniform_zero_smul_vector {X I E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {l : Filter I} {f : I → X → ℝ} (v : E) |
| (hf : TendstoUniformly f (fun _ => 0) l) : |
| TendstoUniformly (fun i x => f i x • v) (fun _ => 0) l := by |
| simpa [Function.comp_def] using |
| ((ContinuousLinearMap.id ℝ ℝ).smulRight v).uniformContinuous.comp_tendstoUniformly hf |
|
|
| lemma rawTwoSpatial_error |
| {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (v : Fin 3 → E) (g r : Fin 5 → ℝ) (z : ℝ) (j : Fin 2) : |
| rawTwoSpatial v g j - z • rawTwoSpatial v r j = |
| rawTwoSpatial v (fun i => g i - r i * z) j := by |
| fin_cases j <;> simp [rawTwoSpatial, sub_smul, mul_smul, smul_add] <;> module |
|
|
| lemma rawTwoSpatial_error_tendsto |
| {X I E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {l : Filter I} (v : Fin 3 → E) (G : I → X → Fin 5 → ℝ) |
| (r : Fin 5 → ℝ) (z : I → X → ℝ) |
| (hG : ∀ i, TendstoUniformly (fun k x => G k x i - r i * z k x) |
| (fun _ => 0) l) (j : Fin 2) : |
| TendstoUniformly (fun k x => rawTwoSpatial v (G k x) j - z k x • rawTwoSpatial v r j) |
| (fun _ => 0) l := by |
| simp_rw [rawTwoSpatial_error] |
| have hc (i : Fin 5) (a : Fin 3) := uniform_zero_smul_vector (v a) (hG i) |
| fin_cases j |
| · convert ((hc 0 0).add (hc 1 1)).add (hc 3 2) using 1 <;> ext <;> simp [rawTwoSpatial] |
| · convert ((hc 1 0).add (hc 2 1)).add (hc 4 2) using 1 <;> ext <;> simp [rawTwoSpatial] |
|
|
| theorem exists_physical_two_flux |
| {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
| (hdim : Module.finrank ℝ E = 3) (μ : Measure E) [μ.IsAddHaarMeasure] |
| (D : E →L[ℝ] (Fin 2 → ℝ)) (hD : Function.Surjective D) |
| (χ : SmoothScalar E) (hχ : HasCompactSupport χ.val) |
| (L : E →L[ℝ] ℝ) (hL : L ≠ 0) |
| (h : SmoothScalar ℝ) (hp : Function.Periodic h.val 1) |
| (hm : ∫ t in (0 : ℝ)..1, h.val t = 0) |
| (R : Fin 2 → E) (hRs : D (R 0) 1 = D (R 1) 0) |
| (hRn : ∀ j, L (R j) = 0) : |
| ∃ F : ℝ → Fin 2 → E → E, |
| (∀ k j, ContDiff ℝ (↑(⊤ : ℕ∞)) (F k j)) ∧ |
| (∀ k j, tsupport (F k j) ⊆ tsupport χ.val) ∧ |
| (∀ k x, D (F k 0 x) 1 = D (F k 1 x) 0) ∧ |
| (∀ k j (ψ : E → ℝ), ContDiff ℝ (↑(⊤ : ℕ∞)) ψ → |
| (∫ x, fderiv ℝ ψ x (F k j x) ∂μ) = 0) ∧ |
| ∀ j, TendstoUniformly |
| (fun k x => F k j x - (χ.val x * h.val (k * L x)) • R j) |
| (fun _ => 0) atTop := by |
| obtain ⟨v, h0, h1, h2⟩ := exists_adapted_spatial_basis hdim D hD |
| obtain ⟨r, hr⟩ := exists_rawTwoSpatial v D h0 h1 h2 R hRs |
| have hξ : (fun i => L (v i)) ≠ 0 := by |
| intro hz |
| apply hL |
| ext x |
| have hx := congrArg L (v.sum_repr x) |
| simp only [map_sum, map_smul] at hx |
| have hv (i : Fin 3) : L (v i) = 0 := congr_fun hz i |
| simpa only [hv, smul_zero, Finset.sum_const_zero, _root_.zero_apply] |
| using hx.symm |
| have hsymbol : divSymbolTwo (fun i => L (v i)) *ᵥ r = 0 := by |
| rw [rawTwoSpatial_symbol, hr] |
| exact funext hRn |
| obtain ⟨G, hdiv, hs, he⟩ := exists_localized_two_flux v χ L h r hχ hξ hsymbol hp hm |
| let F : ℝ → Fin 2 → E → E := fun k j x => rawTwoSpatial v (fun i => (G k i).val x) j |
| refine ⟨F, (fun k j => smooth_rawTwoSpatial v (G k) j), |
| (fun k j => support_rawTwoSpatial v (G k) (isClosed_tsupport _) (hs k) j), |
| (fun k x => rawTwoSpatial_symmetry v D h0 h1 h2 _), ?_, ?_⟩ |
| · intro k j ψ hψ |
| let Ψ : SmoothScalar E := ⟨ψ, hψ⟩ |
| change (∫ x, fderiv ℝ Ψ.val x (rawTwoSpatial v (fun i => (G k i).val x) j) ∂μ) = 0 |
| simp_rw [rawTwoSpatial_pairing] |
| apply smooth_divergence_zero_pairing μ v _ _ _ Ψ |
| · have hc (i : Fin 5) : HasCompactSupport (G k i).val := |
| hχ.of_isClosed_subset (isClosed_tsupport _) (hs k i) |
| intro i |
| fin_cases j <;> fin_cases i <;> exact hc _ |
| · rw [twoFluxColumns_divergence, hdiv] |
| rfl |
| · intro j |
| have ht := rawTwoSpatial_error_tendsto v (fun k x i => (G k i).val x) r |
| (fun k x => χ.val x * h.val (k * L x)) he j |
| simpa only [hr] using ht |
|
|
| end ScalarConductivity |
|
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| end |
|
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| end OAI |
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