| import Mathlib |
| import OAI.Analysis.Conductivity.Fourier.CascadePeriodicity |
| import OAI.Analysis.Conductivity.Branching.EndingCoordinateTransport |
|
|
| namespace OAI |
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|
| noncomputable section |
| namespace ScalarConductivity |
| open Real Set Filter Topology MeasureTheory Matrix |
|
|
| lemma alignedEndingTensor_bounds {lam k J L K c C : ℝ} |
| (hb : ∀ x v : Coord3,c*(v ⬝ᵥ v)≤v ⬝ᵥ (finiteEndingTensor (lam/k) J L K x*ᵥv) ∧ |
| v ⬝ᵥ (finiteEndingTensor (lam/k) J L K x*ᵥv)≤C*(v ⬝ᵥ v)) |
| (x v : Coord3) : |
| c*(v ⬝ᵥ v)≤v ⬝ᵥ (alignedEndingTensor lam k J L K x*ᵥv) ∧ |
| v ⬝ᵥ (alignedEndingTensor lam k J L K x*ᵥv)≤C*(v ⬝ᵥ v) := by |
| have h := hb (permutedDilation angularAxisSwap k x) (fun i => v (angularAxisSwap i)) |
| have hn : (fun i => v (angularAxisSwap i)) ⬝ᵥ (fun i => v (angularAxisSwap i))=v ⬝ᵥ v := |
| Equiv.sum_comp angularAxisSwap (fun i => v i*v i) |
| rw [hn] at h |
| have hq := permutedTensor_quadratic angularAxisSwap |
| (finiteEndingTensor (lam/k) J L K (permutedDilation angularAxisSwap k x)) v |
| change v ⬝ᵥ (alignedEndingTensor lam k J L K x*ᵥv)=_ at hq |
| simpa only [hq] using h |
|
|
| lemma permutedDilation_angularShift (k : ℤ) (n : Fin 2 → ℤ) (x : Coord3) : |
| permutedDilation angularAxisSwap (k:ℝ) (x+angularShift (2*Real.pi) n)= |
| permutedDilation angularAxisSwap (k:ℝ) x+ |
| angularShift (2*Real.pi) ![k*n 1,k*n 0] := by |
| ext i |
| fin_cases i <;> simp [permutedDilation,angularAxisSwap_apply,angularShift] <;> ring |
|
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| lemma alignedEndingValue_angularPeriodic (lam J L K : ℝ) (k : ℤ) : |
| AngularPeriodic (2*Real.pi) (alignedEndingValue lam k J L K) := by |
| intro n x |
| dsimp only [alignedEndingValue] |
| rw [permutedDilation_angularShift] |
| exact finiteEndingValue_angularPeriodic _ _ _ _ _ _ |
|
|
| lemma alignedEndingTensor_angularPeriodic (lam J L K : ℝ) (k : ℤ) : |
| AngularPeriodic (2*Real.pi) (alignedEndingTensor lam k J L K) := by |
| intro n x |
| ext i j |
| dsimp only [alignedEndingTensor] |
| rw [permutedDilation_angularShift,finiteEndingTensor_angularPeriodic] |
|
|
| theorem aligned_positive_finite_ending {lam : ℝ} (hlam : 0<lam) {k : ℤ} (hk : 0<k) : |
| ∃ J L K T c C : ℝ,0<J ∧ 1≤L ∧ 0<K ∧ 0<T ∧ 0<c ∧ c≤C ∧ |
| ContDiff ℝ 2 (alignedEndingValue lam k J L K) ∧ |
| AngularPeriodic (2*Real.pi) (alignedEndingValue lam k J L K) ∧ |
| AngularPeriodic (2*Real.pi) (alignedEndingTensor lam k J L K) ∧ |
| (∀ x v : Coord3,c*(v ⬝ᵥ v)≤v ⬝ᵥ (alignedEndingTensor lam k J L K x*ᵥv) ∧ |
| v ⬝ᵥ (alignedEndingTensor lam k J L K x*ᵥv)≤C*(v ⬝ᵥ v)) ∧ |
| (∀ x : Coord3,(k:ℝ)*x 0≤1 → alignedEndingValue lam k J L K x=exp (-lam*x 0)*cos ((k:ℝ)*x 2)) ∧ |
| (∀ x : Coord3,T≤x 0 → alignedEndingValue lam k J L K x=0) ∧ |
| (∀ x : Coord3,alignedEndingTensor lam k J L K x*ᵥPi.single 0 1=Pi.single 0 1) ∧ |
| ∀ (ψ : SmoothScalar Coord3),HasCompactSupport ψ.val → |
| (∫ x, ∑ i, (alignedEndingTensor lam k J L K x*ᵥ |
| (fun j => direction (Pi.single j 1) (alignedEndingValue lam k J L K) x)) i* |
| (smoothDirection (Pi.single i 1) ψ).val x)=0 := by |
| have hkr : (0:ℝ)<k := by exact_mod_cast hk |
| obtain ⟨J,L,K,c,C,hJ,hL,hK,hc,hcC,hb⟩ := exists_finiteEnding_parameters (div_pos hlam hkr) |
| have hD := cascade_length_positive (lt_of_lt_of_le (by norm_num) hL) hK |
| let T := (J+2+2*cascadeLength L K)/(k:ℝ) |
| refine ⟨J,L,K,T,c,C,hJ,hL,hK,div_pos (by linarith) hkr,hc,hcC, |
| alignedEndingValue_C2 hJ hL hK,alignedEndingValue_angularPeriodic _ _ _ _ _, |
| alignedEndingTensor_angularPeriodic _ _ _ _ _,alignedEndingTensor_bounds hb, |
| alignedEnding_initial hkr hJ,?_,alignedEndingTensor_normal _ _ _ _ _,?_⟩ |
| · intro x hx |
| apply alignedEnding_terminal hJ hL hK |
| have hh := (div_le_iff₀ hkr).mp hx |
| linarith |
| · intro ψ hcψ |
| exact (alignedEnding_weak_equation hkr.ne' hJ hL hK ψ hcψ).2 |
| |
| end ScalarConductivity |
| |
| end |
| |
| end OAI |
| |
