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a458723 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 | import Mathlib
import OAI.Analysis.Conductivity.Fourier.CascadePeriodicity
import OAI.Analysis.Conductivity.Branching.EndingCoordinateTransport
namespace OAI
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
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
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