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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 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 | import Mathlib
import OAI.Analysis.Conductivity.Flux.CascadeStageCurl
import OAI.Analysis.Conductivity.Variational.CrossingGradient
namespace OAI
noncomputable section
namespace ScalarConductivity
open Real Set Filter Topology MeasureTheory Matrix
lemma direction_affineShift {f : Coord3 → ℝ} (hf : Differentiable ℝ f)
(s r : ℝ) (c d v x : Coord3) :
direction v (fun y => s*f (r • (y-c)-d)) x = s*r*direction v f (r • (x-c)-d) := by
have hT : Differentiable ℝ (fun y : Coord3 => f (y-d)) := hf.comp (differentiable_id.sub_const d)
rw [direction_dilation hT]
have hh := direction_dilation hf 1 1 d v (r • (x-c))
simpa only [one_mul,one_smul] using congrArg (fun z => s*r*z) hh
lemma potentialCurl_affineShift {S : Fin 3 → Fin 3 → Coord3 → ℝ}
(hS : ∀ i j, Differentiable ℝ (S i j)) (s r : ℝ) (c d x : Coord3) (i : Fin 3) :
potentialCurl (fun i j y => s*S i j (r • (y-c)-d)) x i =
s*r*potentialCurl S (r • (x-c)-d) i := by
simp only [potentialCurl,direction_affineShift (hS _ _),Finset.mul_sum]
lemma local_scaled_constitution {V v : Coord3 → ℝ}
{T S : Fin 3 → Fin 3 → Coord3 → ℝ}
(hv : Differentiable ℝ v) (hS : ∀ i j, Differentiable ℝ (S i j))
(s r : ℝ) (c d x : Coord3) (A : Matrix (Fin 3) (Fin 3) ℝ)
(hV : V =ᶠ[𝓝 x] fun y => s*v (r • (y-c)-d))
(hT : ∀ i j, T i j =ᶠ[𝓝 x] fun y => s*S i j (r • (y-c)-d))
(hc : A *ᵥ (fun i => direction (Pi.single i 1) v (r • (x-c)-d)) =
potentialCurl S (r • (x-c)-d)) :
A *ᵥ (fun i => direction (Pi.single i 1) V x) = potentialCurl T x := by
ext i
rw [potentialCurl_congr_nhds x hT i,potentialCurl_affineShift hS]
have hg (j : Fin 3) : direction (Pi.single j 1) V x =
(s*r)*direction (Pi.single j 1) v (r • (x-c)-d) := by
rw [direction_congr_nhds hV,direction_affineShift hv]
simp only [mulVec,dotProduct,hg]
calc
_ = s*r*∑ j, A i j*direction (Pi.single j 1) v (r • (x-c)-d) := by
rw [Finset.mul_sum]; apply Finset.sum_congr rfl; intro j _; ring
_ = _ := by rw [←hc]; rfl
lemma cascadeValue_crossing {L K k : ℝ} (hL : 0<L) (hK : 0<K) (hk : k≠0)
(A : ℝ) (x : Coord3) (m : ℕ)
(hs₁ : K+2 < cascadePhase (cascadeLength L K) k m (x 0))
(hs₂ : cascadePhase (cascadeLength L K) k m (x 0) < cascadeLength L K) :
cascadeValue L K k A x =
((A*cascadeRatio L K^m)*(connectorProfile K (K+2)*exp (-3*L)))*
crossingValue3 L (cascadeAxis m) (cascadeAxis (m+1))
((k*2^m) • (x-cascadeCenter (cascadeLength L K) k m)-Pi.single 0 (K+2+L)) := by
rw [cascadeValue_two_modes hL hK A x m (by linarith) hs₂.le]
have h := cascade_overlap_value hL hK hs₁.le hs₂.le (A*cascadeRatio L K^m)
(k*2^m*x (cascadeAxis m)) (k*2^m*x (cascadeAxis (m+1)))
simp only [crossingValue3,Pi.sub_apply,Pi.single_eq_same,
Pi.single_eq_of_ne (cascadeAxis_ne_zero m),Pi.single_eq_of_ne (cascadeAxis_ne_zero (m+1)),
sub_zero,cascade_dilation_time hk,cascade_dilation_coord _ _ _ _ (cascadeAxis_ne_zero m),
cascade_dilation_coord _ _ _ _ (cascadeAxis_ne_zero (m+1))]
rw [cascadeTerm,cascadeTerm,cascadePhase_succ,pow_succ]
convert h using 1 <;> ring_nf
lemma cascade_crossing_nhds {L K k : ℝ} (x : Coord3) (m : ℕ)
(hs₁ : K+2 < cascadePhase (cascadeLength L K) k m (x 0))
(hs₂ : cascadePhase (cascadeLength L K) k m (x 0) < cascadeLength L K) :
{y : Coord3 | K+2 < cascadePhase (cascadeLength L K) k m (y 0) ∧
cascadePhase (cascadeLength L K) k m (y 0) < cascadeLength L K} ∈ 𝓝 x := by
have hc : Continuous (fun y : Coord3 => cascadePhase (cascadeLength L K) k m (y 0)) := by
unfold cascadePhase
fun_prop
exact (hc.isOpen_preimage _ isOpen_Ioo).mem_nhds ⟨hs₁,hs₂⟩
theorem cascade_crossing_constitution {L K k : ℝ} (hL : 0<L) (hK : 0<K) (hk : k≠0)
(A : ℝ) (x : Coord3) (m : ℕ)
(hs₁ : K+2 < cascadePhase (cascadeLength L K) k m (x 0))
(hs₂ : cascadePhase (cascadeLength L K) k m (x 0) < cascadeLength L K) :
crossingTensor L (cascadeAxis m) (cascadeAxis (m+1))
((k*2^m) • (x-cascadeCenter (cascadeLength L K) k m)-Pi.single 0 (K+2+L)) *ᵥ
(fun i => direction (Pi.single i 1) (cascadeValue L K k A) x) =
cascadeFlux L K k A x := by
apply local_scaled_constitution (crossingValue3_diff L _ _) (crossingPotential_diff L _ _)
((A*cascadeRatio L K^m)*(connectorProfile K (K+2)*exp (-3*L))) (k*2^m)
(cascadeCenter (cascadeLength L K) k m) (Pi.single 0 (K+2+L)) x
· filter_upwards [cascade_crossing_nhds x m hs₁ hs₂] with y hy
exact cascadeValue_crossing hL hK hk A y m hy.1 hy.2
· intro i j
filter_upwards [cascade_crossing_nhds x m hs₁ hs₂] with y hy
rw [cascadePotential_two_terms hL hK hk A y (m := m) (by linarith [hy.1]) (by linarith [hy.2])]
exact cascadePotentialTerm_crossing_pair hL hK hk A y m hy.1 hy.2 i j
· exact crossingTensor_constitution hL (cascadeAxis_ne_zero m)
(cascadeAxis_ne_zero (m+1)) (cascadeAxis_ne_succ m) _
end ScalarConductivity
end
end OAI
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