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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 | import Mathlib
import OAI.Analysis.Conductivity.Flux.TwoFluxDivergence
import OAI.Analysis.Conductivity.Variational.LocalizedFirstPotential
namespace OAI
noncomputable section
open MeasureTheory
open scoped ENNReal
namespace ScalarConductivity
open Filter Topology
theorem uniform_zero_const_mul {X I : Type*} {l : Filter I}
{f : I → X → ℝ} (c : ℝ) (hf : TendstoUniformly f (fun _ => 0) l) :
TendstoUniformly (fun i x => c * f i x) (fun _ => 0) l := by
simpa [Function.comp_def] using
(uniformContinuous_const_smul (X := ℝ) c).comp_tendstoUniformly hf
theorem uniform_zero_add {X I : Type*} {l : Filter I} {f g : I → X → ℝ}
(hf : TendstoUniformly f (fun _ => 0) l)
(hg : TendstoUniformly g (fun _ => 0) l) :
TendstoUniformly (fun i x => f i x + g i x) (fun _ => 0) l := by
have hh := hf.add hg
change TendstoUniformly (fun i x => f i x + g i x) (fun _ => (0 : ℝ) + 0) l at hh
simpa only [add_zero] using hh
theorem uniform_zero_sub {X I : Type*} {l : Filter I} {f g : I → X → ℝ}
(hf : TendstoUniformly f (fun _ => 0) l)
(hg : TendstoUniformly g (fun _ => 0) l) :
TendstoUniformly (fun i x => f i x - g i x) (fun _ => 0) l := by
have hh := hf.sub hg
change TendstoUniformly (fun i x => f i x - g i x) (fun _ => (0 : ℝ) - 0) l at hh
simpa only [sub_zero] using hh
def localizedTwoFlux {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
(v : Fin 3 → E) (χ : SmoothScalar E) (L : E →L[ℝ] ℝ)
(H₁ H₂ : SmoothScalar ℝ) (α p q r k : ℝ) : Fin 5 → SmoothScalar E :=
α • airyFlux v (localizedSecondPotential χ L H₂ k) +
transverseFlux v (p • localizedFirstPotential χ L H₁ k)
(q • localizedFirstPotential χ L H₁ k)
(r • localizedFirstPotential χ L H₁ k)
theorem twoFluxDivergence_add {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
(v : Fin 3 → E) (F G : Fin 5 → SmoothScalar E) :
twoFluxDivergence v (F + G) = twoFluxDivergence v F + twoFluxDivergence v G := by
funext i
fin_cases i <;> simp [twoFluxDivergence] <;> abel
theorem twoFluxDivergence_smul {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
(v : Fin 3 → E) (c : ℝ) (F : Fin 5 → SmoothScalar E) :
twoFluxDivergence v (c • F) = c • twoFluxDivergence v F := by
funext i
fin_cases i <;> simp [twoFluxDivergence, smul_add]
theorem localizedTwoFlux_divergence {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
(v : Fin 3 → E) (χ : SmoothScalar E) (L : E →L[ℝ] ℝ)
(H₁ H₂ : SmoothScalar ℝ) (α p q r k : ℝ) :
twoFluxDivergence v (localizedTwoFlux v χ L H₁ H₂ α p q r k) = 0 := by
rw [localizedTwoFlux, twoFluxDivergence_add, twoFluxDivergence_smul,
airyFlux_divergence, transverseFlux_divergence, smul_zero, zero_add]
theorem localizedTwoFlux_error_tendsto {E : Type*} [NormedAddCommGroup E]
[NormedSpace ℝ E] (v : Fin 3 → E) (χ : SmoothScalar E) (L : E →L[ℝ] ℝ)
(h H₁ H₂ : SmoothScalar ℝ) (α p q r : ℝ)
(hχ : HasCompactSupport χ.val)
(h₁ : smoothDirection 1 H₁ = h) (h₂ : smoothDirection 1 H₂ = H₁)
(hB₁ : ∃ C : ℝ, ∀ t, |H₁.val t| ≤ C)
(hB₂ : ∃ C : ℝ, ∀ t, |H₂.val t| ≤ C) (i : Fin 5) :
TendstoUniformly
(fun k x => (localizedTwoFlux v χ L H₁ H₂ α p q r k i).val x -
(α • airySymbol (fun j => L (v j)) +
transverseSymbol (fun j => L (v j)) p q r) i *
(χ.val x * h.val (k * L x))) (fun _ => 0) atTop := by
have hfirst (j : Fin 3) :=
localizedFirstPotential_error_tendsto (v j) χ L H₁ hχ hB₁
simp only [h₁] at hfirst
have hsecond (j l : Fin 3) :=
localizedSecondPotential_error_tendsto (v j) (v l) χ L H₂ hχ hB₂
(by simpa only [h₂] using hB₁)
simp only [h₂, h₁] at hsecond
fin_cases i
· convert uniform_zero_add (uniform_zero_const_mul α (hsecond 1 1))
(uniform_zero_const_mul p (hfirst 2)) using 1
funext k x
simp [localizedTwoFlux, airyFlux, transverseFlux, airySymbol, transverseSymbol,
map_smul, Subalgebra.coe_add]
ring
· convert uniform_zero_add (uniform_zero_const_mul (-α) (hsecond 0 1))
(uniform_zero_const_mul q (hfirst 2)) using 1
funext k x
simp [localizedTwoFlux, airyFlux, transverseFlux, airySymbol, transverseSymbol,
map_smul, Subalgebra.coe_add]
ring
· convert uniform_zero_add (uniform_zero_const_mul α (hsecond 0 0))
(uniform_zero_const_mul r (hfirst 2)) using 1
funext k x
simp [localizedTwoFlux, airyFlux, transverseFlux, airySymbol, transverseSymbol,
map_smul, Subalgebra.coe_add]
ring
· convert uniform_zero_sub (uniform_zero_const_mul (-p) (hfirst 0))
(uniform_zero_const_mul q (hfirst 1)) using 1
funext k x
simp [localizedTwoFlux, airyFlux, transverseFlux, airySymbol, transverseSymbol,
map_smul]
ring
· convert uniform_zero_sub (uniform_zero_const_mul (-q) (hfirst 0))
(uniform_zero_const_mul r (hfirst 1)) using 1
funext k x
simp [localizedTwoFlux, airyFlux, transverseFlux, airySymbol, transverseSymbol,
map_smul]
ring
end ScalarConductivity
end
end OAI
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