| 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 |
|
|
