| import Mathlib |
| import OAI.Analysis.Conductivity.Sources.LocalPiola |
| import OAI.Analysis.Conductivity.Variational.CompactSymmetricMoments |
| import OAI.Analysis.Conductivity.Flux.CompactFluxDivergence |
|
|
| namespace OAI |
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|
| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Matrix |
| open scoped Matrix.Norms.Elementwise |
|
|
| def symmetricSource (H : Coord3 → Mat3) (u : Coord3 → Fin 2 → ℝ) (j : Fin 2) : Coord3 → ℝ := |
| coordinateDivergence (fun x => (H x*gradientColumns (fderiv ℝ u x)).col j) |
|
|
| lemma symmetricFlux_smooth {H : Coord3 → Mat3} {u : Coord3 → Fin 2 → ℝ} |
| (hH : ContDiff ℝ (↑(⊤ : ℕ∞)) H) (hu : ContDiff ℝ (↑(⊤ : ℕ∞)) u) (j : Fin 2) : |
| ContDiff ℝ (↑(⊤ : ℕ∞)) (fun x => (H x*gradientColumns (fderiv ℝ u x)).col j) := by |
| have hm : ContDiff ℝ (↑(⊤ : ℕ∞)) (fun x => H x*gradientColumns (fderiv ℝ u x)) := by |
| rw [contDiff_iff_contDiffAt] |
| intro x |
| exact contDiffAt_matrix_mul hH.contDiffAt (contDiff_gradientColumns u hu).contDiffAt |
| apply contDiff_pi.mpr |
| intro i |
| exact contDiff_pi.mp (contDiff_pi.mp hm i) j |
|
|
| lemma symmetricFlux_compact {H : Coord3 → Mat3} (hH : HasCompactSupport H) |
| (u : Coord3 → Fin 2 → ℝ) (j : Fin 2) : |
| HasCompactSupport (fun x => (H x*gradientColumns (fderiv ℝ u x)).col j) := |
| hH.of_isClosed_subset isClosed_closure (tensorColumn_tsupport H _ j) |
|
|
| theorem symmetric_cutoff_sources |
| (H : Coord3 → Mat3) (hH : ContDiff ℝ (↑(⊤ : ℕ∞)) H) (hs : HasCompactSupport H) |
| (hsy : ∀ x,(H x).IsSymm) |
| (u : Coord3 → Fin 2 → ℝ) (hu : ContDiff ℝ (↑(⊤ : ℕ∞)) u) : |
| (∀ j,ContDiff ℝ (↑(⊤ : ℕ∞)) (symmetricSource H u j)) ∧ |
| (∀ j,HasCompactSupport (symmetricSource H u j)) ∧ |
| (∀ j (ψ : Coord3 → ℝ),ContDiff ℝ (↑(⊤ : ℕ∞)) ψ → |
| (∫ x,fderiv ℝ ψ x ((H x*gradientColumns (fderiv ℝ u x)).col j))= |
| -(∫ x,ψ x*symmetricSource H u j x)) ∧ |
| (∫ x,symmetricSource H u 0 x)=0 ∧ (∫ x,symmetricSource H u 1 x)=0 ∧ |
| (∫ x,u x 1*symmetricSource H u 0 x-u x 0*symmetricSource H u 1 x)=0 := by |
| have hr (j : Fin 2) : ContDiff ℝ (↑(⊤ : ℕ∞)) (symmetricSource H u j) := |
| coordinateDivergence_smooth (symmetricFlux_smooth hH hu j) |
| have hsr (j : Fin 2) : HasCompactSupport (symmetricSource H u j) := |
| coordinateDivergence_compact (symmetricFlux_compact hs u j) |
| have hw (j : Fin 2) (ψ : Coord3 → ℝ) (hψ : ContDiff ℝ (↑(⊤ : ℕ∞)) ψ) : |
| (∫ x,fderiv ℝ ψ x ((H x*gradientColumns (fderiv ℝ u x)).col j))= |
| -(∫ x,ψ x*symmetricSource H u j x) := |
| coordinateDivergence_weak (symmetricFlux_smooth hH hu j) (symmetricFlux_compact hs u j) ψ hψ |
| exact ⟨hr,hsr,hw,compact_symmetric_source_moments u hu H hsy _ hr hsr hw⟩ |
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|
| end ScalarConductivity |
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| end |
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| end OAI |
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