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159357f | 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 | import Mathlib
import OAI.Analysis.Conductivity.Sources.LocalPiola
import OAI.Analysis.Conductivity.Variational.CompactSymmetricMoments
import OAI.Analysis.Conductivity.Flux.CompactFluxDivergence
namespace OAI
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⟩
end ScalarConductivity
end
end OAI
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