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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 | import Mathlib
import OAI.Analysis.Conductivity.Fourier.PeriodicBandCompactness
import OAI.Analysis.Conductivity.Walls.EndingJoinCutoff
namespace OAI
section
noncomputable section
namespace ScalarConductivity
open Set Filter Topology Real MeasureTheory Matrix
open scoped Matrix.Norms.Elementwise
def delayedInitialTensor (lam k J R : ℝ) (x : Coord3) : Mat3 :=
fun i j => pureModeTensor (initialConnectorAngular (lam/k) (1/2) J (k*(x 0-R))) 1
(angularAxisSwap i) (angularAxisSwap j)
lemma delayedInitialTensor_smooth (lam k J R : ℝ) :
ContDiff ℝ (↑(⊤:ℕ∞)) (delayedInitialTensor lam k J R) := by
have hq : ContDiff ℝ (↑(⊤:ℕ∞)) (fun x : Coord3 => initialConnectorAngular (lam/k) (1/2) J (k*(x 0-R))) :=
(initialConnectorAngular_smooth _ _ _).comp (contDiff_const.mul ((contDiff_apply ℝ ℝ 0).sub contDiff_const))
have hh := pureModeTensor_smooth hq (1:Fin 3)
apply contDiff_pi.mpr; intro i
apply contDiff_pi.mpr; intro j
exact contDiff_pi.mp (contDiff_pi.mp hh (angularAxisSwap i)) (angularAxisSwap j)
lemma delayedInitialTensor_symm (lam k J R : ℝ) (x : Coord3) :
(delayedInitialTensor lam k J R x).IsSymm := by
ext i j
exact congrFun (congrFun (pureModeTensor_symm _ _) (angularAxisSwap i)) (angularAxisSwap j)
lemma delayedEndingTensor_initial {lam k J L K R : ℝ} (x : Coord3) (hx : k*(x 0-R)<J+2) :
delayedEndingTensor lam k J L K R x=delayedInitialTensor lam k J R x := by
ext i j
change finiteEndingTensor (lam/k) J L K (permutedDilation angularAxisSwap k (x-Pi.single 0 R))
(angularAxisSwap i) (angularAxisSwap j)=_
have ht : (permutedDilation angularAxisSwap k (x-Pi.single 0 R)) 0=k*(x 0-R) := by
simp [permutedDilation,angularAxisSwap_apply]
rw [finiteEndingTensor,ht,ite_eq_left hx]
rfl
def endingJoinMiddleTensor (s : Fin 3 → ℝ) (lam k J : ℝ) (x : Coord3) : Symmetric3 :=
⟨(1-endingJoinCutoff (x 0)) • flatBackgroundTensor s+
endingJoinCutoff (x 0) • delayedInitialTensor lam k J 7 x,
((flatBackgroundTensor_symm s).smul _).add ((delayedInitialTensor_symm _ _ _ _ _).smul _)⟩
lemma endingJoinMiddleTensor_smooth (s : Fin 3 → ℝ) (lam k J : ℝ) :
ContDiff ℝ (↑(⊤:ℕ∞)) (fun x => (endingJoinMiddleTensor s lam k J x).val) := by
have hχ := endingJoinCutoff_smooth.comp (contDiff_apply ℝ ℝ (0:Fin 3))
exact ((contDiff_const.sub hχ).smul contDiff_const).add (hχ.smul (delayedInitialTensor_smooth _ _ _ _))
lemma endingJoinMiddle_regular_ae (s : Fin 3 → ℝ) (lam J : ℝ) {k : ℝ} (hk : k≠0) :
∀ᵐ x : Coord3,x∈regularRegion (wallCoordinatePair (pureWallMode 1 lam k))
(endingJoinMiddleTensor s lam k J) univ := by
have hu : ContDiff ℝ (↑(⊤:ℕ∞)) (wallCoordinatePair (pureWallMode 1 lam k)) :=
wallCoordinatePair_smooth (pureWallMode_smooth _ _ _)
have hh := regularRegion_ae_of_simple_minor hu (endingJoinMiddleTensor_smooth s lam k J) isOpen_univ
(fun x _ hx => (pureMode_pairMinor_noncritical (by norm_num : (1:ℝ)≠0) hk x).resolve_left (not_not.mpr hx))
filter_upwards [hh] with x hx
exact hx (mem_univ _)
lemma wallCoordinatePair_pure (lam k : ℝ) (x : Coord3) :
wallCoordinatePair (pureWallMode 1 lam k) x=![x 0,exp (-lam*x 0)*cos (k*x 2)] := by
simp [wallCoordinatePair,pureWallMode,boxCoordinates_apply]
end ScalarConductivity
end
end
end OAI
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