| import Mathlib |
| import OAI.Analysis.Conductivity.Sobolev.CentralPairRegularity |
| import OAI.Analysis.Conductivity.Fourier.TorusSpectralContinuation |
| import OAI.Analysis.Conductivity.Branching.TorusEndingCorrection |
| |
| namespace OAI |
|
|
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Set Filter Topology MeasureTheory Matrix UnitAddTorus |
|
|
| variable {E : Type*} [AddCommGroup E] |
|
|
| def branchNormalize (i : Fin 3) (u : Fin 2 → E) : Fin 2 → E := |
| if i=0 then ![-u 0,u 1-u 0] else if i=1 then u else ![u 1,u 0] |
|
|
| lemma branchNormalize_involutive (i : Fin 3) : Function.Involutive (branchNormalize (E:=E) i) := by |
| intro u |
| ext j |
| fin_cases i < |
|
|
| lemma branchNormalize_add (i : Fin 3) (u v : Fin 2 → E) : |
| branchNormalize i (u+v)=branchNormalize i u+branchNormalize i v := by |
| ext j |
| fin_cases i < |
|
|
| variable [Module ℝ E] |
|
|
| def branchNormalizeLinear (i : Fin 3) : (Fin 2 → E) →ₗ[ℝ] (Fin 2 → E) where |
| toFun := branchNormalize i |
| map_add' := branchNormalize_add i |
| map_smul' c u := by |
| ext j |
| fin_cases i < |
|
|
| def branchNormalizeEquiv (i : Fin 3) : (Fin 2 → ℝ) ≃L[ℝ] (Fin 2 → ℝ) := |
| ({ toLinearMap := branchNormalizeLinear i |
| invFun := branchNormalize i |
| left_inv := branchNormalize_involutive i |
| right_inv := branchNormalize_involutive i } : |
| (Fin 2 → ℝ) ≃ₗ[ℝ] (Fin 2 → ℝ)).toContinuousLinearEquiv |
|
|
| @[simp] lemma branchNormalizeEquiv_apply (i : Fin 3) (u : Fin 2 → ℝ) : |
| branchNormalizeEquiv i u=branchNormalize i u := rfl |
|
|
| lemma branchNormalize_basis_slopes (i : Fin 3) : |
| branchNormalize i (fun j => centralBasisSlopes j i)=normalizedSlope := by |
| ext j |
| fin_cases i < |
|
|
| lemma branchNormalize_normalized_slope (i : Fin 3) : |
| branchNormalize i normalizedSlope=(fun j => centralBasisSlopes j i) := by |
| rw [←branchNormalize_basis_slopes i,branchNormalize_involutive i] |
|
|
| lemma branchNormalize_affineField {s : Fin 3 → ℝ} |
| (hs : ∀ x y : ℝ,(1/2)*(x^2+y^2) ≤ s 0*x^2+2*s 1*x*y+s 2*y^2) |
| (i : Fin 3) (κ : Fin 2 → ℝ) (f : Fin 2 → TorusL2) {x : Coord3} (hx : 0<x 0) : |
| branchNormalize i (fun j => torusAffineField s (κ j) (f j) x)= |
| fun j => torusAffineField s (branchNormalize i κ j) (branchNormalize i f j) x := by |
| ext j |
| fin_cases i < |
| simp [branchNormalize,torusAffineField,torusRealContinuation_neg, |
| torusRealContinuation_sub hs _ _ hx] < |
|
|
| lemma branchNormalize_means (i : Fin 3) (f : Fin 2 → TorusL2) : |
| branchNormalize i (fun j => (mFourierCoeff (f j) 0).re)= |
| fun j => (mFourierCoeff (branchNormalize i f j : TorusL2) 0).re := by |
| fin_cases i |
| · ext j |
| fin_cases j |
| · change -(mFourierCoeff (f 0) 0).re=(mFourierCoeff (-f 0 : TorusL2) 0).re |
| rw [mFourierCoeff_Lp_neg,Complex.neg_re] |
| · change (mFourierCoeff (f 1) 0).re-(mFourierCoeff (f 0) 0).re= |
| (mFourierCoeff (f 1-f 0 : TorusL2) 0).re |
| simp only [sub_eq_add_neg] |
| rw [mFourierCoeff_Lp_add,mFourierCoeff_Lp_neg,Complex.add_re,Complex.neg_re] |
| · rfl |
| · ext j |
| fin_cases j < |
|
|
| end ScalarConductivity |
| |
|
|
|
|
| namespace ScalarConductivity |
| open Set Filter Topology MeasureTheory Matrix UnitAddTorus |
| open scoped Matrix.Norms.Elementwise |
| |
| namespace TorusEndingData |
| variable {s : Fin 3 → ℝ} {f : Fin 2 → TorusL2} (z : TorusEndingData s f) |
|
|
| lemma terminal_component (j : Fin 2) (x : Coord3) (hx : z.R≤x 0) : |
| z.w.v x j=normalizedSlope j*x 0+(mFourierCoeff (f j) 0).re := by |
| rw [z.terminal_v x hx] |
| fin_cases j < |
|
|
| lemma terminal_gradient (j : Fin 2) (x : Coord3) (hx : z.R<x 0) : |
| (gradientColumns (fderiv ℝ z.w.v x)).col j=normalizedSlope j • Pi.single 0 1 := by |
| have he : (fun x => z.w.v x j)=ᶠ[𝓝 x] |
| (fun x => normalizedSlope j*x 0+(mFourierCoeff (f j) 0).re) := by |
| filter_upwards [(axial_halfspace_open z.R).mem_nhds hx] with y hy |
| exact z.terminal_component j y hy.le |
| have hd := (((ContinuousLinearMap.proj (0 : Fin 3) : Coord3 →L[ℝ] ℝ).hasFDerivAt |
| (x:=x)).const_mul (normalizedSlope j)).add_const ((mFourierCoeff (f j) 0).re) |
| ext i |
| change fderiv ℝ z.w.v x (Pi.single i 1) j=normalizedSlope j*((Pi.single (0:Fin 3) (1:ℝ) : Coord3) i) |
| rw [←potential_component_fderiv (z.w.smooth.differentiable (by norm_num)) j, |
| he.fderiv_eq (𝕜 := ℝ)] |
| change fderiv ℝ (fun x => normalizedSlope j * |
| (ContinuousLinearMap.proj (0 : Fin 3) : Coord3 →L[ℝ] ℝ) x + |
| (mFourierCoeff (f j) 0).re) x (Pi.single i 1)=_ |
| rw [hd.fderiv] |
| fin_cases i < |
|
|
| lemma terminal_flux (j : Fin 2) (x : Coord3) (hx : z.R<x 0) : |
| (z.w.G x).col j=normalizedSlope j • Pi.single 0 1 := by |
| have ha : (fun x => (z.w.G x).col j)=ᵐ[volume.restrict {x : Coord3 | z.R<x 0}] |
| (fun _ => normalizedSlope j • Pi.single 0 1) := by |
| filter_upwards [ae_restrict_of_ae z.w.constitution, |
| ae_restrict_mem (axial_halfspace_open z.R).measurableSet] with y hy hyR |
| rw [←hy] |
| change z.w.E y*ᵥ(gradientColumns (fderiv ℝ z.w.v y)).col j=_ |
| rw [z.terminal_gradient j y hyR,Matrix.mulVec_smul,z.terminal_E y hyR.le] |
| exact (Measure.eqOn_open_of_ae_eq ha (axial_halfspace_open z.R) |
| (z.w.component_flux_C1 j).continuous.continuousOn continuous_const.continuousOn) hx |
| |
| lemma fluxCorrection_at_zero |
| (hs : ∀ x y : ℝ,(1/2)*(x^2+y^2) ≤ s 0*x^2+2*s 1*x*y+s 2*y^2) |
| (j : Fin 2) (x : Coord3) (hx : x 0=0) : z.fluxCorrection j x=0 := by |
| exact (torusEndingFluxCorrection_initial_zero z.e_pos (fun y hy => z.initial_flux hs j y hy) |
| (by rw [hx]; linarith [z.d_pos,z.e_pos])).eq_of_nhds |
| |
| lemma fluxCorrection_at_terminal (j : Fin 2) (x : Coord3) (hx : z.R<x 0) : |
| z.fluxCorrection j x=normalizedSlope j • Pi.single 0 1- |
| flatModeFlux s (torusAffineField s (normalizedSlope j) (f j)) x := by |
| rw [show z.fluxCorrection j x=(z.w.G x).col j- |
| flatModeFlux s (torusAffineField s (normalizedSlope j) (f j)) x from |
| (torusEndingFluxCorrection_terminal (by linarith [z.e_pos,z.overlap_lt] : z.d+z.e/2<x 0)).eq_of_nhds, |
| z.terminal_flux j x hx] |
| |
| end TorusEndingData |
| end ScalarConductivity |
| |
| end |
| |
| end OAI |
| |
