lean/OAI/Analysis/Conductivity/Sources/PhysicalPatching.lean

import Mathlib
import OAI.Analysis.Conductivity.Sobolev.SobolevGaugeTransform
import OAI.Analysis.Conductivity.Walls.PhysicalTerminalCutoff

namespace OAI

section

noncomputable section
namespace ScalarConductivity
open Set Filter Topology MeasureTheory Matrix UnitAddTorus
attribute [local instance] Classical.propDecidable
namespace PhysicalFiniteEndingData
variable {s : Fin 3 → ℝ} (z : PhysicalFiniteEndingData s)

lemma terminal_band_value
    (hs : ∀ x y : ℝ,(1/2)*(x^2+y^2)≤ s 0*x^2+2*s 1*x*y+s 2*y^2)
    {η : ℝ} (hη : 0<η) (hηc : 2*η<centralThickness)
    (hT : ∀ i : Fin 3,∀ t : ℝ,|t|≤η → (z.ending i).R<z.compression i*(t-terminalBase i))
    (w : Fin 2 → H1) (he : ∀ j i,H1JetOn (w j) (physicalEndRegion i) (z.correctedEndJet j i))
    (j : Fin 2) :
    ∀ᵐ y : Coord3,y∈physicalOpenBlock → y∉terminalCore η →
      weakValue (w j) (WithLp.toLp 2 y)-z.terminalValue 0 j=z.terminalProfile η j y := by
  have hev (i : Fin 3) := (he j i).value_cartesian (fun _ => physicalEndRegion_subset_ball i)
  filter_upwards [ae_all_iff.mpr hev,ae_all_iff.mpr (fun i => z.correctedEndValue_terminal_ae hs i j)]
    with y hv ht hy hn
  have hy' := (sourceOpenBlock_time_iff y).mp hy
  change 0<terminalTime 0 y ∧ ∀ k : Fin 2,terminalTime k.succ y<0 at hy'
  by_cases hp : terminalTime 0 y≤η
  · have he0 : y∈physicalEndRegion 0 := by
      change 0≤terminalTime 0 y ∧ terminalTime 0 y≤centralThickness
      exact ⟨hy'.1.le,by linarith⟩
    have hTa : |terminalTime 0 y|≤η := by rw [abs_of_pos hy'.1]; exact hp
    have hh := (hv 0 he0).trans (ht 0 he0 (hT 0 _ hTa).le)
    rw [z.terminalProfile_parent_band hη hηc j hy'.1.le hp]
    linarith
  · have hn' : ¬∀ k : Fin 2,terminalTime k.succ y≤-η := fun h => hn ⟨(lt_of_not_ge hp).le,h⟩
    push Not at hn'
    obtain ⟨k,hk⟩ := hn'
    have hc : terminalTime k.succ y≤0 := (hy'.2 k).le
    have hek : y∈physicalEndRegion k.succ := by
      change -centralThickness≤terminalTime k.succ y ∧ terminalTime k.succ y≤0
      exact ⟨by linarith,hc⟩
    have hTa : |terminalTime k.succ y|≤η := by rw [abs_of_nonpos hc]; linarith
    have hh := (hv k.succ hek).trans (ht k.succ hek (hT k.succ _ hTa).le)
    rw [z.terminalProfile_child_band hη hηc j k hk.le,min_eq_left hc]
    linarith

theorem source_patch_exists
    (hs : ∀ x y : ℝ,(1/2)*(x^2+y^2)≤ s 0*x^2+2*s 1*x*y+s 2*y^2)
    {η : ℝ} (hη : 0<η) (hηc : 2*η<centralThickness)
    (hT : ∀ i : Fin 3,∀ t : ℝ,|t|≤η → (z.ending i).R<z.compression i*(t-terminalBase i))
    (w q : Fin 2 → H1) (he : ∀ j i,H1JetOn (w j) (physicalEndRegion i) (z.correctedEndJet j i))
    (Z : VoltageJetSpace volume physicalOpenBlock)
    (hZ : Z∈zeroVoltageJets volume physicalOpenBlock)
    (hq : ∀ j,(q j-w j).val=voltageOriginalJetCLM physicalOpenBlock_open.measurableSet j Z)
    (j : Fin 2) :
    ∃ W : H1,W∈H10 ∧ (∀ᵐ x∂ballMeasure,weakValue W x=
      if WithLp.ofLp x∈physicalOpenBlock then weakValue (q j) x-z.terminalValue 0 j
      else z.terminalProfile η j (WithLp.ofLp x)) := by
  obtain ⟨f,hf,hfv⟩ := z.terminalProfile_H10 hη hηc j
  obtain ⟨χ,hχ,hχc,hχs,hχ1⟩ := terminal_cutoff_exists hη
  let E : R3 ≃L[ℝ] Coord3 := PiLp.continuousLinearEquiv 2 ℝ (fun _ : Fin 3 => ℝ)
  have hχd : ContDiff ℝ (↑(⊤:ℕ∞)) (χ ∘ E) := hχ.comp E.contDiff
  have hχcc : HasCompactSupport (χ ∘ E) := hχc.comp_homeomorph E.toHomeomorph
  have hχss : tsupport (χ ∘ E)⊆ball := by
    have hh : tsupport (χ ∘ E)⊆E ⁻¹' tsupport χ :=
      closure_minimal (fun y hy => subset_tsupport χ hy) ((isClosed_tsupport χ).preimage E.continuous)
    intro x hx
    exact physicalOpenBlock_subset_ball (hχs (hh hx))
  let u := w j-constantH1 (z.terminalValue 0 j)-f
  obtain ⟨v,hv,hvv⟩ := original_smooth_mul_exists hχd hχcc hχss u
  have hd : q j-w j∈H10 := by
    change (q j-w j).val∈zeroTraceAmbient
    rw [hq]
    exact voltageOriginalJetCLM_zeroTrace physicalOpenBlock_open.measurableSet
      (fun _ => physicalOpenBlock_subset_ball) j hZ
  have hband := ae_restrict_of_ae (s:=ball) ((PiLp.volume_preserving_ofLp (Fin 3)).quasiMeasurePreserving.ae
    (z.terminal_band_value hs hη hηc hT w he j))
  have hdv : ∀ᵐ x∂ballMeasure,WithLp.ofLp x∉physicalOpenBlock → weakValue (q j-w j) x=0 := by
    have hz := voltageOriginalJetCLM_ae physicalOpenBlock_open.measurableSet j Z
    rw [←hq j] at hz
    filter_upwards [hz] with x hx hxn
    have hh := congrArg (fun f : JetFiber => f 0) hx
    change (q j-w j).val x 0=0
    rw [hx,ite_eq_right hxn]
    rfl
  refine ⟨v+f+(q j-w j),H10.add_mem (H10.add_mem hv hf) hd,?_⟩
  filter_upwards [hvv,hfv,hband,hdv,weakValue_add (v+f) (q j-w j),weakValue_add v f,
    weakValue_sub (w j-constantH1 (z.terminalValue 0 j)) f,
    (H1_sub_constant_spec (w j) (z.terminalValue 0 j)).1,weakValue_sub (q j) (w j)]
    with x hmv hfv hband hdv hsum hadd hu1 hu2 hd1
  rw [hsum]
  simp only [Pi.add_apply]
  rw [hadd]
  simp only [Pi.add_apply]
  have hu : weakValue u x=weakValue (w j) x-z.terminalValue 0 j-z.terminalProfile η j (WithLp.ofLp x) := by
    change weakValue (w j-constantH1 (z.terminalValue 0 j)-f) x=_
    rw [hu1,Pi.sub_apply,hu2,hfv]
  rw [hmv,hu,hfv]
  change χ (WithLp.ofLp x)*(weakValue (w j) x-z.terminalValue 0 j-z.terminalProfile η j (WithLp.ofLp x))+
    z.terminalProfile η j (WithLp.ofLp x)+weakValue (q j-w j) x=_
  by_cases hx : WithLp.ofLp x∈physicalOpenBlock
  · rw [ite_eq_left hx,hd1]
    simp only [Pi.sub_apply]
    by_cases hc : WithLp.ofLp x∈terminalCore η
    · rw [hχ1 _ hc]; ring
    · have hh := hband hx hc
      simp only [WithLp.toLp_ofLp] at hh
      rw [show weakValue (w j) x-z.terminalValue 0 j-z.terminalProfile η j (WithLp.ofLp x)=0 by linarith]
      linarith
  · rw [ite_eq_right hx,hdv hx]
    have hz : χ (WithLp.ofLp x)=0 := image_eq_zero_of_notMem_tsupport (fun hn => hx (hχs hn))
    rw [hz]
    ring

end PhysicalFiniteEndingData
end ScalarConductivity

end
end

end OAI