| import Mathlib |
| import OAI.MathematicalPhysics.SheetFlows.Uniqueness |
| import OAI.MathematicalPhysics.SheetFlows.EffectiveForcing |
|
|
| namespace OAI |
|
|
| /-! SheetFlows sheet program. -/ |
|
|
| section |
| noncomputable section |
| open Set MeasureTheory |
| open scoped BigOperators |
| namespace Solenoidal |
|
|
| theorem sheet_theorem {N : ℕ} (D : SheetData N) (ν : ℝ) |
| (hν : 0 < ν) (hνcomp : ComputableReal ν) : |
| ∃ f u : Field, ∃ X : ℝ → Space → Space, |
| SpatiallyPeriodic f ∧ SpatiallyPeriodic u ∧ |
| Smooth f ∧ Smooth u ∧ Effective f ∧ |
| MeanZero f ∧ DivergenceFree f ∧ OnePeriodic f ∧ |
| ClassicalSolution ν f u (fun _ _ => 0) ∧ |
| (∀ v : Field, ∀ p : Pressure, ClassicalSolution ν f v p → |
| ∀ t, 0 ≤ t → ∀ x, v t x = u t x ∧ p t x = 0) ∧ |
| MaterialFlow u X ∧ |
| (∀ i : Fin N, ∀ y ∈ (D.source i).carrier, |
| toTorus (X 1 (sheet y)) = |
| toTorus (sheet (diagonalMap (D.source i) (D.target i) (D.ratio i) y))) ∧ |
| IntegerCollars u ∧ (∀ t x, advection u t x = 0) ∧ |
| EffectiveBounds f ∧ EffectiveBounds u ∧ |
| (N = 0 → f = 0 ∧ u = 0) := by |
| cases N with |
| | zero => exact sheet_theorem_empty D ν hν |
| | succ N => |
| obtain ⟨δ,hδ,hδsmall,hs,ht⟩ := D.exists_rational_collar |
| have hδsmall' : (δ:ℝ) ≤ 1/10 := by |
| have hc : δ*10 ≤ 1 := (le_div_iff₀ (by norm_num)).mp hδsmall |
| have hc' : (δ:ℝ)*10 ≤ 1 := by exact_mod_cast hc |
| linarith |
| let d : Collar := ⟨(δ:ℝ), by exact_mod_cast hδ, hδsmall'⟩ |
| let P := sheetProgram d D |
| let v : Fin P.length → Space → Space := fun i => (P.get i).field |
| obtain ⟨hfsp,husp,hfs,hus,hfm,_,hfd,_,hfp,hup,hsol,hcol,hadv,_,_⟩ := |
| scheduled_direct_forcing v (fun i => (P.get i).smooth) |
| (fun i => (P.get i).periodic) (fun i => (P.get i).mean_zero) |
| (fun i => (P.get i).divergence_zero) (fun i => (P.get i).nonlinear_zero) ν |
| have hformula : HasRationalFormula (scheduleVelocity v) := |
| sheetVelocity_formula d ⟨δ,rfl⟩ D |
| obtain ⟨heff,hbounds⟩ := hformula.directForce_effective hus husp hup hνcomp |
| obtain ⟨hflow,hend⟩ := schedule_material_flow v (fun i => (P.get i).straight) |
| refine ⟨globalDirectForce ν (scheduleVelocity v), scheduleVelocity v, |
| fun t a => repeatedPath v a t, hfsp,husp,hfs,hus,heff,hfm,hfd,hfp,hsol, |
| ?_,hflow,?_,hcol,hadv,hbounds,hformula.circuit.effectiveBounds hus husp hup,?_⟩ |
| · intro w p hw t ht x |
| exact hsol.unique hw hν.le ht x |
| · intro i y hy |
| dsimp only |
| rw [hend,stageState_list_eq P, sheetProgram_on d D hs ht i hy] |
| · simp |
|
|
| end Solenoidal |
| end |
| end |
|
|
|
|
| end OAI |
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|
