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caec230 | 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 | 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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