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37bcf52 | 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 | import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.LoadedMachine
import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.SlowBundle
namespace OAI
/-! Analytic properties and the fixed-particle observation for the
square-integrable forcing construction. -/
noncomputable section
namespace ForcedComputation
open ShearFlows Recorder Set
open scoped ContDiff NNReal
def fixedParticleVelocity (I : Alternating.MachineInput) (hI : Alternating.ValidInput I) : Velocity :=
slowFromRest (loadedMachineVelocity fixedStart 0 I hI)
theorem fixed_particle_analytic_computation (I : Alternating.MachineInput)
(hI : Alternating.ValidInput I) (ν : ℝ) (hν : 0 < ν) :
SlowFluidProperties 1 ν (fixedParticleVelocity I hI) ∧
∃ Ψ : ℝ → Space → Space, IsMaterialFlow 1 (fixedParticleVelocity I hI) Ψ ∧
(Alternating.Halts I ↔ ∃ t : ℝ, 0 ≤ t ∧
1 / 2 < Ψ t ![1 / 8, 3 / 8, 0] 0 ∧ Ψ t ![1 / 8, 3 / 8, 0] 0 < 1) := by
let loader := shiftedLoader fixedStart (initialPointQ I hI) 0
let body := recorderInputAtHeight I.1 hI.1 0
let V := loadedMachineVelocity fixedStart 0 I hI
have hl : ValidInput loader := shiftedLoader_valid _ _ _
(by intro j; fin_cases j <;> norm_num [fixedStart]) (initialPointQ_bounds I hI)
have hb : ValidInput body := recorderInputAtHeight_valid _ _ _
have hp : loader.period = body.period := rfl
have hbodyPeriod : (body.period : ℝ) = 1 := recorderInputAtHeight_period _ _ _
have hprops := slow_initialized_bundle hl hb hp ν hν.le
rw [hbodyPeriod] at hprops
refine ⟨hprops, ?_⟩
obtain ⟨_, Φ, hΦ, hobs⟩ := loadedMachine_computation fixedStart 0
(by intro j; fin_cases j <;> norm_num [fixedStart]) (by norm_num [fixedStart]) I hI ν hν
simp only [Rat.cast_zero] at hobs
obtain ⟨K, hK⟩ := initializedProgram_spatial_lipschitz hl hb
have hs : ContDiff ℝ ∞ V := initializedProgram_smooth hl hb
have hper : SpatiallyPeriodic 1 V := by
change SpatiallyPeriodic 1 (initializedProgram loader body)
simpa only [hbodyPeriod] using initializedProgram_periodic hp
have hz : ∀ t, t < (1 / 32 : ℝ) → ∀ x, V (t, x) = 0 :=
fun _ ht x => initializedProgram_zero_extend loader body ht x
obtain ⟨B, hB, hbound⟩ := initializedProgram_bounded hl hb hp []
obtain ⟨Ψ, hΨ⟩ := slowFromRest_materialFlow hs hper hz hK hB hbound
refine ⟨Ψ, hΨ, ?_⟩
have he := slowFromRest_reaches_iff (fun x => hz 0 (by norm_num) x) hΦ hΨ hK
(atHeight (fun j => (fixedStart j : ℝ)) 0) {x : Space | 1 / 2 < x 0 ∧ x 0 < 1}
simp only [Reaches, mem_ofPred_eq] at he
have hevent := hobs.trans he.symm
simpa only [Reaches, mem_ofPred_eq, fixedStart, atHeight,
Matrix.cons_val_zero, Matrix.cons_val_one, Rat.cast_div, Rat.cast_one,
Rat.cast_ofNat, Rat.cast_zero] using hevent
end ForcedComputation
end
end OAI
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