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f18e7de | 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 | import OAI.MathematicalPhysics.Boltzmann.Compactness.PeriodicJointBudgetLimit
namespace OAI
/-! # Separate collision integrability from entropy and dissipation budgets -/
noncomputable section
open MeasureTheory
open scoped ENNReal
namespace BoltzmannNonuniqueness.PeriodicApproximation
open CollisionCompactness
variable {X : Type*} [MeasurableSpace X] {μ : Measure X} [SFinite μ]
theorem normalized_collisions_of_entropy {g : X × Velocity → ℝ}
(hgm : Measurable g) (hgn : ∀ p, 0 ≤ g p)
(hE : Integrable (entropyGauge g) (μ.prod volume))
(hD : rateDissipation μ g < ⊤) :
Integrable (fun p => gainDensity g p/(1+g p/maxwellian p.2)) (μ.prod volume) ∧
Integrable (fun p => lossDensity g p/(1+g p/maxwellian p.2)) (μ.prod volume) ∧
∀ᵐ p ∂μ.prod (volume : Measure Velocity),
Integrable (fun q => postRate g (collisionGrouping.symm (p,q)))
((volume : Measure Velocity).prod sphereArea) ∧
Integrable (fun q => preRate g (collisionGrouping.symm (p,q)))
((volume : Measure Velocity).prod sphereArea) := by
have hI := integrable_of_entropyGauge hgm hgn hE
apply normalized_gainLoss_and_fibers hgm hgn hI _ hD
have hL := linearMoment_integrable hgm hgn hE
convert hL.sub hI using 1
funext p
simp only [linearMoment,Pi.sub_apply]
ring
end BoltzmannNonuniqueness.PeriodicApproximation
end
end OAI
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