openAI-math-dataset / lean /OAI /MathematicalPhysics /Boltzmann /Compactness /NormalizedGainCompactness.lean
| import OAI.MathematicalPhysics.Boltzmann.Compactness.LocalizedGainSmoothingAE | |
| import OAI.MathematicalPhysics.Boltzmann.Compactness.StrongCompactPairTests | |
| import Mathlib.Analysis.SpecialFunctions.Pow.Real | |
| import Mathlib.Tactic | |
| namespace OAI | |
| section | |
| /-! # Scalar majorants for the gain discrepancy of two positive profiles -/ | |
| namespace BoltzmannNonuniqueness.Compactness | |
| theorem normalized_difference_abs_le {a b u v : ℝ} (hu : 0 ≤ u) (hv : 0 ≤ v) : | |
| |(a-b)/(1+u+v)| ≤ |a-b| := by | |
| rw [abs_div,abs_of_pos (by positivity : 0 < 1+u+v)] | |
| exact div_le_self (abs_nonneg _) (by linarith) | |
| theorem normalized_difference_source_bound {a b u v : ℝ} | |
| (ha : 0 ≤ a) (hb : 0 ≤ b) (hu : 0 ≤ u) (hv : 0 ≤ v) : | |
| |(a-b)/(1+u+v)| ≤ a/(1+u)+b/(1+v) := by | |
| have hd : 0 < 1+u+v := by positivity | |
| calc | |
| _ = |a-b|/(1+u+v) := by rw [abs_div,abs_of_pos hd] | |
| _ ≤ (a+b)/(1+u+v) := by | |
| apply div_le_div_of_nonneg_right _ hd.le | |
| simpa only [abs_of_nonneg ha,abs_of_nonneg hb] using abs_sub a b | |
| _ = a/(1+u+v)+b/(1+u+v) := add_div _ _ _ | |
| _ ≤ _ := add_le_add | |
| (div_le_div_of_nonneg_left ha (by positivity) (by linarith)) | |
| (div_le_div_of_nonneg_left hb (by positivity) (by linarith)) | |
| end BoltzmannNonuniqueness.Compactness | |
| end | |
| section | |
| /-! # The linear normalized-pair gain is the physical gain divided by spatial mass -/ | |
| noncomputable section | |
| open MeasureTheory | |
| namespace BoltzmannNonuniqueness.CollisionCompactness.DoubleGain | |
| open Compactness | |
| variable {X : Type*} [MeasurableSpace X] | |
| omit [MeasurableSpace X] in | |
| theorem normalizedPair_postRate (f : X × Velocity → ℝ) | |
| (z : (X × (Velocity × Velocity)) × Sphere) : | |
| pairPostRate (normalizedPair volume f) z = massNormalizedPostRate f z := by | |
| change pairPreRate (normalizedPair volume f) (scatterTriple z) = | |
| massNormalizedPreRate f (scatterTriple z) | |
| exact (massNormalizedPreRate_eq_pair f (scatterTriple z)).symm | |
| theorem normalizedPair_gain (f : X × Velocity → ℝ) (p : X × Velocity) : | |
| pairGain (normalizedPair volume f) p = gainDensity f p/(1+fiberMass volume f p.1) := by | |
| unfold pairGain | |
| simp_rw [normalizedPair_postRate] | |
| change (∫ q : Velocity × Sphere, | |
| postRate f (collisionGrouping.symm (p,q))/(1+fiberMass volume f p.1) | |
| ∂(volume : Measure Velocity).prod sphereArea) = _ | |
| simp_rw [div_eq_mul_inv] | |
| rw [integral_mul_const] | |
| rfl | |
| end BoltzmannNonuniqueness.CollisionCompactness.DoubleGain | |
| end | |
| end | |
| section | |
| /-! # Strong convergence of full normalized gains from their localized limits -/ | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped ENNReal Topology | |
| namespace BoltzmannNonuniqueness.CollisionCompactness.DoubleGain | |
| open Compactness PeriodicApproximation | |
| variable {X : Type*} [MeasurableSpace X] {μ : Measure X} [IsFiniteMeasure μ] | |
| theorem normalizedGain_strong_of_localized | |
| {f : ℕ → X × Velocity → ℝ} {g : X × Velocity → ℝ} {C : ℝ} | |
| (hfm : ∀ n, Measurable (f n)) (hf : ∀ n p, 0 ≤ f n p) | |
| (hi : ∀ n, Integrable (entropyGauge (f n)) (μ.prod volume)) | |
| (hb : ∀ n, (∫ p, entropyGauge (f n) p ∂μ.prod volume) ≤ C) | |
| (hgm : Measurable g) (hg : ∀ p, 0 ≤ g p) | |
| (hgi : Integrable (entropyGauge g) (μ.prod volume)) | |
| (hgb : (∫ p, entropyGauge g p ∂μ.prod volume) ≤ C) | |
| {ν : Measure (X × Velocity)} (hν : ν ≤ μ.prod volume) | |
| (hc : ∀ η : C(Velocity,ℝ), HasCompactSupport η → | |
| Tendsto (fun n => eLpNorm | |
| (localizedPhysicalGain η (f n)-localizedPhysicalGain η g) 1 ν) atTop (𝓝 0)) : | |
| Tendsto (fun n => eLpNorm | |
| (pairGain (normalizedPair volume (f n))-pairGain (normalizedPair volume g)) 1 ν) | |
| atTop (𝓝 0) := by | |
| choose η hηc hη0 hη1 hηeq using | |
| (fun j : ℕ => exists_velocity_unit_cutoff (show 0 < (j:ℝ)+1 by positivity)) | |
| have hδ : Tendsto (fun j : ℕ => sphereArea univ*ENNReal.ofReal (8*C/((j:ℝ)+1))) | |
| atTop (𝓝 0) := by | |
| have he : Tendsto (fun j : ℕ => 8*C/((j:ℝ)+1)) atTop (𝓝 0) := by | |
| exact tendsto_const_nhds.div_atTop | |
| (tendsto_natCast_atTop_atTop.atTop_add tendsto_const_nhds) | |
| have ht := ENNReal.Tendsto.const_mul (a := sphereArea univ) (ENNReal.tendsto_ofReal he) | |
| (Or.inr (measure_ne_top sphereArea univ)) | |
| simpa only [ENNReal.ofReal_zero,mul_zero] using ht | |
| apply strongL1_of_two_sided_approximations hδ | |
| (u := fun j n => localizedPhysicalGain (η j) (f n)) | |
| (v := fun j => localizedPhysicalGain (η j) g) | |
| · intro j n | |
| exact (eLpNorm_mono_measure _ hν).trans | |
| (localizedGain_tail_le (by positivity) (η j) (hη0 j) (hη1 j) (hηeq j) | |
| (hfm n) (hf n) (hi n) (hb n)) | |
| · intro j | |
| rw [eLpNorm_sub_comm] | |
| exact (eLpNorm_mono_measure _ hν).trans | |
| (localizedGain_tail_le (by positivity) (η j) (hη0 j) (hη1 j) (hηeq j) hgm hg hgi hgb) | |
| · exact fun j => hc (η j) (hηc j) | |
| end BoltzmannNonuniqueness.CollisionCompactness.DoubleGain | |
| end | |
| end | |
| end OAI | |