openAI-math-dataset / lean /OAI /MathematicalPhysics /Boltzmann /Compactness /LocalizedGainCompactness.lean
| import OAI.MathematicalPhysics.Boltzmann.Branching.GainSmoothingError | |
| import OAI.MathematicalPhysics.Boltzmann.Collision.VelocityCutoffEntropyError | |
| import OAI.MathematicalPhysics.Boltzmann.Compactness.StrongCompactPairTests | |
| namespace OAI | |
| /-! # Removing velocity smoothing from entropy-controlled localized gains -/ | |
| noncomputable section | |
| open MeasureTheory Filter | |
| open scoped ENNReal Topology | |
| namespace BoltzmannNonuniqueness.CollisionCompactness.DoubleGain | |
| open Compactness PeriodicApproximation | |
| variable {X : Type*} [MeasurableSpace X] {μ : Measure X} [IsFiniteMeasure μ] | |
| theorem velocitySmooth_error_eLpNorm_le {χ : Velocity → ℝ} {F : X × Velocity → ℝ} | |
| (hχ : Integrable χ) (hF : Integrable F (μ.prod volume)) {ε : ℝ} | |
| (he : (∫ p, |velocitySmooth χ F p-F p| ∂μ.prod volume) ≤ ε) : | |
| eLpNorm (velocitySmooth χ F-F) 1 (μ.prod volume) ≤ ENNReal.ofReal ε := by | |
| have hi := (velocitySmooth_integrable hχ hF).sub hF | |
| rw [eLpNorm_one_eq_lintegral_enorm hi.aestronglyMeasurable, | |
| ← ofReal_integral_norm_eq_lintegral_enorm hi] | |
| exact ENNReal.ofReal_le_ofReal (by simpa only [Real.norm_eq_abs,Pi.sub_apply] using he) | |
| theorem localizedGain_strong_of_smoothing | |
| {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)) | |
| (η : C(Velocity,ℝ)) (hη : HasCompactSupport η) | |
| {ν : Measure (X × Velocity)} (hν : ν ≤ μ.prod volume) | |
| (hc : ∀ χ : C(Velocity,ℝ), HasCompactSupport χ → | |
| Tendsto (fun n => eLpNorm | |
| (velocitySmooth χ (localizedPhysicalGain η (f n))- | |
| velocitySmooth χ (localizedPhysicalGain η g)) 1 ν) atTop (𝓝 0)) : | |
| Tendsto (fun n => eLpNorm | |
| (localizedPhysicalGain η (f n)-localizedPhysicalGain η g) 1 ν) atTop (𝓝 0) := by | |
| let ε : ℕ → ℝ := fun j => 1/((j:ℝ)+1) | |
| have hε j : 0 < ε j := by dsimp [ε]; positivity | |
| have hchoice j : ∃ χ : C(Velocity,ℝ), HasCompactSupport χ ∧ Integrable χ ∧ | |
| (∀ n, (∫ p, |velocitySmooth χ (localizedPhysicalGain η (f n)) p- | |
| localizedPhysicalGain η (f n) p| ∂μ.prod volume) ≤ ε j) ∧ | |
| (∫ p, |velocitySmooth χ (localizedPhysicalGain η g) p- | |
| localizedPhysicalGain η g p| ∂μ.prod volume) ≤ ε j := by | |
| obtain ⟨d,hd,hds⟩ := localizedGain_uniform_smoothing hfm hf hi hb η hη (ε j) (hε j) | |
| obtain ⟨e,he,hes⟩ := localizedGain_uniform_smoothing | |
| (fun _ : ℕ => hgm) (fun _ => hg) (fun _ => hgi) | |
| (fun _ => le_refl (∫ p, entropyGauge g p ∂μ.prod volume)) η hη (ε j) (hε j) | |
| obtain ⟨χ,hχc,hχi,hχ0,hχm,hχs⟩ := exists_velocity_mollifier (lt_min hd he) | |
| refine ⟨χ,hχc,hχi,?_,?_⟩ | |
| · exact hds χ hχi hχ0 hχm (fun a ha => (hχs a ha).trans_le (min_le_left _ _)) | |
| · exact hes χ hχi hχ0 hχm (fun a ha => (hχs a ha).trans_le (min_le_right _ _)) 0 | |
| choose χ hχc hχi hχf hχg using hchoice | |
| have hδ : Tendsto (fun j => ENNReal.ofReal (ε j)) atTop (𝓝 0) := by | |
| have he : Tendsto ε atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat | |
| simpa only [ENNReal.ofReal_zero,Function.comp_def] using ENNReal.continuous_ofReal.tendsto 0 |>.comp he | |
| have hFi n : Integrable (localizedPhysicalGain η (f n)) (μ.prod volume) := by | |
| apply localizedPhysicalGain_integrable η hη (hfm n) | |
| simpa only [Measure.volume_eq_prod] using weightedNormalizedPair_integrable (hfm n) (hf n) (hi n) | |
| have hGi : Integrable (localizedPhysicalGain η g) (μ.prod volume) := by | |
| apply localizedPhysicalGain_integrable η hη hgm | |
| simpa only [Measure.volume_eq_prod] using weightedNormalizedPair_integrable hgm hg hgi | |
| apply strongL1_of_two_sided_approximations hδ | |
| (u := fun j n => velocitySmooth (χ j) (localizedPhysicalGain η (f n))) | |
| (v := fun j => velocitySmooth (χ j) (localizedPhysicalGain η g)) | |
| · intro j n | |
| rw [eLpNorm_sub_comm] | |
| exact (eLpNorm_mono_measure _ hν).trans | |
| (velocitySmooth_error_eLpNorm_le (hχi j) (hFi n) (hχf j n)) | |
| · intro j | |
| exact (eLpNorm_mono_measure _ hν).trans | |
| (velocitySmooth_error_eLpNorm_le (hχi j) hGi (hχg j)) | |
| · exact fun j => hc (χ j) (hχc j) | |
| end BoltzmannNonuniqueness.CollisionCompactness.DoubleGain | |
| end | |
| end OAI | |