openAI-math-dataset / lean /OAI /MathematicalPhysics /Boltzmann /Compactness /KineticEntropyErrorLimit.lean
| import OAI.MathematicalPhysics.Boltzmann.Approximation.RegularizedLogBounds | |
| import OAI.MathematicalPhysics.Boltzmann.Compactness.EntropyDissipationAverage | |
| namespace OAI | |
| section | |
| /-! # Integrable entropy tests for bounded kinetic densities -/ | |
| noncomputable section | |
| open MeasureTheory | |
| open BoltzmannNonuniqueness.VelocityWeight | |
| namespace BoltzmannNonuniqueness | |
| variable {X : Type*} [MeasurableSpace X] | |
| theorem kinetic_log_product_integrable {μ : Measure X} [IsFiniteMeasure μ] | |
| {q B C δ : ℝ} (hq₁ : 1 < q) (hq₂ : q < 2) (hB : 0 ≤ B) (hC : 0 ≤ C) (hδ : 0 < δ) | |
| {u a : X × Velocity → ℝ} (hum : Measurable u) (ham : Measurable a) | |
| (hl : ∀ p, δ*Real.exp (-2*‖p.2‖^2) ≤ u p) | |
| (hu : ∀ p, u p ≤ B*H q p.2) | |
| (ha : ∀ᵐ p ∂μ.prod (volume : Measure Velocity), |a p| ≤ C*H q p.2) : | |
| Integrable (fun p => Real.log (u p/maxwellian p.2)*a p) (μ.prod volume) := by | |
| obtain ⟨p,hp,hpi⟩ := exists_integrable_rpow_H hq₁ hq₂ | |
| have hmaj := ((logarithmicMajorant_integrable (B := B) (δ := δ) hq₂ hpi).const_mul C).comp_snd μ | |
| have hMm : Measurable (fun p : X × Velocity => maxwellian p.2) := by | |
| unfold maxwellian | |
| fun_prop | |
| apply hmaj.mono' (((hum.div hMm).log).mul ham).aestronglyMeasurable | |
| filter_upwards [ha] with z hz | |
| simp only [Pi.mul_apply] | |
| rw [Real.norm_eq_abs,abs_mul] | |
| calc | |
| _ ≤ |Real.log (u z/maxwellian z.2)| * (C*H q z.2) := | |
| mul_le_mul_of_nonneg_left hz (abs_nonneg _) | |
| _ = C*(|Real.log (u z/maxwellian z.2)| * H q z.2) := by ring | |
| _ ≤ _ := mul_le_mul_of_nonneg_left (logarithm_weight_bound hB hp hδ z.2 (hl z) (hu z)) hC | |
| end BoltzmannNonuniqueness | |
| end | |
| end | |
| section | |
| /-! # A quantitative integral bound for the kinetic logarithmic test -/ | |
| noncomputable section | |
| open MeasureTheory | |
| open BoltzmannNonuniqueness.VelocityWeight | |
| namespace BoltzmannNonuniqueness | |
| variable {X : Type*} [MeasurableSpace X] | |
| theorem kinetic_log_integral_bound {μ : Measure X} [IsFiniteMeasure μ] | |
| {q B C δ p : ℝ} (hq : q < 2) (hB : 0 ≤ B) (hC : 0 ≤ C) (hδ : 0 < δ) | |
| (hp : 1 < p) (hpi : Integrable (fun v : Velocity => H q v^p)) | |
| {u a : X × Velocity → ℝ} | |
| (hl : ∀ z, δ*Real.exp (-2*‖z.2‖^2) ≤ u z) (hu : ∀ z, u z ≤ B*H q z.2) | |
| (ha : ∀ᵐ z ∂μ.prod (volume : Measure Velocity), |a z| ≤ C*H q z.2) : | |
| |∫ z, Real.log (u z/maxwellian z.2)*a z ∂μ.prod volume| ≤ | |
| C*∫ z : X × Velocity, logarithmicMajorant q B p δ z.2 ∂μ.prod volume := by | |
| have hi := ((logarithmicMajorant_integrable (B := B) (δ := δ) hq hpi).const_mul C).comp_snd μ | |
| have hb : ∀ᵐ z ∂μ.prod (volume : Measure Velocity), | |
| ‖Real.log (u z/maxwellian z.2)*a z‖ ≤ C*logarithmicMajorant q B p δ z.2 := by | |
| filter_upwards [ha] with z hz | |
| rw [Real.norm_eq_abs,abs_mul] | |
| calc | |
| _ ≤ |Real.log (u z/maxwellian z.2)| * (C*H q z.2) := | |
| mul_le_mul_of_nonneg_left hz (abs_nonneg _) | |
| _ = C*(|Real.log (u z/maxwellian z.2)| * H q z.2) := by ring | |
| _ ≤ _ := mul_le_mul_of_nonneg_left (logarithm_weight_bound hB hp hδ z.2 (hl z) (hu z)) hC | |
| have h := norm_integral_le_of_norm_le hi hb | |
| simpa only [Real.norm_eq_abs,integral_const_mul] using h | |
| end BoltzmannNonuniqueness | |
| end | |
| end | |
| section | |
| /-! # The Gaussian entropy error is integrable on a finite kinetic base -/ | |
| noncomputable section | |
| open MeasureTheory | |
| open BoltzmannNonuniqueness.VelocityWeight | |
| namespace BoltzmannNonuniqueness | |
| variable {X : Type*} [MeasurableSpace X] | |
| def gaussianRegularizedDensity (δ : ℝ) (f : X × Velocity → ℝ) (p : X × Velocity) : ℝ := | |
| f p+δ*Real.exp (-2*‖p.2‖^2) | |
| theorem gaussianRegularizedDensity_measurable (δ : ℝ) {f : X × Velocity → ℝ} | |
| (hfm : Measurable f) : Measurable (gaussianRegularizedDensity δ f) := by | |
| unfold gaussianRegularizedDensity | |
| fun_prop | |
| theorem kinetic_regularization_error_bound (hcarleman : Published.GPV2009CarlemanInput) | |
| {μ : Measure X} [SFinite μ] {δ A q b : ℝ} | |
| (hδ : 0 ≤ δ) (hδone : δ ≤ 1) (hA : 0 ≤ A) | |
| (hq₁ : 1 < q) (hq₂ : q < 2) (hb : 0 ≤ b) | |
| {f : X × Velocity → ℝ} (hfm : Measurable f) (hfn : ∀ p, 0 ≤ f p) | |
| (hfb : ∀ p, f p ≤ A*H q p.2) : | |
| ∀ᵐ p ∂μ.prod (volume : Measure Velocity), | |
| |CollisionCompactness.collisionSource b (gaussianRegularizedDensity δ f) p- | |
| CollisionCompactness.collisionSource b f p| ≤ | |
| (δ*entropyErrorCoefficient q A b)*H q p.2 := by | |
| have hgm := gaussianRegularizedDensity_measurable δ hfm | |
| have hsm := (CollisionCompactness.collisionSource_measurable hgm b).sub | |
| (CollisionCompactness.collisionSource_measurable hfm b) | |
| have hH : Measurable (fun p : X × Velocity => (δ*entropyErrorCoefficient q A b)*H q p.2) := by | |
| unfold H singularity transverse | |
| fun_prop | |
| apply (Measure.ae_prod_iff_ae_ae (measurableSet_le hsm.abs hH)).mpr | |
| filter_upwards [] with x | |
| have hx := gaussianRegularization_collision_error hcarleman hδ hδone hA hq₁ hq₂ hb | |
| (hfm.comp (measurable_const.prodMk measurable_id)) (fun v => hfn (x,v)) (fun v => hfb (x,v)) | |
| exact hx | |
| theorem kinetic_regularization_entropy_integrable (hcarleman : Published.GPV2009CarlemanInput) | |
| {μ : Measure X} [IsFiniteMeasure μ] {δ A q b : ℝ} | |
| (hδ : 0 < δ) (hδone : δ ≤ 1) (hA : 0 ≤ A) | |
| (hq₁ : 1 < q) (hq₂ : q < 2) (hb : 0 ≤ b) | |
| {f : X × Velocity → ℝ} (hfm : Measurable f) (hfn : ∀ p, 0 ≤ f p) | |
| (hfb : ∀ p, f p ≤ A*H q p.2) : | |
| Integrable (fun p => Real.log (gaussianRegularizedDensity δ f p/maxwellian p.2)* | |
| (CollisionCompactness.collisionSource b (gaussianRegularizedDensity δ f) p- | |
| CollisionCompactness.collisionSource b f p)) (μ.prod volume) := by | |
| have hgm := gaussianRegularizedDensity_measurable δ hfm | |
| apply kinetic_log_product_integrable hq₁ hq₂ (add_nonneg hA tiltConstant_pos.le) | |
| (mul_nonneg hδ.le (entropyErrorCoefficient_nonneg hA hb q)) hδ hgm | |
| ((CollisionCompactness.collisionSource_measurable hgm b).sub | |
| (CollisionCompactness.collisionSource_measurable hfm b)) | |
| · intro p | |
| dsimp [gaussianRegularizedDensity] | |
| linarith [hfn p] | |
| · intro p | |
| exact gaussianRegularization_bound hδ.le hδone (fun v => hfb (p.1,v)) p.2 | |
| · exact kinetic_regularization_error_bound hcarleman hδ.le hδone hA hq₁ hq₂ hb hfm hfn hfb | |
| end BoltzmannNonuniqueness | |
| end | |
| end | |
| section | |
| /-! # Vanishing of the entropy-error majorant after integrating the finite base -/ | |
| noncomputable section | |
| open MeasureTheory Filter | |
| open scoped Topology | |
| open BoltzmannNonuniqueness.VelocityWeight | |
| namespace BoltzmannNonuniqueness | |
| variable {X : Type*} [MeasurableSpace X] | |
| theorem kinetic_entropy_error_tendsto (μ : Measure X) [SFinite μ] | |
| {q B p : ℝ} (C : ℝ) (hq : q < 2) | |
| (hp : Integrable (fun v : Velocity => H q v^p)) : | |
| Tendsto (fun δ : ℝ => δ*C*(∫ z : X × Velocity, | |
| logarithmicMajorant q B p δ z.2 ∂μ.prod volume)) (𝓝 0) (𝓝 0) := by | |
| have h := (tendsto_integrated_entropy_error (B := B) C hq hp).mul_const (μ.real Set.univ) | |
| convert h using 1 | |
| · funext δ | |
| rw [integral_fun_snd,smul_eq_mul] | |
| ring | |
| · ring_nf | |
| end BoltzmannNonuniqueness | |
| end | |
| end | |
| end OAI | |