openAI-math-dataset / lean /OAI /MathematicalPhysics /Boltzmann /Compactness /EntropyRegularizationLimit.lean
| import OAI.MathematicalPhysics.Boltzmann.Branching.BoundedEntropyProduction | |
| import OAI.MathematicalPhysics.Boltzmann.Branching.BoundedCollisionBlocks | |
| import OAI.MathematicalPhysics.Boltzmann.Compactness.EntropyIntegralBalance | |
| import Mathlib.MeasureTheory.Integral.DominatedConvergence | |
| namespace OAI | |
| section | |
| /-! # Integrable bounded entropy tests on the phase and spacetime measures -/ | |
| noncomputable section | |
| open MeasureTheory Filter | |
| open scoped Topology | |
| namespace BoltzmannNonuniqueness | |
| variable {X : Type*} [MeasurableSpace X] {μ : Measure X} | |
| theorem measurable_boundedEntropyTest (n : ℕ) {m u : X → ℝ} | |
| (hm : Measurable m) (hu : Measurable u) : | |
| Measurable (fun x => boundedEntropyTest n (m x) (u x)) := by | |
| have hlog := (hu.div hm).log | |
| exact Measurable.ite (measurableSet_eq_fun hu measurable_const) measurable_const | |
| (Measurable.ite (measurableSet_le hlog.abs measurable_const) hlog measurable_const) | |
| theorem measurable_entropyDual {m u s : X → ℝ} | |
| (hm : Measurable m) (hu : Measurable u) (hs : Measurable s) : | |
| Measurable (fun x => entropyDual (m x) (u x) (s x)) := | |
| (hu.mul hs).sub (hm.mul (hs.exp.sub measurable_const)) | |
| theorem integrable_entropyDual {m u s : X → ℝ} | |
| (hm : Integrable m μ) (hu : Integrable u μ) (hs : Measurable s) | |
| {C : ℝ} (hC : ∀ x, |s x| ≤ C) : | |
| Integrable (fun x => entropyDual (m x) (u x) (s x)) μ := by | |
| have hfirst := hu.bdd_mul hs.aestronglyMeasurable (ae_of_all μ hC) | |
| have hbound : ∀ x, |Real.exp (s x)-1| ≤ Real.exp C+1 := by | |
| intro x | |
| calc | |
| |Real.exp (s x)-1| ≤ |Real.exp (s x)|+|1| := abs_sub _ _ | |
| _ = Real.exp (s x)+1 := by rw [abs_of_pos (Real.exp_pos _),abs_one] | |
| _ ≤ Real.exp C+1 := add_le_add (Real.exp_le_exp.mpr ((le_abs_self _).trans (hC x))) le_rfl | |
| have hsecond := hm.bdd_mul (hs.exp.sub measurable_const).aestronglyMeasurable | |
| (ae_of_all μ hbound) | |
| simpa only [entropyDual,mul_comm,Pi.sub_def,Pi.mul_def] using hfirst.sub hsecond | |
| end BoltzmannNonuniqueness | |
| end | |
| end | |
| section | |
| /-! # Removing Gaussian regularization from the integrated relative entropy -/ | |
| noncomputable section | |
| open MeasureTheory Filter Set | |
| open scoped Topology | |
| open BoltzmannNonuniqueness.VelocityWeight | |
| namespace BoltzmannNonuniqueness | |
| variable {X : Type*} [MeasurableSpace X] | |
| theorem gaussianRegularized_entropy_tendsto {μ : Measure X} [IsFiniteMeasure μ] | |
| {q A p : ℝ} (hq : q < 2) (hA : 0 ≤ A) (hp : 1 < p) | |
| (hpi : Integrable (fun v : Velocity => H q v^p)) | |
| {f : X × Velocity → ℝ} (hfm : Measurable f) (hfn : ∀ z, 0 ≤ f z) | |
| (hfb : ∀ z, f z ≤ A*H q z.2) : | |
| Tendsto (fun δ : ℝ => ∫ z, scalarRelativeEntropy (maxwellian z.2) | |
| (gaussianRegularizedDensity δ f z) ∂μ.prod volume) | |
| (𝓝[>] 0) (𝓝 (∫ z, scalarRelativeEntropy (maxwellian z.2) (f z) ∂μ.prod volume)) := by | |
| apply tendsto_integral_filter_of_dominated_convergence | |
| (fun z : X × Velocity => entropyPowerMajorant q (A+tiltConstant) p z.2) | |
| · exact Eventually.of_forall (fun δ => by | |
| have hm : Measurable (fun z => scalarRelativeEntropy (maxwellian z.2) | |
| (gaussianRegularizedDensity δ f z)) := by | |
| have hg := gaussianRegularizedDensity_measurable δ hfm | |
| unfold scalarRelativeEntropy maxwellian | |
| fun_prop | |
| exact hm.aestronglyMeasurable) | |
| · filter_upwards [Ioo_mem_nhdsGT (show (0:ℝ) < 1 by norm_num)] with δ hδ | |
| filter_upwards [] with z | |
| have hn : 0 ≤ gaussianRegularizedDensity δ f z := | |
| add_nonneg (hfn z) (mul_nonneg hδ.1.le (Real.exp_nonneg _)) | |
| have hb : gaussianRegularizedDensity δ f z ≤ (A+tiltConstant)*H q z.2 := | |
| gaussianRegularization_bound hδ.1.le hδ.2.le (fun v => hfb (z.1,v)) z.2 | |
| rw [Real.norm_of_nonneg (scalarRelativeEntropy_nonneg (maxwellian_pos _) hn)] | |
| exact scalarRelativeEntropy_le_majorant (add_nonneg hA tiltConstant_pos.le) hp z.2 hn hb | |
| · exact (entropyPowerMajorant_integrable (B := A+tiltConstant) hq hpi).comp_snd μ | |
| · filter_upwards [] with z | |
| have hc : Continuous (fun δ : ℝ => scalarRelativeEntropy (maxwellian z.2) | |
| (f z+δ*Real.exp (-2*‖z.2‖^2))) := | |
| (scalarRelativeEntropy_continuous _).comp (continuous_const.add (continuous_id.mul_const _)) | |
| simpa only [gaussianRegularizedDensity,zero_mul,add_zero] using | |
| (hc.tendsto 0).mono_left nhdsWithin_le_nhds | |
| theorem regularizedRelativeEntropyDensity_tendsto {q A p : ℝ} | |
| (hq : q < 2) (hA : 0 ≤ A) (hp : 1 < p) | |
| (hpi : Integrable (fun v : Velocity => H q v^p)) | |
| {f : Density} (hfm : Measurable f) (hfn : ∀ z, 0 ≤ f z) | |
| (hfb : ∀ z, f z ≤ A*H q z.2) : | |
| Tendsto (fun δ : ℝ => ∫ z, regularizedRelativeEntropyDensity δ f z) | |
| (𝓝[>] 0) (𝓝 (∫ z, scalarRelativeEntropy (maxwellian z.2) (f z))) := by | |
| rw [Measure.volume_eq_prod] | |
| exact gaussianRegularized_entropy_tendsto hq hA hp hpi hfm hfn hfb | |
| end BoltzmannNonuniqueness | |
| end | |
| end | |
| end OAI | |