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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 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 | 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
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