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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 | import OAI.MathematicalPhysics.Boltzmann.Collision.KernelInvariantOrder
import OAI.MathematicalPhysics.Boltzmann.Compactness.EntropyIntegralBalance
import Mathlib.MeasureTheory.Integral.Prod
namespace OAI
section
/-! # Recovering the finite absolute-entropy datum class from relative entropy -/
noncomputable section
open MeasureTheory
open BoltzmannNonuniqueness.VelocityWeight
namespace BoltzmannNonuniqueness
theorem FiniteEntropyDatum.of_relativeEntropy_integrable {f : Density}
(hn : ∀ᵐ z, 0 ≤ f z) (hf : Integrable f)
(hE : Integrable (fun z => ‖z.2‖^2*f z))
(hH : Integrable (fun z => scalarRelativeEntropy (maxwellian z.2) (f z))) :
FiniteEntropyDatum f := by
have hlog : Integrable (fun z => f z*Real.log (f z)) := by
have h := ((hH.add (hf.const_mul (1-(3/2:ℝ)*Real.log (2*Real.pi)))).sub
(hE.const_mul (1/2:ℝ))).sub integrable_maxwellian_phase
apply h.congr
filter_upwards [] with z
simp only [Pi.sub_apply,Pi.add_apply,scalarRelativeEntropy,log_maxwellian]
ring
refine ⟨hn,hf.aestronglyMeasurable,?_⟩
apply ((hf.add hE).add hlog.norm).congr
filter_upwards [hn] with z hz
simp only [Pi.add_apply,Real.norm_eq_abs,abs_mul,abs_of_nonneg hz,entropyMoment]
ring
theorem bounded_finiteEntropyDatum {q A p : ℝ} {f : Density}
(hq : q < 2) (hA : 0 ≤ A) (hp : 1 < p)
(hpi : Integrable (fun v : Velocity => H q v^p))
(hfm : Measurable f) (hfn : ∀ z, 0 ≤ f z) (hfb : ∀ z, f z ≤ A*H q z.2) :
FiniteEntropyDatum f := by
have hfi : Integrable f := by
apply (((integrable_H hq).const_mul A).comp_snd (volume : Measure Position)).mono'
hfm.aestronglyMeasurable
filter_upwards [] with z
rw [Real.norm_of_nonneg (hfn z)]
exact hfb z
have hEi : Integrable (fun z => ‖z.2‖^2*f z) := by
have hmajor : Integrable (fun z : Phase => A*(‖z.2‖^2*H q z.2)) := by
rw [Measure.volume_eq_prod]
simpa only [Nat.reduceMul] using
((integrable_even_moment_H hq 1).const_mul A).comp_snd (volume : Measure Position)
apply hmajor.mono' (((continuous_snd.norm.pow 2).measurable).mul hfm).aestronglyMeasurable
filter_upwards [] with z
simp only [Pi.mul_apply,Pi.pow_apply]
rw [Real.norm_of_nonneg (mul_nonneg (sq_nonneg _) (hfn z))]
exact (mul_le_mul_of_nonneg_left (hfb z) (sq_nonneg _)).trans_eq (by ring)
have hHi : Integrable (fun z => scalarRelativeEntropy (maxwellian z.2) (f z)) := by
rw [Measure.volume_eq_prod]
exact bounded_relative_entropy_integrable hq hA hp hpi hfm hfn hfb
exact FiniteEntropyDatum.of_relativeEntropy_integrable (Filter.Eventually.of_forall hfn) hfi hEi hHi
end BoltzmannNonuniqueness
end
end
section
/-! # Finite sums preserve convergence in integral L1 -/
noncomputable section
open MeasureTheory Filter
open scoped Topology
namespace BoltzmannNonuniqueness.Compactness
theorem finite_integral_L1_zero {X ι : Type*} [MeasurableSpace X]
{μ : Measure X} (s : Finset ι) {f : ι → ℕ → X → ℝ}
(hi : ∀ i ∈ s, ∀ n, Integrable (f i n) μ)
(hl : ∀ i ∈ s, Tendsto (fun n => ∫ x, |f i n x| ∂μ) atTop (𝓝 0)) :
Tendsto (fun n => ∫ x, |∑ i ∈ s, f i n x| ∂μ) atTop (𝓝 0) := by
have hb := tendsto_finsetSum s hl
simp only [Finset.sum_const_zero] at hb
apply squeeze_zero (fun _ => integral_nonneg (fun _ => abs_nonneg _)) _ hb
intro n
calc
_ ≤ ∫ x, ∑ i ∈ s, |f i n x| ∂μ := by
apply integral_mono_of_nonneg (ae_of_all _ (fun _ => abs_nonneg _))
(integrable_finsetSum s (fun i hi' => (hi i hi' n).abs))
exact ae_of_all _ (fun x => Finset.abs_sum_le_sum_abs (fun i => f i n x) s)
_ = _ := integral_finsetSum s (fun i hi' => (hi i hi' n).abs)
end BoltzmannNonuniqueness.Compactness
end
end
section
/-! # Finite history comparison on a depth-dependent reachable region -/
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped ProbabilityTheory
namespace BoltzmannNonuniqueness
variable {X : Type*} [MeasurableSpace X]
theorem kernel_pow_le_on_depth (K L : Kernel X X) (D : ℕ → Set X) (N : ℕ)
(hstep : ∀ n < N, ∀ x ∈ D (n+1), ∀ᵐ y ∂K x, y ∈ D n)
(hle : ∀ n < N, ∀ x ∈ D (n+1), K x ≤ L x) :
∀ n ≤ N, ∀ x ∈ D n, (K^n) x ≤ (L^n) x := by
intro n hn
induction n with
| zero => intro x hx; exact le_rfl
| succ n ih =>
intro x hx
have hnN : n < N := Nat.lt_of_succ_le hn
rw [show K^(n+1) = (K^n) ∘ₖ K by simpa only [pow_one] using Kernel.pow_add K n 1,
show L^(n+1) = (L^n) ∘ₖ L by simpa only [pow_one] using Kernel.pow_add L n 1]
apply KernelHistoryOrder.measure_comp_mono_ae _ (hle n hnN x hx)
filter_upwards [hstep n hnN x hx] with y hy
exact ih (Nat.le_of_lt hnN) y hy
end BoltzmannNonuniqueness
end
end
end OAI
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