File size: 4,517 Bytes
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 | 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
|