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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 | 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
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