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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 | import OAI.MathematicalPhysics.Boltzmann.Approximation.PositiveComparisonBound
namespace OAI
/-! # The extended nonnegative cutoff dissipation agrees with positive production -/
noncomputable section
open MeasureTheory
open scoped ENNReal
namespace BoltzmannNonuniqueness.CollisionCompactness
variable {X : Type*} [MeasurableSpace X]
def cutoffExtendedProduction (b : ℝ) (f : X × Velocity → ℝ)
(z : (X × (Velocity × Velocity)) × Sphere) : ℝ≥0∞ :=
if ‖z.1.2.1-z.1.2.2‖ < b then
ENNReal.ofReal (collisionKernel z.1.2.1 z.1.2.2 z.2)*
extendedEntropyProduction
(f (z.1.1,scatterLeft z.1.2.1 z.1.2.2 z.2)*f (z.1.1,scatterRight z.1.2.1 z.1.2.2 z.2))
(f (z.1.1,z.1.2.1)*f (z.1.1,z.1.2.2))
else 0
omit [MeasurableSpace X] in
theorem positiveCutoffProduction_formula (b : ℝ) (f : X × Velocity → ℝ)
(hf : ∀ z, 0 < f z) (z : (X × (Velocity × Velocity)) × Sphere) :
positiveCutoffProduction b f z =
if ‖z.1.2.1-z.1.2.2‖ < b then
(1/4:ℝ)*collisionKernel z.1.2.1 z.1.2.2 z.2*
((f (z.1.1,scatterLeft z.1.2.1 z.1.2.2 z.2)*f (z.1.1,scatterRight z.1.2.1 z.1.2.2 z.2)-
f (z.1.1,z.1.2.1)*f (z.1.1,z.1.2.2))*
Real.log ((f (z.1.1,scatterLeft z.1.2.1 z.1.2.2 z.2)*f (z.1.1,scatterRight z.1.2.1 z.1.2.2 z.2))/
(f (z.1.1,z.1.2.1)*f (z.1.1,z.1.2.2))))
else 0 := by
unfold positiveCutoffProduction
rw [collisionJump_log_relative f hf,collisionJump_log_density f hf]
simp only [truncatedPreRate,truncatedPostRate,Set.indicator_apply,Set.mem_ofPred_eq]
by_cases hc : ‖z.1.2.1-z.1.2.2‖ < b
· simp only [hc,↓reduceIte,postRate_formula,preRate]
ring
· simp only [hc,↓reduceIte,sub_self,mul_zero]
omit [MeasurableSpace X] in
theorem ofReal_positiveCutoffProduction (b : ℝ) (f : X × Velocity → ℝ)
(hf : ∀ z, 0 < f z) (z : (X × (Velocity × Velocity)) × Sphere) :
ENNReal.ofReal (positiveCutoffProduction b f z) = (4:ℝ≥0∞)⁻¹*cutoffExtendedProduction b f z := by
rw [positiveCutoffProduction_formula b f hf]
have hA := mul_pos (hf (z.1.1,scatterLeft z.1.2.1 z.1.2.2 z.2))
(hf (z.1.1,scatterRight z.1.2.1 z.1.2.2 z.2))
have hB := mul_pos (hf (z.1.1,z.1.2.1)) (hf (z.1.1,z.1.2.2))
by_cases hc : ‖z.1.2.1-z.1.2.2‖ < b
· simp only [cutoffExtendedProduction,hc,↓reduceIte]
have he : extendedEntropyProduction
(f (z.1.1,scatterLeft z.1.2.1 z.1.2.2 z.2)*f (z.1.1,scatterRight z.1.2.1 z.1.2.2 z.2))
(f (z.1.1,z.1.2.1)*f (z.1.1,z.1.2.2)) =
ENNReal.ofReal ((f (z.1.1,scatterLeft z.1.2.1 z.1.2.2 z.2)*f (z.1.1,scatterRight z.1.2.1 z.1.2.2 z.2)-
f (z.1.1,z.1.2.1)*f (z.1.1,z.1.2.2))*
Real.log ((f (z.1.1,scatterLeft z.1.2.1 z.1.2.2 z.2)*f (z.1.1,scatterRight z.1.2.1 z.1.2.2 z.2))/
(f (z.1.1,z.1.2.1)*f (z.1.1,z.1.2.2)))) := by
simp [extendedEntropyProduction,hA,hB,hA.ne',hB.ne']
rw [he]
rw [ENNReal.ofReal_mul (mul_nonneg (by norm_num) (collisionKernel_nonneg _ _ _)),
ENNReal.ofReal_mul (by norm_num : (0:ℝ) ≤ 1/4)]
rw [ENNReal.ofReal_div_of_pos (by norm_num : (0:ℝ) < 4)]
norm_num [mul_assoc]
· simp [hc,cutoffExtendedProduction]
def cutoffExtendedDissipation (μ : Measure X) (b : ℝ) (f : X × Velocity → ℝ) : ℝ≥0∞ :=
(4:ℝ≥0∞)⁻¹*∫⁻ z, cutoffExtendedProduction b f z
∂(μ.prod (volume : Measure (Velocity × Velocity))).prod sphereArea
theorem ofReal_integral_positiveCutoffProduction {μ : Measure X} (b : ℝ)
(f : X × Velocity → ℝ) (hf : ∀ z, 0 < f z)
(hi : Integrable (positiveCutoffProduction b f)
((μ.prod (volume : Measure (Velocity × Velocity))).prod sphereArea)) :
ENNReal.ofReal (∫ z, positiveCutoffProduction b f z
∂(μ.prod (volume : Measure (Velocity × Velocity))).prod sphereArea) =
cutoffExtendedDissipation μ b f := by
rw [ofReal_integral_eq_lintegral_ofReal hi
(Filter.Eventually.of_forall (positiveCutoffProduction_nonneg b f hf))]
simp_rw [ofReal_positiveCutoffProduction b f hf]
exact lintegral_const_mul' _ _ (by norm_num)
end BoltzmannNonuniqueness.CollisionCompactness
end
end OAI
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