File size: 3,562 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 | import OAI.MathematicalPhysics.Boltzmann.Approximation.CapEntropyMoments
import OAI.MathematicalPhysics.Boltzmann.Compactness.LocalAdmissibility
import OAI.MathematicalPhysics.Boltzmann.Compactness.EntropyDissipationAverage
namespace OAI
/-! # A strong local solution with the full entropy and moment bounds -/
noncomputable section
open MeasureTheory Set
open scoped ENNReal
namespace BoltzmannNonuniqueness
theorem uniform_entropyMoment_of_entropy_inequality {T : ℝ} {f₀ : Density} {F : Evolution}
(hf₀ : FiniteEntropyDatum f₀) (hf : ∀ t ∈ Icc 0 T, FiniteEntropyDatum (F t))
(hH : ∀ t ∈ Icc 0 T, ENNReal.ofReal (relativeEntropy (F t))+
(∫⁻ s in Icc (0:ℝ) t, entropyDissipation (F s)) ≤ ENNReal.ofReal (relativeEntropy f₀)) :
∃ C : ℝ, ∀ t ∈ Icc 0 T, (∫ z, entropyMoment (F t) z) ≤ C := by
refine ⟨(1+4*entropyMomentCoefficient)*relativeEntropy f₀+
4*entropyMomentCoefficient*(∫ z : Phase, entropyGaussian z.2)+
(∫ z : Phase, maxwellian z.2),?_⟩
intro t ht
have hle : ENNReal.ofReal (relativeEntropy (F t)) ≤ ENNReal.ofReal (relativeEntropy f₀) :=
(le_add_of_nonneg_right zero_le).trans (hH t ht)
have hreal : relativeEntropy (F t) ≤ relativeEntropy f₀ := by
have h := ENNReal.toReal_mono (by finiteness : ENNReal.ofReal (relativeEntropy f₀) ≠ ⊤) hle
simpa only [ENNReal.toReal_ofReal (relativeEntropy_nonneg (hf t ht)),
ENNReal.toReal_ofReal (relativeEntropy_nonneg hf₀)] using h
refine (entropyMoment_integral_control (hf t ht)).trans ?_
have ha : 0 ≤ 1+4*entropyMomentCoefficient := by
nlinarith [entropyMomentCoefficient_nonneg]
linarith [mul_le_mul_of_nonneg_left hreal ha]
namespace JetCopyDatum
theorem TightCapFamily.exists_entropy_local_solution
(hcarleman : Published.GPV2009CarlemanInput) (hGLPS : Published.GLPS1988Proposition3)
(hDP : Published.DunfordPettis1940Theorem321)
{s₀ η K α γ d₀ q : ℝ} (A : TightCapFamily s₀ η K α γ d₀ q)
(hs₀ : 0 < s₀) (hsmax : s₀ ≤ 1/8) (hη : 0 < η) (hηmax : η ≤ Real.log 2)
(hK : 0 < K) (hγ : 0 < γ) (hd₀ : 0 < d₀)
(hwidth : d₀*(1/8:ℝ)^10 ≤ 1/64) (hα : 0 ≤ α) (hαmax : α ≤ 1/4)
(hq₁ : 1 < q) (hq₂ : q < 2) :
∃ F : Evolution, LocalMildSolution A.time (JetPeriodicDatum.datum s₀ η K α γ d₀) F ∧
(∀ t ∈ Icc 0 A.time, FiniteEntropyDatum (F t)) ∧
(∀ t ∈ Icc 0 A.time, ENNReal.ofReal (relativeEntropy (F t))+
(∫⁻ s in Icc (0:ℝ) t, entropyDissipation (F s)) ≤
ENNReal.ofReal (relativeEntropy (JetPeriodicDatum.datum s₀ η K α γ d₀))) ∧
∃ C : ℝ, ∀ t ∈ Icc 0 A.time, (∫ z, entropyMoment (F t) z) ≤ C := by
obtain ⟨g,S,hfinite,hH⟩ := A.exists_finite_entropy_mild_limit hcarleman hGLPS hDP
hs₀ hsmax hη hηmax hK hγ hd₀ hwidth hα hαmax hq₁ hq₂
have hm : Measurable (JetPeriodicDatum.datum s₀ η K α γ d₀) :=
JetPeriodicDatum.measurable_pullback (Measurable.tsum
(fun i => JetDatum.term_measurable s₀ η K α γ d₀ i))
have hdatum := JetPeriodicDatum.finiteEntropy hs₀ hsmax hη hηmax hK hγ hd₀ hwidth hα hαmax
have hn : ∀ p, 0 ≤ JetPeriodicDatum.datum s₀ η K α γ d₀ p :=
fun p => JetDatumMoments.datum_nonneg hs₀ hK.le hd₀.le η α γ _
exact ⟨_,S.localMildSolution A.time_pos hm hdatum.integrable hn,hfinite,hH,
uniform_entropyMoment_of_entropy_inequality hdatum hfinite hH⟩
end JetCopyDatum
end BoltzmannNonuniqueness
end
end OAI
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