openAI-math-dataset / lean /OAI /MathematicalPhysics /Boltzmann /Approximation /CapLocalAdmissibility.lean
| import OAI.MathematicalPhysics.Boltzmann.Approximation.CapEntropyMoments | |
| import OAI.MathematicalPhysics.Boltzmann.Approximation.MildLocalAdmissibility | |
| namespace OAI | |
| /-! # A fully admissible strong local solution for the actual cold datum -/ | |
| noncomputable section | |
| open MeasureTheory | |
| namespace BoltzmannNonuniqueness.JetCopyDatum | |
| theorem TightCapFamily.exists_admissible_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) (hα : 0 < α) (hαmax : α ≤ 1/4) (hγ : 0 < γ) | |
| (hd₀ : 0 < d₀) (hd₀max : d₀ ≤ 1) (hwidth : d₀*(1/8:ℝ)^10 ≤ 1/64) | |
| (hsmall : JetComparisonCoefficient.coefficient K*d₀*s₀^9 ≤ 1) (hq₁ : 1 < q) (hq₂ : q < 2) : | |
| ∃ F : Evolution, AdmissibleLocal A.time (JetPeriodicDatum.datum s₀ η K α γ d₀) F := by | |
| obtain ⟨g,S,hfinite,hH,hE⟩ := A.exists_entropy_moment_mild_limit hcarleman hGLPS hDP | |
| hs₀ hsmax hη hηmax hK hα hαmax hγ hd₀ hd₀max hwidth hsmall hq₁ hq₂ | |
| have hf₀ := JetPeriodicDatum.finiteEntropy hs₀ hsmax hη hηmax hK hγ hd₀ hwidth hα.le hαmax | |
| 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 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.localAdmissible A.time_pos hm hn hf₀ hfinite hH hE⟩ | |
| end BoltzmannNonuniqueness.JetCopyDatum | |
| end | |
| end OAI | |