openAI-math-dataset / lean /OAI /MathematicalPhysics /Boltzmann /Compactness /EntropyLocalConstruction.lean
| 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 | |