openAI-math-dataset / lean /OAI /MathematicalPhysics /Boltzmann /Approximation /CapWeightedNonlinearLimit.lean
| import OAI.MathematicalPhysics.Boltzmann.Branching.ProductKineticCutoff | |
| import OAI.MathematicalPhysics.Boltzmann.Branching.WeightedJetStability | |
| import OAI.MathematicalPhysics.Boltzmann.Compactness.LocalMomentIntegrability | |
| import OAI.MathematicalPhysics.Boltzmann.Compactness.PairPatchIdentity | |
| namespace OAI | |
| section | |
| /-! # Identification of the actual cap-family weak pair limit -/ | |
| noncomputable section | |
| open MeasureTheory Set | |
| open BoltzmannNonuniqueness.TransportCutoff | |
| namespace BoltzmannNonuniqueness.JetCopyDatum | |
| /-- The velocity-averaging step identifies the genuine nonlinear pair product. -/ | |
| theorem TightCapFamily.exists_nonlinear_weak_limit | |
| (hcarleman : Published.GPV2009CarlemanInput) (hGLPS : Published.GLPS1988Proposition3) | |
| (hDP : Published.DunfordPettis1940Theorem321) | |
| {s₀ η K α γ d₀ q : ℝ} (F : TightCapFamily s₀ η K α γ d₀ q) | |
| (hq₁ : 1 < q) (hq₂ : q < 2) : | |
| ∃ (φ : ℕ → ℕ) (g : (ℝ × Position) × Velocity → ℝ), StrictMono φ ∧ | |
| Published.WeakL1Converges ((AnnularEnvelope.timePositionVolume F.time).prod volume) | |
| (fun n => torusPhysical q (F.approximation (φ n)).solution.value) g ∧ | |
| Published.WeakL1Converges ((AnnularEnvelope.timePositionVolume F.time).prod volume) | |
| (fun n => CollisionCompactness.pairProduct | |
| (torusPhysical q (F.approximation (φ n)).solution.value)) | |
| (CollisionCompactness.pairProduct g) ∧ | |
| (∀ᵐ z ∂(AnnularEnvelope.timePositionVolume F.time).prod volume, 0 ≤ g z) := by | |
| obtain ⟨ψ,hψ,hψc,hdet⟩ := Compactness.exists_countable_smooth_determining_tests | |
| choose β hβ hβc _hβn hβtest using fun i => Compactness.exists_velocity_cutoff (hψc i) | |
| obtain ⟨χ,hχ,hχc,hχs,hcover⟩ := Compactness.exists_spacetime_cutoff_cover | |
| TorusKineticChart.interiorRegion_open TorusKineticChart.interiorRegion_nonempty | |
| let cuts : ℕ → Published.KineticPhase → ℝ := fun n => | |
| Compactness.productCutoff (χ (Nat.unpair n).1) (β (Nat.unpair n).2) | |
| let tests : ℕ → Velocity → ℝ := fun n => ψ (Nat.unpair n).2 | |
| have hcuts (n) : ContDiff ℝ (⊤ : ℕ∞) (cuts n) := | |
| Compactness.productCutoff_smooth (hχ _) (hβ _) | |
| have hcutsc (n) : HasCompactSupport (cuts n) := | |
| Compactness.productCutoff_compact (hχc _) (hβc _) | |
| have hcutss (n) : tsupport (cuts n) ⊆ {z | z.1.2 ∈ TorusChart.cube} := | |
| Compactness.productCutoff_in_cell _ (hχs _) | |
| have hcuts0 (n) (p : Velocity × Velocity) : cuts n (TransportCutoff.characteristic p 0) = 0 := | |
| Compactness.productCutoff_initial_zero _ (hχs _) p | |
| obtain ⟨φ,g,W,hφ,hg,hW,hmom,hgn,_hWn⟩ := F.exists_factorized_pair_moments | |
| hcarleman hGLPS hDP hq₁ hq₂ cuts hcuts hcutsc hcutss hcuts0 tests | |
| (fun n => hψ _) (fun n => hψc _) | |
| have hw (s i : ℕ) : averageWeight (cuts (Nat.pair s i)) (tests (Nat.pair s i)) = | |
| fun z => χ s z.1*ψ i z.2 := by | |
| dsimp only [cuts,tests] | |
| rw [Nat.unpair_pair] | |
| exact Compactness.productCutoff_preserves_test (χ s) (hβtest i) | |
| have hgw := TorusKineticChart.weak_density_transfer | |
| (fun n => (F.approximation (φ n)).solution.measurable_value) hg | |
| have hWw := TorusKineticChart.weak_pair_transfer | |
| (fun n => (F.approximation (φ n)).solution.measurable_value) hW | |
| have hscaled (s i j : ℕ) : ∀ᵐ x ∂TorusKineticChart.baseVolume F.time, | |
| (∫ y : Velocity × Velocity, ((χ s x*ψ i y.1)*(χ s x*ψ j y.2))*TorusKineticChart.liftPair W (x,y)) = | |
| (∫ v, (χ s x*ψ i v)*TorusKineticChart.liftDensity g (x,v))* | |
| (∫ w, (χ s x*ψ j w)*TorusKineticChart.liftDensity g (x,w)) := by | |
| apply Compactness.ae_of_all_compact_restrict | |
| intro S hS | |
| have ht := hmom (Nat.pair s i) (Nat.pair s j) S hS | |
| simp only [hw s i,hw s j] at ht | |
| filter_upwards [ht] with x hx | |
| exact hx | |
| have hpatch (s) : ∀ᵐ x ∂TorusKineticChart.baseVolume F.time, χ s x ≠ 0 → | |
| (fun y => TorusKineticChart.liftPair W (x,y)) =ᵐ[volume] | |
| (fun y : Velocity × Velocity => TorusKineticChart.liftDensity g (x,y.1)* | |
| TorusKineticChart.liftDensity g (x,y.2)) := by | |
| apply Compactness.pair_identity_on_patch hψ hψc hdet hWw.1 hgw.1 hWw.2.1 hgw.2.1 (χ s) | |
| exact fun i j => (hscaled s i j).mono (fun _ h _ => h) | |
| have hfib : ∀ᵐ x ∂TorusKineticChart.baseVolume F.time, | |
| (fun y => TorusKineticChart.liftPair W (x,y)) =ᵐ[volume] | |
| (fun y : Velocity × Velocity => TorusKineticChart.liftDensity g (x,y.1)* | |
| TorusKineticChart.liftDensity g (x,y.2)) := by | |
| filter_upwards [ae_all_iff.mpr hpatch,TorusKineticChart.interiorRegion_ae F.time] with x hx hxin | |
| obtain ⟨s,hs⟩ := hcover x hxin | |
| exact hx s hs | |
| have he : TorusKineticChart.liftPair W =ᵐ[TorusKineticChart.pairCellVolume F.time] | |
| CollisionCompactness.pairProduct (TorusKineticChart.liftDensity g) := by | |
| apply (Measure.ae_prod_iff_ae_ae ?_).mpr hfib | |
| exact measurableSet_eq_fun hWw.1 (CollisionCompactness.pairProduct_measurable hgw.1) | |
| have hback := (TorusKineticChart.pair_measurePreserving F.time).quasiMeasurePreserving.ae he | |
| have hEq : W =ᵐ[(AnnularEnvelope.timePositionVolume F.time).prod volume] | |
| CollisionCompactness.pairProduct g := by | |
| filter_upwards [hback] with z hz | |
| simpa only [TorusKineticChart.liftPair,TorusKineticChart.pairChart, | |
| CollisionCompactness.pairProduct,TorusKineticChart.liftDensity, | |
| TorusChart.projection_representative] using hz | |
| refine ⟨φ,g,hφ,hg,?_,hgn⟩ | |
| refine ⟨CollisionCompactness.pairProduct_measurable hg.1,hW.2.1.congr hEq,?_⟩ | |
| intro θ hθ hθB | |
| have ht := hW.2.2 θ hθ hθB | |
| have hi : (∫ z, θ z*W z ∂(AnnularEnvelope.timePositionVolume F.time).prod volume) = | |
| ∫ z, θ z*CollisionCompactness.pairProduct g z | |
| ∂(AnnularEnvelope.timePositionVolume F.time).prod volume := | |
| integral_congr_ae (hEq.mono (fun z hz => congrArg (fun y => θ z*y) hz)) | |
| rw [hi] at ht | |
| exact ht | |
| end BoltzmannNonuniqueness.JetCopyDatum | |
| end | |
| end | |
| section | |
| /-! # The physical pair weight also converges to the genuine nonlinear product -/ | |
| noncomputable section | |
| open MeasureTheory | |
| namespace BoltzmannNonuniqueness.JetCopyDatum | |
| theorem TightCapFamily.exists_weighted_nonlinear_weak_limit | |
| (hcarleman : Published.GPV2009CarlemanInput) (hGLPS : Published.GLPS1988Proposition3) | |
| (hDP : Published.DunfordPettis1940Theorem321) | |
| {s₀ η K α γ d₀ q : ℝ} (F : TightCapFamily s₀ η K α γ d₀ q) | |
| (hq₁ : 1 < q) (hq₂ : q < 2) : | |
| ∃ (φ : ℕ → ℕ) (g : (ℝ × Position) × Velocity → ℝ), StrictMono φ ∧ | |
| Published.WeakL1Converges ((AnnularEnvelope.timePositionVolume F.time).prod volume) | |
| (fun n => torusPhysical q (F.approximation (φ n)).solution.value) g ∧ | |
| Published.WeakL1Converges ((AnnularEnvelope.timePositionVolume F.time).prod volume) | |
| (fun n => CollisionCompactness.pairDensity | |
| (torusPhysical q (F.approximation (φ n)).solution.value)) | |
| (CollisionCompactness.pairDensity g) ∧ | |
| (∀ᵐ z ∂(AnnularEnvelope.timePositionVolume F.time).prod volume, 0 ≤ g z) := by | |
| obtain ⟨a,g,ha,hg,hraw,hgn⟩ := F.exists_nonlinear_weak_limit hcarleman hGLPS hDP hq₁ hq₂ | |
| have hm (n) := CollisionCompactness.pairDensity_measurable | |
| (torusPhysical_measurable (F.approximation (a n)).solution.measurable_value q) | |
| obtain ⟨b,G,hb,hG⟩ := (hDP F.time F.time_pos.le).2 | |
| (fun n => CollisionCompactness.pairDensity (torusPhysical q (F.approximation (a n)).solution.value)) | |
| hm (Compactness.uniformIntegrable_comp F.pairs_ui a) (F.pairs_tight.comp a) | |
| let w : (ℝ × Position) × (Velocity × Velocity) → ℝ := fun z => 1+‖z.2.1‖+‖z.2.2‖ | |
| have hw : Measurable w := by dsimp [w]; fun_prop | |
| have hw1 : ∀ z, 1 ≤ w z := by | |
| intro z | |
| dsimp [w] | |
| linarith [norm_nonneg z.2.1,norm_nonneg z.2.2] | |
| have hfun (u : (ℝ × Position) × Velocity → ℝ) : | |
| (fun z => w z*CollisionCompactness.pairProduct u z) = CollisionCompactness.pairDensity u := by | |
| funext z | |
| simp only [w,CollisionCompactness.pairProduct,CollisionCompactness.pairDensity,mul_assoc] | |
| have hraw' := hraw.comp hb.tendsto_atTop | |
| have hG' : Published.WeakL1Converges ((AnnularEnvelope.timePositionVolume F.time).prod volume) | |
| (fun n z => w z*CollisionCompactness.pairProduct | |
| (torusPhysical q (F.approximation (a (b n))).solution.value) z) G := by | |
| have hseq : (fun n z => w z*CollisionCompactness.pairProduct | |
| (torusPhysical q (F.approximation (a (b n))).solution.value) z) = | |
| (fun n => CollisionCompactness.pairDensity (torusPhysical q | |
| (F.approximation (a (b n))).solution.value)) := funext (fun n => hfun _) | |
| rw [hseq] | |
| exact hG | |
| have he := Published.identify_weighted_weak_limit hraw' hG' hw hw1 | |
| have he' : G =ᵐ[(AnnularEnvelope.timePositionVolume F.time).prod volume] | |
| CollisionCompactness.pairDensity g := by | |
| rw [hfun] at he | |
| exact he | |
| exact ⟨a ∘ b,g,ha.comp hb,hg.comp hb.tendsto_atTop, | |
| hG.congr_limit (CollisionCompactness.pairDensity_measurable hg.1) he',hgn⟩ | |
| end BoltzmannNonuniqueness.JetCopyDatum | |
| end | |
| end | |
| end OAI | |