| import OAI.MathematicalPhysics.Boltzmann.Approximation.PositiveNonlinearLimit |
| import OAI.MathematicalPhysics.Boltzmann.Branching.CompactCapFamily |
| import OAI.MathematicalPhysics.Boltzmann.Collision.CollisionDissipation |
| import OAI.MathematicalPhysics.Boltzmann.Compactness.StrongMassClip |
| import OAI.MathematicalPhysics.Boltzmann.Compactness.WeakDuhamelLimit |
| import OAI.MathematicalPhysics.Boltzmann.Compactness.WeakEntropyMarginal |
| import OAI.MathematicalPhysics.Boltzmann.Compactness.WeakL1Operations |
| import OAI.MathematicalPhysics.Boltzmann.Jets.TorusVelocityWindowAE |
|
|
| namespace OAI |
|
|
| section |
|
|
| /-! # Convergence of the actual right-hand sides of the cap equations -/ |
|
|
| noncomputable section |
|
|
| open MeasureTheory Filter |
|
|
| namespace BoltzmannNonuniqueness.JetCopyDatum |
|
|
| theorem TightCapFamily.exists_cutoff_equation_limit_with_pair |
| (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.collisionSource (φ n+1) |
| (torusPhysical q (F.approximation (φ n)).solution.value)) |
| (fun p => CollisionCompactness.gainDensity g p-CollisionCompactness.lossDensity g p) ∧ |
| Integrable (CollisionCompactness.pairDensity g) |
| ((AnnularEnvelope.timePositionVolume F.time).prod volume) ∧ (∀ z, 0 ≤ g z) ∧ |
| Published.WeakL1Converges ((AnnularEnvelope.timePositionVolume F.time).prod volume) |
| (fun n => CollisionCompactness.pairDensity |
| (torusPhysical q (F.approximation (φ n)).solution.value)) |
| (CollisionCompactness.pairDensity g) := by |
| obtain ⟨φ,g,hφ,hg,hpair,hgn⟩ := F.exists_positive_nonlinear_limit hcarleman hGLPS hDP hq₁ hq₂ |
| have hfm n := torusPhysical_measurable (F.approximation (φ n)).solution.measurable_value q |
| have hfn n := torusPhysical_nonneg |
| (fun z => ((F.approximation (φ n)).solution.value_bounds z).1) q |
| have hf n := memLp_one_iff_integrable.mp (F.pairs_ui.memLp (φ n)) |
| have hb : Tendsto (fun n => (φ n : ℝ)+1) atTop atTop := |
| tendsto_atTop_mono (fun n => by linarith : ∀ n, (φ n : ℝ) ≤ (φ n : ℝ)+1) |
| (tendsto_natCast_atTop_atTop.comp hφ.tendsto_atTop) |
| have hterms := CollisionCompactness.weak_cutoff_gainLoss hfm hfn hg.1 hpair hf hb |
| have hdiff := hterms.1.sub hterms.2 |
| (fun n => memLp_one_iff_integrable.mp (F.gain_ui.memLp (φ n))) |
| (fun n => memLp_one_iff_integrable.mp (F.loss_ui.memLp (φ n))) |
| have he n : (fun p => KineticCollisionNullSets.gain (φ n+1) |
| (torusPhysical q (F.approximation (φ n)).solution.value) p- |
| CollisionCompactness.cutoffLossDensity (φ n+1) |
| (torusPhysical q (F.approximation (φ n)).solution.value) p) =ᵐ[ |
| (AnnularEnvelope.timePositionVolume F.time).prod volume] |
| CollisionCompactness.collisionSource (φ n+1) |
| (torusPhysical q (F.approximation (φ n)).solution.value) := by |
| filter_upwards [CollisionCompactness.cutoff_loss_ae_eq_frequency (hfm n) (hfn n) (hf n) |
| (φ n+1)] with p hp |
| simp only [CollisionCompactness.collisionSource,hp] |
| exact ⟨φ,g,hφ,hg,hdiff.congr_sequence he,hpair.2.1,hgn,hpair⟩ |
|
|
| theorem TightCapFamily.exists_cutoff_equation_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.collisionSource (φ n+1) |
| (torusPhysical q (F.approximation (φ n)).solution.value)) |
| (fun p => CollisionCompactness.gainDensity g p-CollisionCompactness.lossDensity g p) ∧ |
| Integrable (CollisionCompactness.pairDensity g) |
| ((AnnularEnvelope.timePositionVolume F.time).prod volume) ∧ (∀ z, 0 ≤ g z) := by |
| obtain ⟨φ,g,hφ,hg,hr,hpi,hgn,_hpair⟩ := |
| F.exists_cutoff_equation_limit_with_pair hcarleman hGLPS hDP hq₁ hq₂ |
| exact ⟨φ,g,hφ,hg,hr,hpi,hgn⟩ |
|
|
| end BoltzmannNonuniqueness.JetCopyDatum |
|
|
| end |
|
|
| end |
|
|
| section |
|
|
| /-! # The cap approximations satisfy the actual torus initial-value identity -/ |
|
|
| noncomputable section |
|
|
| open MeasureTheory Set |
|
|
| namespace BoltzmannNonuniqueness.JetCopyDatum |
|
|
| theorem periodicInitial_measurable (s₀ η K α γ d₀ q : ℝ) (N : ℕ) (ell : ℝ) : |
| Measurable (periodicInitial s₀ η K α γ d₀ q N ell) := by |
| have hm := (measurable_seed s₀ η K α γ d₀ q N).add |
| ((HotSeedProfile.normalized_measurable q (DatumBounds.scale s₀ η (N+1))).const_mul ell) |
| have he : periodicInitial s₀ η K α γ d₀ q N ell = |
| TorusFlight.initialDensity q (initial s₀ η K α γ d₀ q N |
| (fun p => ell*HotSeedProfile.normalized q (DatumBounds.scale s₀ η (N+1)) p)) := by |
| funext p |
| exact mul_comm _ _ |
| rw [he] |
| exact TorusFlight.initialDensity_measurable hm q |
|
|
| theorem CapApproximation.spacetime_mild (hcarleman : Published.GPV2009CarlemanInput) |
| {s₀ η K α γ d₀ q T : ℝ} {N : ℕ} (F : CapApproximation s₀ η K α γ d₀ q T N) |
| (hq₁ : 1 < q) (hq₂ : q < 2) (hT : 0 ≤ T) : |
| TorusFlight.along (TorusFlight.regroup (torusPhysical q F.solution.value)) =ᵐ[ |
| (volume.restrict (Icc 0 T)).prod (volume : Measure Phase)] |
| (fun z => periodicInitial s₀ η K α γ d₀ q N F.coefficient z.2 + |
| Published.timePrimitive (volume.restrict (Icc 0 T)) |
| (TorusFlight.along (TorusFlight.regroup |
| (CollisionCompactness.collisionSource (N+1) (torusPhysical q F.solution.value)))) z) := by |
| have hm := (measurable_seed s₀ η K α γ d₀ q N).add |
| ((HotSeedProfile.normalized_measurable q (DatumBounds.scale s₀ η (N+1))).const_mul F.coefficient) |
| have h := TorusFlight.spacetime_mild hcarleman F.solution hq₁ hq₂ |
| (by positivity) (by have := F.amplitude_pos; positivity) hT le_rfl le_rfl hm |
| filter_upwards [h] with z hz |
| convert hz using 1 |
| congr 1 |
| exact mul_comm _ _ |
|
|
| end BoltzmannNonuniqueness.JetCopyDatum |
|
|
| end |
|
|
| end |
|
|
| section |
|
|
| /-! # Every-time characteristic traces of the finite cap approximations -/ |
|
|
| noncomputable section |
|
|
| open MeasureTheory Set |
|
|
| namespace BoltzmannNonuniqueness.JetCopyDatum |
|
|
| theorem CapApproximation.characteristic_slice (hcarleman : Published.GPV2009CarlemanInput) |
| {s₀ η K α γ d₀ q T : ℝ} {N : ℕ} (F : CapApproximation s₀ η K α γ d₀ q T N) |
| (hq₁ : 1 < q) (hq₂ : q < 2) {t : ℝ} (ht : t ∈ Icc 0 T) : |
| (fun p => torusPhysical q F.solution.value |
| ((t,(TorusFlight.flight t p).1),(TorusFlight.flight t p).2)) =ᵐ[volume] |
| TorusFlight.characteristicProfile (volume.restrict (Icc 0 T)) |
| (periodicInitial s₀ η K α γ d₀ q N F.coefficient) |
| (TorusFlight.along (TorusFlight.regroup |
| (CollisionCompactness.collisionSource (N+1) (torusPhysical q F.solution.value)))) t := by |
| have hm := (measurable_seed s₀ η K α γ d₀ q N).add |
| ((HotSeedProfile.normalized_measurable q (DatumBounds.scale s₀ η (N+1))).const_mul F.coefficient) |
| have h := TorusFlight.characteristic hcarleman F.solution hq₁ hq₂ |
| (by positivity) (by have := F.amplitude_pos; positivity) ht.1 ht.2 le_rfl hm |
| filter_upwards [h] with p hp |
| change _ = periodicInitial s₀ η K α γ d₀ q N F.coefficient p+ |
| Published.timePrimitive (volume.restrict (Icc 0 T)) _ (t,p) |
| rw [Published.timePrimitive_eq_interval ht] |
| convert hp using 1 |
| congr 1 |
| exact mul_comm _ _ |
|
|
| end BoltzmannNonuniqueness.JetCopyDatum |
|
|
| end |
|
|
| end |
|
|
| section |
|
|
| /-! # The actual cap initial data converge weakly to the prescribed cold datum -/ |
|
|
| noncomputable section |
|
|
| open MeasureTheory |
|
|
| namespace BoltzmannNonuniqueness.JetCopyDatum |
|
|
| theorem periodicInitial_integrable {s₀ η K γ d₀ q : ℝ} |
| (hs₀ : 0 < s₀) (hsmax : s₀ ≤ 1/8) (hη : 0 < η) (hηmax : η ≤ Real.log 2) |
| (hK : 0 ≤ K) (hγ : 0 < γ) (hd₀ : 0 < d₀) (hq : 0 ≤ q) (hqmax : q < 2) |
| (α : ℝ) (N : ℕ) (ell : ℝ) : |
| Integrable (periodicInitial s₀ η K α γ d₀ q N ell) := by |
| have hδ : 0 < DatumBounds.scale s₀ η (N+1) := DatumBounds.scale_pos hs₀ _ |
| have hδmax : DatumBounds.scale s₀ η (N+1) ≤ 1 := by |
| have h := JetGeometry.scale_antitone hs₀.le hη.le (Nat.zero_le (N+1)) |
| rw [show DatumBounds.scale s₀ η 0 = s₀ by simp [DatumBounds.scale]] at h |
| linarith |
| have he : periodicInitial s₀ η K α γ d₀ q N ell = |
| fun p => JetPeriodicDatum.truncated s₀ η K α γ d₀ N p+ |
| ell*HotSeedProfile.periodicDensity q (DatumBounds.scale s₀ η (N+1)) p := |
| funext (periodicInitial_eq hs₀ hsmax hη hηmax K α γ d₀ q N ell) |
| rw [he] |
| exact (JetPeriodicDatum.truncated_integrable hs₀ hsmax hη hηmax hK hγ hd₀ α N).add |
| ((HotSeedProfile.periodicDensity_integrable hq hqmax hδ hδmax).const_mul ell) |
|
|
| theorem CompactCapFamily.initial_weak_limit {s₀ η K α γ d₀ q : ℝ} |
| (F : CompactCapFamily 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) (hq : 0 ≤ q) (hqmax : q < 2) : |
| Published.WeakL1Converges volume |
| (fun n => periodicInitial s₀ η K α γ d₀ q n (F.approximation n).coefficient) |
| (JetPeriodicDatum.datum s₀ η K α γ d₀) := by |
| have hi := JetPeriodicDatum.integrable_pullback |
| (JetDatum.datum_integrable hs₀ hsmax hη hηmax hK hγ hd₀ α) |
| exact Published.weakL1_of_integral_norm_sub |
| (fun n => periodicInitial_integrable hs₀ hsmax hη hηmax hK hγ hd₀ hq hqmax α n _) |
| (JetPeriodicDatum.measurable_pullback (Measurable.tsum |
| (fun i => JetDatum.term_measurable s₀ η K α γ d₀ i))) hi F.initial_l1 |
|
|
| end BoltzmannNonuniqueness.JetCopyDatum |
|
|
| end |
|
|
| end |
|
|
| section |
|
|
| /-! # A genuine local mild solution obtained from the actual positive cap family -/ |
|
|
| noncomputable section |
|
|
| open MeasureTheory Set |
|
|
| namespace BoltzmannNonuniqueness |
|
|
| /-- The characteristic identity uses the full physical collision terms and the |
| prescribed initial density. The spacetime identity determines a continuous-time |
| representative through the Bochner integral of the collision source. -/ |
| structure MildCollisionSolution (T : ℝ) (f₀ : Density) |
| (g : (ℝ × Position) × Velocity → ℝ) : Prop where |
| measurable : Measurable g |
| nonnegative : ∀ z, 0 ≤ g z |
| integrable : Integrable g ((AnnularEnvelope.timePositionVolume T).prod volume) |
| pair_integrable : Integrable (CollisionCompactness.pairDensity g) |
| ((AnnularEnvelope.timePositionVolume T).prod volume) |
| characteristic : TorusFlight.along (TorusFlight.regroup g) =ᵐ[ |
| (volume.restrict (Icc 0 T)).prod (volume : Measure Phase)] |
| (fun z => f₀ z.2+Published.timePrimitive (volume.restrict (Icc 0 T)) |
| (TorusFlight.along (TorusFlight.regroup |
| (fun p => CollisionCompactness.gainDensity g p-CollisionCompactness.lossDensity g p))) z) |
|
|
| namespace JetCopyDatum |
|
|
| theorem TightCapFamily.exists_mild_limit_with_convergence |
| (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) |
| (hs₀ : 0 < s₀) (hsmax : s₀ ≤ 1/8) (hη : 0 < η) (hηmax : η ≤ Real.log 2) |
| (hK : 0 ≤ K) (hγ : 0 < γ) (hd₀ : 0 < d₀) (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 ∧ |
| MildCollisionSolution F.time (JetPeriodicDatum.datum s₀ η K α γ d₀) g ∧ |
| Published.WeakL1Converges ((AnnularEnvelope.timePositionVolume F.time).prod volume) |
| (fun n => CollisionCompactness.pairDensity |
| (torusPhysical q (F.approximation (φ n)).solution.value)) |
| (CollisionCompactness.pairDensity g) ∧ |
| Published.WeakL1Converges ((AnnularEnvelope.timePositionVolume F.time).prod volume) |
| (fun n => CollisionCompactness.collisionSource (φ n+1) |
| (torusPhysical q (F.approximation (φ n)).solution.value)) |
| (fun p => CollisionCompactness.gainDensity g p-CollisionCompactness.lossDensity g p) := by |
| obtain ⟨φ,g,hφ,hg,hr,hpair,hgn,hpairweak⟩ := F.exists_cutoff_equation_limit_with_pair hcarleman hGLPS hDP hq₁ hq₂ |
| let μ : Measure ℝ := volume.restrict (Icc 0 F.time) |
| let f : ℕ → (ℝ × Position) × Velocity → ℝ := |
| fun n => torusPhysical q (F.approximation (φ n)).solution.value |
| let r : ℕ → (ℝ × Position) × Velocity → ℝ := |
| fun n => CollisionCompactness.collisionSource (φ n+1) (f n) |
| let a : ℕ → Phase → ℝ := |
| fun n => periodicInitial s₀ η K α γ d₀ q (φ n) (F.approximation (φ n)).coefficient |
| have hfm n : Measurable (f n) := |
| torusPhysical_measurable (F.approximation (φ n)).solution.measurable_value q |
| have hrm n : Measurable (r n) := CollisionCompactness.collisionSource_measurable (hfm n) _ |
| have hrint n : Integrable (r n) ((AnnularEnvelope.timePositionVolume F.time).prod volume) := |
| memLp_one_iff_integrable.mp (F.toCompactCapFamily.source_ui.memLp (φ n)) |
| have hfr := TorusFlight.weak_regroup hfm hg |
| have hrr := TorusFlight.weak_regroup hrm hr |
| have hfa := TorusFlight.weak_along |
| (fun n => (hfm n).comp MeasurableEquiv.prodAssoc.symm.measurable) hfr |
| have hra := TorusFlight.weak_along |
| (fun n => (hrm n).comp MeasurableEquiv.prodAssoc.symm.measurable) hrr |
| have hrain n : Integrable (TorusFlight.along (TorusFlight.regroup (r n))) |
| (μ.prod (volume : Measure Phase)) := by |
| have hrreg : Integrable (TorusFlight.regroup (r n)) (μ.prod (volume : Measure Phase)) := by |
| have hi := (measurePreserving_prodAssoc μ (volume : Measure Position) |
| (volume : Measure Velocity)).symm.integrable_comp_of_integrable (hrint n) |
| rw [Measure.volume_eq_prod] |
| exact hi |
| exact (TorusFlight.along_measurePreserving μ).integrable_comp_of_integrable hrreg |
| have ham n : Measurable (a n) := periodicInitial_measurable s₀ η K α γ d₀ q _ _ |
| have hai n : Integrable (a n) := periodicInitial_integrable hs₀ hsmax hη hηmax hK hγ hd₀ |
| (by linarith : 0 ≤ q) hq₂ α _ _ |
| have ha := (F.toCompactCapFamily.initial_weak_limit hs₀ hsmax hη hηmax hK hγ hd₀ |
| (by linarith : 0 ≤ q) hq₂).comp hφ.tendsto_atTop |
| have heq n : TorusFlight.along (TorusFlight.regroup (f n)) =ᵐ[μ.prod volume] |
| (fun z => a n z.2+Published.timePrimitive μ |
| (TorusFlight.along (TorusFlight.regroup (r n))) z) := |
| (F.approximation (φ n)).spacetime_mild hcarleman hq₁ hq₂ F.time_pos.le |
| have hlim := TorusFlight.weak_duhamel_limit hfa hra |
| (fun n => (hrm n).comp MeasurableEquiv.prodAssoc.symm.measurable |>.comp |
| (TorusFlight.along_measurePreserving μ).measurable) |
| hrain ha ham hai heq |
| exact ⟨φ,g,hφ,hg,⟨hg.1,hgn,hg.2.1,hpair,hlim⟩,hpairweak,hr⟩ |
|
|
| theorem TightCapFamily.exists_mild_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) |
| (hs₀ : 0 < s₀) (hsmax : s₀ ≤ 1/8) (hη : 0 < η) (hηmax : η ≤ Real.log 2) |
| (hK : 0 ≤ K) (hγ : 0 < γ) (hd₀ : 0 < d₀) (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 ∧ |
| MildCollisionSolution F.time (JetPeriodicDatum.datum s₀ η K α γ d₀) g := by |
| obtain ⟨φ,g,hφ,hg,S,_hpair,_hr⟩ := |
| F.exists_mild_limit_with_convergence hcarleman hGLPS hDP |
| hs₀ hsmax hη hηmax hK hγ hd₀ hq₁ hq₂ |
| exact ⟨φ,g,hφ,hg,S⟩ |
|
|
| end JetCopyDatum |
| end BoltzmannNonuniqueness |
|
|
| end |
|
|
| end |
|
|
| end OAI |
|
|
