openAI-math-dataset / lean /OAI /MathematicalPhysics /Boltzmann /Approximation /DormantBoxLimit.lean
| import OAI.MathematicalPhysics.Boltzmann.Approximation.DormantConstruction | |
| import OAI.MathematicalPhysics.Boltzmann.Branching.SmallDilationGeometry | |
| import OAI.MathematicalPhysics.Boltzmann.Compactness.UniformIntegrableCoordinates | |
| namespace OAI | |
| section | |
| /-! # Uniform small-time bounds on bounded kinetic boxes away from the axis -/ | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped Topology | |
| open BoltzmannNonuniqueness.VelocityWeight | |
| namespace BoltzmannNonuniqueness.JetCopyDatum | |
| def awayKineticBox (B R ε : ℝ) : Set Published.KineticPhase := | |
| ((Icc (0:ℝ) B ×ˢ Metric.closedBall (0:Velocity) R) ×ˢ Metric.closedBall (0:Velocity) R) ∩ | |
| {z | ε ≤ ‖transverse z.2‖} | |
| theorem measurableSet_awayKineticBox (B R ε : ℝ) : MeasurableSet (awayKineticBox B R ε) := | |
| ((measurableSet_Icc.prod measurableSet_closedBall).prod measurableSet_closedBall).inter | |
| (measurableSet_le measurable_const ((ChartKernel.measurable_transverse.comp measurable_snd).norm)) | |
| theorem awayKineticBox_measure_lt_top (B R ε : ℝ) : | |
| volume (awayKineticBox B R ε) < ⊤ := | |
| (measure_mono inter_subset_left).trans_lt | |
| (((isCompact_Icc.prod (isCompact_closedBall (0:Velocity) R)).prod | |
| (isCompact_closedBall (0:Velocity) R)).measure_lt_top) | |
| def dormantBoxBound (s₀ η K α d₀ q ε B R r : ℝ) : ℝ := | |
| JetSupersolution.transverseDecayConstant K d₀ ε*r^8*R^9+ | |
| (1+JetEscapedEnvelope.constant s₀ η α K d₀ (JetEscapedEnvelope.exponent 2))*r*B^2*(1+ε^(-q)) | |
| theorem DormantLimit.scaled_box_bound {s₀ η K α γ d₀ q ε B R : ℝ} | |
| {A : DormantFamily s₀ η K α γ d₀ q} (D : DormantLimit A) | |
| (hs₀ : 0 < s₀) (hsmax : s₀ ≤ 1/8) (hη : 0 < η) (hηmax : η ≤ Real.log 2) | |
| (hK : 0 ≤ K) (hα : 0 ≤ α) (hαmax : α ≤ 1) (hγ : 0 ≤ γ) | |
| (hd₀ : 0 < d₀) (hd₀max : d₀ ≤ 1) | |
| (hsmall : JetComparisonCoefficient.coefficient K*d₀*s₀^9 ≤ 1) | |
| (hq : 0 ≤ q) (hqmax : q < 2) (hε : 0 < ε) (_hB : 0 ≤ B) (_hR : 0 ≤ R) | |
| {r : ℕ → ℝ} (hr : ∀ n, 0 < r n) (hlim : Tendsto r atTop (𝓝 0)) : | |
| ∀ᶠ n in atTop, ∀ᵐ z ∂volume.restrict (awayKineticBox B R ε), | |
| |(r n)⁻¹*TorusKineticChart.liftDensity D.density (kineticDilation (r n) z)| ≤ | |
| dormantBoxBound s₀ η K α d₀ q ε B R (r n) := by | |
| have hc := JetSupersolution.transverseDecayConstant_nonneg (ε := ε) hK hd₀.le | |
| have he := JetEscapedEnvelope.constant_nonneg hs₀ hK hd₀ η α (JetEscapedEnvelope.exponent 2) | |
| filter_upwards [eventually_dilation_in_window hr hlim A.time_pos B R] with n hn | |
| have hb := D.scaled_central_bound hs₀ hsmax hη hηmax hK hα hαmax hγ hd₀ hd₀max | |
| hsmall hqmax hε (hr n) | |
| filter_upwards [ae_restrict_of_ae hb,ae_restrict_mem (measurableSet_awayKineticBox B R ε)] with z hz hbox | |
| have hx : ‖z.1.2‖ ≤ R := by simpa only [Metric.mem_closedBall,dist_zero_right] using hbox.1.1.2 | |
| obtain ⟨ht,hxc⟩ := hn z.1.1 hbox.1.1.1 z.1.2 hx | |
| have hpos : 0 ≤ (r n)⁻¹*TorusKineticChart.liftDensity D.density (kineticDilation (r n) z) := | |
| mul_nonneg (inv_nonneg.mpr (hr n).le) (D.mild.nonnegative _) | |
| rw [abs_of_nonneg hpos] | |
| apply (hz ht hxc hbox.2).trans | |
| have htx : ‖transverse z.1.2‖ ≤ R := by | |
| have hn := norm_sq_split z.1.2 | |
| nlinarith [norm_nonneg (transverse z.1.2),norm_nonneg z.1.2,sq_nonneg (axial z.1.2)] | |
| have hv := H_le_off_axis hq hε hbox.2 | |
| unfold dormantBoxBound | |
| apply add_le_add | |
| · exact mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (norm_nonneg _) htx 9) | |
| (mul_nonneg hc (pow_nonneg (hr n).le 8)) | |
| · have htcoef : 0 ≤ (1+JetEscapedEnvelope.constant s₀ η α K d₀ | |
| (JetEscapedEnvelope.exponent 2))*r n := mul_nonneg (by linarith) (hr n).le | |
| exact mul_le_mul | |
| (mul_le_mul_of_nonneg_left (pow_le_pow_left₀ hbox.1.1.1.1 hbox.1.1.1.2 2) htcoef) | |
| hv (H_nonneg q z.2) (mul_nonneg htcoef (sq_nonneg B)) | |
| end BoltzmannNonuniqueness.JetCopyDatum | |
| end | |
| end | |
| section | |
| /-! # Small-time L1 vanishing of the actual dormant solution away from the axis -/ | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped Topology | |
| namespace BoltzmannNonuniqueness.JetCopyDatum | |
| theorem DormantLimit.scaled_box_L1_tendsto {s₀ η K α γ d₀ q ε B R : ℝ} | |
| {A : DormantFamily s₀ η K α γ d₀ q} (D : DormantLimit A) | |
| (hs₀ : 0 < s₀) (hsmax : s₀ ≤ 1/8) (hη : 0 < η) (hηmax : η ≤ Real.log 2) | |
| (hK : 0 ≤ K) (hα : 0 ≤ α) (hαmax : α ≤ 1) (hγ : 0 ≤ γ) | |
| (hd₀ : 0 < d₀) (hd₀max : d₀ ≤ 1) | |
| (hsmall : JetComparisonCoefficient.coefficient K*d₀*s₀^9 ≤ 1) | |
| (hq : 0 ≤ q) (hqmax : q < 2) (hε : 0 < ε) (hB : 0 ≤ B) (hR : 0 ≤ R) | |
| {r : ℕ → ℝ} (hr : ∀ n, 0 < r n) (hlim : Tendsto r atTop (𝓝 0)) : | |
| Tendsto (fun n => ∫ z in awayKineticBox B R ε, | |
| |(r n)⁻¹*TorusKineticChart.liftDensity D.density (kineticDilation (r n) z)|) | |
| atTop (𝓝 0) := by | |
| let μ : Measure Published.KineticPhase := volume.restrict (awayKineticBox B R ε) | |
| let : IsFiniteMeasure μ := isFiniteMeasure_restrict.mpr (awayKineticBox_measure_lt_top B R ε).ne | |
| have hm (n) : Measurable (fun z => | |
| (r n)⁻¹*TorusKineticChart.liftDensity D.density (kineticDilation (r n) z)) := | |
| ((TorusKineticChart.liftDensity_measurable D.mild.measurable).comp | |
| (kineticDilation_quasiMeasurePreserving (hr n)).measurable).const_mul _ | |
| apply integral_abs_tendsto_of_uniform_bound (μ := μ) (fun n => (hm n).aestronglyMeasurable) | |
| (D.scaled_box_bound hs₀ hsmax hη hηmax hK hα hαmax hγ hd₀ hd₀max hsmall | |
| hq hqmax hε hB hR hr hlim) | |
| have h1 := ((hlim.pow 8).const_mul (JetSupersolution.transverseDecayConstant K d₀ ε)).mul_const (R^9) | |
| have h2 := (((hlim.const_mul | |
| (1+JetEscapedEnvelope.constant s₀ η α K d₀ (JetEscapedEnvelope.exponent 2))).mul_const (B^2)).mul_const | |
| (1+ε^(-q))) | |
| simpa only [dormantBoxBound,zero_pow (by norm_num : 8 ≠ 0),mul_zero,zero_mul,zero_add] using h1.add h2 | |
| end BoltzmannNonuniqueness.JetCopyDatum | |
| end | |
| end | |
| end OAI | |