openAI-math-dataset / lean /OAI /MathematicalPhysics /Boltzmann /Approximation /CutoffLipschitz.lean
| import OAI.MathematicalPhysics.Boltzmann.Branching.GainSmoothingError | |
| namespace OAI | |
| /-! # Local weighted Lipschitz bounds for the finite-cutoff equation -/ | |
| noncomputable section | |
| open MeasureTheory | |
| open BoltzmannNonuniqueness.VelocityWeight | |
| namespace BoltzmannNonuniqueness.CutoffLipschitz | |
| theorem gain_difference (hcarleman : Published.GPV2009CarlemanInput) | |
| {q A B D : ℝ} (hq₁ : 1 < q) (hq₂ : q < 2) (hA : 0 ≤ A) (hB : 0 ≤ B) (hD : 0 ≤ D) | |
| {f g : Velocity → ℝ} (hfm : Measurable f) (hgm : Measurable g) | |
| (hf : ∀ w, 0 ≤ f w) (hg : ∀ w, 0 ≤ g w) | |
| (hfB : ∀ w, f w ≤ A * H q w) (hgB : ∀ w, g w ≤ B * H q w) | |
| (hfg : ∀ w, |f w - g w| ≤ D * H q w) (b : ℝ) : | |
| ∀ᵐ v : Velocity, |CollisionCutoff.gain b f v - CollisionCutoff.gain b g v| ≤ | |
| (D * (A + B) * HotApplication.realBound q) * H q v := by | |
| have hforward := GainUpperDifference.cutoff_gain_bound hcarleman hq₁ hq₂ hA hB hD | |
| hfm hgm hf hg hfB hgB (fun w => (le_abs_self _).trans (hfg w)) b | |
| have hbackward := GainUpperDifference.cutoff_gain_bound hcarleman hq₁ hq₂ hB hA hD | |
| hgm hfm hg hf hgB hfB (fun w => (le_abs_self _).trans (by simpa only [abs_sub_comm] using hfg w)) b | |
| filter_upwards [hforward, hbackward] with v hv hv' | |
| rw [add_comm B A] at hv' | |
| exact abs_le.mpr ⟨by linarith, hv⟩ | |
| def frequencyConstant (q b : ℝ) : ℝ := 2 * Real.pi * b * ∫ w, H q w | |
| theorem frequencyConstant_nonneg {b : ℝ} (hb : 0 ≤ b) (q : ℝ) : 0 ≤ frequencyConstant q b := by | |
| unfold frequencyConstant | |
| have hi : 0 ≤ ∫ w, H q w := integral_nonneg (H_nonneg q) | |
| positivity | |
| theorem integrable_dominated {q A : ℝ} (hq : q < 2) {f : Velocity → ℝ} | |
| (hfm : Measurable f) (hf : ∀ w, 0 ≤ f w) (hfB : ∀ w, f w ≤ A * H q w) : Integrable f := by | |
| apply ((integrable_H hq).const_mul A).mono' hfm.aestronglyMeasurable | |
| filter_upwards [] with w | |
| simpa only [Real.norm_eq_abs, abs_of_nonneg (hf w)] using hfB w | |
| theorem frequency_nonneg {f : Velocity → ℝ} (hf : ∀ w, 0 ≤ f w) (b : ℝ) (v : Velocity) : | |
| 0 ≤ CollisionCutoff.frequency b f v := by | |
| apply mul_nonneg (by positivity) | |
| exact integral_nonneg (fun w => mul_nonneg (CollisionCutoff.rate_nonneg b v w) (hf w)) | |
| theorem frequency_bound {q A b : ℝ} (hq : q < 2) (hb : 0 ≤ b) | |
| {f : Velocity → ℝ} (hfm : Measurable f) (hf : ∀ w, 0 ≤ f w) | |
| (hfB : ∀ w, f w ≤ A * H q w) (v : Velocity) : | |
| CollisionCutoff.frequency b f v ≤ A * frequencyConstant q b := by | |
| have hi := integrable_dominated hq hfm hf hfB | |
| calc | |
| _ ≤ 2 * Real.pi * b * ∫ w, |f w| := | |
| (le_abs_self _).trans (CollisionCutoff.abs_frequency_le hb hi v) | |
| _ ≤ 2 * Real.pi * b * ∫ w, A * H q w := by | |
| apply mul_le_mul_of_nonneg_left _ (by positivity) | |
| exact integral_mono hi.abs ((integrable_H hq).const_mul A) | |
| (fun w => by simpa only [abs_of_nonneg (hf w)] using hfB w) | |
| _ = _ := by rw [integral_const_mul]; unfold frequencyConstant; ring | |
| theorem frequency_difference {q A B D b : ℝ} (hq : q < 2) (hb : 0 ≤ b) | |
| {f g : Velocity → ℝ} (hfm : Measurable f) (hgm : Measurable g) | |
| (hf : ∀ w, 0 ≤ f w) (hg : ∀ w, 0 ≤ g w) | |
| (hfB : ∀ w, f w ≤ A * H q w) (hgB : ∀ w, g w ≤ B * H q w) | |
| (hfg : ∀ w, |f w - g w| ≤ D * H q w) (v : Velocity) : | |
| |CollisionCutoff.frequency b f v - CollisionCutoff.frequency b g v| ≤ D * frequencyConstant q b := by | |
| have hfi := integrable_dominated hq hfm hf hfB | |
| have hgi := integrable_dominated hq hgm hg hgB | |
| have he : CollisionCutoff.frequency b f v - CollisionCutoff.frequency b g v = | |
| CollisionCutoff.frequency b (fun w => f w - g w) v := by | |
| unfold CollisionCutoff.frequency | |
| rw [← mul_sub, ← integral_sub (CollisionCutoff.integrable_frequency hb hfi v) | |
| (CollisionCutoff.integrable_frequency hb hgi v)] | |
| congr 2 | |
| funext w | |
| ring | |
| rw [he] | |
| calc | |
| _ ≤ 2 * Real.pi * b * ∫ w, |f w - g w| := | |
| CollisionCutoff.abs_frequency_le hb (hfi.sub hgi) v | |
| _ ≤ 2 * Real.pi * b * ∫ w, D * H q w := by | |
| apply mul_le_mul_of_nonneg_left _ (by positivity) | |
| exact integral_mono (hfi.sub hgi).abs ((integrable_H hq).const_mul D) hfg | |
| _ = _ := by rw [integral_const_mul]; unfold frequencyConstant; ring | |
| theorem loss_difference {q A B D b : ℝ} (hq : q < 2) (hb : 0 ≤ b) | |
| (hB : 0 ≤ B) (hD : 0 ≤ D) | |
| {f g : Velocity → ℝ} (hfm : Measurable f) (hgm : Measurable g) | |
| (hf : ∀ w, 0 ≤ f w) (hg : ∀ w, 0 ≤ g w) | |
| (hfB : ∀ w, f w ≤ A * H q w) (hgB : ∀ w, g w ≤ B * H q w) | |
| (hfg : ∀ w, |f w - g w| ≤ D * H q w) (v : Velocity) : | |
| |f v * CollisionCutoff.frequency b f v - g v * CollisionCutoff.frequency b g v| ≤ | |
| (D * (A + B) * frequencyConstant q b) * H q v := by | |
| have hνf := frequency_nonneg hf b v | |
| have hνB := frequency_bound hq hb hfm hf hfB v | |
| have hνD := frequency_difference hq hb hfm hgm hf hg hfB hgB hfg v | |
| calc | |
| _ = |(f v - g v) * CollisionCutoff.frequency b f v + | |
| g v * (CollisionCutoff.frequency b f v - CollisionCutoff.frequency b g v)| := by congr 1; ring | |
| _ ≤ |f v - g v| * CollisionCutoff.frequency b f v + | |
| g v * |CollisionCutoff.frequency b f v - CollisionCutoff.frequency b g v| := by | |
| simpa only [abs_mul, abs_of_nonneg hνf, abs_of_nonneg (hg v)] using abs_add_le | |
| ((f v - g v) * CollisionCutoff.frequency b f v) | |
| (g v * (CollisionCutoff.frequency b f v - CollisionCutoff.frequency b g v)) | |
| _ ≤ (D * H q v) * (A * frequencyConstant q b) + (B * H q v) * (D * frequencyConstant q b) := | |
| add_le_add (mul_le_mul (hfg v) hνB hνf (mul_nonneg hD (H_nonneg q v))) | |
| (mul_le_mul (hgB v) hνD (abs_nonneg _) (mul_nonneg hB (H_nonneg q v))) | |
| _ = _ := by ring | |
| def collision (b : ℝ) (f : Velocity → ℝ) (v : Velocity) : ℝ := | |
| CollisionCutoff.gain b f v - f v * CollisionCutoff.frequency b f v | |
| /-- The local difference constant for the actual cutoff collision field. -/ | |
| theorem collision_difference (hcarleman : Published.GPV2009CarlemanInput) | |
| {q A B D b : ℝ} (hq₁ : 1 < q) (hq₂ : q < 2) (hb : 0 ≤ b) | |
| (hA : 0 ≤ A) (hB : 0 ≤ B) (hD : 0 ≤ D) | |
| {f g : Velocity → ℝ} (hfm : Measurable f) (hgm : Measurable g) | |
| (hf : ∀ w, 0 ≤ f w) (hg : ∀ w, 0 ≤ g w) | |
| (hfB : ∀ w, f w ≤ A * H q w) (hgB : ∀ w, g w ≤ B * H q w) | |
| (hfg : ∀ w, |f w - g w| ≤ D * H q w) : | |
| ∀ᵐ v : Velocity, |collision b f v - collision b g v| ≤ | |
| (D * (A + B) * (HotApplication.realBound q + frequencyConstant q b)) * H q v := by | |
| filter_upwards [gain_difference hcarleman hq₁ hq₂ hA hB hD hfm hgm hf hg hfB hgB hfg b] | |
| with v hv | |
| have hl := loss_difference hq₂ hb hB hD hfm hgm hf hg hfB hgB hfg v | |
| calc | |
| _ = |(CollisionCutoff.gain b f v - CollisionCutoff.gain b g v) - | |
| (f v * CollisionCutoff.frequency b f v - g v * CollisionCutoff.frequency b g v)| := by | |
| unfold collision | |
| congr 1 | |
| ring | |
| _ ≤ |CollisionCutoff.gain b f v - CollisionCutoff.gain b g v| + | |
| |f v * CollisionCutoff.frequency b f v - g v * CollisionCutoff.frequency b g v| := abs_sub _ _ | |
| _ ≤ _ := by nlinarith only [hv, hl] | |
| end BoltzmannNonuniqueness.CutoffLipschitz | |
| end | |
| end OAI | |