| import Mathlib |
| import OAI.Analysis.HardSphere.Geometry.NormalizationDerivative |
| import OAI.Analysis.HardSphere.Comparison.RawAngleLiftSumSquares |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped ENNReal Topology BigOperators |
| open scoped BigOperators ComplexConjugate |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped ENNReal NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Real Complex |
| open scoped ENNReal ComplexConjugate |
| open ProbabilityTheory |
| namespace HardSphere |
| open PeriodicCutoff |
|
|
| lemma rawCutoff_gradient_integral_le (n : ℕ) (L : Length) {r M : ℝ} (hr : 0<r) (hM : 0≤M) |
| (hb : ∀ t : ℝ, |deriv Real.smoothTransition t|≤M) (Y : Configuration n L) : |
| (∑ p : Direction (n+1), ∫ y : Torus L, |
| ‖coordDerivative (n+1) L (rawCutoff n L r).toFun p (Fin.cons y Y)‖^2)≤ |
| 48*Real.pi^5*M^2*(n:ℝ)*r := by |
| have hf (p : Direction (n+1)) : Integrable (fun y : Torus L ↦ |
| ‖coordDerivative (n+1) L (rawCutoff n L r).toFun p (Fin.cons y Y)‖^2) volume := by |
| have hc : Continuous (fun y : Torus L ↦ |
| ‖coordDerivative (n+1) L (rawCutoff n L r).toFun p (Fin.cons y Y)‖^2) := |
| (((continuous_coordDerivative (rawCutoff n L r).smooth p).comp |
| ((continuous_id : Continuous (id : Torus L → Torus L)).finCons |
| (continuous_const : Continuous (fun _ : Torus L ↦ Y)))).norm).pow 2 |
| exact hc.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) |
| have hi (i : Fin n) : Integrable ((box L.val (Real.pi*r) (Y i)).indicator (fun _ ↦ (1:ℝ))) volume := |
| (integrable_const (1:ℝ)).indicator (box_measurable L.val _ _) |
| have hh := integral_mono (integrable_finsetSum Finset.univ (fun p _ ↦ hf p)) |
| ((integrable_finsetSum Finset.univ (fun i _ ↦ hi i)).const_mul (6*Real.pi^2*M^2/r^2)) |
| (fun y : Torus L ↦ rawCutoff_gradient_le n L hr hM hb (Fin.cons y Y)) |
| rw [integral_finsetSum _ (fun p _ ↦ hf p),integral_const_mul, |
| integral_finsetSum _ (fun i _ ↦ hi i)] at hh |
| have hv : (∑ i : Fin n, ∫ y : Torus L, (box L.val (Real.pi*r) (Y i)).indicator (fun _ ↦ (1:ℝ)) y)≤ |
| (n:ℝ)*(2*Real.pi*r)^3 := by |
| calc |
| _ = ∑ i : Fin n, volume.real (box L.val (Real.pi*r) (Y i)) := by |
| apply Finset.sum_congr rfl |
| intro i _ |
| exact integral_indicator_one (box_measurable L.val _ _) |
| _ ≤ ∑ i : Fin n, (2*(Real.pi*r))^3 := Finset.sum_le_sum (fun i _ ↦ box_volume_le L.val (by positivity) (Y i)) |
| _ = _ := by simp [mul_assoc] |
| calc |
| _ ≤ (6*Real.pi^2*M^2/r^2)*∑ i : Fin n, ∫ y : Torus L, |
| (box L.val (Real.pi*r) (Y i)).indicator (fun _ ↦ (1:ℝ)) y := hh |
| _ ≤ (6*Real.pi^2*M^2/r^2)*((n:ℝ)*(2*Real.pi*r)^3) := |
| mul_le_mul_of_nonneg_left hv (by positivity) |
| _ = _ := by field_simp [hr.ne']; ring |
| |
| lemma uniform_insertion : ∃ C : ℝ, 0<C ∧ ∀ (n : ℕ) (L : Length) (r : ℝ), 0<r → |
| (n:ℝ)*(2*Real.pi*r)^3≤L.val^3/2 → |
| ∀ u : Jet n L, u∈dirichletDomain n L r → |
| ∃ v : Jet (n+1) L, v∈dirichletDomain (n+1) L r ∧ |
| ‖v.1‖=‖u.1‖ ∧ energy v≤energy u+(C*(n:ℝ)*r/L.val^3)*‖u.1‖^2 := by |
| obtain ⟨M,hM,hb⟩ := exists_transition_bound |
| refine ⟨96*Real.pi^5*M^2,by positivity,fun n L r hr hs u hu ↦ ?_⟩ |
| have hL := L.property |
| have hZ (Y : Configuration n L) : L.val^3/2≤(rawCutoff n L r).fiberMass Y := by |
| linarith [rawCutoff_mass_lower n L hr Y] |
| obtain ⟨v,hv,hn,he⟩ := (rawCutoff n L r).insertion_from_raw |
| (rawCutoff_clearance n L hr) (rawCutoff_real n L r) (show 0<L.val^3/2 by positivity) |
| hZ (rawCutoff_gradient_integral_le n L hr hM.le hb) hu |
| refine ⟨v,hv,hn,?_⟩ |
| convert he using 1 |
| congr 2 |
| ring |
| |
| end HardSphere |
| |
| end |
| |
| end OAI |
| |
