| import Mathlib |
| import OAI.Analysis.HardSphere.Collisions.LocalInsertion |
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| namespace OAI |
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|
| 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 MeasureTheory Set Filter PeriodicCutoff TightCutoff |
| open scoped ComplexConjugate |
| open scoped Classical |
|
|
| lemma closeLabels_cons {n : ℕ} {L : Length} (r : ℝ) (w : SmoothMultiplier 1 L) |
| (x : Torus L) (Y : Configuration n L) : |
| (closeLabels r w (Fin.cons x Y)).card=(closeLabels r w Y).card+ |
| (if ∃ y∈tsupport w.orbital, torusDistance L y x ≤ 2*r then 1 else 0) := by |
| classical |
| simp only [closeLabels,Finset.card_filter] |
| rw [Fin.sum_univ_succ] |
| simp only [Fin.cons_zero,Fin.cons_succ] |
| exact Nat.add_comm _ _ |
|
|
| lemma closeLabels_insert_le {S : Type*} [DecidableEq S] {n : ℕ} {L : Length} (r : ℝ) |
| (w : S → SmoothMultiplier 1 L) |
| (hsep : ∀ z v : S, z≠v → ∀ x : Torus L, (w v).orbital x≠0 → |
| ¬ ∃ y∈tsupport (w z).orbital, torusDistance L y x ≤ 2*r) |
| (v z : S) (X : Configuration (n+1) L) (hx : (w v).orbital (X 0)≠0) : |
| (closeLabels r (w z) X).card ≤ (closeLabels r (w z) (Fin.tail X)).card+(if v=z then 1 else 0) := by |
| classical |
| have he : Fin.cons (X 0) (Fin.tail X)=X := by funext i; cases i using Fin.cases <;> rfl |
| rw [←he,closeLabels_cons] |
| simp only [Fin.tail_cons] |
| by_cases hvz : v=z |
| · subst v |
| simp only [ite_true] |
| split_ifs <;> omega |
| · have hzv : z≠v := Ne.symm hvz |
| rw [ite_eq_right hvz,ite_eq_right (hsep z v hzv (X 0) hx)] |
|
|
| noncomputable def insertionCount {S : Type*} [DecidableEq S] (site : ℕ → S) (t : ℕ) (z : S) : ℕ := |
| ((Finset.range t).filter (fun i ↦ site i=z)).card |
|
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| lemma insertionCount_zero {S : Type*} [DecidableEq S] (site : ℕ → S) (z : S) : |
| insertionCount site 0 z=0 := by simp [insertionCount] |
|
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| lemma insertionCount_succ {S : Type*} [DecidableEq S] (site : ℕ → S) (t : ℕ) (z : S) : |
| insertionCount site (t+1) z=insertionCount site t z+(if site t=z then 1 else 0) := by |
| simp only [insertionCount,Finset.card_filter,Finset.sum_range_succ] |
|
|
| lemma insertionCount_mono {S : Type*} [DecidableEq S] (site : ℕ → S) {t T : ℕ} (ht : t ≤ T) (z : S) : |
| insertionCount site t z ≤ insertionCount site T z := by |
| apply Finset.card_le_card |
| intro i hi |
| obtain ⟨hi,hiz⟩ := Finset.mem_filter.mp hi |
| exact Finset.mem_filter.mpr ⟨Finset.mem_range.mpr ((Finset.mem_range.mp hi).trans_le ht),hiz⟩ |
|
|
| lemma local_insertion_iterated : ∃ C : ℝ, 0<C ∧ |
| ∀ (S : Type) [DecidableEq S] (n : ℕ) (L : Length) (r : ℝ), 0<r → |
| ∀ (w : S → SmoothMultiplier 1 L), |
| (∀ z X, conj ((w z).toFun X)=(w z).toFun X) → |
| (∀ z, (∫ y : Torus L, ‖(w z).orbital y‖^2)=1) → |
| (∀ z v : S, z≠v → ∀ x : Torus L, (w v).orbital x≠0 → |
| ¬ ∃ y∈tsupport (w z).orbital, torusDistance L y x ≤ 2*r) → |
| ∀ (T k m : ℕ) (site : ℕ → S) (B E : ℝ), |
| (∀ z, (w z).orbitalBound ≤ B) → (∀ z, (w z).orbitalEnergy ≤ E) → |
| (∀ z, insertionCount site T z ≤ m) → C*B*(k+m)*r^3 ≤ 1/4 → |
| ∀ u : Jet n L, u∈dirichletDomain n L r → |
| (∀ᵐ Y ∂volume, u.1 Y≠0 → ∀ z, (closeLabels r (w z) Y).card ≤ k) → |
| ∃ v : Jet (n+T) L, v∈dirichletDomain (n+T) L r ∧ ‖v.1‖=‖u.1‖ ∧ |
| energy v ≤ energy u+T*(C*(E+B*(k+m)*r))*‖u.1‖^2 := by |
| obtain ⟨C,hC,hins⟩ := local_insertion_supported |
| refine ⟨C,hC,?_⟩ |
| intro S _ n L r hr w hw hn hsep T k m site B E hB hE hct hs u hu hk |
| have hrec (t : ℕ) (ht : t ≤ T) : ∃ v : Jet (n+t) L, |
| v∈dirichletDomain (n+t) L r ∧ ‖v.1‖=‖u.1‖ ∧ |
| energy v ≤ energy u+t*(C*(E+B*(k+m)*r))*‖u.1‖^2 ∧ |
| (∀ᵐ X ∂volume, v.1 X≠0 → ∀ z, |
| (closeLabels r (w z) X).card ≤ k+insertionCount site t z) := by |
| induction t with |
| | zero => |
| refine ⟨u,hu,rfl,?_,?_⟩ |
| · simp only [Nat.cast_zero,zero_mul,add_zero,le_refl] |
| · simpa only [insertionCount_zero,Nat.add_zero] using hk |
| | succ t ih => |
| obtain ⟨v,hv,hvn,hve,hvk⟩ := ih (Nat.le_trans (Nat.le_succ t) ht) |
| have hmass : C*(w (site t)).orbitalBound*(k+m:ℕ)*r^3 ≤ 1/4 := by |
| apply le_trans _ hs |
| simp only [Nat.cast_add] |
| apply mul_le_mul_of_nonneg_right _ (by positivity) |
| apply mul_le_mul_of_nonneg_right _ (by positivity) |
| exact mul_le_mul_of_nonneg_left (hB _) hC.le |
| have hbudget : ∀ᵐ X ∂volume, v.1 X≠0 → (closeLabels r (w (site t)) X).card ≤ k+m := by |
| filter_upwards [hvk] with X hX |
| intro hne |
| exact (hX hne (site t)).trans (Nat.add_le_add_left |
| ((insertionCount_mono site (Nat.le_trans (Nat.le_succ t) ht) _).trans (hct _)) k) |
| obtain ⟨v',hv',hvn',hve',hvs⟩ := hins (n+t) L r hr (w (site t)) (hw _) (hn _) (k+m) hmass v hv hbudget |
| refine ⟨v',hv',hvn'.trans hvn,?_,?_⟩ |
| · have hbound : C*((w (site t)).orbitalEnergy+(w (site t)).orbitalBound*(k+m:ℕ)*r) ≤ |
| C*(E+B*(k+m)*r) := by |
| apply mul_le_mul_of_nonneg_left _ hC.le |
| apply add_le_add (hE _) |
| rw [Nat.cast_add] |
| exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right (hB _) (by positivity)) hr.le |
| rw [hvn] at hve' |
| have he := hve'.trans (add_le_add le_rfl (mul_le_mul_of_nonneg_right hbound (sq_nonneg _))) |
| rw [Nat.cast_add,Nat.cast_one] |
| nlinarith |
| · filter_upwards [hvs,(configurationTail_quasiPreserving (n+t) L).ae hvk] with X hX hOld |
| intro hne z |
| obtain ⟨hOldne,hOrb⟩ := hX hne |
| have hh := closeLabels_insert_le r w hsep (site t) z X hOrb |
| have hh' := hOld hOldne z |
| rw [insertionCount_succ] |
| omega |
| obtain ⟨v,hv,hvn,hve,_⟩ := hrec T le_rfl |
| exact ⟨v,hv,hvn,hve⟩ |
|
|
| end HardSphere |
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| end |
|
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| end OAI |
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