| import Mathlib |
| import OAI.Analysis.HardSphere.Geometry.IntegralConfigurationCons |
|
|
| 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 MeasureTheory Set Filter |
| open scoped ComplexConjugate |
|
|
| lemma SmoothMultiplier.energy_insertionJet {n : ℕ} {L : Length} |
| (f : SmoothMultiplier (n+1) L) |
| (hn : ∀ Y : Configuration n L, ∫ x : Torus L, ‖f.toFun (Fin.cons x Y)‖^2=1) |
| (ho : ∀ (i : Fin n) (j : Fin 3) (Y : Configuration n L), ∫ x : Torus L, |
| conj (f.toFun (Fin.cons x Y))*coordDerivative (n+1) L f.toFun (i.succ,j) (Fin.cons x Y)=0) |
| (u : Jet n L) : |
| energy (f.insertionJet u)=energy u + ∑ p : Direction (n+1), |
| ‖L2Multiplier.clm _ (f.partial_memLp_top p) (configurationTailCLM n L u.1)‖^2 := by |
| have hz (j : Fin 3) : (f.insertionJet u).2 (0,j)= |
| L2Multiplier.clm _ (f.partial_memLp_top (0,j)) (configurationTailCLM n L u.1) := by |
| change f.insertionScalar 0 + _ = _ |
| rw [map_zero,zero_add] |
| rfl |
| have hs (i : Fin n) (j : Fin 3) : ‖(f.insertionJet u).2 (i.succ,j)‖^2= |
| ‖u.2 (i,j)‖^2+ |
| ‖L2Multiplier.clm _ (f.partial_memLp_top (i.succ,j)) (configurationTailCLM n L u.1)‖^2 := by |
| change ‖f.insertionScalar (u.2 (i,j)) + |
| L2Multiplier.clm _ (f.partial_memLp_top (i.succ,j)) (configurationTailCLM n L u.1)‖^2 = _ |
| rw [@norm_add_sq ℂ, f.insertion_cross_zero (i.succ,j) (ho i j),map_zero, |
| mul_zero,add_zero,f.norm_insertionScalar hn] |
| unfold energy |
| simp only [Fintype.sum_prod_type,Fin.sum_univ_succ,hz,hs,Finset.sum_add_distrib] |
| ring |
|
|
| lemma SmoothMultiplier.insertion_derivative_integrable {n : ℕ} {L : Length} |
| (f : SmoothMultiplier (n+1) L) (u : ScalarL2 n L) (p : Direction (n+1)) : |
| Integrable (fun Y : Configuration n L ↦ ‖u Y‖^2 * |
| ∫ x : Torus L, ‖coordDerivative (n+1) L f.toFun p (Fin.cons x Y)‖^2) volume := by |
| have hm := ((memLp_configurationTail (Lp.memLp u)).mul (f.partial_memLp_top p)).integrable_norm_pow |
| (by decide : (2:ℕ)≠0) |
| let e := MeasurableEquiv.piFinSuccAbove (fun _ : Fin (n+1) ↦ Torus L) 0 |
| have he := (volume_preserving_piFinSuccAbove (fun _ : Fin (n+1) ↦ Torus L) 0).symm |
| have hi := (he.integrable_comp_emb e.symm.measurableEmbedding).mpr hm |
| simp only [MeasurableEquiv.piFinSuccAbove_symm_apply,Fin.insertNthEquiv, |
| Equiv.coe_fn_mk,Fin.insertNth_zero',Function.comp_def,Pi.mul_apply,Fin.tail_cons] at hi |
| convert hi.integral_prod_right using 1 |
| funext Y |
| rw [← integral_const_mul] |
| apply integral_congr_ae |
| exact Eventually.of_forall fun x ↦ by dsimp only |
|
|
| lemma SmoothMultiplier.energy_insertionJet_integral {n : ℕ} {L : Length} |
| (f : SmoothMultiplier (n+1) L) |
| (hn : ∀ Y : Configuration n L, ∫ x : Torus L, ‖f.toFun (Fin.cons x Y)‖^2=1) |
| (ho : ∀ (i : Fin n) (j : Fin 3) (Y : Configuration n L), ∫ x : Torus L, |
| conj (f.toFun (Fin.cons x Y))*coordDerivative (n+1) L f.toFun (i.succ,j) (Fin.cons x Y)=0) |
| (u : Jet n L) : |
| energy (f.insertionJet u)=energy u + ∫ Y : Configuration n L, |
| ‖u.1 Y‖^2 * ∑ p : Direction (n+1), ∫ x : Torus L, |
| ‖coordDerivative (n+1) L f.toFun p (Fin.cons x Y)‖^2 := by |
| rw [f.energy_insertionJet hn ho] |
| simp_rw [norm_weightedTail_sq,Finset.mul_sum] |
| rw [integral_finsetSum _ (fun p _ ↦ f.insertion_derivative_integrable u.1 p)] |
|
|
| lemma SmoothMultiplier.energy_insertionJet_le {n : ℕ} {L : Length} |
| (f : SmoothMultiplier (n+1) L) |
| (hn : ∀ Y : Configuration n L, ∫ x : Torus L, ‖f.toFun (Fin.cons x Y)‖^2=1) |
| (ho : ∀ (i : Fin n) (j : Fin 3) (Y : Configuration n L), ∫ x : Torus L, |
| conj (f.toFun (Fin.cons x Y))*coordDerivative (n+1) L f.toFun (i.succ,j) (Fin.cons x Y)=0) |
| (C : ℝ) (hC : ∀ Y : Configuration n L, |
| ∑ p : Direction (n+1), ∫ x : Torus L, ‖coordDerivative (n+1) L f.toFun p (Fin.cons x Y)‖^2≤C) |
| (u : Jet n L) : energy (f.insertionJet u)≤energy u + C*‖u.1‖^2 := by |
| rw [f.energy_insertionJet_integral hn ho,IndicatorStrong.norm_sq_integral,← integral_const_mul] |
| apply add_le_add_right |
| apply integral_mono |
| · simpa only [Finset.mul_sum] using integrable_finsetSum _ |
| (fun p _ ↦ f.insertion_derivative_integrable u.1 p) |
| · exact (Lp.memLp u.1).integrable_norm_pow (by decide : (2:ℕ)≠0) |>.const_mul C |
| · intro Y |
| simpa only [mul_comm C] using mul_le_mul_of_nonneg_left (hC Y) (sq_nonneg ‖u.1 Y‖) |
| |
| end HardSphere |
| |
| end |
| |
| end OAI |
| |
