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67aee1d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 | import Mathlib
import OAI.Analysis.HardSphere.FormDomain.RawCutoffGradientIntegral
import OAI.Analysis.HardSphere.FormDomain.RmsPermuteFunBosonic
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
lemma iterated_insertion (C : ℝ) (_hC : 0<C)
(hins : ∀ (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)
(n m : ℕ) (L : Length) {r : ℝ} (hr : 0<r)
(hs : ((n+m:ℕ):ℝ)*(2*Real.pi*r)^3≤L.val^3/2)
(u : Jet n L) (hu : u∈dirichletDomain n L r) :
∃ v : Jet (n+m) L, v∈dirichletDomain (n+m) L r ∧ ‖v.1‖=‖u.1‖ ∧
energy v≤energy u+(∑ j∈Finset.range m,C*((n+j:ℕ):ℝ)*r/L.val^3)*‖u.1‖^2 := by
induction m with
| zero => exact ⟨u,hu,rfl,by simp⟩
| succ m ih =>
have hs' : ((n+m:ℕ):ℝ)*(2*Real.pi*r)^3≤L.val^3/2 := by
exact (mul_le_mul_of_nonneg_right
(show ((n+m:ℕ):ℝ)≤((n+(m+1):ℕ):ℝ) by exact_mod_cast Nat.le_succ (n+m))
(by positivity)).trans hs
obtain ⟨v,hv,hn,he⟩ := ih hs'
obtain ⟨w,hw,hnw,hew⟩ := hins (n+m) L r hr hs' v hv
refine ⟨w,hw,hnw.trans hn,?_⟩
rw [hn] at hew
rw [Finset.sum_range_succ]
nlinarith
lemma ground_energy_removal : ∃ C : ℝ, 0<C ∧
∀ (n m : ℕ) (L : Length) (r : ℝ), 0<r →
((n+m:ℕ):ℝ)*(2*Real.pi*r)^3≤L.val^3/2 →
∀ u : Jet (n+m) L, IsGroundVector (n+m) L r u →
∀ v : Jet n L, v∈dirichletDomain n L r →
(energy u-C*(m:ℝ)*((n+m:ℕ):ℝ)*r/L.val^3)*‖v.1‖^2≤energy v := by
obtain ⟨C,hC,hins⟩ := uniform_insertion
refine ⟨C,hC,fun n m L r hr hs u hu v hv ↦ ?_⟩
have hL : 0<L.val := L.property
obtain ⟨w,hw,hn,he⟩ := iterated_insertion C hC hins n m L hr hs v hv
have hmin := hu.rayleigh_all w hw
rw [hn] at hmin
have hsum : (∑ j∈Finset.range m,C*((n+j:ℕ):ℝ)*r/L.val^3)≤
C*(m:ℝ)*((n+m:ℕ):ℝ)*r/L.val^3 := by
calc
_ ≤ ∑ _j∈Finset.range m,C*((n+m:ℕ):ℝ)*r/L.val^3 := by
apply Finset.sum_le_sum
intro j hj
apply div_le_div_of_nonneg_right _ (by positivity)
apply mul_le_mul_of_nonneg_right _ hr.le
apply mul_le_mul_of_nonneg_left _ hC.le
exact_mod_cast Nat.add_le_add_left (Nat.le_of_lt (Finset.mem_range.mp hj)) n
_ = _ := by simp only [Finset.sum_const,Finset.card_range,nsmul_eq_mul]; ring
have := mul_le_mul_of_nonneg_right hsum (sq_nonneg ‖v.1‖)
nlinarith
end HardSphere
end
end OAI
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