File size: 7,024 Bytes
caec230 | 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 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 | import OAI.MathematicalPhysics.NavierStokes.VelocityDetection.ExpandingArrayGate
import OAI.MathematicalPhysics.NavierStokes.VelocityDetection.UniformDerivativesNormSumLeSparse
namespace OAI
noncomputable section
namespace VelocityDetection.ExpandingArray
open scoped BigOperators Topology ContDiff
open Set Function Filter
open Set Function Filter MeasureTheory
open scoped Topology BigOperators ContDiff
open scoped Topology ContDiff BigOperators
open Expanding Stacks FiniteAddresses UniformDerivatives
variable {N b : ℕ} (hb : 0 < b) (table : Fin N → Fin b → Option (Rule (Fin N) b))
(terminal : Fin N) (ν : ℝ) (m : ℕ)
theorem gate_jet_zero_outside (hν : 0 < ν) (n d : ℕ)
(c : Active hb table (capacity b m n)) (q : ℝ × Coord 2) (i : Fin 2)
(hq : ∃ j, 3 * radius ν (K₀ N b m) (D b) n <
|q.2 j - path hb table terminal ν m n c q.1 j|) :
iteratedFDeriv ℝ d (fun q => gate hb table terminal ν m n c q i) q = 0 := by
obtain ⟨j, hj⟩ := hq
have hcont : Continuous (fun p : ℝ × Coord 2 =>
|p.2 j - path hb table terminal ν m n c p.1 j|) :=
((continuous_apply j).comp continuous_snd |>.sub
((continuous_apply j).comp ((contDiff_path hb table terminal ν m n c).continuous.comp
continuous_fst))).abs
have heq : (fun p => gate hb table terminal ν m n c p i) =ᶠ[𝓝 q] 0 := by
filter_upwards [(isOpen_lt continuous_const hcont).mem_nhds hj] with p hp
have h := TranslationGates.field_zero_outside
(lt_of_lt_of_le zero_lt_one (radius_ge_one hν (K₀_nonneg N b m) (D_ge_one hb) n))
(path hb table terminal ν m n c p.1) (deriv (path hb table terminal ν m n c) p.1) p.2 ⟨j, hp.le⟩
exact congrFun h i
simpa using (heq.iteratedFDeriv ℝ d).eq_of_nhds
theorem gate_jets_disjoint (hν : 0 < ν) (hm : 1 ≤ m)
(hdir : IncomingDirection table) (hin : IncomingRule table) (n d : ℕ)
(c e : Active hb table (capacity b m n)) (hce : c ≠ e) (q : ℝ × Coord 2) (i : Fin 2) :
iteratedFDeriv ℝ d (fun q => gate hb table terminal ν m n c q i) q = 0 ∨
iteratedFDeriv ℝ d (fun q => gate hb table terminal ν m n e q i) q = 0 := by
obtain ⟨j, hj⟩ := paths_separated hb table terminal ν m hν hm hdir hin n c e hce q.1
have hR : 0 < radius ν (K₀ N b m) (D b) n :=
lt_of_lt_of_le zero_lt_one (radius_ge_one hν (K₀_nonneg N b m) (D_ge_one hb) n)
by_cases hc : 3 * radius ν (K₀ N b m) (D b) n <
|q.2 j - path hb table terminal ν m n c q.1 j|
· exact Or.inl (gate_jet_zero_outside hb table terminal ν m hν n d c q i ⟨j, hc⟩)
· right
apply gate_jet_zero_outside hb table terminal ν m hν n d e q i
refine ⟨j, ?_⟩
have htri : |path hb table terminal ν m n c q.1 j - path hb table terminal ν m n e q.1 j| ≤
|q.2 j - path hb table terminal ν m n c q.1 j| +
|q.2 j - path hb table terminal ν m n e q.1 j| := by
simpa only [abs_sub_comm (path hb table terminal ν m n c q.1 j) (q.2 j)]
using abs_sub_le (path hb table terminal ν m n c q.1 j) (q.2 j)
(path hb table terminal ν m n e q.1 j)
linarith
theorem stages_uniformly_bounded (hν : 0 < ν) (hm : 1 ≤ m)
(hdir : IncomingDirection table) (hin : IncomingRule table) (i : Fin 2) :
Bounded (fun n (q : ℝ × Coord 2) => stage hb table terminal ν m n q.1 q.2 i) := by
intro d
obtain ⟨C, hC, hc⟩ := gates_uniformly_bounded hb table terminal ν m hν hm i d
refine ⟨C, hC, fun n q => ?_⟩
have heq : (fun q : ℝ × Coord 2 => stage hb table terminal ν m n q.1 q.2 i) =
∑ c : Active hb table (capacity b m n), fun q => gate hb table terminal ν m n c q i := by
ext q
simp only [stage, gate, Finset.sum_apply]
change ‖iteratedFDeriv ℝ d (fun q : ℝ × Coord 2 => stage hb table terminal ν m n q.1 q.2 i) q‖ ≤ C
rw [heq, iteratedFDeriv_sum_apply]
· apply norm_sum_le_sparse _ _ hC
· intro c _
exact hc ⟨n, c⟩ q
· intro c _ e _ hce
exact gate_jets_disjoint hb table terminal ν m hν hm hdir hin n d c e hce q i
· intro c _
exact (contDiff_infty.mp ((contDiff_apply ℝ ℝ i).comp
(contDiff_gate hb table terminal ν m n c)) d).contDiffAt
theorem stage_jet_zero_before (hν : 0 < ν) (n d : ℕ) (q : ℝ × Coord 2) (i : Fin 2)
(hq : q.1 ≤ startTime ν (K₀ N b m) (D b) n) :
iteratedFDeriv ℝ d (fun q => stage hb table terminal ν m n q.1 q.2 i) q = 0 := by
exact jet_zero_before ((contDiff_apply ℝ ℝ i).comp (contDiff_stage hb table terminal ν m n))
(fun t ht X => congrFun (stage_zero_before hb table terminal ν m hν n ht X) i) d hq q.2
theorem stage_jet_zero_after (hν : 0 < ν) (n d : ℕ) (q : ℝ × Coord 2) (i : Fin 2)
(hq : startTime ν (K₀ N b m) (D b) (n + 1) ≤ q.1) :
iteratedFDeriv ℝ d (fun q => stage hb table terminal ν m n q.1 q.2 i) q = 0 := by
exact jet_zero_after ((contDiff_apply ℝ ℝ i).comp (contDiff_stage hb table terminal ν m n))
(fun t ht X => congrFun (stage_zero_after hb table terminal ν m hν n ht X) i) d hq q.2
theorem stage_jets_disjoint (hν : 0 < ν) (n k d : ℕ) (hnk : n ≠ k) (q : ℝ × Coord 2) (i : Fin 2) :
iteratedFDeriv ℝ d (fun q => stage hb table terminal ν m n q.1 q.2 i) q = 0 ∨
iteratedFDeriv ℝ d (fun q => stage hb table terminal ν m k q.1 q.2 i) q = 0 := by
wlog hnk' : n < k generalizing n k
· exact (this k n hnk.symm (by omega)).symm
by_cases hq : q.1 ≤ startTime ν (K₀ N b m) (D b) k
· exact Or.inr (stage_jet_zero_before hb table terminal ν m hν k d q i hq)
· left
apply stage_jet_zero_after hb table terminal ν m hν n d q i
have hm := (startTime_strictMono hν (K₀_nonneg N b m) (D_ge_one hb)).monotone (show n + 1 ≤ k by omega)
linarith
theorem field_uniformly_bounded (hν : 0 < ν) (hm : 1 ≤ m)
(hdir : IncomingDirection table) (hin : IncomingRule table) (i : Fin 2) :
Bounded (fun _ : Unit => fun q : ℝ × Coord 2 => field hb table terminal ν m q.1 q.2 i) := by
intro d
obtain ⟨C, hC, hc⟩ := stages_uniformly_bounded hb table terminal ν m hν hm hdir hin i d
refine ⟨C, hC, fun _ q => ?_⟩
obtain ⟨Q, hQ⟩ := exists_nat_gt q.1
have heq : (fun q : ℝ × Coord 2 => field hb table terminal ν m q.1 q.2 i) =ᶠ[𝓝 q]
∑ n ∈ Finset.range Q, fun q => stage hb table terminal ν m n q.1 q.2 i := by
filter_upwards [(isOpen_lt continuous_fst continuous_const).mem_nhds hQ] with p hp
simpa only [Finset.sum_apply] using congrFun (field_eq_finite hb table terminal ν m hν hp p.2) i
rw [(heq.iteratedFDeriv ℝ d).eq_of_nhds, iteratedFDeriv_sum_apply]
· apply norm_sum_le_sparse _ _ hC
· intro n _
exact hc n q
· intro n _ k _ hnk
exact stage_jets_disjoint hb table terminal ν m hν n k d hnk q i
· intro n _
exact (contDiff_infty.mp ((contDiff_apply ℝ ℝ i).comp
(contDiff_stage hb table terminal ν m n)) d).contDiffAt
end VelocityDetection.ExpandingArray
end
end OAI
|