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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 | import OAI.MathematicalPhysics.RapidForcing.JointSmoothness
namespace OAI
open scoped BigOperators ENNReal Topology NNReal
open Set MeasureTheory
namespace RapidForcing
lemma spatialFDeriv_joint_smooth {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E]
{v : Field E} (hv : ContDiff ℝ (⊤ : ℕ∞) (Function.uncurry v)) :
ContDiff ℝ (⊤ : ℕ∞) (fun p : ℝ × Space => fderiv ℝ (v p.1) p.2) := by
have h : ContDiff ℝ (⊤ : ℕ∞)
(fun p : (ℝ × Space) × Space => v p.1.1 p.2) :=
hv.comp (contDiff_fst.fst.prodMk contDiff_snd)
exact h.fderiv contDiff_snd (by simp)
lemma finite_interval_bounds {v : Field Space}
(hv : ContDiff ℝ (⊤ : ℕ∞) (Function.uncurry v)) (hs : Supported v)
(T : ℝ) : ∃ L C : ℝ≥0, ∀ t ∈ Icc (0 : ℝ) T, ∀ x : Space,
‖v t x‖ ≤ L ∧ ‖fderiv ℝ (v t) x‖ ≤ C := by
have hc : IsCompact (Icc (0 : ℝ) T ×ˢ K) := isCompact_Icc.prod K_compact
obtain ⟨L, hL⟩ := hc.exists_bound_of_continuousOn hv.continuous.continuousOn
obtain ⟨C, hC⟩ := hc.exists_bound_of_continuousOn
(spatialFDeriv_joint_smooth hv).continuous.continuousOn
refine ⟨⟨max L 0, le_max_right _ _⟩, ⟨max C 0, le_max_right _ _⟩, ?_⟩
intro t ht x
by_cases hx : x ∈ K
· exact ⟨(hL (t, x) ⟨ht, hx⟩).trans (le_max_left _ _),
(hC (t, x) ⟨ht, hx⟩).trans (le_max_left _ _)⟩
· have hxt : x ∉ tsupport (v t) := fun h => hx (hs t ht.1 h)
rw [hs.zero_off ht.1 hx, fderiv_of_notMem_tsupport ℝ hxt, norm_zero, norm_zero]
exact ⟨le_max_right _ _, le_max_right _ _⟩
lemma finite_interval_lipschitz {v : Field Space}
(hv : ContDiff ℝ (⊤ : ℕ∞) (Function.uncurry v)) (hs : Supported v)
(T : ℝ) : ∃ L C : ℝ≥0, ∀ t ∈ Icc (0 : ℝ) T,
(∀ x : Space, ‖v t x‖ ≤ L) ∧ LipschitzWith C (v t) := by
obtain ⟨L, C, h⟩ := finite_interval_bounds hv hs T
refine ⟨L, C, fun t ht => ⟨fun x => (h t ht x).1, ?_⟩⟩
apply lipschitzWith_of_nnnorm_fderiv_le (𝕜 := ℝ)
· exact (hv.comp (contDiff_const.prodMk contDiff_id)).differentiable (by simp)
· intro x
exact_mod_cast (h t ht x).2
lemma exists_trajectoryOn {v : Field Space}
(hv : ContDiff ℝ (⊤ : ℕ∞) (Function.uncurry v)) (hs : Supported v)
(a : Space) {T : ℝ} (hT : 0 ≤ T) : ∃ γ : ℝ → Space, TrajectoryOn v a T γ := by
obtain ⟨L, C, hb⟩ := finite_interval_lipschitz hv hs T
let R : ℝ≥0 := L * ⟨T, hT⟩
have hpl : IsPicardLindelof v (⟨0, ⟨le_rfl, hT⟩⟩ : Icc (0 : ℝ) T) a R 0 L C := by
constructor
· intro t ht
exact (hb t ht).2.lipschitzOnWith
· intro x _
exact (hv.continuous.comp (continuous_id.prodMk continuous_const)).continuousOn
· intro t ht x _
exact (hb t ht).1 x
· dsimp [R]
simp only [sub_zero, max_eq_left hT]
rfl
exact hpl.exists_eq_forall_mem_Icc_hasDerivWithinAt₀
lemma trajectoryOn_unique {v : Field Space}
(hv : ContDiff ℝ (⊤ : ℕ∞) (Function.uncurry v)) (hs : Supported v)
{a : Space} {T : ℝ} {γ η : ℝ → Space}
(hγ : TrajectoryOn v a T γ) (hη : TrajectoryOn v a T η) :
EqOn γ η (Icc 0 T) := by
obtain ⟨L, C, hb⟩ := finite_interval_lipschitz hv hs T
apply ODE_solution_unique_of_mem_Icc_right (K := C) (s := fun _ => univ)
· intro t ht
exact (hb t ⟨ht.1, ht.2.le⟩).2.lipschitzOnWith
· intro t ht
exact (hγ.2 t ht).continuousWithinAt
· intro t ht
exact (hγ.2 t ⟨ht.1, ht.2.le⟩).mono_of_mem_nhdsWithin (Icc_mem_nhdsGE_of_mem ht)
· simp
· intro t ht
exact (hη.2 t ht).continuousWithinAt
· intro t ht
exact (hη.2 t ⟨ht.1, ht.2.le⟩).mono_of_mem_nhdsWithin (Icc_mem_nhdsGE_of_mem ht)
· simp
· rw [hγ.1, hη.1]
lemma TrajectoryOn.mono {v : Field Space} {a : Space} {S T : ℝ} {γ : ℝ → Space}
(h : TrajectoryOn v a T γ) (hST : S ≤ T) : TrajectoryOn v a S γ := by
refine ⟨h.1, fun t ht => ?_⟩
exact (h.2 t ⟨ht.1, ht.2.trans hST⟩).mono (Icc_subset_Icc_right hST)
theorem materialFlow_of_smooth_supported {v : Field Space}
(hv : ContDiff ℝ (⊤ : ℕ∞) (Function.uncurry v)) (hs : Supported v) :
∃ X : ℝ → Space → Space, MaterialFlow v X := by
classical
have hex (a : Space) (n : ℕ) : ∃ γ : ℝ → Space, TrajectoryOn v a ((n : ℝ) + 1) γ :=
exists_trajectoryOn hv hs a (by positivity)
choose γ hγ using hex
have hcoh (a : Space) (n m : ℕ) (t : ℝ) (ht : 0 ≤ t)
(hn : t ≤ (n : ℝ) + 1) (hm : t ≤ (m : ℝ) + 1) : γ a n t = γ a m t := by
exact trajectoryOn_unique hv hs ((hγ a n).mono (min_le_left _ _))
((hγ a m).mono (min_le_right _ _)) ⟨ht, le_min hn hm⟩
let X : ℝ → Space → Space := fun t a => γ a (Nat.ceil t) t
have he (a : Space) (n : ℕ) {t : ℝ} (ht : t ∈ Icc (0 : ℝ) ((n : ℝ) + 1)) :
X t a = γ a n t := by
exact hcoh a (Nat.ceil t) n t ht.1 (by have := Nat.le_ceil t; linarith) ht.2
have hi (a : Space) : X 0 a = a := (hγ a (Nat.ceil 0)).1
have hd (a : Space) (t : ℝ) (ht : 0 ≤ t) :
HasDerivWithinAt (fun s => X s a) (v t (X t a)) (Ici 0) t := by
obtain ⟨n, hn⟩ := exists_nat_gt t
have htn : t ∈ Icc (0 : ℝ) ((n : ℝ) + 1) := ⟨ht, by linarith⟩
have hN : Icc (0 : ℝ) ((n : ℝ) + 1) ∈ 𝓝[Ici 0] t := by
filter_upwards [self_mem_nhdsWithin,
mem_nhdsWithin_of_mem_nhds (Iic_mem_nhds (by linarith : t < (n : ℝ) + 1))] with s hs hs'
exact ⟨hs, hs'⟩
have hD := ((hγ a n).2 t htn).mono_of_mem_nhdsWithin hN
rw [← he a n htn] at hD
apply hD.congr_of_eventuallyEq
· filter_upwards [hN] with s hs
exact he a n hs
· exact he a n htn
refine ⟨X, hi, hd, ?_⟩
intro T _ a η hη t ht
have hX : TrajectoryOn v a T (fun s => X s a) :=
⟨hi a, fun s hs => (hd a s hs.1).mono Icc_subset_Ici_self⟩
exact trajectoryOn_unique hv hs hη hX ht
theorem addressed_materialFlow (M : Machine) (w : M.Input) :
∃ X : ℝ → Space → Space, MaterialFlow (addressedVelocity M w) X :=
materialFlow_of_smooth_supported (addressedVelocity_joint_smooth M w)
(addressedVelocity_supported M w)
lemma solution_unique_Icc {v : Field Space}
(hv : ContDiff ℝ (⊤ : ℕ∞) (Function.uncurry v)) (hs : Supported v)
{A B : ℝ} (hA : 0 ≤ A) {γ η : ℝ → Space}
(hγ : ∀ t ∈ Icc A B, HasDerivWithinAt γ (v t (γ t)) (Icc A B) t)
(hη : ∀ t ∈ Icc A B, HasDerivWithinAt η (v t (η t)) (Icc A B) t)
(he : γ A = η A) : EqOn γ η (Icc A B) := by
obtain ⟨L, C, hb⟩ := finite_interval_lipschitz hv hs B
apply ODE_solution_unique_of_mem_Icc_right (K := C) (s := fun _ => univ)
· intro t ht
exact (hb t ⟨hA.trans ht.1, ht.2.le⟩).2.lipschitzOnWith
· intro t ht
exact (hγ t ht).continuousWithinAt
· intro t ht
exact (hγ t ⟨ht.1, ht.2.le⟩).mono_of_mem_nhdsWithin (Icc_mem_nhdsGE_of_mem ht)
· simp
· intro t ht
exact (hη t ht).continuousWithinAt
· intro t ht
exact (hη t ⟨ht.1, ht.2.le⟩).mono_of_mem_nhdsWithin (Icc_mem_nhdsGE_of_mem ht)
· simp
· exact he
end RapidForcing
end OAI
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