| import OAI.MathematicalPhysics.RapidForcing.JointSmoothness |
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| namespace OAI |
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| open scoped BigOperators ENNReal Topology NNReal |
| open Set MeasureTheory |
| namespace RapidForcing |
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| 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) |
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| 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 _ _⟩ |
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| 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 |
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| 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 |
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| 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] |
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| 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) |
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| 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 |
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| 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) |
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| 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 |
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| end RapidForcing |
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| end OAI |
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