openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Flow /CompactPeriodizationDerivatives.lean
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Flow.CompactPeriodization | |
| namespace OAI | |
| /-! On the support chart, differentiation commutes exactly with periodization. | |
| Consequently every mixed rapid estimate transfers with the same constant. -/ | |
| noncomputable section | |
| open Set Filter | |
| open scoped Topology | |
| namespace RapidForcing.CompactEmbedding | |
| theorem chart_closed : IsClosed chart := by | |
| have he : chart = (⋂ i : Fin 3, {x : Space | 1 / 4 ≤ x i}) ∩ | |
| (⋂ i : Fin 3, {x : Space | x i ≤ 3 / 4}) := by | |
| ext x | |
| simp [chart] | |
| rw [he] | |
| exact (isClosed_iInter fun i => isClosed_le continuous_const (EuclideanSpace.proj i).continuous).inter | |
| (isClosed_iInter fun i => isClosed_le (EuclideanSpace.proj i).continuous continuous_const) | |
| def ChartSupported {E : Type} [Zero E] (f : Field E) : Prop := | |
| ∀ t x, x ∉ chart → f t x = 0 | |
| namespace ChartSupported | |
| variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : Field E} | |
| omit [NormedSpace ℝ E] in | |
| theorem tsupport (hf : ChartSupported f) (t : ℝ) : tsupport (f t) ⊆ chart := by | |
| apply closure_minimal _ chart_closed | |
| intro x hx | |
| by_contra hn | |
| exact hx (hf t x hn) | |
| theorem spatialD (hf : ChartSupported f) (i : Fin 3) : | |
| ChartSupported (RapidForcing.spatialD i f) := by | |
| intro t x hx | |
| apply image_eq_zero_of_notMem_tsupport | |
| intro hm | |
| exact hx (hf.tsupport t (tsupport_fderiv_apply_subset ℝ (basis i) hm)) | |
| theorem timeD (hf : ChartSupported f) : ChartSupported (RapidForcing.timeD f) := by | |
| intro t x hx | |
| change derivWithin (fun s => f s x) (Ici 0) t = 0 | |
| rw [show (fun s => f s x) = (fun _ => (0 : E)) from funext (fun s => hf s x hx)] | |
| simp | |
| theorem spatialIter (hf : ChartSupported f) (i : Fin 3) (n : ℕ) : | |
| ChartSupported ((RapidForcing.spatialD i)^[n] f) := by | |
| induction n with | |
| | zero => exact hf | |
| | succ n ih => simpa only [Function.iterate_succ_apply'] using ih.spatialD i | |
| theorem timeIter (hf : ChartSupported f) (n : ℕ) : | |
| ChartSupported (RapidForcing.timeD^[n] f) := by | |
| induction n with | |
| | zero => exact hf | |
| | succ n ih => simpa only [Function.iterate_succ_apply'] using ih.timeD | |
| theorem mixedD (hf : ChartSupported f) (l : ℕ) (α : MultiIndex) : | |
| ChartSupported (RapidForcing.mixedD l α f) := | |
| (((hf.spatialIter 2 (α 2)).spatialIter 1 (α 1)).spatialIter 0 (α 0)).timeIter l | |
| end ChartSupported | |
| section Derivatives | |
| variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : Field E} | |
| theorem periodic_spatialD (hf : ChartSupported f) (i : Fin 3) : | |
| spatialD i (periodic f) = periodic (spatialD i f) := by | |
| funext t x | |
| by_cases hx : ∀ j, x j ≠ (⌊x j⌋ : ℤ) | |
| · have hr := representative_local_translation hx | |
| have he : periodic f t =ᶠ[𝓝 x] | |
| (fun y => f t (y - latticeVector (fun j => ⌊x j⌋))) := | |
| hr.mono (fun y hy => congrArg (f t) hy) | |
| change fderiv ℝ (periodic f t) x (basis i) = | |
| fderiv ℝ (f t) (representative x) (basis i) | |
| rw [he.fderiv_eq, fderiv_comp_sub, hr.eq_of_nhds] | |
| · push Not at hx | |
| obtain ⟨j, hj⟩ := hx | |
| have hr := representative_local_outside hj | |
| have he : periodic f t =ᶠ[𝓝 x] (fun _ => (0 : E)) := | |
| hr.mono (fun y hy => hf t (representative y) hy) | |
| have hd := (hf.spatialD i) t (representative x) hr.self_of_nhds | |
| change fderiv ℝ (periodic f t) x (basis i) = spatialD i f t (representative x) | |
| rw [he.fderiv_eq, hd] | |
| simp | |
| theorem periodic_timeD : timeD (periodic f) = periodic (timeD f) := rfl | |
| theorem periodic_spatialIter (hf : ChartSupported f) (i : Fin 3) (n : ℕ) : | |
| (spatialD i)^[n] (periodic f) = periodic ((spatialD i)^[n] f) := by | |
| induction n with | |
| | zero => rfl | |
| | succ n ih => | |
| rw [Function.iterate_succ_apply', ih, | |
| periodic_spatialD (hf.spatialIter i n), Function.iterate_succ_apply'] | |
| theorem periodic_timeIter (n : ℕ) : timeD^[n] (periodic f) = periodic (timeD^[n] f) := by | |
| induction n with | |
| | zero => rfl | |
| | succ n ih => | |
| rw [Function.iterate_succ_apply', ih, periodic_timeD, Function.iterate_succ_apply'] | |
| theorem periodic_mixedD (hf : ChartSupported f) (l : ℕ) (α : MultiIndex) : | |
| mixedD l α (periodic f) = periodic (mixedD l α f) := by | |
| unfold mixedD spatialMulti | |
| rw [periodic_spatialIter hf, | |
| periodic_spatialIter (hf.spatialIter 2 (α 2)), | |
| periodic_spatialIter ((hf.spatialIter 2 (α 2)).spatialIter 1 (α 1)), periodic_timeIter] | |
| theorem periodic_mixed_bound (hf : ChartSupported f) {J l : ℕ} {α : MultiIndex} {C : ℝ} | |
| (hb : ∀ t, 0 ≤ t → ∀ x, (1 + t) ^ J * ‖mixedD l α f t x‖ ≤ C) : | |
| ∀ t, 0 ≤ t → ∀ x, (1 + t) ^ J * ‖mixedD l α (periodic f) t x‖ ≤ C := by | |
| intro t ht x | |
| rw [periodic_mixedD hf] | |
| exact hb t ht (representative x) | |
| theorem periodic_rapid (hf : ChartSupported f) (hr : Rapid f) : Rapid (periodic f) := by | |
| intro J l α | |
| obtain ⟨C, hC, hb⟩ := hr J l α | |
| exact ⟨C, hC, periodic_mixed_bound hf hb⟩ | |
| end Derivatives | |
| end RapidForcing.CompactEmbedding | |
| end | |
| end OAI | |