| import OAI.MathematicalPhysics.NavierStokes.Material.Curls |
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| namespace OAI |
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| namespace Alternating |
| open scoped Topology BigOperators |
| open MeasureTheory Filter |
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| noncomputable section |
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| theorem spatialD_iterate_smooth {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {g : Space → E} (hg : ContDiff ℝ (⊤ : ℕ∞) g) (i : Fin 3) (n : ℕ) : |
| ContDiff ℝ (⊤ : ℕ∞) ((spatialD i)^[n] g) := by |
| induction n with |
| | zero => exact hg |
| | succ n ih => simpa only [Function.iterate_succ_apply'] using spatialD_smooth ih i |
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| theorem spatialPartial_smooth {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {g : Space → E} (hg : ContDiff ℝ (⊤ : ℕ∞) g) (α : MultiIndex) : |
| ContDiff ℝ (⊤ : ℕ∞) (spatialPartial α g) := |
| spatialD_iterate_smooth (spatialD_iterate_smooth (spatialD_iterate_smooth hg 0 (α 0)) 1 (α 1)) 2 (α 2) |
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| theorem spatialD_finiteCombination {ι : Type*} (s : Finset ι) (c : ι → ℝ) |
| {W : ι → Space → Space} (hW : ∀ n ∈ s, ContDiff ℝ (⊤ : ℕ∞) (W n)) (i : Fin 3) : |
| spatialD i (fun x => ∑ n ∈ s, c n • W n x) = fun x => ∑ n ∈ s, c n • spatialD i (W n) x := by |
| funext x |
| have hd : ∀ n ∈ s, Differentiable ℝ (fun y => c n • W n y) := by |
| intro n hn |
| exact ((hW n hn).differentiable (by simp)).fun_const_smul (c n) |
| simp only [spatialD, fderiv_fun_sum (fun n hn => hd n hn x), sum_apply] |
| apply Finset.sum_congr rfl |
| intro n hn |
| rw [fderiv_fun_const_smul ((hW n hn).differentiable (by simp) x), smul_apply] |
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| theorem spatialD_iterate_finiteCombination {ι : Type*} (s : Finset ι) (c : ι → ℝ) |
| {W : ι → Space → Space} (hW : ∀ n ∈ s, ContDiff ℝ (⊤ : ℕ∞) (W n)) (i : Fin 3) (r : ℕ) : |
| (spatialD i)^[r] (fun x => ∑ n ∈ s, c n • W n x) = |
| fun x => ∑ n ∈ s, c n • (spatialD i)^[r] (W n) x := by |
| induction r with |
| | zero => rfl |
| | succ r ih => |
| rw [Function.iterate_succ_apply', ih, |
| spatialD_finiteCombination s c (fun n hn => spatialD_iterate_smooth (hW n hn) i r) i] |
| simp only [Function.iterate_succ_apply'] |
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| theorem spatialPartial_finiteCombination {ι : Type*} (s : Finset ι) (c : ι → ℝ) |
| {W : ι → Space → Space} (hW : ∀ n ∈ s, ContDiff ℝ (⊤ : ℕ∞) (W n)) (α : MultiIndex) : |
| spatialPartial α (fun x => ∑ n ∈ s, c n • W n x) = |
| fun x => ∑ n ∈ s, c n • spatialPartial α (W n) x := by |
| unfold spatialPartial |
| rw [spatialD_iterate_finiteCombination s c hW 0 (α 0), |
| spatialD_iterate_finiteCombination s c (fun n hn => spatialD_iterate_smooth (hW n hn) 0 (α 0)) 1 (α 1), |
| spatialD_iterate_finiteCombination s c |
| (fun n hn => spatialD_iterate_smooth (spatialD_iterate_smooth (hW n hn) 0 (α 0)) 1 (α 1)) 2 (α 2)] |
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| theorem tsupport_spatialD_iterate_subset {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (g : Space → E) (i : Fin 3) (n : ℕ) : tsupport ((spatialD i)^[n] g) ⊆ tsupport g := by |
| induction n with |
| | zero => exact Set.Subset.rfl |
| | succ n ih => |
| rw [Function.iterate_succ_apply'] |
| exact (tsupport_fderiv_apply_subset ℝ (basisVector i)).trans ih |
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| theorem spatialPartial_hasCompactSupport {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {g : Space → E} (hg : HasCompactSupport g) (α : MultiIndex) : HasCompactSupport (spatialPartial α g) := by |
| apply hg.of_isClosed_subset (isClosed_tsupport _) |
| exact (tsupport_spatialD_iterate_subset _ 2 (α 2)).trans |
| ((tsupport_spatialD_iterate_subset _ 1 (α 1)).trans (tsupport_spatialD_iterate_subset _ 0 (α 0))) |
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| namespace SlotL2 |
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| variable (W : ℕ → Space → Space) (hW : ∀ n, ContDiff ℝ (⊤ : ℕ∞) (W n)) |
| (hc : ∀ n, HasCompactSupport (W n)) |
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| include hW hc in |
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| theorem mem (α : MultiIndex) (n : ℕ) : MemLp (spatialPartial α (W n)) 2 volume := |
| (spatialPartial_smooth (hW n) α).continuous.memLp_of_hasCompactSupport |
| (spatialPartial_hasCompactSupport (hc n) α) |
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| def component (α : MultiIndex) (n : ℕ) : SpatialL2 Space := (mem W hW hc α n).toLp _ |
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| def finiteLift (a : ℝ → ℝ) (α : MultiIndex) (N : ℕ) (t : ℝ) : SpatialL2 Space := |
| ∑ n ∈ Finset.range N, a (t - n) • component W hW hc α n |
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| theorem finiteLift_continuous {a : ℝ → ℝ} (ha : Continuous a) (α : MultiIndex) (N : ℕ) : |
| Continuous (finiteLift W hW hc a α N) := by |
| apply continuous_finsetSum |
| intro n hn |
| exact (ha.comp (continuous_id.sub continuous_const)).smul continuous_const |
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| theorem finiteLift_ae (a : ℝ → ℝ) (α : MultiIndex) (N : ℕ) (t : ℝ) : |
| (fun x : Space => (finiteLift W hW hc a α N t) x) =ᵐ[volume] |
| fun x => ∑ n ∈ Finset.range N, a (t - n) • spatialPartial α (W n) x := by |
| unfold finiteLift |
| induction N with |
| | zero => simp |
| | succ N ih => |
| simp only [Finset.sum_range_succ] |
| exact (Lp.coeFn_add _ _).trans |
| (ih.add ((Lp.coeFn_smul _ _).trans ((mem W hW hc α N).coeFn_toLp.const_smul _))) |
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| theorem finiteLift_hasDerivAt {a : ℝ → ℝ} (ha : Differentiable ℝ a) |
| (α : MultiIndex) (N : ℕ) (t : ℝ) : |
| HasDerivAt (finiteLift W hW hc a α N) |
| (finiteLift W hW hc (deriv a) α N t) t := by |
| apply HasDerivAt.fun_sum |
| intro n hn |
| have hh := (ha (t - n)).hasDerivAt.comp t ((hasDerivAt_id t).sub_const (n : ℝ)) |
| simpa using hh.smul_const (component W hW hc α n) |
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| end SlotL2 |
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| end |
| end Alternating |
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| end OAI |
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