| import OAI.MathematicalPhysics.AlternatingFlow.Model |
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| namespace OAI |
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| open scoped BigOperators ENNReal NNReal Topology ContDiff |
| open MeasureTheory |
| namespace AlternatingNS |
| namespace Spatial |
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| noncomputable def d {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (i : Fin 3) (F : Space → E) (x : Space) : E := fderiv ℝ F x (e i) |
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| lemma coord_smooth (i : Fin 3) : ContDiff ℝ ∞ (fun x : Space => x i) := |
| (PiLp.proj 2 (𝕜 := ℝ) (fun _ : Fin 3 => ℝ) i).contDiff |
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| lemma coord_hasFDeriv (i : Fin 3) (x : Space) : |
| HasFDerivAt (fun x : Space => x i) (PiLp.proj 2 (𝕜 := ℝ) (fun _ : Fin 3 => ℝ) i) x := |
| (PiLp.proj 2 (𝕜 := ℝ) (fun _ : Fin 3 => ℝ) i).hasFDerivAt |
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| lemma e_apply (i j : Fin 3) : e i j = if j = i then 1 else 0 := by |
| simp [e, EuclideanSpace.single, PiLp.single_apply] |
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| lemma d_smooth {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (i : Fin 3) (F : Space → E) (hF : ContDiff ℝ ∞ F) : ContDiff ℝ ∞ (d i F) := by |
| exact (hF.fderiv_right (by simp : (∞ : ℕ∞ω) + 1 ≤ (∞ : ℕ∞ω))).clm_apply contDiff_const |
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| lemma d_tsupport {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (i : Fin 3) (F : Space → E) : tsupport (d i F) ⊆ tsupport F := |
| tsupport_fderiv_apply_subset ℝ (e i) |
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| lemma d_comm (i j : Fin 3) (F : Space → ℝ) (hF : ContDiff ℝ ∞ F) : |
| d i (d j F) = d j (d i F) := by |
| ext x |
| have hd := (hF.fderiv_right (by simp : (∞ : ℕ∞ω) + 1 ≤ (∞ : ℕ∞ω))).differentiable (by simp) |
| have heq (a b : Fin 3) : d a (d b F) x = fderiv ℝ (fderiv ℝ F) x (e a) (e b) := by |
| unfold d |
| rw [fderiv_clm_apply (hd x) (differentiableAt_const _)] |
| simp |
| rw [heq, heq] |
| exact hF.contDiffAt.isSymmSndFDerivAt (by norm_num) _ _ |
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| noncomputable def curl (i j : Fin 3) (φ : Space → ℝ) (x : Space) : Space := |
| d j φ x • e i - d i φ x • e j |
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| lemma curl_smooth (i j : Fin 3) (φ : Space → ℝ) (hφ : ContDiff ℝ ∞ φ) : |
| ContDiff ℝ ∞ (curl i j φ) := |
| ((d_smooth j φ hφ).smul contDiff_const).sub ((d_smooth i φ hφ).smul contDiff_const) |
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| lemma curl_tsupport (i j : Fin 3) (φ : Space → ℝ) : tsupport (curl i j φ) ⊆ tsupport φ := by |
| apply (tsupport_sub _ _).trans |
| apply Set.union_subset |
| · exact (tsupport_smul_subset_left _ _).trans (d_tsupport j φ) |
| · exact (tsupport_smul_subset_left _ _).trans (d_tsupport i φ) |
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| lemma curl_div (i j : Fin 3) (φ : Space → ℝ) (hφ : ContDiff ℝ ∞ φ) (x : Space) : |
| (∑ k : Fin 3, d k (curl i j φ) x k) = 0 := by |
| have hd (k : Fin 3) := (d_smooth k φ hφ).differentiable (by simp) |
| have hc (k : Fin 3) : d k (curl i j φ) x k = |
| d k (d j φ) x * (if k = i then 1 else 0) - |
| d k (d i φ) x * (if k = j then 1 else 0) := by |
| change (fderiv ℝ ((fun y => d j φ y • e i) - (fun y => d i φ y • e j)) x (e k)) k = _ |
| rw [fderiv_sub ((hd j x).smul_const _) ((hd i x).smul_const _), |
| fderiv_smul_const (hd j x), fderiv_smul_const (hd i x)] |
| simp [e_apply, d] |
| simp_rw [hc] |
| rw [Finset.sum_sub_distrib] |
| simp only [mul_ite, mul_one, mul_zero, Finset.sum_ite_eq', Finset.mem_univ, ite_true] |
| rw [d_comm i j φ hφ, sub_self] |
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| def box : Set Space := {x | ∀ i : Fin 3, |x i| ≤ 3} |
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| def plateau : Set Space := {x | ∀ i : Fin 3, |x i| < 2} |
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| noncomputable def cutoff (x : Space) : ℝ := |
| letI := threeAtLeastTwo |
| ∏ i : Fin 3, Real.smoothTransition (3 - x i) * Real.smoothTransition (3 + x i) |
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| lemma cutoff_smooth : ContDiff ℝ ∞ cutoff := by |
| unfold cutoff |
| apply contDiff_prod |
| intro i _ |
| have hi := coord_smooth i |
| fun_prop |
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| lemma isClosed_box : IsClosed box := by |
| have heq : box = ⋂ i : Fin 3, {x : Space | |x i| ≤ 3} := by ext; simp [box] |
| rw [heq] |
| exact isClosed_iInter (fun i => isClosed_le (coord_smooth i).continuous.abs continuous_const) |
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| lemma box_compact : IsCompact box := by |
| apply (isCompact_closedBall (0 : Space) 9).of_isClosed_subset isClosed_box |
| intro x hx |
| rw [Metric.mem_closedBall, dist_zero_right] |
| rw [EuclideanSpace.norm_eq, Real.sqrt_le_iff] |
| refine ⟨by norm_num, ?_⟩ |
| calc |
| ∑ i : Fin 3, ‖x i‖ ^ 2 ≤ ∑ _i : Fin 3, (3 : ℝ) ^ 2 := by |
| apply Finset.sum_le_sum |
| intro i _ |
| exact pow_le_pow_left₀ (norm_nonneg _) (hx i) 2 |
| _ ≤ 9 ^ 2 := by norm_num |
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| lemma plateau_open : IsOpen plateau := by |
| have heq : plateau = ⋂ i : Fin 3, {x : Space | |x i| < 2} := by ext; simp [plateau] |
| rw [heq] |
| exact isOpen_iInter_of_finite (fun i => isOpen_lt (coord_smooth i).continuous.abs continuous_const) |
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| lemma cutoff_one (x : Space) (hx : x ∈ plateau) : cutoff x = 1 := by |
| apply Finset.prod_eq_one |
| intro i _ |
| have hi := abs_lt.1 (hx i) |
| rw [Real.smoothTransition.one_of_one_le (by linarith : 1 ≤ 3 - x i), |
| Real.smoothTransition.one_of_one_le (by linarith : 1 ≤ 3 + x i), one_mul] |
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| lemma cutoff_one_near (x : Space) (hx : x ∈ plateau) : |
| cutoff =ᶠ[𝓝 x] (fun _ => 1) := |
| Filter.eventually_of_mem (plateau_open.mem_nhds hx) (fun y hy => cutoff_one y hy) |
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| lemma cutoff_support : tsupport cutoff ⊆ box := by |
| apply closure_minimal _ isClosed_box |
| intro x hx i |
| by_contra hi |
| have hn : 3 < |x i| := lt_of_not_ge hi |
| have hz : Real.smoothTransition (3 - x i) * Real.smoothTransition (3 + x i) = 0 := by |
| rcases (lt_abs.1 hn) with hn | hn |
| · rw [Real.smoothTransition.zero_of_nonpos (by linarith : 3 - x i ≤ 0), zero_mul] |
| · rw [Real.smoothTransition.zero_of_nonpos (by linarith : 3 + x i ≤ 0), mul_zero] |
| exact hx (Finset.prod_eq_zero (Finset.mem_univ i) hz) |
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| noncomputable def shearPotential (d i j : Fin 3) (L E : ℝ → ℝ) (x : Space) : ℝ := |
| x j * L (x d) - x i * E (x d) |
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| noncomputable def localizedShear (d i j : Fin 3) (L E : ℝ → ℝ) : Space → Space := |
| curl i j (fun x => cutoff x * shearPotential d i j L E x) |
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| lemma shearPotential_smooth (d i j : Fin 3) (L E : ℝ → ℝ) |
| (hL : ContDiff ℝ ∞ L) (hE : ContDiff ℝ ∞ E) : |
| ContDiff ℝ ∞ (shearPotential d i j L E) := |
| ((coord_smooth j).mul (hL.comp (coord_smooth d))).sub |
| ((coord_smooth i).mul (hE.comp (coord_smooth d))) |
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| lemma localizedShear_smooth (d i j : Fin 3) (L E : ℝ → ℝ) |
| (hL : ContDiff ℝ ∞ L) (hE : ContDiff ℝ ∞ E) : |
| ContDiff ℝ ∞ (localizedShear d i j L E) := |
| curl_smooth i j _ (cutoff_smooth.mul (shearPotential_smooth d i j L E hL hE)) |
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| lemma localizedShear_support (d i j : Fin 3) (L E : ℝ → ℝ) : |
| tsupport (localizedShear d i j L E) ⊆ box := |
| (curl_tsupport _ _ _).trans (tsupport_mul_subset_left.trans cutoff_support) |
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| lemma localizedShear_div (d i j : Fin 3) (L E : ℝ → ℝ) |
| (hL : ContDiff ℝ ∞ L) (hE : ContDiff ℝ ∞ E) (x : Space) : |
| (∑ k : Fin 3, Spatial.d k (localizedShear d i j L E) x k) = 0 := |
| curl_div i j _ (cutoff_smooth.mul (shearPotential_smooth d i j L E hL hE)) x |
|
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| lemma localizedShear_plateau (d i j : Fin 3) (hij : i ≠ j) (hdi : d ≠ i) (hdj : d ≠ j) |
| (L E : ℝ → ℝ) (hL : ContDiff ℝ ∞ L) (hE : ContDiff ℝ ∞ E) |
| (x : Space) (hx : x ∈ plateau) : |
| localizedShear d i j L E x = L (x d) • e i + E (x d) • e j := by |
| have heq : (fun y => cutoff y * shearPotential d i j L E y) =ᶠ[𝓝 x] |
| shearPotential d i j L E := by |
| filter_upwards [cutoff_one_near x hx] with y hy |
| rw [hy, one_mul] |
| have hL' := ((hL.differentiable (by simp)) (x d)).hasFDerivAt.comp x (coord_hasFDeriv d x) |
| have hE' := ((hE.differentiable (by simp)) (x d)).hasFDerivAt.comp x (coord_hasFDeriv d x) |
| have hφ := ((coord_hasFDeriv j x).mul hL').sub ((coord_hasFDeriv i x).mul hE') |
| have hφeq := hφ.fderiv |
| change fderiv ℝ (shearPotential d i j L E) x = _ at hφeq |
| simp only [localizedShear, curl, Spatial.d, heq.fderiv_eq, hφeq] |
| simp [ContinuousLinearMap.comp_apply, PiLp.proj_apply, e_apply, hij, hij.symm, hdi, |
| hdj, sub_eq_add_neg] |
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| end Spatial |
| end AlternatingNS |
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| end OAI |
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