openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Programs /BoundedProfiles.lean
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.LogarithmicSpatialForce | |
| import OAI.MathematicalPhysics.NavierStokes.ShearFlows.ForceEvaluation | |
| namespace OAI | |
| /-! The logarithmic estimate applies to an initialized field as soon as all | |
| its original derivatives are bounded. Periodicity in time is unnecessary. -/ | |
| noncomputable section | |
| open scoped ContDiff BigOperators | |
| open ShearFlows | |
| namespace ForcedComputation | |
| theorem boundedMixed_derivative {V : Velocity} (hb : BoundedMixedDerivatives V) | |
| (β : List (Fin 4)) : BoundedMixedDerivatives (mixedDerivative V β) := by | |
| intro α | |
| simpa only [mixedDerivative_append] using hb (α ++ β) | |
| theorem boundedMixed_sub {V W : Velocity} (hV : ContDiff ℝ ∞ V) | |
| (hW : ContDiff ℝ ∞ W) (hVB : BoundedMixedDerivatives V) | |
| (hWB : BoundedMixedDerivatives W) : BoundedMixedDerivatives (V - W) := by | |
| intro α | |
| obtain ⟨A, hA, hbA⟩ := hVB α | |
| obtain ⟨B, hB, hbB⟩ := hWB α | |
| refine ⟨A + B, add_nonneg hA hB, fun y => ?_⟩ | |
| rw [mixedDerivative_sub hV hW] | |
| exact (norm_sub_le _ _).trans (add_le_add (hbA y) (hbB y)) | |
| theorem boundedMixed_sum {ι : Type*} [Fintype ι] (V : ι → Velocity) | |
| (hs : ∀ i, ContDiff ℝ ∞ (V i)) (hb : ∀ i, BoundedMixedDerivatives (V i)) : | |
| BoundedMixedDerivatives (∑ i, V i) := by | |
| intro α | |
| choose C hC hbC using fun i => hb i α | |
| refine ⟨∑ i, C i, Finset.sum_nonneg (fun i _ => hC i), fun y => ?_⟩ | |
| rw [mixedDerivative_sum V hs] | |
| simpa only [Finset.sum_apply] using | |
| (norm_sum_le (s := Finset.univ) (f := fun i => mixedDerivative (V i) α y)).trans | |
| (Finset.sum_le_sum fun i _ => hbC i y) | |
| theorem bounded_uniformTimeBounds {V : Velocity} (hV : ContDiff ℝ ∞ V) | |
| (hb : BoundedMixedDerivatives V) : UniformTimeBounds (fun x s => V (s, x)) := by | |
| intro n | |
| obtain ⟨C, hC, hbound⟩ := hb (List.replicate n 0) | |
| exact ⟨C, hC, fun x t => by rw [iteratedDeriv_eq_mixedTime hV]; exact hbound (t, x)⟩ | |
| theorem bounded_spatialPhase {V : Velocity} (hV : ContDiff ℝ ∞ V) | |
| (hb : BoundedMixedDerivatives V) (α : List (Fin 3)) : | |
| BoundedMixedDerivatives (spatialPhase V α) := by | |
| rw [spatialPhase_eq_mixed hV] | |
| exact boundedMixed_derivative hb _ | |
| theorem bounded_quadraticPhase {V : Velocity} (hV : ContDiff ℝ ∞ V) | |
| (hb : BoundedMixedDerivatives V) (ha : ZeroAdvection V) : | |
| BoundedMixedDerivatives (fun y : SpaceTime => quadraticPhase V y.2 y.1) := by | |
| have he : (fun y : SpaceTime => quadraticPhase V y.2 y.1) = mixedDerivative V [0] - V := by | |
| funext y | |
| simp only [quadraticPhase, ha y.1 y.2, add_zero, timeDerivative_eq hV] | |
| rfl | |
| rw [he] | |
| exact boundedMixed_sub (mixedDerivative_smooth hV _) hV (boundedMixed_derivative hb _) hb | |
| theorem bounded_viscousPhase {V : Velocity} (hV : ContDiff ℝ ∞ V) | |
| (hb : BoundedMixedDerivatives V) : | |
| BoundedMixedDerivatives (fun y : SpaceTime => viscousPhase V y.2 y.1) := by | |
| have he : (fun y : SpaceTime => viscousPhase V y.2 y.1) = | |
| ∑ j : Fin 3, mixedDerivative V [j.succ,j.succ] := by | |
| funext y | |
| simpa only [viscousPhase, Finset.sum_apply] using laplacian_eq_mixed hV y.1 y.2 | |
| rw [he] | |
| exact boundedMixed_sum _ (fun _ => mixedDerivative_smooth hV _) | |
| (fun _ => boundedMixed_derivative hb _) | |
| theorem bounded_profile_mixed_decay {V : Velocity} (hV : ContDiff ℝ ∞ V) | |
| (hb : BoundedMixedDerivatives V) (r n : ℕ) (α : List (Fin 3)) : | |
| ∃ C : ℝ, 0 ≤ C ∧ ∀ x t, 0 ≤ t → | |
| ‖iteratedDeriv n | |
| (fun s => spatialWord α (fun y => logarithmicProfile r (fun q => V (q, y)) s) x) t‖ ≤ | |
| C * (1 + t)⁻¹ ^ (r + n) := by | |
| have hphase (x : Space) : ContDiff ℝ ∞ (fun s => spatialPhase V α (s, x)) := | |
| (spatialPhase_smooth hV α).comp (contDiff_id.prodMk contDiff_const) | |
| obtain ⟨C, hC, hbound⟩ := logarithmicProfile_uniform_decay r n hphase | |
| (bounded_uniformTimeBounds (spatialPhase_smooth hV α) (bounded_spatialPhase hV hb α)) | |
| refine ⟨C, hC, fun x t ht => ?_⟩ | |
| have he : (fun s => spatialWord α (fun y => logarithmicProfile r (fun q => V (q, y)) s) x) = | |
| logarithmicProfile r (fun s => spatialPhase V α (s, x)) := | |
| funext (fun s => spatialWord_logarithmicProfile r V α s x) | |
| rw [he] | |
| exact hbound x t ht | |
| theorem bounded_slowForce_spatial_time_decay (ν : ℝ) {V : Velocity} | |
| (hV : ContDiff ℝ ∞ V) (hb : BoundedMixedDerivatives V) (ha : ZeroAdvection V) | |
| (α : List (Fin 3)) (n : ℕ) : | |
| ∃ C : ℝ, 0 ≤ C ∧ ∀ x t, 0 ≤ t → | |
| ‖iteratedDeriv n (fun s => spatialWord α (fun y => slowForce ν V (s, y)) x) t‖ ≤ | |
| C * (1 + t)⁻¹ ^ (1 + n) := by | |
| let Q : Velocity := fun y => quadraticPhase V y.2 y.1 | |
| let D : Velocity := fun y => viscousPhase V y.2 y.1 | |
| have hQ : ContDiff ℝ ∞ Q := quadraticPhase_smooth hV | |
| have hD : ContDiff ℝ ∞ D := viscousPhase_smooth hV | |
| obtain ⟨A, hA, hQA⟩ := bounded_profile_mixed_decay hQ (bounded_quadraticPhase hV hb ha) 2 n α | |
| obtain ⟨B, hB, hDB⟩ := bounded_profile_mixed_decay hD (bounded_viscousPhase hV hb) 1 n α | |
| refine ⟨A + ‖ν‖ * B, by positivity, fun x t ht0 => ?_⟩ | |
| have hQx : ContDiff ℝ ∞ (fun s => spatialPhase Q α (s, x)) := | |
| (spatialPhase_smooth hQ α).comp (contDiff_id.prodMk contDiff_const) | |
| have hDx : ContDiff ℝ ∞ (fun s => spatialPhase D α (s, x)) := | |
| (spatialPhase_smooth hD α).comp (contDiff_id.prodMk contDiff_const) | |
| have hqt := logarithmicProfile_contDiffAt 2 hQx (show -1 < t by linarith) | |
| have hdt := logarithmicProfile_contDiffAt 1 hDx (show -1 < t by linarith) | |
| have he : (fun s => spatialWord α (fun y => slowForce ν V (s, y)) x) = | |
| fun s => logarithmicProfile 2 (fun q => spatialPhase Q α (q, x)) s - | |
| ν • logarithmicProfile 1 (fun q => spatialPhase D α (q, x)) s := | |
| funext (fun s => spatialWord_slowForce hV ν α s x) | |
| rw [he, iteratedDeriv_fun_sub (hqt.of_le (by simp)) | |
| ((hdt.const_smul ν).of_le (by simp)), iteratedDeriv_fun_const_smul_field] | |
| have hQA' := hQA x t ht0 | |
| have hDB' := hDB x t ht0 | |
| simp_rw [spatialWord_logarithmicProfile] at hQA' hDB' | |
| have hinv : 0 ≤ (1 + t)⁻¹ := by positivity | |
| have hinvle : (1 + t)⁻¹ ≤ 1 := | |
| (inv_le_one₀ (by linarith : 0 < 1 + t)).mpr (by linarith) | |
| have hpow : (1 + t)⁻¹ ^ (2 + n) ≤ (1 + t)⁻¹ ^ (1 + n) := by | |
| calc | |
| _ = (1 + t)⁻¹ ^ (1 + n) * (1 + t)⁻¹ := by rw [← pow_succ]; congr 1; omega | |
| _ ≤ (1 + t)⁻¹ ^ (1 + n) * 1 := | |
| mul_le_mul_of_nonneg_left hinvle (pow_nonneg hinv _) | |
| _ = _ := mul_one _ | |
| calc | |
| _ ≤ ‖iteratedDeriv n (logarithmicProfile 2 (fun q => spatialPhase Q α (q, x))) t‖ + | |
| ‖ν • iteratedDeriv n (logarithmicProfile 1 (fun q => spatialPhase D α (q, x))) t‖ := | |
| norm_sub_le _ _ | |
| _ ≤ A * (1 + t)⁻¹ ^ (2 + n) + ‖ν‖ * (B * (1 + t)⁻¹ ^ (1 + n)) := by | |
| rw [norm_smul] | |
| exact add_le_add hQA' (mul_le_mul_of_nonneg_left hDB' (norm_nonneg ν)) | |
| _ ≤ A * (1 + t)⁻¹ ^ (1 + n) + ‖ν‖ * (B * (1 + t)⁻¹ ^ (1 + n)) := | |
| add_le_add (mul_le_mul_of_nonneg_left hpow hA) le_rfl | |
| _ = _ := by ring | |
| end ForcedComputation | |
| end | |
| end OAI | |