openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /VelocityDetection /ExpandingArrayForce.lean
| import OAI.MathematicalPhysics.NavierStokes.VelocityDetection.ArrayBounds | |
| import OAI.MathematicalPhysics.NavierStokes.VelocityDetection.JointCalculusHorizontalLinear | |
| namespace OAI | |
| noncomputable section | |
| namespace VelocityDetection.ExpandingArray | |
| open scoped BigOperators Topology ContDiff | |
| open Set Function Filter | |
| open Set Function Filter MeasureTheory | |
| open scoped Topology BigOperators ContDiff | |
| open scoped Topology ContDiff BigOperators | |
| open Expanding Stacks FiniteAddresses UniformDerivatives JointCalculus SmoothProfiles | |
| variable {N b : ℕ} (hb : 0 < b) (table : Fin N → Fin b → Option (Rule (Fin N) b)) | |
| (terminal : Fin N) (ν : ℝ) (m : ℕ) | |
| def force (p : Coord 2) : VectorField 3 := fun t X => | |
| lift (residual ν (uncurry (field hb table terminal ν m))) (uncurry (impulse p)) (t, X) | |
| @[fun_prop] theorem contDiff_force (hν : 0 < ν) (p : Coord 2) : | |
| ContDiff ℝ ∞ (uncurry (force hb table terminal ν m p)) := | |
| contDiff_lift (contDiff_residual ν (contDiff_field hb table terminal ν m hν)) | |
| (contDiff_impulse p) | |
| theorem force_uniformly_bounded (hν : 0 < ν) (hm : 1 ≤ m) | |
| (hdir : IncomingDirection table) (hin : IncomingRule table) (p : Coord 2) : | |
| Bounded (fun _ : Unit => uncurry (force hb table terminal ν m p)) := by | |
| apply bounded_lift (contDiff_residual ν (contDiff_field hb table terminal ν m hν)) | |
| (contDiff_impulse p) | |
| · exact Bounded.residual ν (contDiff_field hb table terminal ν m hν) | |
| (field_uniformly_bounded hb table terminal ν m hν hm hdir hin) | |
| · exact compact (contDiff_impulse p) (compactSupport_impulse p) | |
| theorem force_vertical_independent (p : Coord 2) (t : ℝ) (X : Coord 3) (z : ℝ) : | |
| force hb table terminal ν m p t (update X 2 z) = force hb table terminal ν m p t X := by | |
| simp only [force, lift, projection_apply, horizontal_update_two] | |
| theorem force_eq_triangular (hν : 0 < ν) (p : Coord 2) {t : ℝ} (ht : 0 ≤ t) (X : Coord 3) : | |
| force hb table terminal ν m p t X = | |
| triangularForce ν (field hb table terminal ν m) (impulse p) t X := by | |
| simp only [force, lift, projection_apply, | |
| residual_eq_horizontalResidual ν (a := field hb table terminal ν m) (contDiff_field hb table terminal ν m hν) ht, | |
| triangularForce, liftVelocity, uncurry_apply_pair] | |
| theorem force_finiteCylinderSupport (hν : 0 < ν) (p : Coord 2) (T : ℝ) : | |
| ∃ K : Set (Coord 2), IsCompact K ∧ ∀ t, 0 ≤ t → t ≤ T → | |
| ∀ X : Coord 3, horizontal X ∉ K → force hb table terminal ν m p t X = 0 := by | |
| obtain ⟨Q, hQ⟩ := exists_nat_gt T | |
| let a : ℝ × Coord 2 → Coord 2 := | |
| fun q => ∑ n ∈ Finset.range Q, stage hb table terminal ν m n q.1 q.2 | |
| have ha : HasCompactSupport a := by | |
| simpa only [Finset.sum_fn, a] using HasCompactSupport.finset_sum | |
| (fun n (_ : n ∈ Finset.range Q) => compactSupport_stage hb table terminal ν m hν n) | |
| let K : Set (Coord 2) := (Prod.snd '' tsupport (residual ν a)) ∪ tsupport (spatialPulse p) | |
| refine ⟨K, ((compactSupport_residual ν ha).image continuous_snd).union | |
| (compactSupport_spatialPulse p), ?_⟩ | |
| intro t ht0 ht X hX | |
| have hev : uncurry (field hb table terminal ν m) =ᶠ[𝓝 (t, horizontal X)] a := by | |
| filter_upwards [(isOpen_lt continuous_fst continuous_const).mem_nhds (ht.trans_lt hQ)] with q hq | |
| exact field_eq_finite hb table terminal ν m hν hq q.2 | |
| have hr : residual ν a (t, horizontal X) = 0 := by | |
| apply image_eq_zero_of_notMem_tsupport | |
| intro h | |
| exact hX (Or.inl ⟨(t, horizontal X), h, rfl⟩) | |
| have hg : spatialPulse p (horizontal X) = 0 := by | |
| apply image_eq_zero_of_notMem_tsupport | |
| intro h | |
| exact hX (Or.inr h) | |
| simp only [force, lift, projection_apply, | |
| (eventuallyEq_residual ν hev).eq_of_nhds, hr, Pi.zero_apply, uncurry_apply_pair, impulse, hg, | |
| mul_zero] | |
| ext i | |
| fin_cases i <;> rfl | |
| theorem force_NavierStokes (hν : 0 < ν) (p : Coord 2) (ρ : ScalarField 2) | |
| (hρ : ∀ t, 0 ≤ t → ∀ X, timeD ρ t X + advection (field hb table terminal ν m) ρ t X = | |
| ν * laplacian ρ t X + impulse p t X) (hρ0 : ∀ X, ρ 0 X = 0) : | |
| NavierStokes ν (liftVelocity (field hb table terminal ν m) ρ) 0 | |
| (force hb table terminal ν m p) := by | |
| have hn := triangular_reduction ν (field hb table terminal ν m) ρ (impulse p) | |
| (fun t _ => divergence_field hb table terminal ν m hν t) hρ | |
| (field_at_rest hb table terminal ν m hν (by norm_num)) hρ0 | |
| refine ⟨?_, hn.2⟩ | |
| intro t ht X i | |
| rw [force_eq_triangular hb table terminal ν m hν p ht] | |
| exact hn.1 t ht X i | |
| end VelocityDetection.ExpandingArray | |
| end | |
| end OAI | |