openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Programs /CompactRapidDifferential.lean
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.CompactRapidEmbedding | |
| namespace OAI | |
| /-! The similarity has length scale 1/8. Thus viscosity 64ν before the | |
| similarity gives viscosity ν afterwards. All identities are proved directly | |
| from the coordinate derivatives used in the rapid construction. -/ | |
| noncomputable section | |
| open Set | |
| open scoped BigOperators | |
| namespace RapidForcing.CompactEmbedding | |
| theorem spatialD_field {u : Field Space} {t : ℝ} | |
| (hu : Differentiable ℝ (u t)) (i : Fin 3) (x : Space) : | |
| spatialD i (field u) t x = (8 * coordinateSign i) • | |
| linear (spatialD i u t (unembed x)) := by | |
| have hd := linear.hasFDerivAt.comp x | |
| ((hu (unembed x)).hasFDerivAt.comp x (unembed_hasFDerivAt x)) | |
| simp only [Function.comp_def] at hd | |
| change fderiv ℝ (fun y => linear (u t (unembed y))) x (basis i) = _ | |
| rw [hd.fderiv] | |
| simp only [ContinuousLinearMap.comp_apply, inverseLinear_basis, map_smul, spatialD] | |
| theorem divergence_field {u : Field Space} {t : ℝ} | |
| (hu : Differentiable ℝ (u t)) (x : Space) : | |
| divergence (field u) t x = divergence u t (unembed x) := by | |
| unfold divergence | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| rw [spatialD_field hu, PiLp.smul_apply, linear_apply] | |
| by_cases hi : i = 0 | |
| all_goals simp [coordinateSign, hi] | |
| all_goals ring | |
| theorem advection_field {u : Field Space} {t : ℝ} | |
| (hu : Differentiable ℝ (u t)) (x : Space) : | |
| advection (field u) t x = linear (advection u t (unembed x)) := by | |
| unfold advection | |
| rw [map_sum] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| rw [spatialD_field hu, field, linear_apply, smul_smul, map_smul] | |
| congr 1 | |
| by_cases hi : i = 0 <;> simp [coordinateSign, hi] <;> ring | |
| theorem spatialD_const_smul {u : Field Space} {t : ℝ} | |
| (hu : Differentiable ℝ (u t)) (a : ℝ) (i : Fin 3) (x : Space) : | |
| spatialD i (fun t x => a • u t x) t x = a • spatialD i u t x := by | |
| change fderiv ℝ (a • u t) x (basis i) = _ | |
| rw [((hu x).hasFDerivAt.const_smul a).fderiv] | |
| rfl | |
| theorem spatialD_field_differentiable {u : Field Space} {t : ℝ} | |
| (hu : Differentiable ℝ (u t)) : Differentiable ℝ (field u t) := | |
| linear.differentiable.comp (hu.comp (unembed_contDiff.differentiable (by simp))) | |
| theorem spatialD_twice_field {u : Field Space} {t : ℝ} | |
| (hu : Differentiable ℝ (u t)) | |
| (hdu : ∀ i, Differentiable ℝ (spatialD i u t)) (i : Fin 3) (x : Space) : | |
| spatialD i (spatialD i (field u)) t x = | |
| (64 : ℝ) • linear (spatialD i (spatialD i u) t (unembed x)) := by | |
| have he : spatialD i (field u) t = | |
| fun y => (8 * coordinateSign i) • field (spatialD i u) t y := by | |
| funext y | |
| exact spatialD_field hu i y | |
| change fderiv ℝ (spatialD i (field u) t) x (basis i) = _ | |
| rw [he] | |
| change spatialD i (fun s y => (8 * coordinateSign i) • field (spatialD i u) s y) t x = _ | |
| rw [spatialD_const_smul (spatialD_field_differentiable (hdu i)), | |
| spatialD_field (hdu i), smul_smul] | |
| congr 1 | |
| by_cases hi : i = 0 <;> norm_num [coordinateSign, hi] | |
| theorem laplace_field {u : Field Space} {t : ℝ} | |
| (hu : Differentiable ℝ (u t)) | |
| (hdu : ∀ i, Differentiable ℝ (spatialD i u t)) (x : Space) : | |
| laplace (field u) t x = (64 : ℝ) • linear (laplace u t (unembed x)) := by | |
| unfold laplace | |
| rw [map_sum, Finset.smul_sum] | |
| exact Finset.sum_congr rfl (fun i _ => spatialD_twice_field hu hdu i x) | |
| theorem timeD_field {u : Field Space} {t : ℝ} (ht : 0 ≤ t) (x : Space) | |
| (hu : DifferentiableWithinAt ℝ (fun s => u s (unembed x)) (Ici 0) t) : | |
| timeD (field u) t x = linear (timeD u t (unembed x)) := by | |
| have hd := linear.hasFDerivAt.comp_hasDerivWithinAt t hu.hasDerivWithinAt | |
| exact hd.derivWithin (uniqueDiffOn_Ici 0 t ht) | |
| theorem residual_field {u : Field Space} {t : ℝ} (ht : 0 ≤ t) | |
| (hu : Differentiable ℝ (u t)) | |
| (hdu : ∀ i, Differentiable ℝ (spatialD i u t)) | |
| (htime : ∀ x, DifferentiableWithinAt ℝ (fun s => u s x) (Ici 0) t) | |
| (ν : ℝ) (x : Space) : | |
| residual ν (field u) t x = linear (residual (64 * ν) u t (unembed x)) := by | |
| unfold residual | |
| rw [timeD_field ht x (htime _), advection_field hu, laplace_field hu hdu, | |
| map_sub, map_add, map_smul, smul_smul] | |
| congr 1 | |
| congr 1 | |
| ring | |
| theorem navierStokes_field {u f : Field Space} (ν : ℝ) | |
| (hNS : NavierStokes (64 * ν) f u (fun _ _ => 0)) | |
| (hu : ∀ t, 0 ≤ t → Differentiable ℝ (u t)) | |
| (hdu : ∀ t, 0 ≤ t → ∀ i, Differentiable ℝ (spatialD i u t)) | |
| (htime : ∀ t, 0 ≤ t → ∀ x, | |
| DifferentiableWithinAt ℝ (fun s => u s x) (Ici 0) t) : | |
| NavierStokes ν (field f) (field u) (fun _ _ => 0) := by | |
| refine ⟨?_, ?_, ?_⟩ | |
| · intro t ht x | |
| have hres : residual (64 * ν) u t (unembed x) = f t (unembed x) := by | |
| have he := hNS.1 t ht (unembed x) | |
| simp [gradient, spatialD] at he | |
| unfold residual | |
| rw [he] | |
| abel | |
| have he := residual_field ht (hu t ht) (hdu t ht) (htime t ht) ν x | |
| rw [hres] at he | |
| change timeD (field u) t x + advection (field u) t x = _ | |
| simp only [gradient, spatialD, fderiv_const_apply, zero_apply, | |
| zero_smul, Finset.sum_const_zero, neg_zero, zero_add] | |
| change timeD (field u) t x + advection (field u) t x = | |
| ν • laplace (field u) t x + linear (f t (unembed x)) | |
| unfold residual at he | |
| rw [← he] | |
| abel | |
| · intro t ht x | |
| rw [divergence_field (hu t ht)] | |
| exact hNS.2.1 t ht (unembed x) | |
| · intro x | |
| simp [field, hNS.2.2] | |
| end RapidForcing.CompactEmbedding | |
| end | |
| end OAI | |