openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Flow /CompactPeriodizationNS.lean
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Flow.CompactPeriodizationDerivatives | |
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.CompactRapidDifferential | |
| namespace OAI | |
| /-! Local periodization preserves the full residual equation. A pressure | |
| projection, when supplied by the torus Poisson solver, is a separate algebraic | |
| step and does not change the velocity or its trajectories. -/ | |
| noncomputable section | |
| open Set | |
| open scoped BigOperators | |
| namespace RapidForcing.CompactEmbedding | |
| theorem field_chartSupported {u : Field Space} (hu : Supported u) | |
| (hneg : ∀ t, t < 0 → ∀ x, u t x = 0) : ChartSupported (field u) := by | |
| intro t x hx | |
| by_cases ht : 0 ≤ t | |
| · have hu' : unembed x ∉ K := by | |
| intro h | |
| exact hx (by simpa only [embed_unembed] using embed_mem_chart h) | |
| simp [field, hu.zero_off ht hu'] | |
| · simp [field, hneg t (lt_of_not_ge ht)] | |
| theorem periodic_divergence {u : Field Space} (hu : ChartSupported u) (t : ℝ) (x : Space) : | |
| divergence (periodic u) t x = divergence u t (representative x) := by | |
| unfold divergence | |
| simp_rw [periodic_spatialD hu] | |
| rfl | |
| theorem periodic_advection {u : Field Space} (hu : ChartSupported u) (t : ℝ) (x : Space) : | |
| advection (periodic u) t x = advection u t (representative x) := by | |
| unfold advection | |
| simp_rw [periodic_spatialD hu] | |
| rfl | |
| theorem periodic_laplace {u : Field Space} (hu : ChartSupported u) (t : ℝ) (x : Space) : | |
| laplace (periodic u) t x = laplace u t (representative x) := by | |
| unfold laplace | |
| simp_rw [periodic_spatialD hu] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| rw [periodic_spatialD (hu.spatialD i)] | |
| rfl | |
| theorem periodic_residual {u : Field Space} (hu : ChartSupported u) (ν t : ℝ) (x : Space) : | |
| residual ν (periodic u) t x = residual ν u t (representative x) := by | |
| unfold residual | |
| rw [periodic_timeD, periodic_advection hu, periodic_laplace hu] | |
| rfl | |
| theorem periodic_navierStokes {u f : Field Space} {ν : ℝ} | |
| (hu : ChartSupported u) (hNS : NavierStokes ν f u (fun _ _ => 0)) : | |
| NavierStokes ν (periodic f) (periodic u) (fun _ _ => 0) := by | |
| refine ⟨?_, ?_, ?_⟩ | |
| · intro t ht x | |
| have h := hNS.1 t ht (representative x) | |
| simp only [gradient, spatialD, fderiv_const_apply, zero_apply, | |
| zero_smul, Finset.sum_const_zero, neg_zero, zero_add] at h ⊢ | |
| rw [periodic_timeD, periodic_advection hu, periodic_laplace hu] | |
| exact h | |
| · intro t ht x | |
| rw [periodic_divergence hu] | |
| exact hNS.2.1 t ht (representative x) | |
| · intro x | |
| exact hNS.2.2 (representative x) | |
| theorem navierStokes_pressure_projection {u g : Field Space} {ν : ℝ} | |
| {φ : Field ℝ} (hNS : NavierStokes ν g u (fun _ _ => 0)) : | |
| NavierStokes ν (fun t x => g t x - gradient φ t x) u (fun t x => -φ t x) := by | |
| refine ⟨?_, hNS.2⟩ | |
| intro t ht x | |
| have h := hNS.1 t ht x | |
| simp only [gradient, spatialD, fderiv_const_apply, zero_apply, | |
| zero_smul, Finset.sum_const_zero, neg_zero, zero_add] at h | |
| have hg : gradient (fun t x => -φ t x) t x = -gradient φ t x := by | |
| unfold gradient spatialD | |
| change (∑ i, fderiv ℝ (-(φ t)) x (basis i) • basis i) = _ | |
| rw [fderiv_neg] | |
| simp only [neg_apply, neg_smul, Finset.sum_neg_distrib] | |
| rw [hg, neg_neg, h] | |
| module | |
| end RapidForcing.CompactEmbedding | |
| end | |
| end OAI | |