openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Programs /ForceMean.lean
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.AutonomousStartup | |
| namespace OAI | |
| /-! | |
| # The mean of the full startup force | |
| For a periodic incompressible field, the convective acceleration is the | |
| divergence of its momentum flux, and therefore has zero spatial mean. This is | |
| the additional term needed in the autonomous construction; a shear-only force | |
| identity would not suffice. | |
| -/ | |
| noncomputable section | |
| open Set MeasureTheory | |
| open scoped ContDiff | |
| open ShearFlows | |
| namespace ForcedComputation | |
| theorem advection_mean_zero {L : ℝ} (hL : 0 ≤ L) {W : Space → Space} | |
| (hW : ContDiff ℝ ∞ W) (hp : CubePeriodic L W) | |
| (hd : ∀ x, divergence W x = 0) : | |
| (∫ x in fundamentalCube L, advection W x) = 0 := by | |
| have hW₁ : ContDiff ℝ 1 W := hW.of_le (by simp) | |
| have hi : IntegrableOn (advection W) (fundamentalCube L) := | |
| ((hW.continuous_fderiv (by simp)).clm_apply hW.continuous).integrableOn_Icc | |
| ext j | |
| change (ContinuousLinearMap.proj j : Space →L[ℝ] ℝ) | |
| (∫ x in fundamentalCube L, advection W x) = 0 | |
| rw [← (ContinuousLinearMap.proj j : Space →L[ℝ] ℝ).integral_comp_comm hi] | |
| change (∫ x in fundamentalCube L, advection W x j) = 0 | |
| have hcoord : ContDiff ℝ 1 (fun y => W y j) := | |
| (ContinuousLinearMap.proj j : Space →L[ℝ] ℝ).contDiff.comp hW₁ | |
| have he (x : Space) : divergence (fun y => W y j • W y) x = advection W x j := by | |
| rw [divergence_scalar_mul (hcoord.differentiable (by norm_num)) | |
| (hW₁.differentiable (by norm_num)), hd x, mul_zero, zero_add] | |
| exact fderiv_coordinate (hW.differentiable (by simp)) x (W x) j | |
| have hz : (∫ x in fundamentalCube L, divergence (fun y => W y j • W y) x) = 0 := | |
| integral_periodic_divergence hL (hcoord.smul hW₁) | |
| (show CubePeriodic L (fun y => W y j • W y) from fun x n => by dsimp only; rw [hp]) | |
| simpa only [he] using hz | |
| theorem ramp_force_mean_zero {L : ℝ} (hL : 0 ≤ L) {W : Space → Space} | |
| (hW : ContDiff ℝ ∞ W) (hp : CubePeriodic L W) | |
| (hd : ∀ x, divergence W x = 0) | |
| (hm : (∫ x in fundamentalCube L, W x) = 0) | |
| {α : ℝ → ℝ} (hα : Differentiable ℝ α) (ν : ℝ) : | |
| HasZeroMean L (residual ν (rampVelocity α W)) := by | |
| have hw : IntegrableOn W (fundamentalCube L) := hW.continuous.integrableOn_Icc | |
| have ha : IntegrableOn (advection W) (fundamentalCube L) := | |
| ((hW.continuous_fderiv (by simp)).clm_apply hW.continuous).integrableOn_Icc | |
| have hl : IntegrableOn (laplacian W) (fundamentalCube L) := | |
| (laplacian_continuous (hW.of_le (by simp))).integrableOn_Icc | |
| have hla : (∫ x in fundamentalCube L, laplacian W x) = 0 := | |
| laplacian_mean_zero hL (hW.comp contDiff_snd) | |
| (show SpatiallyPeriodic L (fun y : SpaceTime => W y.2) from fun _ => hp) 0 | |
| intro t | |
| simp_rw [residual_ramp ν α W (hα t)] | |
| have hsum := integral_add (hw.smul (deriv α t)) (ha.smul (α t ^ 2)) | |
| simp only [Pi.smul_apply, integral_smul, hm, advection_mean_zero hL hW hp hd, | |
| smul_zero, add_zero] at hsum | |
| have hdiff := integral_sub ((hw.smul (deriv α t)).add (ha.smul (α t ^ 2))) | |
| ((hl.smul (α t)).smul ν) | |
| simpa only [Pi.add_apply, Pi.smul_apply, hsum, integral_smul, hla, smul_zero, sub_zero] using hdiff | |
| theorem startup_force_mean_zero {L : ℝ} (hL : 0 ≤ L) {W : Space → Space} | |
| (hW : ContDiff ℝ ∞ W) (hp : CubePeriodic L W) | |
| (hd : ∀ x, divergence W x = 0) | |
| (hm : (∫ x in fundamentalCube L, W x) = 0) (ν : ℝ) : | |
| HasZeroMean L (residual ν (rampVelocity startupRamp W)) := | |
| ramp_force_mean_zero hL hW hp hd hm (startupRamp_smooth.differentiable (by simp)) ν | |
| end ForcedComputation | |
| end | |
| end OAI | |