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37bcf52 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 | 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
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