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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 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 | import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Detector.TriangularFluid
import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Flow.CompactPlanarFlow
namespace OAI
/-! A compact vertical cutoff of a planar incompressible processor. Its
horizontal trajectories at a cutoff plateau remain exact, and the cutoff
introduces no divergence because the lifted field has zero third component. -/
noncomputable section
namespace ForcedComputation.CompactLift
open ShearFlows VelocityDetector Set
open scoped ContDiff BigOperators
def spatial (χ : ℝ → ℝ) (a : Plane → Plane) (x : Space) : Space :=
χ (x 2) • triangularLift a (fun _ => 0) x
def velocity (χ : ℝ → ℝ) (V : ℝ → Plane → Plane) : Velocity :=
fun y => spatial χ (V y.1) y.2
theorem spatial_smooth {χ : ℝ → ℝ} {a : Plane → Plane}
(hχ : ContDiff ℝ ∞ χ) (ha : ContDiff ℝ ∞ a) :
ContDiff ℝ ∞ (spatial χ a) :=
(hχ.comp (contDiff_apply ℝ ℝ (2 : Fin 3))).smul
(triangularLift_smooth ha contDiff_const)
theorem velocity_smooth {χ : ℝ → ℝ} {V : ℝ → Plane → Plane}
(hχ : ContDiff ℝ ∞ χ) (hV : ContDiff ℝ ∞ (Function.uncurry V)) :
ContDiff ℝ ∞ (velocity χ V) := by
have hi : ContDiff ℝ ∞ (fun y : SpaceTime => (y.1, horizontalLinear y.2)) :=
contDiff_fst.prodMk (horizontalLinear.contDiff.comp contDiff_snd)
have hW : ContDiff ℝ ∞ (fun y : SpaceTime => planeInclusion (V y.1 (horizontalLinear y.2))) :=
planeInclusion.contDiff.comp (hV.comp hi)
have he : velocity χ V = fun y => χ (y.2 2) • planeInclusion (V y.1 (horizontalLinear y.2)) := by
funext y
simp [velocity, spatial, triangularLift]
rw [he]
exact (hχ.comp ((contDiff_apply ℝ ℝ (2 : Fin 3)).comp contDiff_snd)).smul hW
theorem spatial_third (χ : ℝ → ℝ) (a : Plane → Plane) (x : Space) :
spatial χ a x 2 = 0 := by
simp [spatial, triangularLift_eq, atHeight]
theorem spatial_atHeight {χ : ℝ → ℝ} (a : Plane → Plane) {z : ℝ}
(hz : χ z = 1) (x : Plane) : spatial χ a (atHeight x z) = planeInclusion (a x) := by
simp only [spatial, triangularLift, horizontalLinear_eq, atHeight_horizontal,
zero_smul, add_zero]
change χ z • planeInclusion (a x) = _
rw [hz, one_smul]
theorem height_fderiv {χ : ℝ → ℝ} (hχ : ContDiff ℝ ∞ χ) (x : Space) (j : Fin 3) :
fderiv ℝ (fun y : Space => χ (y 2)) x (basis j) = deriv χ (x 2) * basis j 2 := by
have hd := ((hχ.differentiable (by simp) (x 2)).hasDerivAt).comp_hasFDerivAt x
(ContinuousLinearMap.proj (2 : Fin 3) : Space →L[ℝ] ℝ).hasFDerivAt
simpa only [Function.comp_def, smul_apply,
ContinuousLinearMap.proj_apply, smul_eq_mul] using
congrArg (fun L => L (basis j)) hd.fderiv
theorem spatial_divergence {χ : ℝ → ℝ} {a : Plane → Plane}
(hχ : ContDiff ℝ ∞ χ) (ha : ContDiff ℝ ∞ a) (x : Space) :
divergence (spatial χ a) x = χ (x 2) * PlanarHamiltonian.divergence a (horizontal x) := by
let W := triangularLift a (fun _ => 0)
have hw : ContDiff ℝ ∞ W := triangularLift_smooth ha contDiff_const
have hc : ContDiff ℝ ∞ (fun y : Space => χ (y 2)) := hχ.comp (contDiff_apply ℝ ℝ (2 : Fin 3))
have hd (j : Fin 3) : derivative (spatial χ a) j x j =
χ (x 2) * derivative W j x j +
(deriv χ (x 2) * basis j 2) * W x j := by
unfold derivative
rw [show spatial χ a = (fun y : Space => χ (y 2) • W y) from rfl,
fderiv_fun_smul (hc.differentiable (by simp) x) (hw.differentiable (by simp) x)]
simp only [add_apply, smul_apply,
ContinuousLinearMap.smulRight_apply, Pi.add_apply, Pi.smul_apply, smul_eq_mul,
height_fderiv hχ]
have hz : W x 2 = 0 := by simp [W, triangularLift_eq, atHeight]
have he : divergence (spatial χ a) x = χ (x 2) * divergence W x := by
rw [divergence, divergence, Fin.sum_univ_three, Fin.sum_univ_three]
rw [hd, hd, hd]
simp only [basis, Pi.single_eq_of_ne (by decide : (2 : Fin 3) ≠ 0),
Pi.single_eq_of_ne (by decide : (2 : Fin 3) ≠ 1), Pi.single_eq_same,
hz, mul_zero, add_zero]
ring
rw [he, triangularLift_divergence ha contDiff_const]
theorem atHeight_hasDerivAt {γ : ℝ → Plane} {v : Plane} {t z : ℝ}
(hγ : HasDerivAt γ v t) :
HasDerivAt (fun s => atHeight (γ s) z) (planeInclusion v) t := by
have hd := (planeInclusion.hasFDerivAt.comp_hasDerivAt t hγ).add_const (z • basis 2)
convert hd using 1
· funext s
ext j
fin_cases j <;> simp [planeInclusion, atHeight, basis]
theorem trajectory_atHeight {χ : ℝ → ℝ} {V : ℝ → Plane → Plane}
{γ : ℝ → Plane} {a t z : ℝ} (hz : χ z = 1)
(hγ : HasDerivAt γ (V (a + t) (γ t)) t) :
HasDerivAt (fun s => atHeight (γ s) z)
(velocity χ V (a + t, atHeight (γ t) z)) t := by
simpa only [velocity, spatial_atHeight _ hz] using atHeight_hasDerivAt (z := z) hγ
end ForcedComputation.CompactLift
end
end OAI
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