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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 | 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
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