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caec230 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 | import OAI.MathematicalPhysics.RapidForcing.VelocitySyntax
import OAI.MathematicalPhysics.RapidForcing.JetBounds
namespace OAI
section
open scoped BigOperators Topology ENNReal ContDiff
open Set Filter
namespace RapidForcing.EffectiveProfile.Formula
def spIndex (i : Fin 3) : Fin 10 := ⟨i.val+1, by omega⟩
def etaIndex (i : Fin 3) : Fin 10 := ⟨i.val+4, by omega⟩
def xiIndex (i : Fin 3) : Fin 10 := ⟨i.val+7, by omega⟩
noncomputable def profileLinear : (ℝ × Space) →L[ℝ] (Fin 10 → ℝ) :=
ContinuousLinearMap.pi ![ContinuousLinearMap.fst ℝ ℝ Space,
(EuclideanSpace.proj 0).comp (ContinuousLinearMap.snd ℝ ℝ Space),
(EuclideanSpace.proj 1).comp (ContinuousLinearMap.snd ℝ ℝ Space),
(EuclideanSpace.proj 2).comp (ContinuousLinearMap.snd ℝ ℝ Space), 0,0,0,0,0,0]
noncomputable def profileParams (η ξ : Space) : Fin 10 → ℝ :=
![0,0,0,0,η 0,η 1,η 2,ξ 0,ξ 1,ξ 2]
noncomputable def profilePack (η ξ : Space) (t : ℝ) (x : Space) : Fin 10 → ℝ :=
profileLinear (t,x) + profileParams η ξ
lemma profilePack_eq (η ξ : Space) (t : ℝ) (x : Space) :
profilePack η ξ t x = ![t,x 0,x 1,x 2,η 0,η 1,η 2,ξ 0,ξ 1,ξ 2] := by
ext i
fin_cases i <;> dsimp [profilePack, profileLinear, profileParams] <;> simp
lemma profileLinear_norm : ‖profileLinear‖ ≤ 1 := by
apply ContinuousLinearMap.opNorm_le_bound _ zero_le_one
intro p
rw [one_mul]
apply (pi_norm_le_iff_of_nonneg (norm_nonneg p)).mpr
intro i
fin_cases i
· exact norm_fst_le p
· exact (PiLp.norm_apply_le p.2 0).trans (norm_snd_le p)
· exact (PiLp.norm_apply_le p.2 1).trans (norm_snd_le p)
· exact (PiLp.norm_apply_le p.2 2).trans (norm_snd_le p)
all_goals simp [profileLinear]
lemma profilePack_line (η ξ : Space) (t : ℝ) (x : Space) (i : Fin 3) (s : ℝ) :
profilePack η ξ t (x + s • basis i) = profilePack η ξ t x + s • Pi.single (spIndex i) 1 := by
ext j
fin_cases i <;> fin_cases j <;> simp [profilePack_eq, spIndex, basis]
noncomputable def profileField (a : Fin 3 → Formula 10) (η ξ : Space) : Field Space :=
fun t x => WithLp.toLp 2 (fun i => (a i).value (profilePack η ξ t x))
lemma profileField_smooth (a : Fin 3 → Formula 10) (η ξ : Space) :
ContDiff ℝ (⊤ : ℕ∞) (Function.uncurry (profileField a η ξ)) := by
apply (contDiff_piLp 2).mpr
intro i
exact (a i).contDiff.comp (profileLinear.contDiff.add contDiff_const)
lemma profile_partial (a : Formula 10) (η ξ : Space) (t : ℝ) (x : Space) (i : Fin 3) :
fderiv ℝ (fun y => a.value (profilePack η ξ t y)) x (basis i) =
(a.diff (spIndex i)).value (profilePack η ξ t x) := by
have hline : HasDerivAt (fun s : ℝ => x + s • basis i) (basis i) 0 := by
simpa using ((hasDerivAt_id (0 : ℝ)).smul_const (basis i)).const_add x
have hsm : ContDiff ℝ (⊤ : ℕ∞) (fun y => a.value (profilePack η ξ t y)) :=
a.contDiff.comp ((profileLinear.contDiff.comp (contDiff_const.prodMk contDiff_id)).add contDiff_const)
have hp := a.hasDerivAt_line (spIndex i) (profilePack η ξ t x) 0
have he : (fun s : ℝ => a.value (profilePack η ξ t x + s • Pi.single (spIndex i) 1)) =
fun s => a.value (profilePack η ξ t (x+s • basis i)) := by
funext s
rw [← profilePack_line]
rw [he] at hp
have hf : HasFDerivAt (fun y => a.value (profilePack η ξ t y))
(fderiv ℝ (fun y => a.value (profilePack η ξ t y)) x) (x+(0:ℝ) • basis i) := by
simpa using (hsm.differentiable (by simp) x).hasFDerivAt
have hc := hf.comp_hasDerivAt 0 hline
simpa only [zero_smul, add_zero] using hc.unique hp
def profileCurl (a : Fin 3 → Formula 10) : Fin 3 → Formula 10 :=
![((a 2).diff (spIndex 1)).sub ((a 1).diff (spIndex 2)),
((a 0).diff (spIndex 2)).sub ((a 2).diff (spIndex 0)),
((a 1).diff (spIndex 0)).sub ((a 0).diff (spIndex 1))]
lemma profileCurl_value (a : Fin 3 → Formula 10) (η ξ : Space) (t : ℝ) (x : Space) :
profileField (profileCurl a) η ξ t x = curl (profileField a η ξ t) x := by
have hs (i j : Fin 3) : (fderiv ℝ (profileField a η ξ t) x (basis i)) j =
((a j).diff (spIndex i)).value (profilePack η ξ t x) := by
have hd : DifferentiableAt ℝ (profileField a η ξ t) x :=
((profileField_smooth a η ξ).comp
(contDiff_const.prodMk contDiff_id)).differentiable (by simp) x
rw [← fderiv_coordinate hd]
exact profile_partial (a j) η ξ t x i
ext i
fin_cases i <;> simp [profileField, profileCurl, curl, vec, hs, value_sub]
def scaledFormula : Fin 3 → Formula 10 :=
let y := fun i => (Formula.var (spIndex i) : Formula 10).sub
(Formula.mul (Formula.var 0).theta (Formula.var (etaIndex i)))
let speed := fun i => Formula.mul (Formula.var 0).thetaDot (Formula.var (xiIndex i))
profileCurl (fun i => Formula.mul (zeta y) (Formula.mul (Formula.const (1/2)) (crossF speed y i)))
lemma scaledFormula_value (η ξ : Space) (t : ℝ) (x : Space) :
profileField scaledFormula η ξ t x = scaledCurl η ξ t x := by
unfold scaledFormula
rw [profileCurl_value]
unfold scaledCurl
congr 1
funext y
have hy : WithLp.toLp 2 (fun i => ((Formula.var (spIndex i) : Formula 10).sub
(Formula.mul (Formula.var 0).theta (Formula.var (etaIndex i)))).value (profilePack η ξ t y)) =
y - θ t • η := by
ext i
fin_cases i <;> simp [value_sub, value, value_theta, profilePack_eq, spIndex, etaIndex]
have hv : WithLp.toLp 2 (fun i => (Formula.mul (Formula.var 0).thetaDot
(Formula.var (xiIndex i)) : Formula 10).value (profilePack η ξ t y)) = deriv θ t • ξ := by
ext i
fin_cases i <;> simp [value, value_thetaDot, profilePack_eq, xiIndex]
have hcross := value_crossF (fun i => (Formula.mul (Formula.var 0).thetaDot (Formula.var (xiIndex i)) : Formula 10))
(fun i => (Formula.var (spIndex i) : Formula 10).sub (Formula.mul (Formula.var 0).theta (Formula.var (etaIndex i))))
(profilePack η ξ t y)
rw [hy, hv] at hcross
ext i
change (zeta _).value _ * ((1/2:ℚ) * (crossF _ _ i).value _) = _
rw [value_zeta, hy]
have hc := congrArg (fun v : Space => v i) hcross
dsimp at hc
rw [hc]
norm_num [smul_apply, smul_eq_mul]
def curlJetBound (k : ℕ) : ℚ := ∑ i : Fin 3, (scaledFormula i).jetBound 3 k
lemma curlJetBound_nonneg (k : ℕ) : 0 ≤ curlJetBound k :=
Finset.sum_nonneg (fun _ _ => jetBound_nonneg _ _ _)
lemma norm_jet_affine_le {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E]
(a : Formula 10) (L : E →L[ℝ] (Fin 10 → ℝ)) (hL : ‖L‖ ≤ 1)
(b : Fin 10 → ℝ) (k : ℕ) (x : E) :
‖iteratedFDeriv ℝ k (fun y => a.value (L y+b)) x‖ ≤
‖iteratedFDeriv ℝ k a.value (L x+b)‖ := by
change ‖iteratedFDeriv ℝ k ((fun z => a.value (z+b)) ∘ L) x‖ ≤ _
have hsm : ContDiff ℝ (⊤ : ℕ∞) (fun z => a.value (z+b)) :=
a.contDiff.comp (contDiff_id.add contDiff_const)
rw [L.iteratedFDeriv_comp_right hsm _ (finite_order k),
iteratedFDeriv_comp_add_right]
have hb := (iteratedFDeriv ℝ k a.value (L x+b)).norm_compContinuousLinearMap_le (fun _ : Fin k => L)
simp only [Finset.prod_const, Finset.card_univ, Fintype.card_fin] at hb
exact hb.trans (by
simpa using mul_le_mul_of_nonneg_left
(pow_le_one₀ (norm_nonneg L) hL : ‖L‖ ^ k ≤ 1) (norm_nonneg (iteratedFDeriv ℝ k a.value (L x+b))))
lemma curlJetBound_spec (k : ℕ) (η ξ : Space) (hη : ‖η‖ ≤ 2) (hξ : ‖ξ‖ ≤ 2) (p : ℝ × Space) :
‖iteratedFDeriv ℝ k (Function.uncurry (scaledCurl η ξ)) p‖ ≤ (curlJetBound k : ℝ) := by
by_cases hp : p ∈ Icc (0:ℝ) 1 ×ˢ Metric.closedBall (0:Space) 3
· have hx : ∀ i, |profilePack η ξ p.1 p.2 i| ≤ |((3:ℚ):ℝ)| := by
have ht : |p.1| ≤ 3 := by rw [abs_of_nonneg hp.1.1]; linarith [hp.1.2]
have hy : ‖p.2‖ ≤ 3 := by simpa using hp.2
have hi (v : Space) (hv : ‖v‖ ≤ 3) (i : Fin 3) : |v i| ≤ 3 := (PiLp.norm_apply_le v i).trans hv
intro i
fin_cases i <;> norm_num [profilePack_eq]
· exact ht
· exact hi _ hy 0
· exact hi _ hy 1
· exact hi _ hy 2
· exact hi _ (hη.trans (by norm_num)) 0
· exact hi _ (hη.trans (by norm_num)) 1
· exact hi _ (hη.trans (by norm_num)) 2
· exact hi _ (hξ.trans (by norm_num)) 0
· exact hi _ (hξ.trans (by norm_num)) 1
· exact hi _ (hξ.trans (by norm_num)) 2
have he : Function.uncurry (scaledCurl η ξ) = fun q : ℝ × Space => ∑ i : Fin 3,
(scaledFormula i).value (profilePack η ξ q.1 q.2) • basis i := by
funext q
change scaledCurl η ξ q.1 q.2 = _
rw [← scaledFormula_value]
ext j
fin_cases j <;> simp [profileField, basis, Fin.sum_univ_three]
rw [he]
apply (jet_sum_le _ _ (fun i _ => ((scaledFormula i).contDiff.comp
(profileLinear.contDiff.add contDiff_const)).smul contDiff_const) k p).trans
rw [curlJetBound, Rat.cast_sum]
apply Finset.sum_le_sum
intro i _
have hb : ‖basis i‖ = 1 := by simp [basis]
apply (jet_smul_const_le ((scaledFormula i).contDiff.comp
(profileLinear.contDiff.add contDiff_const)) (basis i) k p).trans
rw [hb, one_mul]
exact (norm_jet_affine_le (scaledFormula i) profileLinear profileLinear_norm _ _ _).trans
(norm_jet_le 3 k (scaledFormula i) _ hx)
· have hz : iteratedFDeriv ℝ k (Function.uncurry (scaledCurl η ξ)) p = 0 :=
image_eq_zero_of_notMem_tsupport (fun h => hp (scaledCurl_tsupport_subset η ξ hη
(tsupport_iteratedFDeriv_subset k h)))
rw [hz, norm_zero]
exact_mod_cast curlJetBound_nonneg k
end RapidForcing.EffectiveProfile.Formula
end
end OAI
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