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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 | import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Flow.FlowPointwise
import Mathlib.Analysis.Calculus.ContDiff.Comp
import Mathlib.Analysis.Normed.Module.FiniteDimension
import Mathlib.Topology.UniformSpace.HeineCantor
namespace OAI
/-! Differentiation of a smooth finite-dimensional field on compact path space.
Compactness makes the derivative remainder uniform over the path; this is the
analytic ingredient needed to apply the implicit function theorem to Picard's
integral equation. -/
noncomputable section
namespace ForcedComputation.Flow
open Set Filter Metric
open scoped Topology ContDiff
variable {K E F : Type} [TopologicalSpace K] [CompactSpace K]
[NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
def postcompose (f : C(E, F)) (u : C(K, E)) : C(K, F) := f.comp u
omit [CompactSpace K] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [NormedSpace ℝ F] in
theorem postcompose_continuous (f : C(E, F)) :
Continuous (postcompose (K := K) f) := f.continuous_postcomp
theorem postcompose_hasFDerivAt (f : C(E, F))
(hf : ContDiff ℝ 1 (f : E → F)) (u : C(K, E)) :
HasFDerivAt (postcompose f)
(applyPath ⟨fun t => fderiv ℝ f (u t),
(hf.continuous_fderiv (by simp)).comp u.continuous⟩) u := by
rw [hasFDerivAt_iff_isLittleO, Asymptotics.isLittleO_iff]
intro ε hε
have hdf := hf.continuous_fderiv (by simp)
have huc : UniformContinuousOn (fderiv ℝ f)
(closedBall (0 : E) (‖u‖ + 1)) :=
(isCompact_closedBall (0 : E) (‖u‖ + 1)).uniformContinuousOn_of_continuous
hdf.continuousOn
obtain ⟨δ, hδ, hcontrol⟩ := Metric.uniformContinuousOn_iff.mp huc ε hε
filter_upwards [Metric.ball_mem_nhds u (lt_min hδ zero_lt_one)] with v hv
have hvnorm : ‖v - u‖ < min δ 1 := by simpa only [mem_ball, dist_eq_norm] using hv
apply (ContinuousMap.norm_le _
(mul_nonneg hε.le (norm_nonneg (v - u)))).mpr
intro t
change ‖f (v t) - f (u t) - fderiv ℝ f (u t) (v t - u t)‖ ≤ ε * ‖v - u‖
have hu : u t ∈ closedBall (0 : E) (‖u‖ + 1) := by
rw [mem_closedBall, dist_zero_right]
exact (u.norm_coe_le_norm t).trans (by linarith)
have hdiff : ‖v t - u t‖ ≤ ‖v - u‖ := (v - u).norm_coe_le_norm t
have hv' : v t ∈ closedBall (0 : E) (‖u‖ + 1) := by
rw [mem_closedBall, dist_zero_right]
calc
‖v t‖ ≤ ‖v t - u t‖ + ‖u t‖ := by
simpa only [sub_add_cancel] using norm_add_le (v t - u t) (u t)
_ ≤ ‖v - u‖ + ‖u‖ := add_le_add hdiff (u.norm_coe_le_norm t)
_ ≤ ‖u‖ + 1 := by linarith [lt_of_lt_of_le hvnorm (min_le_right δ 1)]
have hsegment := (convex_closedBall (0 : E) (‖u‖ + 1)).segment_subset hu hv'
have hr := remainder_bound (fun z => (hf.differentiable (by simp) z).hasFDerivAt)
(u t) (v t) (ε := ε) (by
intro z hz
have hzδ : dist z (u t) < δ := by
rw [dist_eq_norm]
exact (norm_sub_le_of_mem_segment hz).trans_lt
(hdiff.trans_lt (lt_of_lt_of_le hvnorm (min_le_left δ 1)))
simpa only [dist_eq_norm] using (hcontrol z (hsegment hz) (u t) hu hzδ).le)
exact hr.trans (mul_le_mul_of_nonneg_left hdiff hε.le)
theorem postcompose_contDiff_nat (n : ℕ) (f : C(E, F))
(hf : ContDiff ℝ n (f : E → F)) :
ContDiff ℝ n (postcompose (K := K) f) := by
induction n generalizing F with
| zero =>
exact contDiff_zero.mpr (postcompose_continuous f)
| succ n ih =>
have h1 : ContDiff ℝ 1 (f : E → F) := hf.of_le (by norm_cast; omega)
let df : C(E, E →L[ℝ] F) :=
⟨fderiv ℝ f, h1.continuous_fderiv (by simp)⟩
have hdf : ContDiff ℝ n (df : E → E →L[ℝ] F) :=
hf.fderiv_right (m := n) (by simp)
apply contDiff_succ_iff_hasFDerivAt.mpr
refine ⟨fun u => applyPathL (postcompose df u), ?_, ?_⟩
· exact (applyPathL (K := K) (E := E) (F := F)).contDiff.comp (ih df hdf)
· intro u
exact postcompose_hasFDerivAt f h1 u
theorem postcompose_contDiff (f : C(E, F)) (hf : ContDiff ℝ ∞ (f : E → F)) :
ContDiff ℝ ∞ (postcompose (K := K) f) := by
rw [contDiff_infty]
intro n
exact postcompose_contDiff_nat n f (hf.of_le (by simp))
end ForcedComputation.Flow
end
end OAI
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