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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.FlowPicardSolution
import Mathlib.Analysis.ODE.ExistUnique
namespace OAI
/-! Global smooth dependence for a finite-dimensional autonomous flow.
The proof iterates the short-time implicit-function construction. -/
noncomputable section
namespace ForcedComputation.Flow
open Set
open scoped ContDiff NNReal
variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E]
[FiniteDimensional ℝ E] [CompleteSpace E]
omit [FiniteDimensional ℝ E] [CompleteSpace E] in
theorem flow_add {Z : E → E} {L : ℝ≥0} (hL : LipschitzWith L Z)
{Ψ : ℝ → E → E}
(hΨ : ∀ t x, HasDerivAt (fun s => Ψ s x) (Z (Ψ t x)) t)
(hzero : ∀ x, Ψ 0 x = x) (s t : ℝ) (x : E) :
Ψ (s + t) x = Ψ t (Ψ s x) := by
have hd (r : ℝ) : HasDerivAt (fun u => Ψ (s + u) x)
(Z (Ψ (s + r) x)) r := by
simpa only [Function.comp_def, one_smul] using
(hΨ (s + r) x).scomp r ((hasDerivAt_id r).const_add s)
have he := ODE_solution_unique_univ (s := fun _ => univ)
(fun _ => hL.lipschitzOnWith)
(fun r => ⟨hd r, mem_univ _⟩)
(fun r => ⟨hΨ r (Ψ s x), mem_univ _⟩)
(show Ψ (s + 0) x = Ψ 0 (Ψ s x) by rw [add_zero, hzero])
exact congrFun he t
def flowSteps (Ψ : ℝ → E → E) : ℕ → ℝ → E → E
| 0, _, x => x
| n + 1, τ, x => Ψ τ (flowSteps Ψ n τ x)
omit [FiniteDimensional ℝ E] [CompleteSpace E] in
theorem flowSteps_eq {Z : E → E} {L : ℝ≥0} (hL : LipschitzWith L Z)
{Ψ : ℝ → E → E}
(hΨ : ∀ t x, HasDerivAt (fun s => Ψ s x) (Z (Ψ t x)) t)
(hzero : ∀ x, Ψ 0 x = x) (n : ℕ) (τ : ℝ) (x : E) :
flowSteps Ψ n τ x = Ψ ((n : ℝ) * τ) x := by
induction n with
| zero => simp only [flowSteps, Nat.cast_zero, zero_mul, hzero]
| succ n ih =>
rw [flowSteps, ih, ← flow_add hL hΨ hzero]
congr 2
push_cast
ring
theorem flowSteps_contDiffAt (Z : C(E, E))
(hZ : ContDiff ℝ ∞ (Z : E → E)) {L : ℝ≥0} (hL : LipschitzWith L Z)
{Ψ : ℝ → E → E}
(hΨ : ∀ t x, HasDerivAt (fun s => Ψ s x) (Z (Ψ t x)) t)
(hzero : ∀ x, Ψ 0 x = x) (n : ℕ) (τ : ℝ) (x : E)
(hτ : |τ| * (L : ℝ) < 1) :
ContDiffAt ℝ ∞ (fun p : ℝ × E => flowSteps Ψ n p.1 p.2) (τ, x) := by
induction n with
| zero => exact contDiffAt_snd
| succ n ih =>
exact (flow_contDiffAt_short Z hZ hL hΨ hzero τ
(flowSteps Ψ n τ x) hτ).comp (τ, x) (contDiffAt_fst.prodMk ih)
/-- An actual global flow of a smooth globally Lipschitz finite-dimensional
field is jointly smooth in elapsed time and the initial point. -/
theorem flow_contDiff (Z : C(E, E))
(hZ : ContDiff ℝ ∞ (Z : E → E)) {L : ℝ≥0} (hL : LipschitzWith L Z)
{Ψ : ℝ → E → E}
(hΨ : ∀ t x, HasDerivAt (fun s => Ψ s x) (Z (Ψ t x)) t)
(hzero : ∀ x, Ψ 0 x = x) :
ContDiff ℝ ∞ (fun p : ℝ × E => Ψ p.1 p.2) := by
apply contDiff_iff_contDiffAt.mpr
rintro ⟨t, x⟩
obtain ⟨n, hn⟩ := exists_nat_gt (|t| * (L : ℝ) + 1)
have hnpos : (0 : ℝ) < n := by
have hp := mul_nonneg (abs_nonneg t) L.coe_nonneg
linarith
have hsmall : |t / (n : ℝ)| * (L : ℝ) < 1 := by
rw [abs_div, abs_of_pos hnpos, div_mul_eq_mul_div, div_lt_one hnpos]
linarith
have hs := flowSteps_contDiffAt Z hZ hL hΨ hzero n (t / (n : ℝ)) x hsmall
have hc : ContDiffAt ℝ ∞ (fun p : ℝ × E => (p.1 / (n : ℝ), p.2)) (t, x) :=
(contDiffAt_fst.div_const _).prodMk contDiffAt_snd
have he : (fun p : ℝ × E => flowSteps Ψ n (p.1 / (n : ℝ)) p.2) =
fun p => Ψ p.1 p.2 := by
funext p
rw [flowSteps_eq hL hΨ hzero]
congr 2
field_simp [ne_of_gt hnpos]
have hcomp := hs.comp (t, x) hc
simpa only [Function.comp_def, he] using hcomp
end ForcedComputation.Flow
end
end OAI
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