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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 | import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Flow.FlowSmooth
namespace OAI
/-! Smooth dependence on the initial phase for a nonautonomous equation,
obtained by adjoining phase as an autonomous coordinate. -/
noncomputable section
namespace ForcedComputation.Flow
open scoped ContDiff NNReal
variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E]
[FiniteDimensional ℝ E] [CompleteSpace E]
theorem transition_contDiff (V : C(ℝ × E, E))
(hV : ContDiff ℝ ∞ (V : ℝ × E → E)) {L : ℝ≥0}
(hL : LipschitzWith L V) {Ω : ℝ → ℝ → E → E}
(hode : ∀ a t x, HasDerivAt (fun s => Ω a s x) (V (a + t, Ω a t x)) t)
(hzero : ∀ a x, Ω a 0 x = x) :
ContDiff ℝ ∞ (fun p : ℝ × (ℝ × E) => Ω p.1 p.2.1 p.2.2) := by
let Z : C(ℝ × E, ℝ × E) :=
⟨fun p => (1, V p), continuous_const.prodMk V.continuous⟩
have hZ : ContDiff ℝ ∞ (Z : ℝ × E → ℝ × E) := contDiff_const.prodMk hV
have hZL : LipschitzWith L Z := by
change LipschitzWith L (fun p : ℝ × E => ((1 : ℝ), V p))
simpa only [max_eq_right (show (0 : ℝ≥0) ≤ L from bot_le)] using
(LipschitzWith.const (1 : ℝ)).prodMk hL
let Ψ : ℝ → (ℝ × E) → ℝ × E := fun t p => (p.1 + t, Ω p.1 t p.2)
have hΨ (t : ℝ) (p : ℝ × E) :
HasDerivAt (fun s => Ψ s p) (Z (Ψ t p)) t :=
((hasDerivAt_id t).const_add p.1).prodMk (hode p.1 t p.2)
have hi (p : ℝ × E) : Ψ 0 p = p := by simp only [Ψ, add_zero, hzero]
have hs := flow_contDiff Z hZ hZL hΨ hi
have hq : ContDiff ℝ ∞ (fun p : ℝ × (ℝ × E) => (p.2.1, (p.1, p.2.2))) :=
contDiff_snd.fst.prodMk (contDiff_fst.prodMk contDiff_snd.snd)
exact hs.snd.comp hq
theorem transition_backward_contDiff (V : C(ℝ × E, E))
(hV : ContDiff ℝ ∞ (V : ℝ × E → E)) {L : ℝ≥0}
(hL : LipschitzWith L V) {Ω : ℝ → ℝ → E → E}
(hode : ∀ a t x, HasDerivAt (fun s => Ω a s x) (V (a + t, Ω a t x)) t)
(hzero : ∀ a x, Ω a 0 x = x) :
ContDiff ℝ ∞ (fun p : ℝ × E => Ω p.1 (-p.1) p.2) :=
(transition_contDiff V hV hL hode hzero).comp
(contDiff_fst.prodMk (contDiff_fst.neg.prodMk contDiff_snd))
end ForcedComputation.Flow
end
end OAI
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