openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Flow /FlowNonautonomous.lean
| 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 | |