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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 | import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Flow.CompactPlanarFlow
import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Flow.PlanarReverseVariations
namespace OAI
/-! Quantitative estimates for the compact planar extension, assuming the
variational equations for the actual periodic transition map. -/
noncomputable section
namespace ForcedComputation.Recorder.Planar
open ShearFlows PlanarHamiltonian Set Filter
open scoped Topology
theorem compactTransition_germ (I : Alternating.MachineInput)
(hI : Alternating.ValidInput I) {Ψ Ω : ℝ → ℝ → Plane → Plane}
(hΨ : IsPlanarTransition (compactVelocity I hI) Ψ)
(hΩ : IsPlanarTransition (planarSlice (normalizedHamiltonian I hI)) Ω)
{x : Plane} (hx : ∀ j : Fin 2, 0 < x j ∧ x j < 1) (a t : ℝ) :
Ψ a t =ᶠ[𝓝 x] Ω a t := by
have ho : IsOpen {y : Plane | ∀ j : Fin 2, 0 < y j ∧ y j < 1} := by
have he : {y : Plane | ∀ j : Fin 2, 0 < y j ∧ y j < 1} =
⋂ j : Fin 2, (fun y : Plane => y j) ⁻¹' Ioo 0 1 := by
ext y
simp only [mem_ofPred_eq, mem_iInter, mem_preimage, mem_Ioo]
rw [he]
exact isOpen_iInter_of_finite (fun j => isOpen_Ioo.preimage (continuous_apply j))
filter_upwards [ho.mem_nhds hx] with y hy
exact compactTransition_eq_slice I hI hΨ hΩ hy a t
theorem compactTransition_germ_id (I : Alternating.MachineInput)
(hI : Alternating.ValidInput I) {Ψ : ℝ → ℝ → Plane → Plane}
(hΨ : IsPlanarTransition (compactVelocity I hI) Ψ)
{x : Plane} (hx : x ∉ commonSupport (normalizedPulse I hI)) (a t : ℝ) :
Ψ a t =ᶠ[𝓝 x] id := by
have ho := (commonSupport_compact (normalizedPulse I hI)).isClosed.isOpen_compl
filter_upwards [ho.mem_nhds hx] with y hy
exact compactTransition_fixed I hI hΨ hy a t
theorem compactTransition_derivative_bounds (I : Alternating.MachineInput)
(hI : Alternating.ValidInput I) {Ψ Ω : ℝ → ℝ → Plane → Plane}
(hΨ : IsPlanarTransition (compactVelocity I hI) Ψ)
(hΩ : IsPlanarTransition (planarSlice (normalizedHamiltonian I hI)) Ω)
(hv : PlanarVariations (planarSlice (normalizedHamiltonian I hI)) Ω)
(a : ℝ) (x : Plane) {t : ℝ} (ht : 0 ≤ t) :
‖fderiv ℝ (Ψ a t) x‖ ≤ Real.exp ((planarFlowBound (normalizedHamiltonian I hI) : ℝ) * t) ∧
‖fderiv ℝ (fderiv ℝ (Ψ a t)) x‖ ≤
Real.exp (2 * (planarFlowBound (normalizedHamiltonian I hI) : ℝ) * t) -
Real.exp ((planarFlowBound (normalizedHamiltonian I hI) : ℝ) * t) := by
by_cases hx : x ∈ commonSupport (normalizedPulse I hI)
· have hg := compactTransition_germ I hI hΨ hΩ
(commonSupport_in_unit (normalizedPulse_inUnit I hI) hx) a t
rw [hg.fderiv.self_of_nhds, hg.fderiv.fderiv.self_of_nhds]
exact planar_quantitative_variations (normalizedHamiltonian_valid I hI)
(normalizedHamiltonian_noTime I hI) hv a x ht
· have hg := compactTransition_germ_id I hI hΨ hx a t
have hi : fderiv ℝ (id : Plane → Plane) = fun _ => ContinuousLinearMap.id ℝ Plane :=
funext fun _ => fderiv_id
have hz : fderiv ℝ (fderiv ℝ (id : Plane → Plane)) x = 0 := by
rw [hi]
exact fderiv_const_apply (ContinuousLinearMap.id ℝ Plane)
have hL : (0 : ℝ) ≤ planarFlowBound (normalizedHamiltonian I hI) := Nat.cast_nonneg _
constructor
· rw [hg.fderiv.self_of_nhds, fderiv_id]
exact (ContinuousLinearMap.norm_id_le).trans (Real.one_le_exp (mul_nonneg hL ht))
· rw [hg.fderiv.fderiv.self_of_nhds, hz]
have hn : ‖(0 : Plane →L[ℝ] (Plane →L[ℝ] Plane))‖ = 0 :=
ContinuousLinearMap.opNorm_zero
rw [hn]
apply sub_nonneg.mpr
apply Real.exp_le_exp.mpr
nlinarith
end ForcedComputation.Recorder.Planar
end
end OAI
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