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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 | import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.StationaryBounds
namespace OAI
/-! A common compact spatial support controls all derivatives, including time
jets. Periodicity is used only to reduce the time variable to one closed cell. -/
noncomputable section
namespace ForcedComputation
open ShearFlows Set Filter
open scoped ContDiff Topology
theorem mixedDerivative_common_support {V : Velocity} {K : Set Space}
(hK : IsClosed K) (hV : ∀ t, tsupport (fun x => V (t, x)) ⊆ K)
(α : List (Fin 4)) (t : ℝ) :
tsupport (fun x => mixedDerivative V α (t, x)) ⊆ K := by
apply closure_minimal _ hK
intro x hx
by_contra hn
have ho : {y : SpaceTime | y.2 ∉ K} ∈ 𝓝 (t, x) :=
(hK.isOpen_compl.preimage continuous_snd).mem_nhds hn
have he : V =ᶠ[𝓝 (t, x)] fun _ => 0 := by
filter_upwards [ho] with y hy
exact image_eq_zero_of_notMem_tsupport (f := fun x : Space => V (y.1, x))
(fun hm => hy (hV y.1 hm))
have hz := (mixedDerivative_eventuallyEq he α).self_of_nhds
rw [mixedDerivative_zero] at hz
exact hx hz
theorem boundedMixed_of_compact_periodic {V : Velocity} {K : Set Space}
(hV : ContDiff ℝ ∞ V) (hK : IsCompact K)
(hs : ∀ t, tsupport (fun x => V (t, x)) ⊆ K) (hp : TimePeriodic V) :
BoundedMixedDerivatives V := by
intro α
obtain ⟨C, hC⟩ := (isCompact_Icc.prod hK).bddAbove_image
(mixedDerivative_smooth hV α).continuous.norm.continuousOn
refine ⟨max C 0, le_max_right _ _, ?_⟩
rintro ⟨t, x⟩
by_cases hx : x ∈ K
· have ht : Function.Periodic (fun s => mixedDerivative V α (s, x)) 1 :=
fun s => mixedDerivative_time_periodic hV hp α s x
have he : mixedDerivative V α (Int.fract t, x) = mixedDerivative V α (t, x) := by
simpa only [Int.fract, mul_one] using ht.sub_int_mul_eq (x := t) ⌊t⌋
rw [← he]
exact (hC (mem_image_of_mem _
(show (Int.fract t, x) ∈ Icc (0 : ℝ) 1 ×ˢ K from
⟨⟨Int.fract_nonneg t, (Int.fract_lt_one t).le⟩, hx⟩))).trans (le_max_left _ _)
· have hz : mixedDerivative V α (t, x) = 0 :=
image_eq_zero_of_notMem_tsupport (f := fun x : Space => mixedDerivative V α (t, x))
(fun hm => hx (mixedDerivative_common_support hK.isClosed hs α t hm))
rw [hz, norm_zero]
exact le_max_right _ _
theorem boundedMixed_of_compact_tail {F G : Velocity} {K : Set Space}
(hF : ContDiff ℝ ∞ F) (hG : ContDiff ℝ ∞ G) (hK : IsCompact K)
(hsF : ∀ t, tsupport (fun x => F (t, x)) ⊆ K)
(hsG : ∀ t, tsupport (fun x => G (t, x)) ⊆ K) (hpG : TimePeriodic G)
(hleft : ∀ t, t < 0 → ∀ x, F (t, x) = 0)
(hright : ∀ t, 1 < t → ∀ x, F (t, x) = G (t, x)) :
BoundedMixedDerivatives F := by
intro α
obtain ⟨C, hC⟩ := (isCompact_Icc.prod hK).bddAbove_image
(mixedDerivative_smooth hF α).continuous.norm.continuousOn
obtain ⟨D, hD, hbD⟩ := boundedMixed_of_compact_periodic hG hK hsG hpG α
refine ⟨max C D, hD.trans (le_max_right _ _), ?_⟩
rintro ⟨t, x⟩
by_cases ht0 : t < 0
· have he : F =ᶠ[𝓝 (t, x)] fun _ => 0 := by
filter_upwards [(continuous_fst.tendsto (t, x)).eventually (eventually_lt_nhds ht0)] with y hy
exact hleft y.1 hy y.2
rw [(mixedDerivative_eventuallyEq he α).self_of_nhds, mixedDerivative_zero, norm_zero]
exact hD.trans (le_max_right _ _)
by_cases ht1 : 1 < t
· have he : F =ᶠ[𝓝 (t, x)] G := by
filter_upwards [(continuous_fst.tendsto (t, x)).eventually (eventually_gt_nhds ht1)] with y hy
exact hright y.1 hy y.2
rw [(mixedDerivative_eventuallyEq he α).self_of_nhds]
exact (hbD (t, x)).trans (le_max_right _ _)
by_cases hx : x ∈ K
· exact (hC (mem_image_of_mem _
(show (t, x) ∈ Icc (0 : ℝ) 1 ×ˢ K from
⟨⟨le_of_not_gt ht0, le_of_not_gt ht1⟩, hx⟩))).trans (le_max_left _ _)
· have hz : mixedDerivative F α (t, x) = 0 :=
image_eq_zero_of_notMem_tsupport (f := fun x : Space => mixedDerivative F α (t, x))
(fun hm => hx (mixedDerivative_common_support hK.isClosed hsF α t hm))
rw [hz, norm_zero]
exact hD.trans (le_max_right _ _)
end ForcedComputation
end
end OAI
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