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