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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 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 | import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.BalancedPath
import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.BalancedEuclidean
namespace OAI
/-! The compact balanced application: a machine-only repeated field, a
finite initial loader, and the fixed open-cube material observation. -/
noncomputable section
namespace ForcedComputation.Balanced
open ShearFlows Set Filter Recorder.Planar RapidForcing.CompactEmbedding
open scoped ContDiff Topology
def velocity (I : Alternating.MachineInput) (hI : Alternating.ValidInput I) : Velocity :=
BalancedVelocity.velocity I hI
def force (I : Alternating.MachineInput) (hI : Alternating.ValidInput I) (ν : ℝ) : Velocity :=
residual ν (velocity I hI)
def initialLabel : AlternatingNS.Space := piCoordinates.symm ![4,0,0]
def observer : Set AlternatingNS.Space := {x | ∀ j : Fin 3, -1 < x j ∧ x j < 2}
theorem force_tail (I : Alternating.MachineInput) (hI : Alternating.ValidInput I)
(ν : ℝ) {t : ℝ} (ht : 1 ≤ t) (x : Space) :
force I hI ν (t,x) = residual ν (BalancedBody.velocity I.1 hI.1) (t,x) := by
have he : EqOn (fun s => force I hI ν (s,x))
(fun s => residual ν (BalancedBody.velocity I.1 hI.1) (s,x)) (Ioi 1) := by
intro s hs
apply residual_congr_germ
filter_upwards [(continuous_fst.tendsto (s,x)).eventually
(eventually_gt_nhds hs)] with y hy
exact BalancedVelocity.tail I hI hy.le y.2
have h₁ : Continuous (fun s => force I hI ν (s,x)) :=
(residual_smooth (BalancedVelocity.smooth I hI) ν).continuous.comp
(continuous_id.prodMk continuous_const)
have h₂ : Continuous (fun s => residual ν (BalancedBody.velocity I.1 hI.1) (s,x)) :=
(residual_smooth (BalancedBody.velocity_smooth I.1 hI.1) ν).continuous.comp
(continuous_id.prodMk continuous_const)
exact he.closure h₁ h₂ (by
rw [closure_Ioi]
change 1 ≤ t
exact ht)
theorem force_tail_periodic (I : Alternating.MachineInput) (hI : Alternating.ValidInput I)
(ν : ℝ) {t : ℝ} (ht : 1 ≤ t) (x : Space) :
force I hI ν (t+1,x) = force I hI ν (t,x) := by
rw [force_tail I hI ν (by linarith), force_tail I hI ν ht]
exact BalancedVelocity.residual_periodic
(BalancedBody.velocity_smooth I.1 hI.1)
(BalancedBody.velocity_periodic I.1 hI.1) ν t x
/-- Every clause refers to the named prescribed force and its actual
Euclidean Navier--Stokes solution. -/
def FluidProperties (I : Alternating.MachineInput) (hI : Alternating.ValidInput I)
(ν : ℝ) : Prop :=
let V := velocity I hI
let f := force I hI ν
let U := CompactEuclidean.velocity V
let F := CompactEuclidean.velocity f
ContDiff ℝ ∞ V ∧ ContDiff ℝ ∞ f ∧
BoundedMixedDerivatives V ∧ BoundedMixedDerivatives f ∧
(∀ t, tsupport (fun x => V (t,x)) ⊆ BalancedVelocity.support ∧
tsupport (fun x => f (t,x)) ⊆ BalancedVelocity.support) ∧
(∀ t, 1 ≤ t → ∀ x, V (t,x) = BalancedBody.velocity I.1 hI.1 (t,x) ∧
V (t+1,x) = V (t,x) ∧
f (t,x) = residual ν (BalancedBody.velocity I.1 hI.1) (t,x) ∧
f (t+1,x) = f (t,x)) ∧
AlternatingNS.NavierStokes ν F U (fun _ _ => 0) ∧
AlternatingNS.EnergyClass U (fun _ _ => 0) ∧
(∀ v p, CompactEuclidean.IsEnergySolutionModuloConstants ν F v p →
∃ c : ℝ → ℝ, ∀ t, 0 ≤ t → ∀ x, v t x = U t x ∧ p t x = c t) ∧
∃ X : AlternatingNS.Space → ℝ → AlternatingNS.Space,
(∀ a, AlternatingNS.IsTrajectory U a (X a)) ∧
(∀ a γ, AlternatingNS.IsTrajectory U a γ → ∀ t, 0 ≤ t → γ t = X a t) ∧
((∃ t, 0 ≤ t ∧ X initialLabel t ∈ observer) ↔ Alternating.Halts I)
theorem fluid_properties (I : Alternating.MachineInput) (hI : Alternating.ValidInput I)
{ν : ℝ} (hν : 0 < ν) : FluidProperties I hI ν := by
have hs := BalancedVelocity.smooth I hI
have hsupport (t : ℝ) : Function.support (fun x => velocity I hI (t,x)) ⊆
BalancedVelocity.support :=
(subset_tsupport _).trans (BalancedVelocity.supported I hI t)
obtain ⟨hsol,hE,huniq⟩ := CompactEuclidean.compact_prescribed_solution hν hs
BalancedVelocity.support_compact hsupport
(fun x => BalancedVelocity.zero_before I hI (by norm_num) x)
(BalancedVelocity.solenoidal I hI)
obtain ⟨X,hX,hXuniq⟩ := CompactEuclidean.velocity_global_material_flow hs
BalancedVelocity.support_compact hsupport
refine ⟨hs, residual_smooth hs ν, BalancedVelocity.bounded I hI,
BalancedVelocity.force_bounded I hI ν,
fun t => ⟨BalancedVelocity.supported I hI t, BalancedVelocity.force_supported I hI ν t⟩,
fun t ht x => ⟨BalancedVelocity.tail I hI ht x,
by
change BalancedVelocity.velocity I hI (t+1,x) = BalancedVelocity.velocity I hI (t,x)
rw [BalancedVelocity.tail I hI (by linarith), BalancedVelocity.tail I hI ht,
BalancedBody.velocity_periodic],
force_tail I hI ν ht x, force_tail_periodic I hI ν ht x⟩,
hsol,hE,huniq,X,hX,hXuniq,?_⟩
obtain ⟨Ψ,hΨ⟩ := compactVelocity_transitions (BalancedPlanar.referenceInput I.1)
(BalancedPlanar.referenceValid hI.1)
obtain ⟨Ω,hΩ⟩ := exists_planarTransition
(normalizedHamiltonian_valid (BalancedPlanar.referenceInput I.1)
(BalancedPlanar.referenceValid hI.1))
(normalizedHamiltonian_noTime (BalancedPlanar.referenceInput I.1)
(BalancedPlanar.referenceValid hI.1))
let γ : ℝ → AlternatingNS.Space := fun t => piCoordinates.symm (BalancedPath.path I hI Ψ t)
have hγ : AlternatingNS.IsTrajectory (CompactEuclidean.velocity (velocity I hI))
initialLabel γ := by
refine ⟨congrArg piCoordinates.symm (BalancedPath.path_initial I hI Ψ),?_⟩
intro t _
let : AddCommGroup AlternatingNS.Space := WithLp.instAddCommGroup 2 (Fin 3 → ℝ)
let : Module ℝ AlternatingNS.Space := WithLp.instModule 2 ℝ (Fin 3 → ℝ)
let : TopologicalSpace AlternatingNS.Space := PiLp.topologicalSpace 2 (fun _ : Fin 3 => ℝ)
let lift : Space →L[ℝ] AlternatingNS.Space :=
{ toFun := fun x => WithLp.toLp 2 x
map_add' := fun _ _ => rfl
map_smul' := fun _ _ => rfl
cont := PiLp.continuous_toLp 2 (fun _ : Fin 3 => ℝ) }
have hd := lift.hasFDerivAt.comp_hasDerivAt t (BalancedPath.path_deriv I hI hΨ t)
convert hd.hasDerivWithinAt (s := Ici 0) using 1 <;>
simp only [γ, lift, Function.comp_def, CompactEuclidean.velocity, velocity,
piCoordinates, ContinuousLinearEquiv.apply_symm_apply] <;> rfl
have he (t : ℝ) (ht : 0 ≤ t) : γ t = X initialLabel t := hXuniq _ _ hγ t ht
have hob := BalancedPath.path_observation I hI hΨ hΩ
constructor
· rintro ⟨t,ht,hobs⟩
apply hob.mp
refine ⟨t,ht,?_⟩
intro j
have hj := hobs j
rw [← he t ht] at hj
change -1 < piCoordinates.symm (BalancedPath.path I hI Ψ t) j ∧
piCoordinates.symm (BalancedPath.path I hI Ψ t) j < 2 at hj
exact hj
· intro hh
obtain ⟨t,ht,hobs⟩ := hob.mpr hh
refine ⟨t,ht,?_⟩
intro j
rw [← he t ht]
exact hobs j
end ForcedComputation.Balanced
end
end OAI
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