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namespace OAI
/-! Periodic incompressible fields, classical solutions and bounded smooth forcing. -/
namespace Solenoidal
namespace Fluid
open scoped ContDiff
abbrev Space := Fin 3 → ℝ
abbrev SpaceTime := Fin 4 → ℝ
abbrev Field := SpaceTime → Space
abbrev Pressure := SpaceTime → ℝ
def point (t : ℝ) (x : Space) : SpaceTime := Fin.cons t x
def translateTime (h : ℝ) (p : SpaceTime) : SpaceTime :=
Function.update p 0 (p 0 + h)
def translateSpace (k : Fin 3 → ℤ) (p : SpaceTime) : SpaceTime :=
point (p 0) (fun i => p i.succ + (k i : ℝ))
def SpatiallyPeriodic {E : Type*} (F : SpaceTime → E) : Prop :=
∀ (p : SpaceTime) (k : Fin 3 → ℤ), F (translateSpace k p) = F p
noncomputable def partialD (i : Fin 4) (F : Field) : Field :=
fun p => fderiv ℝ F p (Pi.single i 1)
noncomputable def mixedD : List (Fin 4) → Field → Field
| [], F => F
| i :: l, F => partialD i (mixedD l F)
noncomputable def diverge (F : Field) (p : SpaceTime) : ℝ :=
∑ i : Fin 3, partialD i.succ F p i
noncomputable def laplacian (F : Field) (p : SpaceTime) : Space :=
∑ i : Fin 3, partialD i.succ (partialD i.succ F) p
noncomputable def advection (F : Field) (p : SpaceTime) : Space :=
∑ i : Fin 3, (F p i) • partialD i.succ F p
noncomputable def force (ν : ℝ) (U : Field) : Field :=
fun p => partialD 0 U p - ν • laplacian U p
def cell : Set Space := Set.Icc 0 (fun _ => 1)
noncomputable def spatialMean (F : Field) (t : ℝ) : Space :=
∫ x in cell, F (point t x)
noncomputable def pressureMean (p : Pressure) (t : ℝ) : ℝ :=
∫ x in cell, p (point t x)
noncomputable def energy (U : Field) (t : ℝ) : ℝ :=
(1 / 2 : ℝ) * ∫ x in cell, ∑ i : Fin 3, (U (point t x) i) ^ 2
def BoundedMixed (F : Field) : Prop :=
∀ a : List (Fin 4), ∃ C : ℝ, 0 ≤ C ∧
∀ p : SpaceTime, 0 ≤ p 0 → ‖mixedD a F p‖ ≤ C
def PeriodicAfterOne (F : Field) : Prop :=
∀ p : SpaceTime, 1 ≤ p 0 → F (translateTime 1 p) = F p
def ZeroNearIntegers (U f : Field) : Prop :=
∀ n : ℤ, ∃ ε : ℝ, 0 < ε ∧ ∀ p : SpaceTime,
0 ≤ p 0 → |p 0 - (n : ℝ)| < ε → U p = 0 ∧ f p = 0
structure ClassicalSolutionOn (ν : ℝ) (f v : Field) (p : Pressure) (T : ℝ) : Prop where
periodic_v : ∀ t ∈ Set.Icc 0 T, ∀ (x : Space) (k : Fin 3 → ℤ),
v (point t (fun i => x i + (k i : ℝ))) = v (point t x)
periodic_p : ∀ t ∈ Set.Icc 0 T, ∀ (x : Space) (k : Fin 3 → ℤ),
p (point t (fun i => x i + (k i : ℝ))) = p (point t x)
continuous_v : ContinuousOn (fun tx : ℝ × Space => v (point tx.1 tx.2))
(Set.Icc 0 T ×ˢ Set.univ)
continuous_p : ContinuousOn (fun tx : ℝ × Space => p (point tx.1 tx.2))
(Set.Icc 0 T ×ˢ Set.univ)
zero_initial : ∀ x : Space, v (point 0 x) = 0
zero_pressure_mean : ∀ t ∈ Set.Icc 0 T, pressureMean p t = 0
derivatives : ∃ (vTime : Field) (vSpace : Fin 3 → Field)
(vSpace2 : Fin 3 → Fin 3 → Field) (pSpace : Fin 3 → Pressure),
(ContinuousOn (fun tx : ℝ × Space => vTime (point tx.1 tx.2))
(Set.Icc 0 T ×ˢ Set.univ)) ∧
(∀ i, ContinuousOn (fun tx : ℝ × Space => vSpace i (point tx.1 tx.2))
(Set.Icc 0 T ×ˢ Set.univ)) ∧
(∀ i j, ContinuousOn (fun tx : ℝ × Space => vSpace2 i j (point tx.1 tx.2))
(Set.Icc 0 T ×ˢ Set.univ)) ∧
(∀ i, ContinuousOn (fun tx : ℝ × Space => pSpace i (point tx.1 tx.2))
(Set.Icc 0 T ×ˢ Set.univ)) ∧
(∀ t ∈ Set.Icc 0 T, ∀ x : Space,
HasDerivWithinAt (fun s => v (point s x)) (vTime (point t x)) (Set.Icc 0 T) t) ∧
(∀ i, ∀ t ∈ Set.Icc 0 T, ∀ x : Space,
HasDerivAt (fun s : ℝ => v (point t (x + s • Pi.single i 1)))
(vSpace i (point t x)) 0) ∧
(∀ i j, ∀ t ∈ Set.Icc 0 T, ∀ x : Space,
HasDerivAt (fun s : ℝ => vSpace i (point t (x + s • Pi.single j 1)))
(vSpace2 i j (point t x)) 0) ∧
(∀ i, ∀ t ∈ Set.Icc 0 T, ∀ x : Space,
HasDerivAt (fun s : ℝ => p (point t (x + s • Pi.single i 1)))
(pSpace i (point t x)) 0) ∧
(∀ t ∈ Set.Icc 0 T, ∀ x : Space, ∑ i : Fin 3, vSpace i (point t x) i = 0) ∧
(∀ t ∈ Set.Icc 0 T, ∀ x : Space,
vTime (point t x) + ∑ i : Fin 3, v (point t x) i • vSpace i (point t x) =
-(fun i => pSpace i (point t x)) +
ν • (∑ i : Fin 3, vSpace2 i i (point t x)) + f (point t x))
def FluidAssertions (ν : ℝ) (U f : Field) : Prop :=
ContDiff ℝ ∞ U ∧ ContDiff ℝ ∞ f ∧ SpatiallyPeriodic U ∧ SpatiallyPeriodic f ∧
(∀ q : SpaceTime, 0 ≤ q 0 → diverge f q = 0) ∧
(∀ t : ℝ, 0 ≤ t → spatialMean f t = 0) ∧
(∀ q : SpaceTime, 0 ≤ q 0 →
partialD 0 U q + advection U q = ν • laplacian U q + f q) ∧
(∀ q : SpaceTime, 0 ≤ q 0 → diverge U q = 0) ∧
(∀ x : Space, U (point 0 x) = 0) ∧
(∀ T : ℝ, 0 < T → ∀ (v : Field) (p : Pressure), ClassicalSolutionOn ν f v p T →
∀ t ∈ Set.Icc 0 T, ∀ x : Space, v (point t x) = U (point t x) ∧ p (point t x) = 0)
def BoundedPeriodicAssertions (U f : Field) : Prop :=
BoundedMixed U ∧ BoundedMixed f ∧
(∃ C : ℝ, 0 ≤ C ∧ ∀ t : ℝ, 0 ≤ t → energy U t ≤ C) ∧
PeriodicAfterOne U ∧ PeriodicAfterOne f ∧ ZeroNearIntegers U f
end Fluid
end Solenoidal
end OAI
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