| import Mathlib |
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| namespace OAI |
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| /-! Periodic incompressible fields, classical solutions and bounded smooth forcing. -/ |
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| namespace Solenoidal |
| namespace Fluid |
| open scoped ContDiff |
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| abbrev Space := Fin 3 → ℝ |
| abbrev SpaceTime := Fin 4 → ℝ |
| abbrev Field := SpaceTime → Space |
| abbrev Pressure := SpaceTime → ℝ |
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| def point (t : ℝ) (x : Space) : SpaceTime := Fin.cons t x |
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| def translateTime (h : ℝ) (p : SpaceTime) : SpaceTime := |
| Function.update p 0 (p 0 + h) |
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| def translateSpace (k : Fin 3 → ℤ) (p : SpaceTime) : SpaceTime := |
| point (p 0) (fun i => p i.succ + (k i : ℝ)) |
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| def SpatiallyPeriodic {E : Type*} (F : SpaceTime → E) : Prop := |
| ∀ (p : SpaceTime) (k : Fin 3 → ℤ), F (translateSpace k p) = F p |
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| noncomputable def partialD (i : Fin 4) (F : Field) : Field := |
| fun p => fderiv ℝ F p (Pi.single i 1) |
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| noncomputable def mixedD : List (Fin 4) → Field → Field |
| | [], F => F |
| | i :: l, F => partialD i (mixedD l F) |
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| noncomputable def diverge (F : Field) (p : SpaceTime) : ℝ := |
| ∑ i : Fin 3, partialD i.succ F p i |
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| noncomputable def laplacian (F : Field) (p : SpaceTime) : Space := |
| ∑ i : Fin 3, partialD i.succ (partialD i.succ F) p |
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| noncomputable def advection (F : Field) (p : SpaceTime) : Space := |
| ∑ i : Fin 3, (F p i) • partialD i.succ F p |
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| |
| noncomputable def force (ν : ℝ) (U : Field) : Field := |
| fun p => partialD 0 U p - ν • laplacian U p |
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| |
| def cell : Set Space := Set.Icc 0 (fun _ => 1) |
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| noncomputable def spatialMean (F : Field) (t : ℝ) : Space := |
| ∫ x in cell, F (point t x) |
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| noncomputable def pressureMean (p : Pressure) (t : ℝ) : ℝ := |
| ∫ x in cell, p (point t x) |
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| noncomputable def energy (U : Field) (t : ℝ) : ℝ := |
| (1 / 2 : ℝ) * ∫ x in cell, ∑ i : Fin 3, (U (point t x) i) ^ 2 |
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| |
| def BoundedMixed (F : Field) : Prop := |
| ∀ a : List (Fin 4), ∃ C : ℝ, 0 ≤ C ∧ |
| ∀ p : SpaceTime, 0 ≤ p 0 → ‖mixedD a F p‖ ≤ C |
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| def PeriodicAfterOne (F : Field) : Prop := |
| ∀ p : SpaceTime, 1 ≤ p 0 → F (translateTime 1 p) = F p |
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| 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 |
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| 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)) |
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| 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) |
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| 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 |
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| end OAI |
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