| import Mathlib |
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| namespace OAI |
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| /-! Periodic classical solutions and material trajectories. -/ |
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| noncomputable section |
| open scoped ContDiff |
| namespace PrefixFlows |
| 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 := ![t, x 0, x 1, x 2] |
| def position (z : Spacetime) : Space := ![z 1, z 2, z 3] |
| def future : Set Spacetime := {z | 0 ≤ z 0} |
| def cylinder (T : ℝ) : Set Spacetime := {z | 0 ≤ z 0 ∧ z 0 ≤ T} |
| def unitCube : Set Space := Set.pi Set.univ (fun _ => Set.Icc (0 : ℝ) 1) |
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| def axis (i : Fin 3) : Space := Pi.single i 1 |
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| def SpacePeriodic {E : Type*} (f : Spacetime → E) : Prop := |
| ∀ t, 0 ≤ t → ∀ (x : Space) (k : Fin 3 → ℤ), |
| f (point t (x + fun i => (k i : ℝ))) = f (point t x) |
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| noncomputable def spatialMean {E : Type*} [NormedAddCommGroup E] |
| [NormedSpace ℝ E] [CompleteSpace E] (f : Spacetime → E) (t : ℝ) : E := |
| ∫ x in unitCube, f (point t x) |
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| def MeanZero {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [CompleteSpace E] (f : Spacetime → E) : Prop := |
| ∀ t, 0 ≤ t → spatialMean f t = 0 |
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| |
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| structure ClassicalJet where |
| dt : Field |
| dx : Fin 3 → Field |
| dxx : Fin 3 → Fin 3 → Field |
| dp : Fin 3 → Pressure |
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| |
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| def ClassicalRegularity (v : Field) (p : Pressure) (J : ClassicalJet) : Prop := |
| (∀ t, 0 ≤ t → ∀ x, |
| HasDerivWithinAt (fun s => v (point s x)) (J.dt (point t x)) (Set.Ici 0) t) ∧ |
| (∀ t, 0 ≤ t → ∀ x i, |
| HasDerivAt (fun (s : ℝ) => v (point t (x + s • axis i))) (J.dx i (point t x)) 0) ∧ |
| (∀ t, 0 ≤ t → ∀ x i j, |
| HasDerivAt (fun (s : ℝ) => J.dx i (point t (x + s • axis j))) |
| (J.dxx i j (point t x)) 0) ∧ |
| (∀ t, 0 ≤ t → ∀ x i, |
| HasDerivAt (fun (s : ℝ) => p (point t (x + s • axis i))) (J.dp i (point t x)) 0) ∧ |
| (∀ T, 0 ≤ T → |
| ContinuousOn v (cylinder T) ∧ ContinuousOn p (cylinder T) ∧ |
| ContinuousOn J.dt (cylinder T) ∧ |
| (∀ i, ContinuousOn (J.dx i) (cylinder T)) ∧ |
| (∀ i j, ContinuousOn (J.dxx i j) (cylinder T)) ∧ |
| (∀ i, ContinuousOn (J.dp i) (cylinder T))) |
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| |
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| def ClassicalSolution (ν : ℝ) (f v : Field) (p : Pressure) : Prop := |
| SpacePeriodic v ∧ SpacePeriodic p ∧ MeanZero p ∧ |
| (∀ x, v (point 0 x) = 0) ∧ |
| ∃ J : ClassicalJet, ClassicalRegularity v p J ∧ |
| (∀ t, 0 ≤ t → ∀ x, ∑ i : Fin 3, J.dx i (point t x) i = 0) ∧ |
| (∀ t, 0 ≤ t → ∀ x, |
| J.dt (point t x) + |
| (∑ i : Fin 3, v (point t x) i • J.dx i (point t x)) - |
| ν • (∑ i : Fin 3, J.dxx i i (point t x)) + |
| (fun i => J.dp i (point t x)) = f (point t x)) |
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| def GloballySmooth {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (f : Spacetime → E) : Prop := ContDiffOn ℝ ∞ f future |
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| noncomputable def kineticEnergy (v : Field) (t : ℝ) : ℝ := |
| (1 / 2 : ℝ) * ∫ x in unitCube, ∑ i : Fin 3, (v (point t x) i) ^ 2 |
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| def BoundedEnergy (v : Field) : Prop := |
| ∃ C : ℝ, 0 ≤ C ∧ ∀ t, 0 ≤ t → kineticEnergy v t ≤ C |
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| def initialLabel : Space := ![1 / 8, 1 / 4, 1 / 4] |
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| def MaterialTrajectory (u : Field) (X : ℝ → Space) : Prop := |
| X 0 = initialLabel ∧ ContinuousOn X (Set.Ici 0) ∧ |
| ∀ t, 0 ≤ t → HasDerivWithinAt X (u (point t (X t))) (Set.Ici 0) t |
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| |
| def observed (x : Space) : Prop := |
| ∃ k : ℤ, (1 / 2 : ℝ) < x 0 - k ∧ x 0 - k < 1 |
| end PrefixFlows |
| end |
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| end OAI |
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