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caec230 | 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 | import OAI.MathematicalPhysics.Transonic.Phase.RadialFlux
namespace OAI
section
noncomputable section
namespace SepticProfile
open Set
open scoped ContDiff BigOperators
abbrev PhysicalFunction := ℝ → PhysicalSpace → ℝ
def partialTime (s : PhysicalFunction) (t : ℝ) (X : PhysicalSpace) : ℝ :=
deriv (fun q => s q X) t
def partialSpace (s : PhysicalFunction) (j : Fin 4) (t : ℝ) (X : PhysicalSpace) : ℝ :=
deriv (fun r => s t (coordinateLine X j r)) 0
def minkowskiPair (s f : PhysicalFunction) (t : ℝ) (X : PhysicalSpace) : ℝ :=
partialTime s t X*partialTime f t X-
∑ j : Fin 4, partialSpace s j t X*partialSpace f j t X
def phaseTransport (s : PhysicalFunction) (t : ℝ) (X : PhysicalSpace) : ℝ :=
partialTime (fun q Z => (minkowskiSquare s q Z)^((1:ℝ)/3)*partialTime s q Z) t X-
∑ j : Fin 4, partialSpace (fun q Z =>
(minkowskiSquare s q Z)^((1:ℝ)/3)*partialSpace s j q Z) j t X
def GlobalProfile.w0 (P : GlobalProfile) (t : ℝ) (X : PhysicalSpace) : ℝ :=
(minkowskiSquare P.s0 t X)^((1:ℝ)/6)
def GlobalProfile.linearizedPhase (P : GlobalProfile) (f : PhysicalFunction)
(t : ℝ) (X : PhysicalSpace) : ℝ :=
partialTime (fun q Z => (P.w0 q Z)^2*partialTime f q Z+
(2/3:ℝ)*(P.w0 q Z)^(-4:ℤ)*minkowskiPair P.s0 f q Z*partialTime P.s0 q Z) t X-
∑ j : Fin 4, partialSpace (fun q Z => (P.w0 q Z)^2*partialSpace f j q Z+
(2/3:ℝ)*(P.w0 q Z)^(-4:ℤ)*minkowskiPair P.s0 f q Z*partialSpace P.s0 j q Z) j t X
def InProfileSpace (a : ℝ) (f : PhysicalFunction) : Prop :=
∃ (N : ℕ) (F : ℕ → ℝ → ℝ),
(∀ j≤N, ContDiffOn ℝ ∞ (F j) (Ici 0)) ∧
∀ t : ℝ, 0<t → ∀ X : PhysicalSpace,
f t X=t^a*∑ j∈Finset.range (N+1), (Real.log t)^j*F j (radiusSq X/t^2)
def ProfileSurjectivity (P : GlobalProfile) : Prop :=
∀ (a : ℝ) (f : PhysicalFunction), InProfileSpace (a-2-2*P.beta/3) f →
∃ phi : PhysicalFunction, InProfileSpace a phi ∧
∀ t : ℝ, 0<t → ∀ X : PhysicalSpace, P.linearizedPhase phi t X=f t X
def FullProfileInputExists : Prop :=
∃ P : GlobalProfile, (∀ t : ℝ, 0<t → ∀ X : PhysicalSpace,
0 < minkowskiSquare P.s0 t X ∧ phaseTransport P.s0 t X=0) ∧ ProfileSurjectivity P
end SepticProfile
end
end
end OAI
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