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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 | import OAI.MathematicalPhysics.Transonic.Phase.Transport
import OAI.MathematicalPhysics.Transonic.Profile.GlobalExistence
namespace OAI
section
noncomputable section
namespace SepticProfile
open Set
open scoped ContDiff
/-- A positive smooth radial phase with timelike gradient and vanishing phase
transport, together with its positive septic amplitude, in four spatial dimensions. -/
theorem exists_global_phase_input :
∃ P : GlobalProfile,
(∀ y : ℝ, 0<P.H y) ∧
(∃ Hrad : ℝ → ℝ, ContDiffOn ℝ ∞ Hrad (Ici 0) ∧
∀ y : ℝ, P.H y=Hrad (y^2)) ∧
(∀ t : ℝ, 0<t → ∀ X : PhysicalSpace,
0 < minkowskiSquare P.s0 t X ∧ phaseTransport P.s0 t X=0 ∧
0<P.w0 t X) := by
obtain ⟨P⟩ := globalProfile_of_exists SonicShooting.exists_global_source_profile
exact ⟨P,P.H_pos,⟨P.radialH,P.radialH_smooth,P.H_radial⟩,
fun t ht X => ⟨P.s0_timelike ht X,P.s0_transport ht X,P.w0_pos ht X⟩⟩
end SepticProfile
end
end
end OAI
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