| 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 | |