| import OAI.MathematicalPhysics.Transonic.Profile.FiniteRadialProfile | |
| import OAI.MathematicalPhysics.Transonic.Shooting.MatchedExistence | |
| namespace OAI | |
| noncomputable section | |
| namespace SepticProfile.SonicShooting | |
| open Set Filter SourceFamily | |
| open scoped Topology ContDiff | |
| theorem exists_finite_source_profile : ∃ (beta R : ℝ) (g : ℝ → ℝ), | |
| 1<beta ∧ beta*(ell+Real.sqrt ell)<3 ∧ | |
| 0<sonicRadius beta ∧ sonicRadius beta<R ∧ R<1 ∧ | |
| ContDiffOn ℝ ∞ g (Icc 0 (R^2)) ∧ | |
| let v : ℝ → ℝ := fun y => y*g (y^2) | |
| (∀ y ∈ Icc (0:ℝ) R, |v y|<1) ∧ | |
| (∀ y ∈ Icc (0:ℝ) R, profileDenom ell y (v y)*deriv v y=profileNumer ell beta y (v y)) ∧ | |
| (∀ y ∈ Ioc (0:ℝ) R, v y<y ∧ y*v y<1) ∧ | |
| (∀ y ∈ Icc (0:ℝ) R, velocityToU y (v y)=sonicSpeed ↔ y=sonicRadius beta) ∧ | |
| 0<deriv (fun y => velocityToU y (v y)) (sonicRadius beta) := by | |
| obtain ⟨M⟩ := exists_matched_pair | |
| obtain ⟨W⟩ := M.exists_physicalWidth | |
| refine ⟨M.beta,W.endpoint,M.radial,M.beta_bounds.1,M.beta_bounds.2,M.radius_bounds.1, | |
| W.endpoint_bounds.1,W.endpoint_bounds.2,fun x hx => (W.radial_smooth hx).contDiffWithinAt,?_⟩ | |
| let v : ℝ → ℝ := fun y => y*M.radial (y^2) | |
| have hEq (y : ℝ) (hy : 0≤y) : v y=M.velocity y := (M.velocity_radial hy).symm | |
| have hNear (y : ℝ) (hy : 0<y) : v =ᶠ[𝓝 y] M.velocity := by | |
| filter_upwards [Ioi_mem_nhds hy] with t ht | |
| exact hEq t ht.le | |
| refine ⟨fun y hy => ?_,fun y hy => ?_,fun y hy => ?_,fun y hy => ?_,?_⟩ | |
| · change |v y|<1 | |
| rw [hEq y hy.1] | |
| exact W.velocity_abs hy | |
| · change profileDenom ell y (v y)*deriv v y=profileNumer ell M.beta y (v y) | |
| rw [hEq y hy.1] | |
| by_cases hz : y=0 | |
| · subst y | |
| rw [MatchedPair.velocity,M.sonicU_zero] | |
| norm_num [velocityToU,profileDenom,profileNumer] | |
| · rw [(hNear y (lt_of_le_of_ne hy.1 (Ne.symm hz))).deriv_eq] | |
| exact W.velocity_equation hy | |
| · change v y<y ∧ y*v y<1 | |
| rw [hEq y hy.1.le] | |
| exact W.velocity_below hy | |
| · change velocityToU y (v y)=sonicSpeed ↔ y=M.radius | |
| rw [hEq y hy.1] | |
| exact W.unique_sonic hy | |
| · change 0<deriv (fun y => velocityToU y (v y)) M.radius | |
| have hn : (fun y => velocityToU y (v y)) =ᶠ[𝓝 M.radius] (fun y => velocityToU y (M.velocity y)) := by | |
| filter_upwards [hNear M.radius M.radius_bounds.1] with y hy | |
| rw [hy] | |
| rw [hn.deriv_eq] | |
| exact W.sonic_slope | |
| end SepticProfile.SonicShooting | |
| end | |
| end OAI | |