File size: 2,459 Bytes
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 58 59 60 61 62 63 | 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
|