| import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus |
| import Mathlib.Analysis.SpecialFunctions.ExpDeriv |
| import Mathlib.Analysis.Calculus.ContDiff.Deriv |
| import OAI.MathematicalPhysics.Transonic.Profile.Existence |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace SepticProfile |
| open Set Filter |
| open scoped ContDiff Topology |
|
|
| structure GlobalProfile where |
| beta : ℝ |
| g : ℝ → ℝ |
| beta_gt : 1 < beta |
| beta_lt : beta*(ell+Real.sqrt ell)<3 |
| smooth : ContDiffOn ℝ ∞ g (Ici 0) |
| range : ∀ y : ℝ, 0≤y → |y*g (y^2)|<1 |
| equation : ∀ y : ℝ, 0≤y → |
| profileDenom ell y (y*g (y^2))*deriv (fun q => q*g (q^2)) y= |
| profileNumer ell beta y (y*g (y^2)) |
| below : ∀ y : ℝ, 0<y → y*g (y^2)<y ∧ y*(y*g (y^2))<1 |
| radius_pos : 0<sonicRadius beta |
| radius_lt : sonicRadius beta<1 |
| sonic_unique : ∀ y : ℝ, 0≤y → |
| (velocityToU y (y*g (y^2))=sonicSpeed ↔ y=sonicRadius beta) |
| sonic_slope : 0<deriv (fun y => velocityToU y (y*g (y^2))) (sonicRadius beta) |
|
|
| lemma globalProfile_of_exists (h : RelativisticProfileExists) : Nonempty GlobalProfile := by |
| obtain ⟨β,g,hβ,hup,hg,hv,he,hb,hr,hr1,hu,hs⟩ := h |
| exact ⟨⟨β,g,hβ,hup,hg,hv,he,hb,hr,hr1,hu,hs⟩⟩ |
|
|
| def GlobalProfile.b (P : GlobalProfile) : ℝ := P.beta-1 |
| def GlobalProfile.velocity (P : GlobalProfile) (y : ℝ) : ℝ := y*P.g (y^2) |
| def GlobalProfile.integrand (P : GlobalProfile) (y : ℝ) : ℝ := |
| P.velocity y/(1-y*P.velocity y) |
| def GlobalProfile.primitive (P : GlobalProfile) (y : ℝ) : ℝ := |
| ∫ q in (0:ℝ)..y, P.integrand q |
|
|
| def GlobalProfile.H (P : GlobalProfile) (y : ℝ) : ℝ := |
| (1/P.b)*Real.exp (P.b*P.primitive y) |
|
|
| lemma GlobalProfile.b_pos (P : GlobalProfile) : 0<P.b := sub_pos.mpr P.beta_gt |
| lemma GlobalProfile.velocity_zero (P : GlobalProfile) : P.velocity 0=0 := by simp [GlobalProfile.velocity] |
| lemma GlobalProfile.velocity_neg (P : GlobalProfile) (y : ℝ) : P.velocity (-y)= -P.velocity y := by |
| simp [GlobalProfile.velocity] |
|
|
| lemma GlobalProfile.velocity_smooth (P : GlobalProfile) : ContDiff ℝ ∞ P.velocity := by |
| have hg : ContDiff ℝ ∞ (fun y : ℝ => P.g (y^2)) := by |
| rw [← contDiffOn_univ] |
| exact P.smooth.comp (contDiff_id.pow 2).contDiffOn (fun y _ => sq_nonneg y) |
| exact contDiff_id.mul hg |
|
|
| lemma GlobalProfile.pole_pos (P : GlobalProfile) (y : ℝ) : 0<1-y*P.velocity y := by |
| rcases lt_trichotomy y 0 with hy|hy|hy |
| · have h := (P.below (-y) (neg_pos.mpr hy)).2 |
| change (-y)*P.velocity (-y)<1 at h |
| rw [P.velocity_neg] at h |
| nlinarith only [h] |
| · subst y |
| simp |
| · exact sub_pos.mpr (P.below y hy).2 |
|
|
| lemma GlobalProfile.velocity_range (P : GlobalProfile) (y : ℝ) : |P.velocity y|<1 := by |
| by_cases hy : 0≤y |
| · exact P.range y hy |
| · have h : |P.velocity (-y)|<1 := P.range (-y) (neg_nonneg.mpr (le_of_not_ge hy)) |
| simpa only [P.velocity_neg,abs_neg] using h |
|
|
| lemma GlobalProfile.integrand_smooth (P : GlobalProfile) : ContDiff ℝ ∞ P.integrand := |
| P.velocity_smooth.div (contDiff_const.sub (contDiff_id.mul P.velocity_smooth)) |
| (fun x => ne_of_gt (P.pole_pos x)) |
|
|
| lemma GlobalProfile.primitive_derivative (P : GlobalProfile) (y : ℝ) : |
| HasDerivAt P.primitive (P.integrand y) y := by |
| have hc := P.integrand_smooth.continuous |
| exact intervalIntegral.integral_hasDerivAt_right (hc.intervalIntegrable _ _) |
| hc.aestronglyMeasurable.stronglyMeasurableAtFilter hc.continuousAt |
|
|
| lemma GlobalProfile.primitive_smooth (P : GlobalProfile) : ContDiff ℝ ∞ P.primitive := by |
| rw [contDiff_infty_iff_deriv] |
| refine ⟨fun y => (P.primitive_derivative y).differentiableAt,?_⟩ |
| have he : deriv P.primitive=P.integrand := funext fun y => (P.primitive_derivative y).deriv |
| rw [he] |
| exact P.integrand_smooth |
|
|
| lemma GlobalProfile.H_pos (P : GlobalProfile) (y : ℝ) : 0<P.H y := |
| mul_pos (one_div_pos.mpr P.b_pos) (Real.exp_pos _) |
| lemma GlobalProfile.H_smooth (P : GlobalProfile) : ContDiff ℝ ∞ P.H := |
| contDiff_const.mul (contDiff_const.mul P.primitive_smooth).exp |
| lemma GlobalProfile.H_derivative (P : GlobalProfile) (y : ℝ) : |
| HasDerivAt P.H (P.b*P.H y*P.integrand y) y := by |
| convert (((P.primitive_derivative y).const_mul P.b).exp).const_mul (1/P.b) using 1 <;> |
| first | rfl | (unfold GlobalProfile.H; ring) |
|
|
| def GlobalProfile.A (P : GlobalProfile) (y : ℝ) : ℝ := P.b*P.H y+y*deriv P.H y |
| def GlobalProfile.B (P : GlobalProfile) (y : ℝ) : ℝ := deriv P.H y |
| def GlobalProfile.M0 (P : GlobalProfile) (y : ℝ) : ℝ := (P.A y)^2-(P.B y)^2 |
|
|
| lemma GlobalProfile.A_eq (P : GlobalProfile) (y : ℝ) : |
| P.A y=P.b*P.H y/(1-y*P.velocity y) := by |
| rw [GlobalProfile.A,(P.H_derivative y).deriv,GlobalProfile.integrand] |
| field_simp [ne_of_gt (P.pole_pos y)] |
| ring |
| lemma GlobalProfile.B_eq (P : GlobalProfile) (y : ℝ) : P.B y=P.A y*P.velocity y := by |
| rw [GlobalProfile.B,(P.H_derivative y).deriv,P.A_eq,GlobalProfile.integrand] |
| ring |
| lemma GlobalProfile.A_pos (P : GlobalProfile) (y : ℝ) : 0<P.A y := by |
| rw [P.A_eq] |
| exact div_pos (mul_pos P.b_pos (P.H_pos y)) (P.pole_pos y) |
| lemma GlobalProfile.M0_eq (P : GlobalProfile) (y : ℝ) : |
| P.M0 y=P.b^2*P.H y^2*(1-P.velocity y^2)/(1-y*P.velocity y)^2 := by |
| rw [GlobalProfile.M0,P.B_eq,P.A_eq] |
| field_simp [ne_of_gt (P.pole_pos y)] |
| lemma GlobalProfile.M0_pos (P : GlobalProfile) (y : ℝ) : 0<P.M0 y := by |
| have hv : 0<1-(P.velocity y)^2 := by |
| have hr := abs_lt.mp (P.velocity_range y) |
| nlinarith only [hr.1,hr.2,sq_nonneg (P.velocity y)] |
| rw [P.M0_eq] |
| exact div_pos (mul_pos (mul_pos (sq_pos_of_pos P.b_pos) (sq_pos_of_pos (P.H_pos y))) hv) |
| (sq_pos_of_pos (P.pole_pos y)) |
|
|
| end SepticProfile |
|
|
| end |
| end |
|
|
| end OAI |
|
|
