| import OAI.MathematicalPhysics.Transonic.Profile.PhysicalEntry |
| import OAI.MathematicalPhysics.Transonic.Exterior.WindowSound |
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| namespace OAI |
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| section |
| noncomputable section |
| namespace SepticProfile.PhysicalExterior |
| open Set SourceFamily |
| open |
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| lemma /3)=1 := by |
| unfold sonicSpeed ell |
| exact one_div_mul_cancel (ne_of_gt (Real.sqrt_pos.mpr (by norm_num))) |
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| lemma scaled_abs {v : ℝ} (hv : 0<v ∧ v<Real.sqrt (5/ || |
| rw |
| simpa |
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| lemma ℝ ℝ |
| (W |
| 0<sonicRadius /256) ∧ |
| sonicRadius (ShootingParameters.beta p.val)*(1+d/ |
| have |
| have |
| have /256) := |
| mul_pos hr (by linarith [div_pos W.dpos (by norm_num : (0:ℝ)<256)]) |
| refine ⟨hp,?_⟩ |
| have hs := W.subsonic |
| rw [← source_sigma p,sigma_eq_radius_sq (ne_of_gt (source_q_pos p))] at hs |
| nlinarith |
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| lemma window_global {p : Parameter} {d ue : ℝ} {u : PowerSeries ℝ} |
| (W : AdmissibleWindow (sig p) (kap p) d u) |
| (he : (lowerPoly u d).eval 1≤ue) (heu : ue<Real.sqrt (5/ |
| let |
| let /256) |
| let m := velocityToU a (sonicSpeed*(lowerPoly u d).eval 1) |
| ∃ v : ℝ → ℝ, v a=velocityToU a (sonicSpeed*ue) ∧ |
| (∀ y∈Ici a, -1<v y ∧ v y≤ m) ∧ |
| (∀ y∈Ici a, HasDerivWithinAt v (field ell beta (y,v y)) (Ici a) y) ∧ |
| ContDiffOn ℝ ∞ v (Ici a) ∧ |
| (∀ y∈Ici a, v y<y ∧ y*v y<1) := by |
| dsimp only |
| let beta := ShootingParameters.beta p.val |
| let a := sonicRadius beta*(1+d/ |
| let |
| let |
| have |
| have ∧ |
| have ||;exact ha.2 |
| have hP : 1<P ∧ P<6/5 := W.range 1 ⟨by norm_num,le_rfl⟩ |
| have hq : (6/5:ℝ)<Real.sqrt (5/3) := by |
| have he := Real.sq_sqrt (by norm_num : (0:ℝ)≤5/3) |
| have hn := Real.sqrt_nonneg (5/3:ℝ) |
| nlinarith |
| have hU : |sonicSpeed*P|<1 := scaled_abs ⟨by linarith [hP.1],hP.2.trans hq⟩ |
| have hUE : |sonicSpeed*ue|<1 := scaled_abs ⟨by linarith [hP.1],heu⟩ |
| have hpos (v:ℝ) (hv:|sonicSpeed*v|<1) : 0<1-a*(sonicSpeed*v) := by |
| have hu := (abs_lt.mp hv).2 |
| nlinarith [mul_nonneg ha.1.le (sub_nonneg.mpr hu.le)] |
| have hm : -1< m ∧ m<1 := abs_lt.mp (velocityToU_range hy hU) |
| have hfirst := sonic_exit_factors hy hU hP.1 |
| have hcrit : 0<beta*ell*(1-m^2)*a-3*m*(1-a*m) := by |
| apply critical_exit hb.1 hb.2 rfl hy hU |
| simpa only [beta,source_kappa] using W.exit |
| have hstart : -1<velocityToU a (sonicSpeed*ue) ∧ velocityToU a (sonicSpeed*ue)≤ m := by |
| refine ⟨(abs_lt.mp (velocityToU_range hy hUE)).1,?_⟩ |
| exact chart_antitone hy (hpos P hU) (hpos ue hUE) |
| (mul_le_mul_of_nonneg_left he sonicSpeed_bounds.1.le) |
| have hqs : 1<Real.sqrt ell := by |
| change 1<Real.sqrt (5/3) |
| linarith |
| have hsq : (Real.sqrt ell)^2=ell := Real.sq_sqrt (by norm_num [ell]) |
| obtain ⟨v,hv,hvr,hvd⟩ := global_continuation (beta:=beta) hqs (by linarith [hb.1]) ha.1 |
| hm hfirst.1 hfirst.2 (by simpa only [hsq] using hcrit) hstart |
| have hvs := solution_smooth hqs ha.1 hm hfirst.1 hfirst.2 hvr hvd |
| have hrec := solution_below_reciprocal hqs hb.1 ha.2 hm.2 |
| (fun y hy => (hvr y hy).2) hvd |
| refine ⟨v,hv,hvr,?_,?_,?_⟩ |
| · simpa only [hsq] using hvd |
| · simpa only [hsq] using hvs |
| · intro y hy |
| have hma : m<a := velocityToU_below (show |a|<1 from by rw [abs_of_pos ha.1];exact ha.2) (mul_pos sonicSpeed_bounds.1 (by linarith [hP.1])) (hpos P hU) |
| constructor |
| · exact ((hvr y hy).2.trans_lt hma).trans_le hy |
| · by_cases hy1 : 1≤y |
| · have hh := mul_lt_mul_of_pos_left (hrec y hy1) (by linarith : 0<y) |
| simpa only [mul_one_div_cancel (by linarith : y≠0)] using hh |
| · have hv1 := ((hvr y hy).2).trans_lt hm.2 |
| have hyp : 0<y := ha.1.trans_le hy |
| nlinarith [mul_nonneg hyp.le (sub_nonneg.mpr hv1.le)] |
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| end SepticProfile.PhysicalExterior |
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| end |
| end |
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| end OAI |
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