| import OAI.MathematicalPhysics.Transonic.Profile.SourceParameters |
| import OAI.MathematicalPhysics.Transonic.Profile.PhysicalProfile |
| import OAI.MathematicalPhysics.Transonic.Shooting.FiniteProfile |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace SepticProfile.SonicShooting |
| open Set Filter SourceFamily AxisBarriers |
| open scoped Topology ContDiff |
|
|
| def MatchedPair.beta (M : MatchedPair) : ℝ := ShootingParameters.beta M.parameter.val |
| def MatchedPair.radius (M : MatchedPair) : ℝ := sonicRadius M.beta |
| def MatchedPair.sonicU (M : MatchedPair) (y : ℝ) : ℝ := sonicSpeed*M.localProfile (y/M.radius) |
| def MatchedPair.velocity (M : MatchedPair) (y : ℝ) : ℝ := velocityToU y (M.sonicU y) |
|
|
| lemma MatchedPair.beta_bounds (M : MatchedPair) : 1<M.beta ∧ M.beta*(ell+Real.sqrt ell)<3 := |
| ShootingParameters.standing_beta_bound M.parameter.property |
| lemma MatchedPair.radius_bounds (M : MatchedPair) : 0<M.radius ∧ M.radius<1 := by |
| have h := source_parameter_bounds M.beta_bounds.1 M.beta_bounds.2 |
| exact ⟨h.2.2.2.2.1,h.2.2.2.2.2.1⟩ |
| lemma sonicSpeed_bounds : 0<sonicSpeed ∧ sonicSpeed<1 := by |
| have hp : 0<sonicSpeed := by unfold sonicSpeed ell;positivity |
| refine ⟨hp,?_⟩ |
| nlinarith [sonicSpeed_sq] |
|
|
| lemma MatchedPair.local_zero (M : MatchedPair) : M.localProfile 0=0 := by |
| rw [(M.local_eventually_inner (by linarith [(endRadius_bounds M.sonic.e_pos M.sonic.e_lt).1] : 0<endRadius M.sonic.e)).eq_of_nhds, |
| (M.inner_eventually_axis M.axis.δ_pos).eq_of_nhds] |
| simp only [MatchedPair.axisGerm,AxisFamily.UniformGerm.velocity,zero_mul] |
|
|
| lemma MatchedPair.sonicU_smooth (M : MatchedPair) {y : ℝ} |
| (hy : y/M.radius ∈ Icc (0:ℝ) (1+M.sonic.width)) : ContDiffAt ℝ ∞ M.sonicU y := by |
| exact contDiffAt_const.mul ((M.local_smooth_at hy).comp y (contDiffAt_id.div_const M.radius)) |
|
|
| lemma MatchedPair.sonicU_derivative (M : MatchedPair) {y : ℝ} |
| (hy : y/M.radius ∈ Icc (0:ℝ) (1+M.sonic.width)) : |
| HasDerivAt M.sonicU (SepticProfile.q M.beta/3*deriv M.localProfile (y/M.radius)) y := by |
| have hd := ((M.local_smooth_at hy).differentiableAt (by norm_num)).hasDerivAt |
| have h := (hd.comp (h:=fun x : ℝ => x/M.radius) y ((hasDerivAt_id y).div_const M.radius)).const_mul sonicSpeed |
| apply h.congr_deriv |
| have hr : M.radius=3*sonicSpeed/SepticProfile.q M.beta := sonicRadius_eq M.beta |
| have hq : SepticProfile.q M.beta≠0 := ne_of_gt (source_q_pos M.parameter) |
| have hc : sonicSpeed≠0 := ne_of_gt sonicSpeed_bounds.1 |
| rw [hr] |
| field_simp |
|
|
| lemma MatchedPair.sonicU_equation (M : MatchedPair) {y : ℝ} |
| (hy : y/M.radius ∈ Icc (0:ℝ) (1+M.sonic.width)) : |
| sonicDenom ell y (M.sonicU y)*deriv M.sonicU y=sonicNumer ell M.beta y (M.sonicU y) := by |
| have he := M.local_equation hy |
| have hp : normalizedDenom M.beta (y/M.radius) (M.localProfile (y/M.radius))*deriv M.localProfile (y/M.radius)= |
| normalizedNumer M.beta (y/M.radius) (M.localProfile (y/M.radius)) := by |
| simpa only [normalizedDenom,normalizedNumer,MatchedPair.beta,source_sigma,source_kappa,AxisBarriers.D,AxisBarriers.N] using he |
| have ht := (normalized_equation_iff (ne_of_gt (source_q_pos M.parameter)) |
| (y/M.radius) (M.localProfile (y/M.radius)) (deriv M.localProfile (y/M.radius))).mpr hp |
| have hr : sonicRadius M.beta*(y/M.radius)=y := by |
| change M.radius*(y/M.radius)=y |
| exact mul_div_cancel₀ y (ne_of_gt M.radius_bounds.1) |
| change sonicDenom ell (sonicRadius M.beta*(y/M.radius)) (M.sonicU y)* |
| (SepticProfile.q M.beta/3*deriv M.localProfile (y/M.radius))= |
| sonicNumer ell M.beta (sonicRadius M.beta*(y/M.radius)) (M.sonicU y) at ht |
| rw [hr,← (M.sonicU_derivative hy).deriv] at ht |
| exact ht |
|
|
| lemma MatchedPair.sonicU_zero (M : MatchedPair) : M.sonicU 0=0 := by |
| simp only [MatchedPair.sonicU,zero_div,M.local_zero,mul_zero] |
| lemma MatchedPair.sonicU_at_radius (M : MatchedPair) : M.sonicU M.radius=sonicSpeed := by |
| simp only [MatchedPair.sonicU,div_self (ne_of_gt M.radius_bounds.1),M.local_value,mul_one] |
|
|
| lemma MatchedPair.exists_physical_width (M : MatchedPair) : ∃ w : ℝ, |
| 0<w ∧ w≤M.sonic.width ∧ M.radius*(1+w)<1 ∧ |
| ∀ z ∈ Icc (0:ℝ) (1+w), |sonicSpeed*M.localProfile z|<1 := by |
| have hs := (M.local_smooth_at (show (1:ℝ) ∈ Icc (0:ℝ) (1+M.sonic.width) from |
| ⟨by norm_num,by linarith [M.sonic.width_pos]⟩)).continuousAt |
| have hc : ContinuousAt (fun z => |sonicSpeed*M.localProfile z|) 1 := |
| (continuousAt_const.mul hs).abs |
| have hn : ∀ᶠ z in 𝓝 (1:ℝ), |sonicSpeed*M.localProfile z|<1 := by |
| apply hc.eventually (Iio_mem_nhds _) |
| simpa only [M.local_value,mul_one,abs_of_pos sonicSpeed_bounds.1] using sonicSpeed_bounds.2 |
| obtain ⟨ε,hε,hball⟩ := Metric.mem_nhds_iff.mp hn |
| let w := min M.sonic.width (min (ε/2) ((1/M.radius-1)/2)) |
| have hw : 0<w := by |
| apply lt_min M.sonic.width_pos |
| apply lt_min (by linarith) |
| have hdiv : 1<1/M.radius := (one_lt_div M.radius_bounds.1).mpr M.radius_bounds.2 |
| linarith |
| refine ⟨w,hw,min_le_left _ _,?_,?_⟩ |
| · have hh : w≤(1/M.radius-1)/2 := (min_le_right _ _).trans (min_le_right _ _) |
| have hd : M.radius*(1/M.radius)=1 := mul_div_cancel₀ 1 (ne_of_gt M.radius_bounds.1) |
| nlinarith [mul_le_mul_of_nonneg_left hh M.radius_bounds.1.le,M.radius_bounds.2] |
| · intro z hz |
| by_cases hl : z<1 |
| · by_cases hz0 : z=0 |
| · simp only [hz0,M.local_zero,mul_zero,abs_zero];norm_num |
| · have hu := M.local_range ⟨lt_of_le_of_ne hz.1 (Ne.symm hz0),hl⟩ |
| rw [abs_of_pos (mul_pos sonicSpeed_bounds.1 hu.1)] |
| exact (mul_lt_mul_of_pos_left hu.2 sonicSpeed_bounds.1).trans (by simpa using sonicSpeed_bounds.2) |
| · apply hball |
| rw [Metric.mem_ball,Real.dist_eq,abs_of_nonneg (sub_nonneg.mpr (le_of_not_gt hl))] |
| have hh : w≤ε/2 := (min_le_right _ _).trans (min_le_left _ _) |
| linarith [hz.2] |
|
|
| structure PhysicalWidth (M : MatchedPair) where |
| w : ℝ |
| pos : 0<w |
| le_width : w≤M.sonic.width |
| end_lt : M.radius*(1+w)<1 |
| range : ∀ z ∈ Icc (0:ℝ) (1+w), |sonicSpeed*M.localProfile z|<1 |
|
|
| lemma MatchedPair.exists_physicalWidth (M : MatchedPair) : Nonempty (PhysicalWidth M) := by |
| obtain ⟨w,hp,hl,he,hr⟩ := M.exists_physical_width |
| exact ⟨⟨w,hp,hl,he,hr⟩⟩ |
|
|
| def PhysicalWidth.endpoint {M : MatchedPair} (W : PhysicalWidth M) : ℝ := M.radius*(1+W.w) |
|
|
| lemma PhysicalWidth.endpoint_bounds {M : MatchedPair} (W : PhysicalWidth M) : |
| M.radius<W.endpoint ∧ W.endpoint<1 := by |
| refine ⟨?_,W.end_lt⟩ |
| unfold PhysicalWidth.endpoint |
| nlinarith [mul_pos M.radius_bounds.1 W.pos] |
|
|
| lemma PhysicalWidth.rescale {M : MatchedPair} (W : PhysicalWidth M) {y : ℝ} |
| (hy : y ∈ Icc (0:ℝ) W.endpoint) : y/M.radius ∈ Icc (0:ℝ) (1+W.w) := by |
| exact ⟨div_nonneg hy.1 M.radius_bounds.1.le, |
| (div_le_iff₀ M.radius_bounds.1).mpr (by simpa only [PhysicalWidth.endpoint,mul_comm] using hy.2)⟩ |
| lemma PhysicalWidth.rescale_full {M : MatchedPair} (W : PhysicalWidth M) {y : ℝ} |
| (hy : y ∈ Icc (0:ℝ) W.endpoint) : y/M.radius ∈ Icc (0:ℝ) (1+M.sonic.width) := |
| ⟨(W.rescale hy).1,(W.rescale hy).2.trans (by linarith [W.le_width])⟩ |
| lemma PhysicalWidth.y_abs {M : MatchedPair} (W : PhysicalWidth M) {y : ℝ} |
| (hy : y ∈ Icc (0:ℝ) W.endpoint) : |y|<1 := by |
| rw [abs_of_nonneg hy.1] |
| exact hy.2.trans_lt W.end_lt |
| lemma PhysicalWidth.U_abs {M : MatchedPair} (W : PhysicalWidth M) {y : ℝ} |
| (hy : y ∈ Icc (0:ℝ) W.endpoint) : |M.sonicU y|<1 := W.range _ (W.rescale hy) |
| lemma PhysicalWidth.chart_pos {M : MatchedPair} (W : PhysicalWidth M) {y : ℝ} |
| (hy : y ∈ Icc (0:ℝ) W.endpoint) : 0<1-y*M.sonicU y := by |
| have hu := (abs_lt.mp (W.U_abs hy)).2 |
| have hy1 : y<1 := hy.2.trans_lt W.end_lt |
| have hm : y*M.sonicU y≤y := by nlinarith [mul_nonneg hy.1 (sub_nonneg.mpr hu.le)] |
| linarith |
|
|
| lemma PhysicalWidth.velocity_smooth {M : MatchedPair} (W : PhysicalWidth M) {y : ℝ} |
| (hy : y ∈ Icc (0:ℝ) W.endpoint) : ContDiffAt ℝ ∞ M.velocity y := by |
| have hU := M.sonicU_smooth (W.rescale_full hy) |
| exact (contDiffAt_id.sub hU).div (contDiffAt_const.sub (contDiffAt_id.mul hU)) |
| (ne_of_gt (W.chart_pos hy)) |
|
|
| lemma PhysicalWidth.velocity_equation {M : MatchedPair} (W : PhysicalWidth M) {y : ℝ} |
| (hy : y ∈ Icc (0:ℝ) W.endpoint) : |
| profileDenom ell y (M.velocity y)*deriv M.velocity y=profileNumer ell M.beta y (M.velocity y) := by |
| have hu := ((M.sonicU_smooth (W.rescale_full hy)).differentiableAt (by norm_num)).hasDerivAt |
| have hv := hasDerivAt_velocityToU hu (ne_of_gt (W.chart_pos hy)) |
| change HasDerivAt M.velocity _ y at hv |
| rw [hv.deriv] |
| exact inverse_profile_equation (ne_of_gt (W.chart_pos hy)) (M.sonicU_equation (W.rescale_full hy)) |
|
|
| lemma PhysicalWidth.velocity_abs {M : MatchedPair} (W : PhysicalWidth M) {y : ℝ} |
| (hy : y ∈ Icc (0:ℝ) W.endpoint) : |M.velocity y|<1 := |
| velocityToU_range (W.y_abs hy) (W.U_abs hy) |
|
|
| lemma MatchedPair.local_positive (M : MatchedPair) {z : ℝ} |
| (hz : z ∈ Ioc (0:ℝ) (1+M.sonic.width)) : 0<M.localProfile z := by |
| rcases lt_trichotomy z 1 with hl|he|hr |
| · exact (M.local_range ⟨hz.1,hl⟩).1 |
| · simp only [he,M.local_value];norm_num |
| · have hE : endRadius M.sonic.e<z := (endRadius_bounds M.sonic.e_pos M.sonic.e_lt).2.trans hr |
| rw [(M.local_eventually_germ hE).eq_of_nhds] |
| linarith [M.sonic.germArc_above M.parameter ⟨hr,hz.2⟩] |
|
|
| lemma PhysicalWidth.velocity_below {M : MatchedPair} (W : PhysicalWidth M) {y : ℝ} |
| (hy : y ∈ Ioc (0:ℝ) W.endpoint) : M.velocity y<y ∧ y*M.velocity y<1 := by |
| have hu : 0<M.sonicU y := mul_pos sonicSpeed_bounds.1 |
| (M.local_positive ⟨div_pos hy.1 M.radius_bounds.1,(W.rescale_full ⟨hy.1.le,hy.2⟩).2⟩) |
| refine ⟨velocityToU_below (W.y_abs ⟨hy.1.le,hy.2⟩) hu (W.chart_pos ⟨hy.1.le,hy.2⟩),?_⟩ |
| have hv := (abs_lt.mp (W.velocity_abs ⟨hy.1.le,hy.2⟩)).2 |
| have hy1 : y<1 := hy.2.trans_lt W.end_lt |
| nlinarith [mul_pos hy.1 (sub_pos.mpr hv)] |
|
|
| lemma PhysicalWidth.recover_U {M : MatchedPair} (W : PhysicalWidth M) {y : ℝ} |
| (hy : y ∈ Icc (0:ℝ) W.endpoint) : velocityToU y (M.velocity y)=M.sonicU y := by |
| apply velocityToU_involutive _ (ne_of_gt (W.chart_pos hy)) |
| have hh := W.y_abs hy |
| have hp : 0<1-y^2 := by nlinarith [(abs_lt.mp hh).1,(abs_lt.mp hh).2] |
| exact ne_of_gt hp |
|
|
| lemma PhysicalWidth.unique_sonic {M : MatchedPair} (W : PhysicalWidth M) {y : ℝ} |
| (hy : y ∈ Icc (0:ℝ) W.endpoint) : |
| velocityToU y (M.velocity y)=sonicSpeed ↔ y=M.radius := by |
| rw [W.recover_U hy] |
| by_cases hz : y=0 |
| · simp only [hz,M.sonicU_zero] |
| constructor |
| · intro h;linarith [sonicSpeed_bounds.1] |
| · intro h;linarith [M.radius_bounds.1] |
| · have hl : 0<y := lt_of_le_of_ne hy.1 (Ne.symm hz) |
| have hh := M.local_unique_sonic ⟨div_pos hl M.radius_bounds.1,(W.rescale_full hy).2⟩ |
| change sonicSpeed*M.localProfile (y/M.radius)=sonicSpeed ↔ _ |
| rw [mul_eq_left₀ (ne_of_gt sonicSpeed_bounds.1),hh,div_eq_one_iff_eq (ne_of_gt M.radius_bounds.1)] |
|
|
| lemma PhysicalWidth.sonic_slope {M : MatchedPair} (W : PhysicalWidth M) : |
| 0<deriv (fun y => velocityToU y (M.velocity y)) M.radius := by |
| have he : (fun y => velocityToU y (M.velocity y)) =ᶠ[𝓝 M.radius] M.sonicU := by |
| filter_upwards [Ioo_mem_nhds M.radius_bounds.1 W.endpoint_bounds.1] with y hy |
| exact W.recover_U ⟨hy.1.le,hy.2.le⟩ |
| rw [he.deriv_eq,(M.sonicU_derivative (W.rescale_full ⟨M.radius_bounds.1.le,W.endpoint_bounds.1.le⟩)).deriv] |
| rw [div_self (ne_of_gt M.radius_bounds.1)] |
| exact mul_pos (div_pos (source_q_pos M.parameter) (by norm_num)) M.local_slope |
|
|
| end SepticProfile.SonicShooting |
|
|
| end |
| end |
|
|
| end OAI |
|
|
