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fac6777 | 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 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 | import OAI.MathematicalPhysics.DefocusingNLS.Profile.NormalizedSlowDerivative
namespace OAI
/-! # Continuous extension of the normalized slow solution at inverse radius zero -/
open Filter Topology Set
namespace DefocusingNLS
noncomputable def compactifiedSlowSolution (q : ℂ) (m : ℕ) (x : ℂ) (t : ℝ) : ℂ :=
if t = 0 then 1 else normalizedSlowSolution q m (x / (t : ℂ))
@[simp] theorem compactifiedSlowSolution_zero (q : ℂ) (m : ℕ) (x : ℂ) :
compactifiedSlowSolution q m x 0 = 1 := by simp [compactifiedSlowSolution]
theorem compactifiedSlowSolution_pos (q : ℂ) (m : ℕ) (x : ℂ) {t : ℝ} (ht : 0 < t) :
compactifiedSlowSolution q m x t = normalizedSlowSolution q m (x / (t : ℂ)) := by
simp [compactifiedSlowSolution, ht.ne']
theorem slow_div_real_re_nonneg {x : ℂ} {t : ℝ} (hx : 0 ≤ x.re) (ht : 0 < t) :
0 ≤ (x / (t : ℂ)).re := by
simp only [Complex.div_re, Complex.ofReal_re, Complex.ofReal_im, mul_zero, zero_div, add_zero]
exact div_nonneg (mul_nonneg hx ht.le) (Complex.normSq_nonneg _)
theorem continuousWithinAt_compactifiedSlowSolution_zero (q : ℂ) (m : ℕ) (x : ℂ)
(hq : -1 < q.re) (hx : 0 ≤ x.re) (hx0 : x ≠ 0) :
ContinuousWithinAt (compactifiedSlowSolution q m x) (Ioi 0) 0 := by
obtain ⟨A, hA, hb⟩ := regularizedSlowSolution_first_remainder q m hq
have hxn : 0 < ‖x‖ := norm_pos_iff.mpr hx0
have hbound : ∀ᶠ t : ℝ in 𝓝[>] 0,
‖compactifiedSlowSolution q m x t - 1‖ ≤ (A / ‖x‖) * t := by
filter_upwards [self_mem_nhdsWithin,
(eventually_lt_nhds hxn).filter_mono nhdsWithin_le_nhds] with t ht htx
rw [compactifiedSlowSolution_pos q m x ht]
have hn : 1 ≤ ‖x / (t : ℂ)‖ := by
rw [norm_div, Complex.norm_of_nonneg ht.le]
exact (le_div_iff₀ ht).mpr (by linarith)
have hh := hb (x / (t : ℂ)) (slow_div_real_re_nonneg hx ht) hn
apply hh.trans_eq
rw [norm_div, Complex.norm_of_nonneg ht.le]
field_simp
have hz : Tendsto (fun t : ℝ => (A / ‖x‖) * t) (𝓝[>] 0) (𝓝 0) := by
have hc : Continuous (fun t : ℝ => (A / ‖x‖) * t) := continuous_const.mul continuous_id
simpa only [mul_zero] using (hc.tendsto 0).mono_left nhdsWithin_le_nhds
have hh := (squeeze_zero_norm' hbound hz).add_const (1 : ℂ)
rw [ContinuousWithinAt, compactifiedSlowSolution_zero]
simpa only [sub_add_cancel, zero_add] using hh
theorem continuousAt_compactifiedSlowSolution_pos (q : ℂ) (m : ℕ) (x : ℂ)
(hq : -1 < q.re) (hx : 0 ≤ x.re) (hx0 : x ≠ 0) {t : ℝ} (ht : 0 < t) :
ContinuousAt (compactifiedSlowSolution q m x) t := by
have hy := slow_div_real_re_nonneg hx ht
have hy0 : x / (t : ℂ) ≠ 0 := div_ne_zero hx0 (Complex.ofReal_ne_zero.mpr ht.ne')
have hnorm := (hasDerivAt_normalizedSlowSolution q m (x / (t : ℂ)) hq hy hy0).continuousAt
have harg : ContinuousAt (fun u : ℝ => x / (u : ℂ)) t :=
continuousAt_const.div Complex.continuous_ofReal.continuousAt
(Complex.ofReal_ne_zero.mpr ht.ne')
have he : compactifiedSlowSolution q m x =ᶠ[𝓝 t]
(fun u : ℝ => normalizedSlowSolution q m (x / (u : ℂ))) := by
filter_upwards [eventually_gt_nhds ht] with u hu
exact compactifiedSlowSolution_pos q m x hu
exact (ContinuousAt.comp (f := fun u : ℝ => x / (u : ℂ))
(g := normalizedSlowSolution q m) hnorm harg).congr
(by simpa only [Function.comp_def] using he.symm)
theorem continuousOn_compactifiedSlowSolution (q : ℂ) (m : ℕ) (x : ℂ)
(hq : -1 < q.re) (hx : 0 ≤ x.re) (hx0 : x ≠ 0) :
ContinuousOn (compactifiedSlowSolution q m x) (Ici 0) := by
intro t ht
change 0 ≤ t at ht
rcases ht.eq_or_lt with he | ht
· subst t
rw [ContinuousWithinAt, ← Ioi_union_left, nhdsWithin_union]
have h0 := continuousWithinAt_compactifiedSlowSolution_zero q m x hq hx hx0
rw [ContinuousWithinAt] at h0
exact h0.sup (by simpa only [nhdsWithin_singleton] using
tendsto_pure_nhds (compactifiedSlowSolution q m x) (0 : ℝ))
· exact (continuousAt_compactifiedSlowSolution_pos q m x hq hx hx0 ht).continuousWithinAt
end DefocusingNLS
end OAI
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