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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 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 | import OAI.MathematicalPhysics.DefocusingNLS.Profile.SlowUniformGrowth
namespace OAI
/-! Joint continuity on the small real parameter rectangle used for the free profile. -/
open MeasureTheory Filter Topology
namespace DefocusingNLS
/-- A fixed rectangle containing the free-profile parameter disk. -/
def freeParameterRectangle : Set (ℝ × ℝ) :=
{p | |p.1| ≤ 1 ∧ 1 ≤ p.2 ∧ p.2 ≤ 4}
noncomputable def freeShiftedQ (a : ℝ) (p : ℝ × ℝ) : ℂ := a-Complex.I*p.1
noncomputable def freeSpatialArgument (p : ℝ × ℝ) : ℂ := -Complex.I*p.2
private theorem continuous_freeShiftedQ (a : ℝ) : Continuous (freeShiftedQ a) := by
unfold freeShiftedQ
fun_prop
private theorem continuous_freeSpatialArgument : Continuous freeSpatialArgument := by
unfold freeSpatialArgument
fun_prop
private theorem freeSpatialArgument_slit {p : ℝ × ℝ} (hp : p ∈ freeParameterRectangle) :
freeSpatialArgument p ∈ Complex.slitPlane := by
simp only [Complex.mem_slitPlane_iff, freeSpatialArgument, Complex.mul_im,
Complex.mul_re, Complex.neg_re, Complex.neg_im, Complex.I_re, Complex.I_im,
Complex.ofReal_re, Complex.ofReal_im]
right
dsimp [freeParameterRectangle] at hp
linarith
private theorem free_parameter_mass_bound (m : ℕ) (a : ℝ)
{p : ℝ × ℝ} (hp : p ∈ freeParameterRectangle) :
‖(m : ℂ)-1-freeShiftedQ a p‖ ≤ |(m : ℝ)-1-a|+1 := by
have he : (m : ℂ)-1-freeShiftedQ a p =
(((m : ℝ)-1-a : ℝ) : ℂ)+Complex.I*p.1 := by
simp only [freeShiftedQ, Complex.ofReal_sub, Complex.ofReal_natCast, Complex.ofReal_one]
ring
rw [he]
have hn := norm_add_le ((((m : ℝ)-1-a : ℝ) : ℂ)) (Complex.I*p.1)
simp only [Complex.norm_real, Real.norm_eq_abs, norm_mul, Complex.norm_I, one_mul] at hn
exact hn.trans (add_le_add le_rfl hp.1)
private theorem continuousOn_free_regularizedSlowKernel (a : ℝ) (m : ℕ)
{u : ℝ} (hu : 0 < u) :
ContinuousOn (fun p => regularizedSlowKernel (freeShiftedQ a p) m
(freeSpatialArgument p) u) freeParameterRectangle := by
have hq : ContinuousOn (freeShiftedQ a) freeParameterRectangle :=
(continuous_freeShiftedQ a).continuousOn
have hx : ContinuousOn freeSpatialArgument freeParameterRectangle :=
continuous_freeSpatialArgument.continuousOn
have hx0 (p : ℝ × ℝ) (hp : p ∈ freeParameterRectangle) : freeSpatialArgument p ≠ 0 :=
Complex.slitPlane_ne_zero (freeSpatialArgument_slit hp)
have hbase : ContinuousOn (fun p => 1+(u : ℂ)/freeSpatialArgument p)
freeParameterRectangle := continuousOn_const.add (continuousOn_const.div hx hx0)
have hexp : ContinuousOn (fun p => (m : ℂ)-1-freeShiftedQ a p) freeParameterRectangle :=
continuousOn_const.sub hq
have hp := hbase.cpow hexp (fun p hp =>
one_add_real_div_mem_slitPlane (freeSpatialArgument p) (by simp [freeSpatialArgument]) hu.le)
have huq : ContinuousOn (fun p => (u : ℂ)^(freeShiftedQ a p-1)) freeParameterRectangle :=
(hq.sub continuousOn_const).const_cpow
(Or.inl (Complex.ofReal_ne_zero.mpr hu.ne'))
unfold regularizedSlowKernel regularizingBracket
convert! (continuousOn_const.mul huq).mul (hp.sub continuousOn_const) using 1
/-- Joint continuity of the actual integral H, including the q=0 normalization. -/
theorem continuousOn_free_regularizedSlowSolution (a : ℝ) (ha : 0 ≤ a) (m : ℕ) :
ContinuousOn (fun p => regularizedSlowSolution (freeShiftedQ a p) m
(freeSpatialArgument p)) freeParameterRectangle := by
let M : ℝ := |(m : ℝ)-1-a|+1
let K : ℝ := M*Real.exp (Real.pi*M)/1*(1+(1:ℝ)⁻¹)^(M+1)
have hM : 0 ≤ M := by dsimp [M]; positivity
have hint : ContinuousOn (fun p => ∫ u : ℝ in Set.Ioi 0,
regularizedSlowKernel (freeShiftedQ a p) m (freeSpatialArgument p) u)
freeParameterRectangle := by
apply continuousOn_of_dominated
(bound := fun u => K*slowParameterMajorant a a M u)
· intro p _
exact (measurable_regularizedSlowKernel _ _ _).aestronglyMeasurable.restrict
· intro p hp
filter_upwards [ae_restrict_mem measurableSet_Ioi] with u hu
apply regularizedSlowKernel_uniform_spatial_bound (freeShiftedQ a p) m
(freeSpatialArgument p) a a M 1 hM (by norm_num)
· simp [freeShiftedQ]
· simp [freeShiftedQ]
· exact free_parameter_mass_bound m a hp
· simp [freeSpatialArgument]
· simpa [freeSpatialArgument, norm_mul, Real.norm_eq_abs, abs_of_nonneg (by linarith [hp.2.1] : 0 ≤ p.2)] using hp.2.1
· exact hu
· exact (integrable_slowParameterMajorant a a M (by linarith) le_rfl hM).const_mul K
· filter_upwards [ae_restrict_mem measurableSet_Ioi] with u hu
exact continuousOn_free_regularizedSlowKernel a m hu
have hq : ContinuousOn (freeShiftedQ a) freeParameterRectangle :=
(continuous_freeShiftedQ a).continuousOn
have hx : ContinuousOn freeSpatialArgument freeParameterRectangle :=
continuous_freeSpatialArgument.continuousOn
have hpow := hx.cpow hq.neg (fun _ hp => freeSpatialArgument_slit hp)
have hgamma : ContinuousOn (fun p => (Complex.Gamma (freeShiftedQ a p))⁻¹)
freeParameterRectangle := Complex.differentiable_one_div_Gamma.continuous.continuousOn.comp
hq (Set.mapsTo_univ _ _)
unfold regularizedSlowSolution
convert! hpow.mul (continuousOn_const.add (hgamma.mul hint)) using 1
end DefocusingNLS
end OAI
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