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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 | import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Topology.Order.LeftRightNhds
namespace OAI
/-!
# A nonzero self-similar amplitude prevents continuous continuation
A positive self-similar amplitude excludes continuous continuation through
the singular time.
-/
open Filter Topology
namespace DefocusingNLS
/-- A positive power times a function with a finite left limit tends to zero. -/
theorem vanishing_rescaled_norm_of_left_limit {E : Type*} [NormedAddCommGroup E]
(u : ℝ → E) (T a : ℝ) (ha : 0 < a) (v : E)
(hu : Tendsto u (𝓝[<] T) (𝓝 v)) :
Tendsto (fun t => (T - t) ^ a * ‖u t‖) (𝓝[<] T) (𝓝 0) := by
have hbase : Tendsto (fun t : ℝ => T - t) (𝓝[<] T) (𝓝 0) := by
have hconst : Tendsto (fun _ : ℝ => T) (𝓝[<] T) (𝓝 T) := tendsto_const_nhds
simpa using (hconst.sub (tendsto_id.mono_left nhdsWithin_le_nhds :
Tendsto (fun t : ℝ => t) (𝓝[<] T) (𝓝 T)))
have hpow : Tendsto (fun t : ℝ => (T - t) ^ a) (𝓝[<] T) (𝓝 0) := by
simpa [Function.comp_def, Real.zero_rpow ha.ne'] using
(Real.continuous_rpow_const ha.le).continuousAt.tendsto.comp hbase
simpa using hpow.mul hu.norm
/-- In particular, the positive rescaled limit in the paper excludes a finite left limit. -/
theorem no_left_limit_of_positive_rescaled_norm {E : Type*} [NormedAddCommGroup E]
(u : ℝ → E) (T a c : ℝ) (ha : 0 < a) (hc : 0 < c)
(hblowup : Tendsto (fun t => (T - t) ^ a * ‖u t‖) (𝓝[<] T) (𝓝 c)) :
¬ ∃ v, Tendsto u (𝓝[<] T) (𝓝 v) := by
rintro ⟨v, hv⟩
have hzero := vanishing_rescaled_norm_of_left_limit u T a ha v hv
have heq : c = 0 := tendsto_nhds_unique hblowup hzero
exact hc.ne' heq
/-- Continuous observations of a continuously extendible trajectory have a finite left limit.
Hence a positive self-similar pointwise amplitude rules out such an extension. -/
theorem no_continuous_extension_of_positive_rescaled_norm
{E F : Type*} [TopologicalSpace E] [NormedAddCommGroup F]
(u : ℝ → E) (observe : E → F) (T a c : ℝ)
(ha : 0 < a) (hc : 0 < c) (hobs : Continuous observe)
(hblowup : Tendsto (fun t => (T - t) ^ a * ‖observe (u t)‖)
(𝓝[<] T) (𝓝 c)) :
¬ ContinuousWithinAt u (Set.Iio T) T := by
intro hu
apply no_left_limit_of_positive_rescaled_norm (fun t => observe (u t)) T a c ha hc
hblowup
exact ⟨observe (u T), hobs.continuousAt.tendsto.comp hu⟩
end DefocusingNLS
end OAI
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