File size: 2,724 Bytes
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 OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffProfilePhysicalLimit
import OAI.MathematicalPhysics.DefocusingNLS.Linear.HomogeneousPhysicalWeakLimit
import OAI.MathematicalPhysics.DefocusingNLS.Linear.HomogeneousLocalizationPhysical
import OAI.MathematicalPhysics.DefocusingNLS.Linear.WeakLinearCoordinates
namespace OAI
/-! # Finite coordinates of the actual sampled profile converge to its whole-space coordinates -/
open Filter Topology
open scoped SchwartzMap ContDiff
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
local notation "Radius" => {L : ℝ // 1 ≤ L}
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F]
theorem cutoffProfile_finite_coordinate_limit (a k : ℝ) (ha : 0 < a)
(ha1 : a < 1) (hk : 8 < k) (χ : 𝓢(E, ℂ))
(hχ : HasCompactSupport (χ : E → ℂ))
(hχzero : ∀ y : E, 1 ≤ ‖y‖ → χ y = 0)
(hχone : ∀ y : E, ‖y‖ ≤ 1 / 2 → χ y = 1)
(Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q)
(q : HomogeneousY a k)
(hq : ∀ y, homogeneousPhysicalCLM a k ha ha1 hk q y = Q y)
(B : ℝ) (_hB : 0 ≤ B)
(hb : ∀ (L : ℝ) (hL : 1 ≤ L),
‖physicalSchwartzTorusSamplingCLM a k L ha1 hk hL
(cutoffProfileSchwartz L (by linarith) χ hχ Q hQ)‖ ≤ B)
(L : ℕ → Radius) (hLinf : Tendsto (fun n => (L n).1) atTop atTop)
(π : HomogeneousY a k →L[ℝ] F) :
Tendsto (fun n => π (homogeneousLocalizationCLM a k (L n).1 ha ha1 hk (L n).2 χ
(physicalSchwartzTorusSamplingCLM a k (L n).1 ha1 hk (L n).2
(cutoffProfileSchwartz (L n).1 (by linarith [(L n).2]) χ hχ Q hQ))))
atTop (𝓝 (π q)) := by
obtain ⟨D, hD, hd⟩ := exists_homogeneousLocalization_bound a k ha ha1 hk χ
apply tendsto_finite_coordinates_of_weak π _ q
apply homogeneousY_weak_of_physical_limit a k (D * B) ha ha1 hk
· intro n
exact (hd (L n).1 (L n).2 _).trans (mul_le_mul_of_nonneg_left (hb (L n).1 (L n).2) hD)
· intro y
rw [hq]
apply tendsto_const_nhds.congr'
filter_upwards [hLinf.eventually (eventually_ge_atTop (2 * ‖y‖))] with n hn
rw [homogeneousLocalization_physical, expandingPhysicalLocalization]
have hc : χ ((L n).1⁻¹ • y) = 1 := by
apply hχone
rw [norm_smul, Real.norm_eq_abs, abs_of_pos (inv_pos.mpr (by linarith [(L n).2]))]
exact (inv_mul_le_iff₀ (by linarith [(L n).2] : 0 < (L n).1)).mpr (by linarith)
rw [hc, one_mul]
simpa only [expandingPhysicalContinuous_apply, physicalSchwartzTorusSamplingCLM_apply] using
(sampledCutoffProfile_physical_eq a k (L n).1 ha ha1 hk (L n).2 χ hχ
hχzero hχone Q hQ y hn).symm
end DefocusingNLS
end OAI
|