openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Nonlinear /CutoffProfilePhysicalLimit.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffMovingPath | |
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.SchwartzPhysicalSampling | |
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingSmoothFilter | |
| namespace OAI | |
| /-! # The sampled cutoff coefficient agrees locally with the whole-space profile -/ | |
| open Set Filter Topology | |
| open scoped SchwartzMap ContDiff | |
| namespace DefocusingNLS | |
| local notation "E" => EuclideanSpace ℝ (Fin 12) | |
| theorem sampledCutoffProfile_physical_eq (a k L : ℝ) | |
| (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) | |
| (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) (y : E) (hy : 2 * ‖y‖ ≤ L) : | |
| expandingPhysicalContinuous a k L ha ha1 hk hL | |
| (schwartzTorusSample a k L ha1 hk hL (radianFourierKernel | |
| (cutoffProfileSchwartz L (by linarith) χ hχ Q hQ))) y = Q y := by | |
| have hpos : 0 < L := by linarith | |
| rw [expandingPhysicalContinuous_apply, | |
| schwartzTorusSample_eq_on_ball a k L ha ha1 hk hL] | |
| · rw [cutoffProfileSchwartz_apply, hχone, one_mul] | |
| rw [norm_smul, Real.norm_eq_abs, abs_of_pos (inv_pos.mpr hpos)] | |
| apply (inv_mul_le_iff₀ hpos).mpr | |
| linarith | |
| · intro x hx | |
| rw [cutoffProfileSchwartz_apply, hχzero, zero_mul] | |
| rw [norm_smul, Real.norm_eq_abs, abs_of_pos (inv_pos.mpr hpos)] | |
| apply (le_inv_mul_iff₀ hpos).mpr | |
| linarith | |
| · linarith [norm_nonneg y] | |
| theorem tendsto_sampledCutoffProfile_physical (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) | |
| (L : ℕ → ℝ) (hL : ∀ n, 1 ≤ L n) (hLinf : Tendsto L atTop atTop) (y : E) : | |
| Tendsto (fun n => expandingPhysicalContinuous a k (L n) ha ha1 hk (hL n) | |
| (schwartzTorusSample a k (L n) ha1 hk (hL n) (radianFourierKernel | |
| (cutoffProfileSchwartz (L n) (by linarith [hL n]) χ hχ Q hQ))) y) | |
| atTop (𝓝 (Q y)) := by | |
| apply tendsto_const_nhds.congr' | |
| filter_upwards [hLinf.eventually (eventually_ge_atTop (2 * ‖y‖))] with n hn | |
| exact (sampledCutoffProfile_physical_eq a k (L n) ha ha1 hk (hL n) | |
| χ hχ hχzero hχone Q hQ y hn).symm | |
| theorem tendsto_sampledCutoffProfilePath_physical (a k T : 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) | |
| (L : ℕ → ℝ) (hL : ∀ n, 1 ≤ L n) (hLinf : Tendsto L atTop atTop) | |
| (s : Icc (0 : ℝ) T) (y : E) : | |
| Tendsto (fun n => expandingPhysicalContinuous a k (expandingRadius (L n) s) | |
| ha ha1 hk ((hL n).trans (expandingRadius_ge (L n) s (hL n) s.2.1)) | |
| (sampledCutoffProfilePath a k (L n) T ha ha1 hk (hL n) χ hχ Q hQ s) y) | |
| atTop (𝓝 (Q y)) := by | |
| exact tendsto_sampledCutoffProfile_physical a k ha ha1 hk χ hχ hχzero hχone Q hQ | |
| (fun n => expandingRadius (L n) s) | |
| (fun n => (hL n).trans (expandingRadius_ge (L n) s (hL n) s.2.1)) | |
| (hLinf.atTop_mul_const (Real.exp_pos _)) y | |
| end DefocusingNLS | |
| end OAI | |