openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Nonlinear /CutoffProfileModeDerivative.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Certificates.CompactFamilyIntegral | |
| import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffMovingPath | |
| namespace OAI | |
| /-! # Time derivative of the actual cutoff-profile Fourier coefficients | |
| The physical volume factor disappears after rescaling to the fixed cutoff. | |
| Differentiating then takes place on a fixed compact support. | |
| -/ | |
| open MeasureTheory | |
| open scoped SchwartzMap ContDiff | |
| namespace DefocusingNLS | |
| local notation "E" => EuclideanSpace ℝ (Fin 12) | |
| theorem cutoffProfile_coefficient_fixed_scale (R : ℝ) (hR : 0 < R) | |
| (χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ)) | |
| (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) (n : frequencyLattice) : | |
| schwartzLatticeCoefficient R (radianFourierKernel (cutoffProfileSchwartz R hR χ hχ Q hQ)) n = | |
| schwartzLatticeCoefficient 1 (radianFourierKernel (compactDilationFamily χ hχ Q hQ R)) n := by | |
| have he : cutoffProfileSchwartz R hR χ hχ Q hQ = | |
| schwartzPhysicalDilation 0 R hR (compactDilationFamily χ hχ Q hQ R) := by | |
| ext x | |
| simp only [cutoffProfileSchwartz_apply, schwartzPhysicalDilation_apply, | |
| compactDilationFamily_apply, mul_zero, Real.rpow_zero, Complex.ofReal_one, | |
| one_mul, smul_smul, mul_inv_cancel₀ hR.ne', one_smul] | |
| rw [he, radianFourierKernel_schwartzPhysicalDilation, | |
| schwartzLatticeCoefficient_homogeneousDilation 0 R R hR hR] | |
| simp only [mul_zero, Real.rpow_zero, Complex.ofReal_one, | |
| one_mul, div_self hR.ne'] | |
| private theorem hasDerivAt_radius (L s : ℝ) : | |
| HasDerivAt (expandingRadius L) (expandingRadius L s / 2) s := by | |
| change HasDerivAt (fun t : ℝ => L * Real.exp (t / 2)) _ s | |
| apply (((hasDerivAt_id s).div_const 2).exp.const_mul L).congr_deriv | |
| dsimp only [expandingRadius, id_eq] | |
| ring | |
| private theorem positive_radius (L t : ℝ) (hL : 0 < L) : 0 < expandingRadius L t := | |
| mul_pos hL (Real.exp_pos _) | |
| /-- An exact scalar Fourier-mode derivative at every real time. -/ | |
| theorem hasDerivAt_cutoffProfile_coefficient (L s : ℝ) (hL : 0 < L) | |
| (χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ)) | |
| (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) (n : frequencyLattice) : | |
| HasDerivAt | |
| (fun t => schwartzLatticeCoefficient (expandingRadius L t) | |
| (radianFourierKernel (cutoffProfileSchwartz (expandingRadius L t) | |
| (positive_radius L _ hL) χ hχ Q hQ)) n) | |
| ((((2 * Real.pi) ^ (12 : ℕ))⁻¹ : ℝ) * | |
| radianFourierIntegral (fun x => χ x * | |
| fderiv ℝ Q ((expandingRadius L s) • x) ((expandingRadius L s / 2) • x)) | |
| (n : E)) s := by | |
| let F : ℝ → E → ℂ := fun t x => χ x * Q ((expandingRadius L t) • x) | |
| let Fd : ℝ → E → ℂ := fun t x => χ x * | |
| fderiv ℝ Q ((expandingRadius L t) • x) ((expandingRadius L t / 2) • x) | |
| have hRc : Continuous (expandingRadius L) := by unfold expandingRadius; fun_prop | |
| have harg : Continuous (fun p : ℝ × E => (expandingRadius L p.1) • p.2) := | |
| (hRc.comp continuous_fst).smul continuous_snd | |
| have hFc : Continuous (Function.uncurry F) := | |
| (χ.continuous.comp continuous_snd).mul (hQ.continuous.comp harg) | |
| have hFdc : Continuous (Function.uncurry Fd) := | |
| (χ.continuous.comp continuous_snd).mul | |
| (((contDiff_infty_iff_fderiv.mp hQ).2.continuous.comp harg).clm_apply | |
| (((hRc.comp continuous_fst).div_const 2).smul continuous_snd)) | |
| have hd (t : ℝ) (x : E) : HasDerivAt (fun u => F u x) (Fd t x) t := | |
| (((hQ.differentiable (by simp) _).hasFDerivAt.comp_hasDerivAt t | |
| ((hasDerivAt_radius L t).smul_const x)).const_mul (χ x)) | |
| have hz (_t : ℝ) (x : E) (hx : x ∉ tsupport (χ : E → ℂ)) : χ x = 0 := | |
| image_eq_zero_of_notMem_tsupport hx | |
| have h := (hasDerivAt_compactFamily_fourier F Fd hFc hFdc | |
| (tsupport (χ : E → ℂ)) hχ | |
| (fun t x hx => by dsimp [F]; rw [hz t x hx, zero_mul]) | |
| (fun t x hx => by dsimp [Fd]; rw [hz t x hx, zero_mul]) hd s (n : E)).const_mul | |
| ((((2 * Real.pi) ^ (12 : ℕ))⁻¹ : ℝ) : ℂ) | |
| have he (t : ℝ) : | |
| schwartzLatticeCoefficient (expandingRadius L t) | |
| (radianFourierKernel (cutoffProfileSchwartz (expandingRadius L t) | |
| (positive_radius L _ hL) χ hχ Q hQ)) n = | |
| ((((2 * Real.pi) ^ (12 : ℕ))⁻¹ : ℝ) : ℂ) * radianFourierIntegral (F t) (n : E) := by | |
| rw [cutoffProfile_coefficient_fixed_scale] | |
| simp only [schwartzLatticeCoefficient, mul_one, inv_one, one_smul, | |
| radianFourierKernel_apply, F] | |
| rfl | |
| simpa only [he, Fd] using h | |
| end DefocusingNLS | |
| end OAI | |