openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Nonlinear /CutoffApproximateMode.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffFourierDefect | |
| namespace OAI | |
| /-! # The actual sampled cutoff profile satisfies the forced mode equation | |
| The smooth stationary-profile premise is explicit. The forcing is the already | |
| constructed three-term cutoff residual, with the same physical normalization. | |
| -/ | |
| open scoped SchwartzMap ContDiff | |
| namespace DefocusingNLS | |
| local notation "E" => EuclideanSpace ℝ (Fin 12) | |
| theorem cutoffProfileSchwartz_dilation_zero (R : ℝ) (hR : 0 < R) | |
| (χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ)) | |
| (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) : | |
| 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] | |
| theorem schwartzOddPower_dilation_zero (R : ℝ) (hR : 0 < R) | |
| (ψ : 𝓢(E, ℂ)) (m : ℕ) : | |
| schwartzOddPower m (schwartzPhysicalDilation 0 R hR ψ) = | |
| schwartzPhysicalDilation 0 R hR (schwartzOddPower m ψ) := by | |
| ext x | |
| simp only [schwartzOddPower_apply, schwartzPhysicalDilation_apply, mul_zero, | |
| Real.rpow_zero, Complex.ofReal_one, one_mul] | |
| theorem cutoffProfileSchwartz_zero_outside (R : ℝ) (hR : 0 < R) | |
| (χ : 𝓢(E, ℝ)) (hχ : HasCompactSupport (χ : E → ℝ)) | |
| (hχzero : ∀ x : E, 2 < ‖x‖ → χ x = 0) | |
| (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) (x : E) (hx : 2 * R < ‖x‖) : | |
| cutoffProfileSchwartz R hR (χ.postcompCLM Complex.ofRealCLM) | |
| (hasCompactSupport_complexCutoff χ hχ) Q hQ x = 0 := by | |
| have hr : 2 < ‖R⁻¹ • x‖ := by | |
| rw [norm_smul, Real.norm_eq_abs, abs_inv, abs_of_pos hR] | |
| exact (lt_inv_mul_iff₀' hR).mpr hx | |
| simp only [cutoffProfileSchwartz_apply, SchwartzMap.postcompCLM_apply, | |
| Complex.ofRealCLM_apply, hχzero _ hr, Complex.ofReal_zero, zero_mul] | |
| theorem cutoff_normalized_nonlinearity_coefficient (a k R : ℝ) | |
| (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hR : 1 ≤ R) (m : ℕ) | |
| (χ : 𝓢(E, ℝ)) (hχ : HasCompactSupport (χ : E → ℝ)) | |
| (hχzero : ∀ x : E, 2 < ‖x‖ → χ x = 0) | |
| (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) (n : frequencyLattice) : | |
| physicalSchwartzCoefficient 1 n (schwartzOddPower m | |
| (compactDilationFamily (χ.postcompCLM Complex.ofRealCLM) | |
| (hasCompactSupport_complexCutoff χ hχ) Q hQ R)) = | |
| expandingFourierCoefficient a k R | |
| (expandingOddPower a k R ha ha1 hk hR m | |
| (schwartzTorusSample a k R ha1 hk hR (radianFourierKernel | |
| (cutoffProfileSchwartz R (lt_of_lt_of_le zero_lt_one hR) | |
| (χ.postcompCLM Complex.ofRealCLM) | |
| (hasCompactSupport_complexCutoff χ hχ) Q hQ)))) n := by | |
| rw [sampled_oddPower_coefficient a k R ha ha1 hk hR m _ | |
| (cutoffProfileSchwartz_zero_outside R (lt_of_lt_of_le zero_lt_one hR) | |
| χ hχ hχzero Q hQ)] | |
| change _ = physicalSchwartzCoefficient R n _ | |
| rw [cutoffProfileSchwartz_dilation_zero, schwartzOddPower_dilation_zero, | |
| physicalSchwartzCoefficient_dilation] | |
| theorem cutoff_normalized_defect_coefficient (a b R : ℝ) (hR : 0 < R) | |
| (m : ℕ) (ham : 2 * a * (m : ℝ) = 1) | |
| (χ : 𝓢(E, ℝ)) (hχ : HasCompactSupport (χ : E → ℝ)) | |
| (hχone : ∀ x : E, ‖x‖ < 1 / 2 → χ x = 1) | |
| (hχzero : ∀ x : E, 2 < ‖x‖ → χ x = 0) | |
| (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) | |
| (hstationary : ∀ y, stationarySimilarityDefect a b m Q y = 0) (n : frequencyLattice) : | |
| physicalSchwartzCoefficient 1 n (normalizedCutoffDefectSchwartz a b R m | |
| (χ.postcompCLM Complex.ofRealCLM) (hasCompactSupport_complexCutoff χ hχ) Q hQ) = | |
| physicalSchwartzCoefficient R n | |
| (cutoffResidualSchwartz χ hχ hχone hχzero a R hR m Q hQ) := by | |
| have he : schwartzPhysicalDilation 0 R hR | |
| (normalizedCutoffDefectSchwartz a b R m | |
| (χ.postcompCLM Complex.ofRealCLM) (hasCompactSupport_complexCutoff χ hχ) Q hQ) = | |
| cutoffResidualSchwartz χ hχ hχone hχzero a R hR m Q hQ := by | |
| ext y | |
| rw [schwartzPhysicalDilation_apply, | |
| normalizedCutoffDefectSchwartz_apply a b R hR m χ hχ Q hQ, | |
| hstationary, | |
| cutoffResidualSchwartz_apply χ hχ hχone hχzero a R hR m ham Q hQ] | |
| simp only [mul_zero, zero_add, Real.rpow_zero, Complex.ofReal_one, one_mul, | |
| smul_smul, mul_inv_cancel₀ hR.ne', one_smul] | |
| rw [← he, physicalSchwartzCoefficient_dilation] | |
| noncomputable def cutoffSampledMode (L : ℝ) (χ : 𝓢(E, ℂ)) | |
| (hχ : HasCompactSupport (χ : E → ℂ)) (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) | |
| (n : frequencyLattice) (t : ℝ) : ℂ := | |
| physicalSchwartzCoefficient 1 n (compactDilationFamily χ hχ Q hQ (expandingRadius L t)) | |
| theorem cutoffSampledMode_eq_physical (L : ℝ) (hL : 0 < L) | |
| (χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ)) | |
| (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) (n : frequencyLattice) (t : ℝ) : | |
| cutoffSampledMode L χ hχ Q hQ n t = | |
| physicalSchwartzCoefficient (expandingRadius L t) n | |
| (cutoffProfileSchwartz (expandingRadius L t) | |
| (mul_pos hL (Real.exp_pos _)) χ hχ Q hQ) := by | |
| exact (cutoffProfile_coefficient_fixed_scale _ (mul_pos hL (Real.exp_pos _)) | |
| χ hχ Q hQ n).symm | |
| theorem hasDerivAt_cutoffSampledMode (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 (cutoffSampledMode L χ hχ Q hQ n) | |
| (physicalSchwartzCoefficient 1 n (cutoffVelocitySchwartz χ hχ Q hQ (expandingRadius L s))) s := by | |
| have hd := hasDerivAt_cutoffProfile_coefficient L s hL χ hχ Q hQ n | |
| have he : (fun t => schwartzLatticeCoefficient (expandingRadius L t) | |
| (radianFourierKernel (cutoffProfileSchwartz (expandingRadius L t) | |
| (mul_pos hL (Real.exp_pos _)) χ hχ Q hQ)) n) = | |
| cutoffSampledMode L χ hχ Q hQ n := by | |
| funext t | |
| exact (cutoffSampledMode_eq_physical L hL χ hχ Q hQ n t).symm | |
| rw [he] at hd | |
| convert hd using 1 | |
| simp only [physicalSchwartzCoefficient_apply, schwartzLatticeCoefficient, | |
| mul_one, inv_one, one_smul, radianFourierKernel_apply] | |
| rfl | |
| /-- The sampled profile has the actual Fourier mode derivative, including the | |
| same sampled residual used in the decaying Duhamel estimate. -/ | |
| theorem hasDerivAt_cutoffSampledMode_equation (a b k L s : ℝ) | |
| (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 0 < L) | |
| (hRs : 1 ≤ expandingRadius L s) (m : ℕ) (ham : 2 * a * (m : ℝ) = 1) | |
| (χ : 𝓢(E, ℝ)) (hχ : HasCompactSupport (χ : E → ℝ)) | |
| (hχone : ∀ x : E, ‖x‖ < 1 / 2 → χ x = 1) | |
| (hχzero : ∀ x : E, 2 < ‖x‖ → χ x = 0) | |
| (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) | |
| (hstationary : ∀ y, stationarySimilarityDefect a b m Q y = 0) (n : frequencyLattice) : | |
| let R := expandingRadius L s | |
| let ψ := cutoffProfileSchwartz R (lt_of_lt_of_le zero_lt_one hRs) | |
| (χ.postcompCLM Complex.ofRealCLM) (hasCompactSupport_complexCutoff χ hχ) Q hQ | |
| let U := schwartzTorusSample a k R ha1 hk hRs (radianFourierKernel ψ) | |
| let residual := cutoffResidualSchwartz χ hχ hχone hχzero a R | |
| (lt_of_lt_of_le zero_lt_one hRs) m Q hQ | |
| HasDerivAt | |
| (cutoffSampledMode L (χ.postcompCLM Complex.ofRealCLM) | |
| (hasCompactSupport_complexCutoff χ hχ) Q hQ n) | |
| (((-(a : ℂ)) + Complex.I * ((b : ℂ) - | |
| (((R ^ (2 : ℕ))⁻¹ : ℝ) : ℂ) * ((‖n‖ ^ 2 : ℝ) : ℂ))) * | |
| cutoffSampledMode L (χ.postcompCLM Complex.ofRealCLM) | |
| (hasCompactSupport_complexCutoff χ hχ) Q hQ n s - | |
| Complex.I * expandingFourierCoefficient a k R | |
| (expandingOddPower a k R ha ha1 hk hRs m U) n - | |
| Complex.I * physicalSchwartzCoefficient R n residual) s := by | |
| intro R ψ U residual | |
| have hd := hasDerivAt_cutoffSampledMode L s hL | |
| (χ.postcompCLM Complex.ofRealCLM) (hasCompactSupport_complexCutoff χ hχ) Q hQ n | |
| have he := physicalSchwartzCoefficient_cutoffDefect a b R m | |
| (χ.postcompCLM Complex.ofRealCLM) (hasCompactSupport_complexCutoff χ hχ) Q hQ n | |
| rw [cutoff_normalized_defect_coefficient a b R | |
| (lt_of_lt_of_le zero_lt_one hRs) m ham χ hχ hχone hχzero Q hQ hstationary n, | |
| cutoff_normalized_nonlinearity_coefficient a k R ha ha1 hk hRs m | |
| χ hχ hχzero Q hQ n] at he | |
| apply hd.congr_deriv | |
| change physicalSchwartzCoefficient 1 n | |
| (cutoffVelocitySchwartz (χ.postcompCLM Complex.ofRealCLM) | |
| (hasCompactSupport_complexCutoff χ hχ) Q hQ R) = _ | |
| change physicalSchwartzCoefficient R n residual = | |
| Complex.I * physicalSchwartzCoefficient 1 n | |
| (cutoffVelocitySchwartz (χ.postcompCLM Complex.ofRealCLM) | |
| (hasCompactSupport_complexCutoff χ hχ) Q hQ R) + | |
| (-(((R ^ (2 : ℕ))⁻¹ : ℝ) : ℂ) * ((‖n‖ ^ 2 : ℝ) : ℂ) + | |
| Complex.I * (a : ℂ) + (b : ℂ)) * | |
| cutoffSampledMode L (χ.postcompCLM Complex.ofRealCLM) | |
| (hasCompactSupport_complexCutoff χ hχ) Q hQ n s - | |
| expandingFourierCoefficient a k R (expandingOddPower a k R ha ha1 hk hRs m U) n at he | |
| linear_combination Complex.I * he + | |
| (physicalSchwartzCoefficient 1 n | |
| (cutoffVelocitySchwartz (χ.postcompCLM Complex.ofRealCLM) | |
| (hasCompactSupport_complexCutoff χ hχ) Q hQ R) + | |
| (a : ℂ) * cutoffSampledMode L (χ.postcompCLM Complex.ofRealCLM) | |
| (hasCompactSupport_complexCutoff χ hχ) Q hQ n s) * Complex.I_sq | |
| end DefocusingNLS | |
| end OAI | |