openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Linear /ExpandingEnergyComponents.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingFreeStep | |
| namespace OAI | |
| /-! # The two exact energy components on an expanding torus | |
| The normalized Fourier space has a fixed Hilbert norm. These two bounded | |
| observations recover the low and high terms of the radius-dependent norm. | |
| -/ | |
| open scoped ENNReal | |
| namespace DefocusingNLS | |
| noncomputable def expandingLowFraction (a k L : ℝ) (n : frequencyLattice) : ℝ := | |
| L ^ (2 * a) * (1 + ‖n‖ ^ 2) ^ (6 - a) / expandingSobolevWeightSq a k L n | |
| theorem expandingLowFraction_bounds (a k L : ℝ) (hL : 1 ≤ L) | |
| (n : frequencyLattice) : 0 ≤ expandingLowFraction a k L n ∧ | |
| expandingLowFraction a k L n ≤ 1 := by | |
| constructor | |
| · exact div_nonneg (by positivity) (expandingSobolevWeightSq_pos a k L hL n).le | |
| · apply (div_le_one (expandingSobolevWeightSq_pos a k L hL n)).mpr | |
| unfold expandingSobolevWeightSq | |
| exact le_add_of_nonneg_right (by positivity) | |
| theorem expandingHighFraction_eq (a k L : ℝ) (hL : 1 ≤ L) | |
| (n : frequencyLattice) : | |
| 1 - expandingLowFraction a k L n = | |
| L ^ (12 - 2 * k) * ‖n‖ ^ (2 * k) / expandingSobolevWeightSq a k L n := by | |
| have hp := (expandingSobolevWeightSq_pos a k L hL n).ne' | |
| unfold expandingLowFraction | |
| field_simp | |
| unfold expandingSobolevWeightSq | |
| ring | |
| noncomputable def fourierEnergyComponent (w : frequencyLattice → ℝ) | |
| (hw : ∀ n, 0 ≤ w n ∧ w n ≤ 1) : FourierL2 →L[ℂ] FourierL2 := | |
| let F : FourierL2 → FourierL2 := fun f => | |
| ⟨fun n => (Real.sqrt (w n) : ℂ) * f n, by | |
| apply (lp.memℓp f).mono' | |
| intro n | |
| rw [norm_mul, Complex.norm_real, Real.norm_eq_abs, | |
| abs_of_nonneg (Real.sqrt_nonneg _)] | |
| exact mul_le_of_le_one_left (norm_nonneg _) (Real.sqrt_le_one.mpr (hw n).2)⟩ | |
| LinearMap.mkContinuous | |
| { toFun := F | |
| map_add' := by intro f g; ext n; exact mul_add _ _ _ | |
| map_smul' := by intro c f; ext n; change _ * (c * _) = c * (_ * _); ring } | |
| 1 (by | |
| intro f | |
| rw [one_mul] | |
| apply lp.norm_mono (by norm_num : (2 : ℝ≥0∞) ≠ 0) | |
| intro n | |
| change ‖(Real.sqrt (w n) : ℂ) * f n‖ ≤ ‖f n‖ | |
| rw [norm_mul, Complex.norm_real, Real.norm_eq_abs, | |
| abs_of_nonneg (Real.sqrt_nonneg _)] | |
| exact mul_le_of_le_one_left (norm_nonneg _) (Real.sqrt_le_one.mpr (hw n).2)) | |
| @[simp] theorem fourierEnergyComponent_apply (w : frequencyLattice → ℝ) | |
| (hw : ∀ n, 0 ≤ w n ∧ w n ≤ 1) (f : FourierL2) (n : frequencyLattice) : | |
| fourierEnergyComponent w hw f n = (Real.sqrt (w n) : ℂ) * f n := rfl | |
| theorem fourierEnergyComponent_norm_sq (w : frequencyLattice → ℝ) | |
| (hw : ∀ n, 0 ≤ w n ∧ w n ≤ 1) (f : FourierL2) : | |
| ‖fourierEnergyComponent w hw f‖ ^ 2 = ∑' n, w n * ‖f n‖ ^ 2 := by | |
| have he := lp.norm_rpow_eq_tsum (p := 2) (by norm_num) | |
| (fourierEnergyComponent w hw f) | |
| simp only [ENNReal.toReal_ofNat, Real.rpow_two] at he | |
| rw [he] | |
| apply tsum_congr | |
| intro n | |
| rw [fourierEnergyComponent_apply, norm_mul, Complex.norm_real, Real.norm_eq_abs, | |
| abs_of_nonneg (Real.sqrt_nonneg _), mul_pow, Real.sq_sqrt (hw n).1] | |
| theorem summable_fourierEnergyComponent (w : frequencyLattice → ℝ) | |
| (hw : ∀ n, 0 ≤ w n ∧ w n ≤ 1) (f : FourierL2) : | |
| Summable (fun n => w n * ‖f n‖ ^ 2) := by | |
| have h := lp.hasSum_norm (p := 2) (by norm_num) (fourierEnergyComponent w hw f) | |
| simp only [ENNReal.toReal_ofNat, Real.rpow_two] at h | |
| convert h.summable using 1 | |
| funext n | |
| rw [fourierEnergyComponent_apply, norm_mul, Complex.norm_real, Real.norm_eq_abs, | |
| abs_of_nonneg (Real.sqrt_nonneg _), mul_pow, Real.sq_sqrt (hw n).1] | |
| noncomputable def expandingLowEnergy (a k L : ℝ) (hL : 1 ≤ L) : | |
| FourierL2 →L[ℂ] FourierL2 := | |
| fourierEnergyComponent (expandingLowFraction a k L) (expandingLowFraction_bounds a k L hL) | |
| noncomputable def expandingHighEnergy (a k L : ℝ) (hL : 1 ≤ L) : | |
| FourierL2 →L[ℂ] FourierL2 := | |
| fourierEnergyComponent (fun n => 1 - expandingLowFraction a k L n) | |
| (fun n => ⟨sub_nonneg.mpr (expandingLowFraction_bounds a k L hL n).2, | |
| sub_le_self _ (expandingLowFraction_bounds a k L hL n).1⟩) | |
| theorem expandingEnergy_norm_sq (a k L : ℝ) (hL : 1 ≤ L) (f : FourierL2) : | |
| ‖f‖ ^ 2 = ‖expandingLowEnergy a k L hL f‖ ^ 2 + | |
| ‖expandingHighEnergy a k L hL f‖ ^ 2 := by | |
| have hl := summable_fourierEnergyComponent (expandingLowFraction a k L) | |
| (expandingLowFraction_bounds a k L hL) f | |
| have hh := summable_fourierEnergyComponent (fun n => 1 - expandingLowFraction a k L n) | |
| (fun n => ⟨sub_nonneg.mpr (expandingLowFraction_bounds a k L hL n).2, | |
| sub_le_self _ (expandingLowFraction_bounds a k L hL n).1⟩) f | |
| rw [expandingLowEnergy, expandingHighEnergy, fourierEnergyComponent_norm_sq, | |
| fourierEnergyComponent_norm_sq, ← hl.tsum_add hh] | |
| have he := lp.norm_rpow_eq_tsum (p := 2) (by norm_num) f | |
| simp only [ENNReal.toReal_ofNat, Real.rpow_two] at he | |
| rw [he] | |
| apply tsum_congr | |
| intro n | |
| ring | |
| end DefocusingNLS | |
| end OAI | |