openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Linear /ExpandingDuhamelLinear.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingDuhamel | |
| namespace OAI | |
| /-! # Linearity of the moving-scale inhomogeneous term -/ | |
| open Set MeasureTheory | |
| namespace DefocusingNLS | |
| attribute [local irreducible] expandingFreeStep | |
| @[simp] theorem expandingDuhamel_zero (a b k L : ℝ) | |
| (ha : 0 < a) (hk : 8 < k) (hL : 1 ≤ L) (t : ℝ) : | |
| expandingDuhamel a b k L ha hk hL t (fun _ => 0) = 0 := by | |
| unfold expandingDuhamel | |
| have hz : expandingDuhamelIntegrand a b k L ha hk hL t (fun _ => 0) = 0 := by | |
| funext τ | |
| unfold expandingDuhamelIntegrand | |
| split_ifs <;> simp only [map_zero, Pi.zero_apply] | |
| simp only [hz, Pi.zero_apply, integral_zero] | |
| theorem expandingDuhamel_add (a b k L : ℝ) | |
| (ha : 0 < a) (hk : 8 < k) (hL : 1 ≤ L) | |
| (t : ℝ) (r q : ℝ → FourierL2) | |
| (hr : ContinuousOn r (Icc 0 t)) (hq : ContinuousOn q (Icc 0 t)) : | |
| expandingDuhamel a b k L ha hk hL t (fun τ => r τ + q τ) = | |
| expandingDuhamel a b k L ha hk hL t r + | |
| expandingDuhamel a b k L ha hk hL t q := by | |
| unfold expandingDuhamel | |
| rw [← integral_add (integrableOn_expandingDuhamelIntegrand a b k L ha hk hL t r hr) | |
| (integrableOn_expandingDuhamelIntegrand a b k L ha hk hL t q hq)] | |
| apply integral_congr_ae | |
| filter_upwards [ae_restrict_mem measurableSet_Icc] with τ hτ | |
| simp only [expandingDuhamelIntegrand_of_mem _ _ _ _ _ _ _ _ _ _ hτ, map_add] | |
| theorem expandingDuhamel_real_smul (a b k L : ℝ) | |
| (ha : 0 < a) (hk : 8 < k) (hL : 1 ≤ L) | |
| (t c : ℝ) (r : ℝ → FourierL2) : | |
| expandingDuhamel a b k L ha hk hL t (fun τ => c • r τ) = | |
| c • expandingDuhamel a b k L ha hk hL t r := by | |
| unfold expandingDuhamel | |
| rw [← integral_smul] | |
| apply integral_congr_ae | |
| filter_upwards [ae_restrict_mem measurableSet_Icc] with τ hτ | |
| simp only [expandingDuhamelIntegrand_of_mem _ _ _ _ _ _ _ _ _ _ hτ, | |
| ContinuousLinearMap.map_smul_of_tower] | |
| end DefocusingNLS | |
| end OAI | |