openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Linear /ExpandingDuhamelContinuity.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingDuhamel | |
| import Mathlib.Topology.Order.ProjIcc | |
| import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | |
| namespace OAI | |
| /-! # Continuity of the moving-scale Duhamel map | |
| Rescaling the integration time to the fixed unit interval gives strong | |
| continuity, including at time zero, without operator-norm continuity. | |
| -/ | |
| open Set MeasureTheory | |
| namespace DefocusingNLS | |
| attribute [local irreducible] expandingFreeStep expandingFreeFamily | |
| private noncomputable def unitTime (u : ℝ) : Icc (0 : ℝ) 1 := | |
| projIcc 0 1 zero_le_one u | |
| private noncomputable def expandingDuhamelParameters (p : ExpandingFreeParameters) | |
| (u : ℝ) : ExpandingFreeParameters := | |
| (⟨expandingRadius p.1.1 ((p.2 : ℝ) * unitTime u), | |
| p.1.2.trans (expandingRadius_ge _ _ p.1.2 (mul_nonneg p.2.2 (unitTime u).2.1))⟩, | |
| ⟨(p.2 : ℝ) - (p.2 : ℝ) * unitTime u, | |
| sub_nonneg.mpr (mul_le_of_le_one_right p.2.2 (unitTime u).2.2)⟩) | |
| private theorem continuous_expandingDuhamelParameters : | |
| Continuous (fun z : ExpandingFreeParameters × ℝ => expandingDuhamelParameters z.1 z.2) := by | |
| have hL : Continuous (fun z : ExpandingFreeParameters × ℝ => z.1.1.1) := | |
| continuous_subtype_val.comp (continuous_fst.comp continuous_fst) | |
| have hs : Continuous (fun z : ExpandingFreeParameters × ℝ => (z.1.2 : ℝ)) := | |
| continuous_subtype_val.comp (continuous_snd.comp continuous_fst) | |
| have hu : Continuous (fun z : ExpandingFreeParameters × ℝ => (unitTime z.2 : ℝ)) := | |
| continuous_subtype_val.comp (continuous_projIcc.comp continuous_snd) | |
| apply Continuous.prodMk | |
| · apply Continuous.subtype_mk | |
| exact hL.mul (Real.continuous_exp.comp ((hs.mul hu).div_const 2)) | |
| · apply Continuous.subtype_mk | |
| exact hs.sub (hs.mul hu) | |
| private noncomputable def expandingUnitDuhamelIntegrand (a b k : ℝ) | |
| (ha : 0 < a) (hk : 8 < k) (r : ℝ → FourierL2) | |
| (p : ExpandingFreeParameters) (u : ℝ) : FourierL2 := | |
| expandingFreeFamily a b k ha hk (expandingDuhamelParameters p u) | |
| (r ((p.2 : ℝ) * unitTime u)) | |
| private theorem continuous_expandingUnitDuhamelIntegrand (a b k : ℝ) | |
| (ha : 0 < a) (hk : 8 < k) (r : ℝ → FourierL2) (hr : Continuous r) : | |
| Continuous (fun z : ExpandingFreeParameters × ℝ => | |
| expandingUnitDuhamelIntegrand a b k ha hk r z.1 z.2) := by | |
| have harg : Continuous (fun z : ExpandingFreeParameters × ℝ => | |
| (expandingDuhamelParameters z.1 z.2, r ((z.1.2 : ℝ) * unitTime z.2))) := by | |
| apply continuous_expandingDuhamelParameters.prodMk | |
| apply hr.comp | |
| exact (continuous_subtype_val.comp (continuous_snd.comp continuous_fst)).mul | |
| (continuous_subtype_val.comp (continuous_projIcc.comp continuous_snd)) | |
| unfold expandingUnitDuhamelIntegrand | |
| exact (continuous_expandingFreeFamily_uncurry a b k ha hk).comp harg | |
| private theorem expandingUnitDuhamelIntegrand_eq (a b k : ℝ) | |
| (ha : 0 < a) (hk : 8 < k) (r : ℝ → FourierL2) | |
| (p : ExpandingFreeParameters) (u : ℝ) (hu : u ∈ Icc 0 1) : | |
| expandingUnitDuhamelIntegrand a b k ha hk r p u = | |
| expandingDuhamelIntegrand a b k p.1.1 ha hk p.1.2 p.2 r ((p.2 : ℝ) * u) := by | |
| have htu : (p.2 : ℝ) * u ∈ Icc (0 : ℝ) (p.2 : ℝ) := | |
| ⟨mul_nonneg p.2.2 hu.1, mul_le_of_le_one_right p.2.2 hu.2⟩ | |
| rw [expandingDuhamelIntegrand_of_mem _ _ _ _ _ _ _ _ _ _ htu] | |
| simp only [expandingUnitDuhamelIntegrand, expandingFreeFamily, expandingDuhamelParameters, | |
| unitTime, projIcc_of_mem _ hu] | |
| rfl | |
| /-- A fixed unit-interval integral represents the moving-scale Duhamel term exactly. -/ | |
| theorem expandingDuhamel_eq_unitIntegral (a b k : ℝ) | |
| (ha : 0 < a) (hk : 8 < k) (r : ℝ → FourierL2) | |
| (p : ExpandingFreeParameters) : | |
| expandingDuhamel a b k p.1.1 ha hk p.1.2 p.2 r = | |
| (p.2 : ℝ) • ∫ u in Icc (0 : ℝ) 1, | |
| expandingUnitDuhamelIntegrand a b k ha hk r p u := by | |
| have hrescale := intervalIntegral.smul_integral_comp_mul_left | |
| (a := 0) (b := 1) | |
| (expandingDuhamelIntegrand a b k p.1.1 ha hk p.1.2 p.2 r) (p.2 : ℝ) | |
| simp only [mul_zero, mul_one] at hrescale | |
| rw [intervalIntegral.integral_of_le zero_le_one, | |
| intervalIntegral.integral_of_le (show (0 : ℝ) ≤ (p.2 : ℝ) from p.2.2), | |
| ← integral_Icc_eq_integral_Ioc, ← integral_Icc_eq_integral_Ioc] at hrescale | |
| rw [expandingDuhamel, ← hrescale] | |
| congr 1 | |
| apply setIntegral_congr_fun measurableSet_Icc | |
| intro u hu | |
| exact (expandingUnitDuhamelIntegrand_eq a b k ha hk r p u hu).symm | |
| /-- The inhomogeneous solution depends strongly continuously on starting scale and time. -/ | |
| theorem continuous_expandingDuhamel (a b k : ℝ) | |
| (ha : 0 < a) (hk : 8 < k) (r : ℝ → FourierL2) (hr : Continuous r) : | |
| Continuous (fun p : ExpandingFreeParameters => | |
| expandingDuhamel a b k p.1.1 ha hk p.1.2 p.2 r) := by | |
| let hLc : LocallyCompactSpace {L : ℝ // 1 ≤ L} := | |
| (isClosed_Ici : IsClosed (Ici (1 : ℝ))).locallyCompactSpace | |
| let hSc : LocallyCompactSpace {s : ℝ // 0 ≤ s} := | |
| (isClosed_Ici : IsClosed (Ici (0 : ℝ))).locallyCompactSpace | |
| let : LocallyCompactSpace ExpandingFreeParameters := | |
| .locallyCompactSpace _ _ _ _ hLc hSc | |
| simp only [expandingDuhamel_eq_unitIntegral] | |
| exact (continuous_subtype_val.comp continuous_snd).smul | |
| (continuous_parametric_integral_of_continuous | |
| (continuous_expandingUnitDuhamelIntegrand a b k ha hk r hr) isCompact_Icc) | |
| /-- Continuous forcing on a closed time slab yields a continuous inhomogeneous solution. -/ | |
| theorem continuousOn_expandingDuhamel (a b k L T : ℝ) | |
| (ha : 0 < a) (hk : 8 < k) (hL : 1 ≤ L) (hT : 0 ≤ T) | |
| (r : ℝ → FourierL2) (hr : ContinuousOn r (Icc 0 T)) : | |
| ContinuousOn (fun t => expandingDuhamel a b k L ha hk hL t r) (Icc 0 T) := by | |
| let r' : ℝ → FourierL2 := fun t => r (projIcc 0 T hT t) | |
| have hr' : Continuous r' := hr.domRestrict.comp continuous_projIcc | |
| have hp : Continuous (fun t : Icc (0 : ℝ) T => | |
| ((⟨L, hL⟩, ⟨t.1, t.2.1⟩) : ExpandingFreeParameters)) := | |
| continuous_const.prodMk (continuous_subtype_val.subtype_mk _) | |
| rw [continuousOn_iff_continuous_domRestrict] | |
| convert (continuous_expandingDuhamel a b k ha hk r' hr').comp hp using 1 | |
| apply funext | |
| intro t | |
| change expandingDuhamel a b k L ha hk hL t r = | |
| expandingDuhamel a b k L ha hk hL t r' | |
| unfold expandingDuhamel | |
| apply setIntegral_congr_fun measurableSet_Icc | |
| intro τ hτ | |
| rw [expandingDuhamelIntegrand_of_mem _ _ _ _ _ _ _ _ _ _ hτ, | |
| expandingDuhamelIntegrand_of_mem _ _ _ _ _ _ _ _ _ _ hτ] | |
| have hτT : τ ∈ Icc 0 T := ⟨hτ.1, hτ.2.trans t.2.2⟩ | |
| simp only [r', projIcc_of_mem _ hτT] | |
| end DefocusingNLS | |
| end OAI | |