openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Nonlinear /CutoffGlobalOrbit.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingSlabTrajectory | |
| import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffPathContinuity | |
| namespace OAI | |
| /-! # A global cutoff orbit from compatible nonlinear steps | |
| The profile is the actual sampled cutoff of the smooth stationary function. | |
| Compatible step solutions with geometric perturbation bounds give a global | |
| nonlinear trajectory, with vanishing error relative to that same profile. | |
| The stable graph supplies these endpoint data in the subsequent application. | |
| -/ | |
| open Set Filter Topology | |
| open scoped SchwartzMap ContDiff | |
| namespace DefocusingNLS | |
| local notation "E" => EuclideanSpace ℝ (Fin 12) | |
| noncomputable def sampledCutoffProfileAtRadius (a k : ℝ) (ha1 : a < 1) (hk : 8 < k) | |
| (χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ)) | |
| (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) (R : {R : ℝ // 1 ≤ R}) : FourierL2 := | |
| schwartzTorusSample a k R.1 ha1 hk R.2 | |
| (radianFourierKernel (cutoffProfileSchwartz R.1 | |
| (lt_of_lt_of_le zero_lt_one R.2) χ hχ Q hQ)) | |
| noncomputable def sampledCutoffProfileOrbit (a k L : ℝ) | |
| (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) | |
| (χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ)) | |
| (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) (s : ℝ) : FourierL2 := | |
| sampledCutoffProfileAtRadius a k ha1 hk χ hχ Q hQ | |
| ⟨expandingRadius L (max 0 s), | |
| hL.trans (expandingRadius_ge L _ hL (le_max_left _ _))⟩ | |
| theorem continuous_sampledCutoffProfileOrbit (a k L : ℝ) | |
| (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) | |
| (χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ)) | |
| (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) : | |
| Continuous (sampledCutoffProfileOrbit a k L ha1 hk hL χ hχ Q hQ) := by | |
| have hR : Continuous (fun s : ℝ => | |
| (⟨expandingRadius L (max 0 s), | |
| hL.trans (expandingRadius_ge L _ hL (le_max_left _ _))⟩ : {R : ℝ // 1 ≤ R})) := by | |
| apply Continuous.subtype_mk | |
| unfold expandingRadius | |
| fun_prop | |
| exact (continuous_cutoffProfile_sample a k ha ha1 hk χ hχ Q hQ).comp hR | |
| theorem sampledCutoffProfileOrbit_at (a k L M : ℝ) | |
| (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) (hM : 0 < M) | |
| (χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ)) | |
| (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) (n : ℕ) (s : Icc (0 : ℝ) M) : | |
| sampledCutoffProfileOrbit a k L ha1 hk hL χ hχ Q hQ ((n : ℝ) * M + s) = | |
| sampledCutoffProfilePath a k (expandingRadius L ((n : ℝ) * M)) M ha ha1 hk | |
| (hL.trans (expandingRadius_ge L _ hL (by positivity))) χ hχ Q hQ s := by | |
| have hpos : 0 ≤ (n : ℝ) * M + s := add_nonneg (by positivity) s.2.1 | |
| unfold sampledCutoffProfileOrbit | |
| simp only [max_eq_right hpos] | |
| change sampledCutoffProfileAtRadius a k ha1 hk χ hχ Q hQ _ = | |
| sampledCutoffProfileAtRadius a k ha1 hk χ hχ Q hQ _ | |
| congr 1 | |
| apply Subtype.ext | |
| exact (expandingRadius_add L ((n : ℝ) * M) s).symm | |
| theorem exists_cutoffGlobal_orbit_of_steps (a b k L M : ℝ) | |
| (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) (hM : 0 < M) (m : ℕ) | |
| (χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ)) | |
| (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) | |
| (x : ℕ → FourierL2) (V : ℕ → C(Icc (0 : ℝ) M, FourierL2)) | |
| (hVM : ∀ n, V n ⟨M, hM.le, le_rfl⟩ = x (n + 1)) | |
| (hsol : ∀ n : ℕ, | |
| let Ln := expandingRadius L ((n : ℝ) * M) | |
| let hLn := hL.trans (expandingRadius_ge L ((n : ℝ) * M) hL (by positivity)) | |
| let q := sampledCutoffProfilePath a k Ln M ha ha1 hk hLn χ hχ Q hQ | |
| q + V n = expandingPicard a b k Ln M ha hk hLn hM.le | |
| (expandingNonlinearReaction a k Ln M ha ha1 hk hLn m) | |
| (q ⟨0, le_rfl, hM.le⟩ + x n) (q + V n)) | |
| (C r : ℝ) (hr : 0 ≤ r) (hr1 : r < 1) | |
| (hVbound : ∀ n, ‖V n‖ ≤ C * r ^ n) : | |
| ∃ u : ℝ → FourierL2, ∃ hu : ContinuousOn u (Ici 0), | |
| u 0 = sampledCutoffProfileOrbit a k L ha1 hk hL χ hχ Q hQ 0 + x 0 ∧ | |
| (∀ (S : ℝ) (hS : 0 < S), expandingSlabRestriction u hu S = | |
| expandingPicard a b k L S ha hk hL hS.le | |
| (expandingNonlinearReaction a k L S ha ha1 hk hL m) (u 0) | |
| (expandingSlabRestriction u hu S)) ∧ | |
| Tendsto (fun s => ‖u s - sampledCutoffProfileOrbit a k L ha1 hk hL χ hχ Q hQ s‖) | |
| atTop (𝓝 0) := by | |
| let q : ℕ → C(Icc (0 : ℝ) M, FourierL2) := fun n => | |
| sampledCutoffProfilePath a k (expandingRadius L ((n : ℝ) * M)) M ha ha1 hk | |
| (hL.trans (expandingRadius_ge L _ hL (by positivity))) χ hχ Q hQ | |
| let W := fun n => q n + V n | |
| let qglob := sampledCutoffProfileOrbit a k L ha1 hk hL χ hχ Q hQ | |
| have hV0 (n : ℕ) : V n ⟨0, le_rfl, hM.le⟩ = x n := by | |
| have hz := congrArg (fun f : C(Icc (0 : ℝ) M, FourierL2) => f ⟨0, le_rfl, hM.le⟩) | |
| (hsol n) | |
| simp only [ContinuousMap.add_apply, expandingPicard_initial] at hz | |
| exact add_left_cancel hz | |
| have hq (n : ℕ) (s : Icc (0 : ℝ) M) : | |
| qglob ((n : ℝ) * M + s) = q n s := | |
| sampledCutoffProfileOrbit_at a k L M ha ha1 hk hL hM χ hχ Q hQ n s | |
| have hqjoin (n : ℕ) : q n ⟨M, hM.le, le_rfl⟩ = q (n + 1) ⟨0, le_rfl, hM.le⟩ := by | |
| rw [← hq n, ← hq (n + 1)] | |
| congr 1 | |
| push_cast | |
| ring | |
| have hjoinV (n : ℕ) : V n ⟨M, hM.le, le_rfl⟩ = V (n + 1) ⟨0, le_rfl, hM.le⟩ := | |
| (hVM n).trans (hV0 (n + 1)).symm | |
| have hjoinW (n : ℕ) : W n ⟨M, hM.le, le_rfl⟩ = W (n + 1) ⟨0, le_rfl, hM.le⟩ := by | |
| change q n _ + V n _ = q (n + 1) _ + V (n + 1) _ | |
| rw [hqjoin n, hjoinV n] | |
| let u := gluedSlabPath M hM W hjoinW | |
| let v := gluedSlabPath M hM V hjoinV | |
| have hu : ContinuousOn u (Ici 0) := continuousOn_gluedSlabPath M hM W hjoinW | |
| have hWsol (n : ℕ) : | |
| W n = expandingPicard a b k (expandingRadius L ((n : ℝ) * M)) M ha hk | |
| (hL.trans (expandingRadius_ge L _ hL (by positivity))) hM.le | |
| (expandingNonlinearReaction a k (expandingRadius L ((n : ℝ) * M)) M ha ha1 hk | |
| (hL.trans (expandingRadius_ge L _ hL (by positivity))) m) | |
| (W n ⟨0, le_rfl, hM.le⟩) (W n) := by | |
| simpa only [W, ContinuousMap.add_apply, hV0] using hsol n | |
| refine ⟨u, hu, ?_, ?_, ?_⟩ | |
| · have hu0 := gluedSlabPath_at M hM W hjoinW 0 ⟨0, le_rfl, hM.le⟩ | |
| have hq0 := hq 0 ⟨0, le_rfl, hM.le⟩ | |
| simp only [Nat.cast_zero, zero_mul, add_zero] at hu0 hq0 | |
| change u 0 = W 0 ⟨0, le_rfl, hM.le⟩ at hu0 | |
| change u 0 = qglob 0 + x 0 | |
| rw [hu0] | |
| change q 0 _ + V 0 _ = _ | |
| rw [hV0, hq0] | |
| · intro S hS | |
| exact expandingGluedSlabs_picard a b k L M ha ha1 hk hL hM m W hjoinW hWsol S hS | |
| · have hv : Tendsto v atTop (𝓝 0) := gluedSlabPath_tendsto_zero M hM V hjoinV C r hr hr1 | |
| (fun n s => (ContinuousMap.norm_coe_le_norm (V n) s).trans (hVbound n)) | |
| have heq : (fun s => u s - qglob s) =ᶠ[atTop] v := by | |
| filter_upwards [eventually_ge_atTop (0 : ℝ)] with t ht | |
| let n := ⌊t / M⌋₊ | |
| let s : Icc (0 : ℝ) M := ⟨t - (n : ℝ) * M, by | |
| have hh := equalSlabInterval_floor M hM t ht | |
| constructor <;> dsimp [n] <;> linarith [hh.1, hh.2]⟩ | |
| have he : (n : ℝ) * M + s = t := by dsimp [s]; ring | |
| have hut := gluedSlabPath_at M hM W hjoinW n s | |
| have hvt := gluedSlabPath_at M hM V hjoinV n s | |
| have hqt := hq n s | |
| rw [he] at hut hvt hqt | |
| change u t = W n s at hut | |
| change v t = V n s at hvt | |
| change u t - qglob t = v t | |
| rw [hut, hvt, hqt] | |
| change q n s + V n s - q n s = V n s | |
| abel | |
| simpa using (hv.congr' heq.symm).norm | |
| end DefocusingNLS | |
| end OAI | |