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) (Ο‡ : 𝓒(E, β„‚)) (hΟ‡ : HasCompactSupport (Ο‡ : E β†’ β„‚)) (Q : E β†’ β„‚) (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) Ο‡ hΟ‡ Q hQ)) noncomputable def sampledCutoffProfileOrbit (a k L : ℝ) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≀ L) (Ο‡ : 𝓒(E, β„‚)) (hΟ‡ : HasCompactSupport (Ο‡ : E β†’ β„‚)) (Q : E β†’ β„‚) (hQ : ContDiff ℝ ∞ Q) (s : ℝ) : FourierL2 := sampledCutoffProfileAtRadius a k ha1 hk Ο‡ hΟ‡ 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) (Ο‡ : 𝓒(E, β„‚)) (hΟ‡ : HasCompactSupport (Ο‡ : E β†’ β„‚)) (Q : E β†’ β„‚) (hQ : ContDiff ℝ ∞ Q) : Continuous (sampledCutoffProfileOrbit a k L ha1 hk hL Ο‡ hΟ‡ 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 Ο‡ hΟ‡ 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) (Ο‡ : 𝓒(E, β„‚)) (hΟ‡ : HasCompactSupport (Ο‡ : E β†’ β„‚)) (Q : E β†’ β„‚) (hQ : ContDiff ℝ ∞ Q) (n : β„•) (s : Icc (0 : ℝ) M) : sampledCutoffProfileOrbit a k L ha1 hk hL Ο‡ hΟ‡ 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))) Ο‡ hΟ‡ 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 Ο‡ hΟ‡ Q hQ _ = sampledCutoffProfileAtRadius a k ha1 hk Ο‡ hΟ‡ 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 : β„•) (Ο‡ : 𝓒(E, β„‚)) (hΟ‡ : HasCompactSupport (Ο‡ : E β†’ β„‚)) (Q : E β†’ β„‚) (hQ : ContDiff ℝ ∞ Q) (x : β„• β†’ FourierL2) (V : β„• β†’ C(Icc (0 : ℝ) M, FourierL2)) (hVM : βˆ€ n, V n ⟨M, hM.le, le_rfl⟩ = x (n + 1)) (hsol : βˆ€ n : β„•, 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 Ο‡ hΟ‡ 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 : βˆ€ n, β€–V nβ€– ≀ C * r ^ n) : βˆƒ u : ℝ β†’ FourierL2, βˆƒ hu : ContinuousOn u (Ici 0), u 0 = sampledCutoffProfileOrbit a k L ha1 hk hL Ο‡ hΟ‡ Q hQ 0 + x 0 ∧ (βˆ€ (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 => β€–u s - sampledCutoffProfileOrbit a k L ha1 hk hL Ο‡ hΟ‡ Q hQ sβ€–) atTop (𝓝 0) := by let q : β„• β†’ 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))) Ο‡ hΟ‡ Q hQ let W := fun n => q n + V n let qglob := sampledCutoffProfileOrbit a k L ha1 hk hL Ο‡ hΟ‡ Q hQ have hV0 (n : β„•) : 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 : β„•) (s : Icc (0 : ℝ) M) : qglob ((n : ℝ) * M + s) = q n s := sampledCutoffProfileOrbit_at a k L M ha ha1 hk hL hM Ο‡ hΟ‡ Q hQ n s have hqjoin (n : β„•) : 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 : β„•) : 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 : β„•) : 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 : β„•) : 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, ?_, ?_, ?_⟩ Β· 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] Β· intro S hS exact expandingGluedSlabs_picard a b k L M ha ha1 hk hL hM m W hjoinW hWsol S hS Β· 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) =αΆ [atTop] v := by filter_upwards [eventually_ge_atTop (0 : ℝ)] with t ht let n := ⌊t / MβŒ‹β‚Š 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