openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Nonlinear /CutoffMildEquation.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffForcingHistory | |
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingModeConverse | |
| namespace OAI | |
| /-! # The genuine sampled cutoff profile is a forced mild solution -/ | |
| open Set Filter Topology | |
| open scoped SchwartzMap ContDiff | |
| namespace DefocusingNLS | |
| local notation "E" => EuclideanSpace ℝ (Fin 12) | |
| theorem expandingModeRate_eq_radius (a b L t : ℝ) (hL : 0 < L) (n : frequencyLattice) : | |
| expandingModeRate a b L t n = -(a : ℂ) + Complex.I * ((b : ℂ) - | |
| ((((expandingRadius L t) ^ (2 : ℕ))⁻¹ : ℝ) : ℂ) * ((‖n‖ ^ 2 : ℝ) : ℂ)) := by | |
| have hR : 0 < expandingRadius L t := mul_pos hL (Real.exp_pos _) | |
| have hp : (expandingRadius L t) ^ (-2 : ℝ) = ((expandingRadius L t) ^ (2 : ℕ))⁻¹ := by | |
| rw [Real.rpow_neg hR.le, Real.rpow_two] | |
| have he : ((expandingRadius L t) ^ (2 : ℕ))⁻¹ = L ^ (-2 : ℝ) * Real.exp (-t) := by | |
| have h := expandingRadius_residual_power 0 L t hL | |
| simp only [sub_zero, add_zero] at h | |
| rw [hp] at h | |
| simpa only [show -(2 : ℝ) * t / 2 = -t by ring] using h | |
| unfold expandingModeRate | |
| rw [he] | |
| push_cast | |
| ring | |
| theorem hasDerivAt_sampledCutoffProfileHistory_coefficient (a b k L T : ℝ) | |
| (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) (hT : 0 ≤ T) | |
| (m : ℕ) (ham : 2 * a * (m : ℝ) = 1) | |
| (χ : 𝓢(E, ℝ)) (hχ : HasCompactSupport (χ : E → ℝ)) | |
| (hχone : ∀ x : E, ‖x‖ < 1 / 2 → χ x = 1) | |
| (hχzero : ∀ x : E, 2 < ‖x‖ → χ x = 0) | |
| (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) | |
| (hstationary : ∀ y, stationarySimilarityDefect a b m Q y = 0) | |
| (t : ℝ) (ht : t ∈ Ioo 0 T) (n : frequencyLattice) : | |
| let u := sampledCutoffProfileHistory a k L T ha ha1 hk hL hT | |
| (χ.postcompCLM Complex.ofRealCLM) (hasCompactSupport_complexCutoff χ hχ) Q hQ | |
| let r := sampledCutoffForcingHistory a k L T ha ha1 hk hL hT | |
| χ hχ hχone hχzero m Q hQ | |
| HasDerivAt (fun s => expandingFourierCoefficient a k (expandingRadius L s) (u s) n) | |
| (expandingModeRate a b L t n * | |
| expandingFourierCoefficient a k (expandingRadius L t) (u t) n + | |
| expandingFourierCoefficient a k (expandingRadius L t) (r t) n) t := by | |
| intro u r | |
| have hLp : 0 < L := lt_of_lt_of_le zero_lt_one hL | |
| have hRt : 1 ≤ expandingRadius L t := hL.trans (expandingRadius_ge L t hL ht.1.le) | |
| have hd := hasDerivAt_cutoffSampledMode_equation a b k L t ha ha1 hk hLp hRt | |
| m ham χ hχ hχone hχzero Q hQ hstationary n | |
| have he : (fun s => expandingFourierCoefficient a k (expandingRadius L s) (u s) n) =ᶠ[𝓝 t] | |
| cutoffSampledMode L (χ.postcompCLM Complex.ofRealCLM) | |
| (hasCompactSupport_complexCutoff χ hχ) Q hQ n := by | |
| filter_upwards [Ioo_mem_nhds ht.1 ht.2] with s hs | |
| exact sampledCutoffProfileHistory_coefficient a k L T ha ha1 hk hL hT | |
| (χ.postcompCLM Complex.ofRealCLM) (hasCompactSupport_complexCutoff χ hχ) Q hQ s | |
| ⟨hs.1.le, hs.2.le⟩ n | |
| apply (hd.congr_of_eventuallyEq he).congr_deriv | |
| rw [expandingModeRate_eq_radius a b L t hLp n, | |
| sampledCutoffProfileHistory_coefficient a k L T ha ha1 hk hL hT | |
| (χ.postcompCLM Complex.ofRealCLM) (hasCompactSupport_complexCutoff χ hχ) | |
| Q hQ t ⟨ht.1.le, ht.2.le⟩ n] | |
| have hforce : expandingFourierCoefficient a k (expandingRadius L t) (r t) n = | |
| (-Complex.I) * expandingFourierCoefficient a k (expandingRadius L t) | |
| (expandingOddPower a k (expandingRadius L t) ha ha1 hk hRt m (u t)) n + | |
| (-Complex.I) * physicalSchwartzCoefficient (expandingRadius L t) n | |
| (cutoffResidualSchwartz χ hχ hχone hχzero a (expandingRadius L t) | |
| (lt_of_lt_of_le zero_lt_one hRt) m Q hQ) := by | |
| change expandingCoefficientCLM a k (expandingRadius L t) n (r t) = _ | |
| rw [show r t = _ from sampledCutoffForcingHistory_of_mem a k L T ha ha1 hk hL hT | |
| χ hχ hχone hχzero m Q hQ t ⟨ht.1.le, ht.2.le⟩, | |
| map_add, map_smul, map_smul] | |
| simp only [expandingCoefficientCLM_apply, smul_eq_mul] | |
| rw [sampledCutoffResidualHistory_of_mem a k L T ha ha1 hk hL hT | |
| χ hχ hχone hχzero m Q hQ t ⟨ht.1.le, ht.2.le⟩, | |
| schwartzTorusSample_coefficient] | |
| rfl | |
| have hu := sampledCutoffProfileHistory_of_mem a k L T ha ha1 hk hL hT | |
| (χ.postcompCLM Complex.ofRealCLM) (hasCompactSupport_complexCutoff χ hχ) | |
| Q hQ t ⟨ht.1.le, ht.2.le⟩ | |
| change u t = _ at hu | |
| rw [hforce, hu] | |
| ring | |
| theorem sampledCutoffProfile_mild (a b k L T : ℝ) | |
| (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) (hT : 0 ≤ T) | |
| (m : ℕ) (ham : 2 * a * (m : ℝ) = 1) | |
| (χ : 𝓢(E, ℝ)) (hχ : HasCompactSupport (χ : E → ℝ)) | |
| (hχone : ∀ x : E, ‖x‖ < 1 / 2 → χ x = 1) | |
| (hχzero : ∀ x : E, 2 < ‖x‖ → χ x = 0) | |
| (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) | |
| (hstationary : ∀ y, stationarySimilarityDefect a b m Q y = 0) | |
| (t : ℝ) (ht : t ∈ Icc 0 T) : | |
| let u := sampledCutoffProfileHistory a k L T ha ha1 hk hL hT | |
| (χ.postcompCLM Complex.ofRealCLM) (hasCompactSupport_complexCutoff χ hχ) Q hQ | |
| let r := sampledCutoffForcingHistory a k L T ha ha1 hk hL hT | |
| χ hχ hχone hχzero m Q hQ | |
| u t = expandingFreeStep a b k L t ha hk hL ht.1 (u 0) + | |
| expandingDuhamel a b k L ha hk hL t r := by | |
| intro u r | |
| exact expandingMild_of_mode_equation a b k L T ha hk hL u r | |
| (continuous_sampledCutoffProfileHistory a k L T ha ha1 hk hL hT | |
| (χ.postcompCLM Complex.ofRealCLM) (hasCompactSupport_complexCutoff χ hχ) Q hQ).continuousOn | |
| (continuous_sampledCutoffForcingHistory a k L T ha ha1 hk hL hT | |
| χ hχ hχone hχzero m Q hQ).continuousOn | |
| (fun s hs n => hasDerivAt_sampledCutoffProfileHistory_coefficient a b k L T | |
| ha ha1 hk hL hT m ham χ hχ hχone hχzero Q hQ hstationary s hs n) t ht | |
| end DefocusingNLS | |
| end OAI | |