openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Nonlinear /CutoffForcingHistory.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffResidualDuhamel | |
| import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffApproximateMode | |
| namespace OAI | |
| /-! # Continuous histories for the actual approximate nonlinear solution -/ | |
| open Set | |
| open scoped SchwartzMap ContDiff | |
| namespace DefocusingNLS | |
| local notation "E" => EuclideanSpace ℝ (Fin 12) | |
| noncomputable def sampledCutoffProfileHistory (a k L T : ℝ) | |
| (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) (hT : 0 ≤ T) | |
| (χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ)) | |
| (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) : ℝ → FourierL2 := | |
| fun t => sampledCutoffProfilePath a k L T ha ha1 hk hL χ hχ Q hQ (projIcc 0 T hT t) | |
| theorem continuous_sampledCutoffProfileHistory (a k L T : ℝ) | |
| (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) (hT : 0 ≤ T) | |
| (χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ)) | |
| (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) : | |
| Continuous (sampledCutoffProfileHistory a k L T ha ha1 hk hL hT χ hχ Q hQ) := | |
| (sampledCutoffProfilePath a k L T ha ha1 hk hL χ hχ Q hQ).continuous.comp continuous_projIcc | |
| theorem sampledCutoffProfileHistory_of_mem (a k L T : ℝ) | |
| (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) (hT : 0 ≤ T) | |
| (χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ)) | |
| (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) (t : ℝ) (ht : t ∈ Icc 0 T) : | |
| sampledCutoffProfileHistory a k L T ha ha1 hk hL hT χ hχ Q hQ t = | |
| schwartzTorusSample a k (expandingRadius L t) ha1 hk | |
| (hL.trans (expandingRadius_ge L t hL ht.1)) | |
| (radianFourierKernel (cutoffProfileSchwartz (expandingRadius L t) | |
| (mul_pos (lt_of_lt_of_le zero_lt_one hL) (Real.exp_pos _)) χ hχ Q hQ)) := by | |
| unfold sampledCutoffProfileHistory | |
| rw [projIcc_of_mem _ ht] | |
| rfl | |
| noncomputable def sampledCutoffNonlinearPath (a k L T : 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) (m : ℕ) : C(Icc (0 : ℝ) T, FourierL2) where | |
| toFun t := expandingOddPower a k (expandingRadiusCurve L T hL t).1 | |
| ha ha1 hk (expandingRadiusCurve L T hL t).2 m | |
| (sampledCutoffProfilePath a k L T ha ha1 hk hL χ hχ Q hQ t) | |
| continuous_toFun := (continuous_expandingOddPower_scale a k ha ha1 hk m).comp | |
| ((expandingRadiusCurve L T hL).continuous.prodMk | |
| (sampledCutoffProfilePath a k L T ha ha1 hk hL χ hχ Q hQ).continuous) | |
| noncomputable def sampledCutoffForcingPath (a k L T : ℝ) | |
| (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) | |
| (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) | |
| (m : ℕ) (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) : C(Icc (0 : ℝ) T, FourierL2) := | |
| (-Complex.I) • sampledCutoffNonlinearPath a k L T ha ha1 hk hL | |
| (χ.postcompCLM Complex.ofRealCLM) (hasCompactSupport_complexCutoff χ hχ) Q hQ m + | |
| (-Complex.I) • sampledCutoffResidualPath a k L T ha ha1 hk hL χ hχ hχone hχzero m Q hQ | |
| noncomputable def sampledCutoffForcingHistory (a k L T : ℝ) | |
| (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) (hT : 0 ≤ T) | |
| (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) | |
| (m : ℕ) (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) : ℝ → FourierL2 := | |
| fun t => sampledCutoffForcingPath a k L T ha ha1 hk hL χ hχ hχone hχzero m Q hQ | |
| (projIcc 0 T hT t) | |
| theorem continuous_sampledCutoffForcingHistory (a k L T : ℝ) | |
| (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) (hT : 0 ≤ T) | |
| (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) | |
| (m : ℕ) (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) : | |
| Continuous (sampledCutoffForcingHistory a k L T ha ha1 hk hL hT χ hχ hχone hχzero m Q hQ) := | |
| (sampledCutoffForcingPath a k L T ha ha1 hk hL χ hχ hχone hχzero m Q hQ).continuous.comp | |
| continuous_projIcc | |
| theorem sampledCutoffForcingHistory_of_mem (a k L T : ℝ) | |
| (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) (hT : 0 ≤ T) | |
| (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) | |
| (m : ℕ) (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) (t : ℝ) (ht : t ∈ Icc 0 T) : | |
| sampledCutoffForcingHistory a k L T ha ha1 hk hL hT χ hχ hχone hχzero m Q hQ t = | |
| (-Complex.I) • expandingOddPower a k (expandingRadius L t) ha ha1 hk | |
| (hL.trans (expandingRadius_ge L t hL ht.1)) m | |
| (sampledCutoffProfileHistory a k L T ha ha1 hk hL hT | |
| (χ.postcompCLM Complex.ofRealCLM) (hasCompactSupport_complexCutoff χ hχ) Q hQ t) + | |
| (-Complex.I) • sampledCutoffResidualHistory a k L T ha ha1 hk hL hT | |
| χ hχ hχone hχzero m Q hQ t := by | |
| unfold sampledCutoffForcingHistory sampledCutoffProfileHistory sampledCutoffResidualHistory | |
| rw [projIcc_of_mem _ ht] | |
| rfl | |
| theorem sampledCutoffProfileHistory_coefficient (a k L T : ℝ) | |
| (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) (hT : 0 ≤ T) | |
| (χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ)) | |
| (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) (t : ℝ) (ht : t ∈ Icc 0 T) | |
| (n : frequencyLattice) : | |
| expandingFourierCoefficient a k (expandingRadius L t) | |
| (sampledCutoffProfileHistory a k L T ha ha1 hk hL hT χ hχ Q hQ t) n = | |
| cutoffSampledMode L χ hχ Q hQ n t := by | |
| rw [sampledCutoffProfileHistory_of_mem a k L T ha ha1 hk hL hT χ hχ Q hQ t ht, | |
| schwartzTorusSample_coefficient] | |
| exact (cutoffSampledMode_eq_physical L (lt_of_lt_of_le zero_lt_one hL) | |
| χ hχ Q hQ n t).symm | |
| theorem sampledCutoffResidualHistory_of_mem (a k L T : ℝ) | |
| (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) (hT : 0 ≤ T) | |
| (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) | |
| (m : ℕ) (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) (t : ℝ) (ht : t ∈ Icc 0 T) : | |
| sampledCutoffResidualHistory a k L T ha ha1 hk hL hT χ hχ hχone hχzero m Q hQ t = | |
| schwartzTorusSample a k (expandingRadius L t) ha1 hk | |
| (hL.trans (expandingRadius_ge L t hL ht.1)) (radianFourierKernel | |
| (cutoffResidualSchwartz χ hχ hχone hχzero a (expandingRadius L t) | |
| (mul_pos (lt_of_lt_of_le zero_lt_one hL) (Real.exp_pos _)) m Q hQ)) := by | |
| unfold sampledCutoffResidualHistory | |
| rw [projIcc_of_mem _ ht] | |
| rfl | |
| end DefocusingNLS | |
| end OAI | |