openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Nonlinear /CutoffFamilyContinuity.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.SchwartzCompactContinuity | |
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.SchwartzSamplingContinuity | |
| import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffResidualSampling | |
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.SchwartzDyadicCutoff | |
| namespace OAI | |
| /-! # Continuous cutoff-profile and residual paths | |
| These are the actual sampled physical functions. The fixed-support rescaling | |
| makes their Schwartz and Hilbert continuity independent of any symbol bounds. | |
| The symbol bounds remain responsible for the separate uniform norm estimates. | |
| -/ | |
| open scoped SchwartzMap ContDiff | |
| namespace DefocusingNLS | |
| local notation "E" => EuclideanSpace ℝ (Fin 12) | |
| local notation "Scale" => {L : ℝ // 1 ≤ L} | |
| noncomputable def compactDilationFamily (κ : 𝓢(E, ℂ)) | |
| (hκ : HasCompactSupport (κ : E → ℂ)) (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) | |
| (L : ℝ) : 𝓢(E, ℂ) := | |
| compactSmoothFamily κ hκ (fun p => Q (p.1 • p.2)) | |
| (hQ.comp (contDiff_fst.smul contDiff_snd)) L | |
| @[simp] theorem compactDilationFamily_apply (κ : 𝓢(E, ℂ)) | |
| (hκ : HasCompactSupport (κ : E → ℂ)) (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) | |
| (L : ℝ) (x : E) : compactDilationFamily κ hκ Q hQ L x = κ x * Q (L • x) := rfl | |
| theorem continuous_compactDilationFamily (κ : 𝓢(E, ℂ)) | |
| (hκ : HasCompactSupport (κ : E → ℂ)) (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) : | |
| Continuous (compactDilationFamily κ hκ Q hQ) := by | |
| unfold compactDilationFamily | |
| exact continuous_compactSmoothFamily κ hκ _ _ | |
| private theorem continuous_scale_rpow (b : ℝ) : | |
| Continuous (fun L : Scale => L.1 ^ b) := | |
| continuous_subtype_val.rpow_const | |
| (fun L => Or.inl (ne_of_gt (lt_of_lt_of_le zero_lt_one L.2))) | |
| noncomputable def normalizedCompactProfile (a : ℝ) (κ : 𝓢(E, ℂ)) | |
| (hκ : HasCompactSupport (κ : E → ℂ)) (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) | |
| (L : Scale) : 𝓢(E, ℂ) := | |
| (L.1 ^ (2 * a) : ℝ) • compactDilationFamily κ hκ Q hQ L.1 | |
| @[simp] theorem normalizedCompactProfile_apply (a : ℝ) (κ : 𝓢(E, ℂ)) | |
| (hκ : HasCompactSupport (κ : E → ℂ)) (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) | |
| (L : Scale) (x : E) : | |
| normalizedCompactProfile a κ hκ Q hQ L x = | |
| κ x * ((L.1 ^ (2 * a) : ℝ) * Q (L.1 • x)) := by | |
| simp only [normalizedCompactProfile, smul_apply, compactDilationFamily_apply, | |
| Complex.real_smul] | |
| ring | |
| theorem continuous_normalizedCompactProfile (a : ℝ) (κ : 𝓢(E, ℂ)) | |
| (hκ : HasCompactSupport (κ : E → ℂ)) (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) : | |
| Continuous (normalizedCompactProfile a κ hκ Q hQ) := | |
| (continuous_scale_rpow (2 * a)).smul | |
| ((continuous_compactDilationFamily κ hκ Q hQ).comp continuous_subtype_val) | |
| section Residual | |
| variable (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) | |
| include hχone hχzero in | |
| private theorem family_gradient_compact (j : Fin 12) : | |
| HasCompactSupport (cutoffGradientCoefficient χ j : E → ℂ) := | |
| hasCompactSupport_of_cutoff_annulus _ (cutoffGradientCoefficient_annulus χ hχone hχzero j) | |
| include hχone hχzero in | |
| private theorem family_laplacian_compact : | |
| HasCompactSupport (cutoffLaplacianCoefficient χ : E → ℂ) := | |
| hasCompactSupport_of_cutoff_annulus _ (cutoffLaplacianCoefficient_annulus χ hχone hχzero) | |
| include hχone hχzero in | |
| private theorem family_power_compact (m : ℕ) : | |
| HasCompactSupport (cutoffPowerCoefficient χ hχ m : E → ℂ) := | |
| hasCompactSupport_of_cutoff_annulus _ (cutoffPowerCoefficient_annulus χ hχ m hχone hχzero) | |
| /-- The normalized three-term residual is a continuous Schwartz family. -/ | |
| theorem continuous_normalizedCutoffResidualSchwartz (a : ℝ) (m : ℕ) | |
| (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) : | |
| Continuous (fun L : Scale => | |
| normalizedCutoffResidualSchwartz χ hχ hχone hχzero a L.1 m Q hQ) := by | |
| let Qj (j : Fin 12) : E → ℂ := | |
| fun x => fderiv ℝ Q x ((EuclideanSpace.basisFun (Fin 12) ℝ) j) | |
| have hQj (j : Fin 12) : ContDiff ℝ ∞ (Qj j) := | |
| (contDiff_infty_iff_fderiv.mp hQ).2.clm_apply contDiff_const | |
| have hN : ContDiff ℝ ∞ (fun x => oddPowerNonlinearity m (Q x)) := | |
| ((contDiff_oddPowerNonlinearity m).of_le (by simp)).comp hQ | |
| let A (j : Fin 12) := compactDilationFamily (cutoffGradientCoefficient χ j) | |
| (family_gradient_compact χ hχone hχzero j) (Qj j) (hQj j) | |
| let B := compactDilationFamily (cutoffLaplacianCoefficient χ) | |
| (family_laplacian_compact χ hχone hχzero) Q hQ | |
| let C := compactDilationFamily (cutoffPowerCoefficient χ hχ m) | |
| (family_power_compact χ hχ hχone hχzero m) | |
| (fun x => oddPowerNonlinearity m (Q x)) hN | |
| have hcA (j : Fin 12) : Continuous (fun L : Scale => | |
| (L.1 ^ (2 * a + 1) : ℝ) • A j L.1) := | |
| (continuous_scale_rpow _).smul | |
| ((continuous_compactDilationFamily _ _ _ _).comp continuous_subtype_val) | |
| have hcB : Continuous (fun L : Scale => (L.1 ^ (2 * a) : ℝ) • B L.1) := | |
| (continuous_scale_rpow _).smul | |
| ((continuous_compactDilationFamily _ _ _ _).comp continuous_subtype_val) | |
| have hcC : Continuous (fun L : Scale => | |
| ((L.1 ^ (2 * a)) ^ (2 * m + 1) : ℝ) • C L.1) := | |
| ((continuous_scale_rpow _).pow _).smul | |
| ((continuous_compactDilationFamily _ _ _ _).comp continuous_subtype_val) | |
| have he (L : Scale) : normalizedCutoffResidualSchwartz χ hχ hχone hχzero a L.1 m Q hQ = | |
| (2 : ℂ) • (∑ j, (L.1 ^ (2 * a + 1) : ℝ) • A j L.1) + | |
| (L.1 ^ (2 * a) : ℝ) • B L.1 + | |
| ((L.1 ^ (2 * a)) ^ (2 * m + 1) : ℝ) • C L.1 := by | |
| ext x | |
| rw [normalizedCutoffResidualSchwartz_apply χ hχ hχone hχzero a L.1 m | |
| (lt_of_lt_of_le zero_lt_one L.2) Q hQ] | |
| simp only [add_apply, smul_apply, sum_apply, A, B, C, compactDilationFamily_apply, | |
| cutoffGradientCoefficient_apply, cutoffLaplacianCoefficient_apply, | |
| cutoffPowerCoefficient_apply, Complex.real_smul, Qj, normalizedCutoffResidual, | |
| oddPowerNonlinearity_real_mul] | |
| congr 1 | |
| · congr 1 | |
| · congr 1 | |
| apply Finset.sum_congr rfl | |
| intro j _ | |
| ring | |
| · ring | |
| · ring | |
| simp_rw [he] | |
| exact ((continuous_finsetSum _ (fun j _ => hcA j)).const_smul (2 : ℂ)).add hcB |>.add hcC | |
| end Residual | |
| /-- The actual cutoff-profile sample varies continuously with the physical radius. -/ | |
| theorem continuous_cutoffProfile_sample (a k : ℝ) (ha : 0 < a) | |
| (ha1 : a < 1) (hk : 8 < k) (χ : 𝓢(E, ℂ)) | |
| (hχ : HasCompactSupport (χ : E → ℂ)) (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) : | |
| Continuous (fun L : Scale => schwartzTorusSample a k L.1 ha1 hk L.2 | |
| (radianFourierKernel | |
| (cutoffProfileSchwartz L.1 (lt_of_lt_of_le zero_lt_one L.2) χ hχ Q hQ))) := by | |
| have he (L : Scale) : | |
| cutoffProfileSchwartz L.1 (lt_of_lt_of_le zero_lt_one L.2) χ hχ Q hQ = | |
| schwartzPhysicalDilation a L.1 (lt_of_lt_of_le zero_lt_one L.2) | |
| (normalizedCompactProfile a χ hχ Q hQ L) := by | |
| ext x | |
| rw [cutoffProfileSchwartz_apply, schwartzPhysicalDilation_apply, | |
| normalizedCompactProfile_apply, smul_smul, | |
| mul_inv_cancel₀ (ne_of_gt (lt_of_lt_of_le zero_lt_one L.2)), one_smul] | |
| have hp : L.1 ^ (-2 * a) * L.1 ^ (2 * a) = 1 := by | |
| rw [← Real.rpow_add (lt_of_lt_of_le zero_lt_one L.2)] | |
| simp | |
| symm | |
| calc | |
| _ = (L.1 ^ (-2 * a) * L.1 ^ (2 * a) : ℝ) * | |
| (χ (L.1⁻¹ • x) * Q x) := by push_cast; ring | |
| _ = _ := by rw [hp]; simp | |
| simp_rw [he] | |
| exact continuous_dilatedPhysicalSampling a k ha ha1 hk id continuous_id | |
| (normalizedCompactProfile a χ hχ Q hQ) | |
| (continuous_normalizedCompactProfile a χ hχ Q hQ) | |
| /-- The exact PDE residual is a continuous forcing path in the sampled Hilbert model. -/ | |
| theorem continuous_cutoffResidual_sample (a k : ℝ) (ha : 0 < a) | |
| (ha1 : a < 1) (hk : 8 < k) (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 (fun L : Scale => schwartzTorusSample a k L.1 ha1 hk L.2 | |
| (radianFourierKernel (cutoffResidualSchwartz χ hχ hχone hχzero a L.1 | |
| (lt_of_lt_of_le zero_lt_one L.2) m Q hQ))) := by | |
| have he (L : Scale) : | |
| schwartzTorusSample a k L.1 ha1 hk L.2 | |
| (radianFourierKernel (cutoffResidualSchwartz χ hχ hχone hχzero a L.1 | |
| (lt_of_lt_of_le zero_lt_one L.2) m Q hQ)) = | |
| ((L.1 ^ (-2 : ℝ) : ℝ) : ℂ) • schwartzTorusSample a k L.1 ha1 hk L.2 | |
| (radianFourierKernel (schwartzPhysicalDilation a L.1 | |
| (lt_of_lt_of_le zero_lt_one L.2) | |
| (normalizedCutoffResidualSchwartz χ hχ hχone hχzero a L.1 m Q hQ))) := by | |
| exact map_smul (physicalSchwartzTorusSamplingLinear a k L.1 ha1 hk L.2) | |
| ((L.1 ^ (-2 : ℝ) : ℝ) : ℂ) _ | |
| simp_rw [he] | |
| exact (Complex.continuous_ofReal.comp (continuous_scale_rpow (-2))).smul | |
| (continuous_dilatedPhysicalSampling a k ha ha1 hk id continuous_id _ | |
| (continuous_normalizedCutoffResidualSchwartz χ hχ hχone hχzero a m Q hQ)) | |
| end DefocusingNLS | |
| end OAI | |