| import Mathlib |
| import OAI.Analysis.Conductivity.Fourier.SpectralTrace |
| import OAI.Analysis.Conductivity.Fourier.CylinderFourier |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology |
| open scoped ENNReal |
|
|
| abbrev EndAxisMeasure : Measure ℝ := volume.restrict (Ioi 0) |
| abbrev EndAxisL2 := Lp ℂ 2 EndAxisMeasure |
|
|
| lemma complexLp_norm_sq {X : Type*} [MeasurableSpace X] {μ : Measure X} |
| (f : Lp ℂ 2 μ) : ‖f‖^2=∫ x,‖f x‖^2 ∂μ := by |
| rw [←real_inner_self_eq_norm_sq,L2.inner_def] |
| simp only [real_inner_self_eq_norm_sq] |
|
|
| def axialRateField (m : ℝ) (a : ℂ) (t : ℝ) : ℂ := |
| (m*Real.exp (-m*t):ℝ)*a |
|
|
| lemma axialRateField_continuous (m : ℝ) (a : ℂ) : |
| Continuous (axialRateField m a) := by unfold axialRateField; fun_prop |
|
|
| lemma axialRateField_norm_sq (m : ℝ) (a : ℂ) (t : ℝ) : |
| ‖axialRateField m a t‖^2=(2*m^2*Real.exp (-2*m*t)*‖a‖^2)/2 := by |
| unfold axialRateField |
| rw [norm_mul,mul_pow,Complex.norm_real,Real.norm_eq_abs,sq_abs,mul_pow, |
| ←Real.exp_nat_mul] |
| ring_nf |
|
|
| lemma axialRateField_memLp {m : ℝ} (hm : 0 ≤ m) (a : ℂ) : |
| MemLp (axialRateField m a) 2 EndAxisMeasure := by |
| apply (memLp_two_iff_integrable_sq_norm (axialRateField_continuous m a).aestronglyMeasurable).mpr |
| simpa only [axialRateField_norm_sq] using ((poisson_mode_energy hm a).1.div_const 2) |
|
|
| def axialRateLp (m : ℝ) (hm : 0 ≤ m) (a : ℂ) : EndAxisL2 := |
| (axialRateField_memLp hm a).toLp _ |
|
|
| lemma axialRateLp_ae (m : ℝ) (hm : 0 ≤ m) (a : ℂ) : |
| axialRateLp m hm a =ᵐ[EndAxisMeasure] axialRateField m a := |
| (axialRateField_memLp hm a).coeFn_toLp |
|
|
| lemma axialRateLp_norm_sq (m : ℝ) (hm : 0 ≤ m) (a : ℂ) : |
| ‖axialRateLp m hm a‖^2=(m*‖a‖^2)/2 := by |
| rw [complexLp_norm_sq] |
| calc |
| (∫ t,‖axialRateLp m hm a t‖^2 ∂EndAxisMeasure)= |
| ∫ t,‖axialRateField m a t‖^2 ∂EndAxisMeasure := |
| by |
| apply integral_congr_ae |
| filter_upwards [axialRateLp_ae m hm a] with t ht |
| rw [ht] |
| _=(∫ t in Ioi (0:ℝ),2*m^2*Real.exp (-2*m*t)*‖a‖^2)/2 := by |
| simp only [axialRateField_norm_sq,integral_div,EndAxisMeasure] |
| _=(m*‖a‖^2)/2 := by rw [(poisson_mode_energy hm a).2] |
|
|
| lemma axialRateLp_add (m : ℝ) (hm : 0 ≤ m) (a b : ℂ) : |
| axialRateLp m hm (a+b)=axialRateLp m hm a+axialRateLp m hm b := by |
| apply Lp.ext |
| filter_upwards [axialRateLp_ae m hm (a+b),axialRateLp_ae m hm a,axialRateLp_ae m hm b, |
| Lp.coeFn_add (axialRateLp m hm a) (axialRateLp m hm b)] with t h h1 h2 h3 |
| rw [h,h3] |
| simp only [Pi.add_apply] |
| rw [h1,h2] |
| exact mul_add _ _ _ |
|
|
| lemma axialRateLp_smul (m : ℝ) (hm : 0 ≤ m) (c a : ℂ) : |
| axialRateLp m hm (c*a)=c•axialRateLp m hm a := by |
| apply Lp.ext |
| filter_upwards [axialRateLp_ae m hm (c*a),axialRateLp_ae m hm a, |
| Lp.coeFn_smul c (axialRateLp m hm a)] with t h h1 h2 |
| rw [h,h2] |
| simp only [Pi.smul_apply,smul_eq_mul] |
| rw [h1] |
| unfold axialRateField |
| ring |
|
|
| lemma axialRateLp_zero (m : ℝ) (hm : 0 ≤ m) : axialRateLp m hm 0=0 := by |
| apply norm_eq_zero.mp |
| have he := axialRateLp_norm_sq m hm 0 |
| simp only [norm_zero,zero_pow (by decide : (2:ℕ)≠0),mul_zero,zero_div] at he |
| nlinarith only [he,norm_nonneg (axialRateLp m hm 0)] |
|
|
| def axialRateSequence {ι : Type*} (m : ι → ℝ) (hm : ∀ i,0 ≤ m i) |
| (a : ι → ℂ) (ha : Summable (fun i => m i*‖a i‖^2)) : |
| lp (fun _ : ι => EndAxisL2) 2 := |
| ⟨fun i => axialRateLp (m i) (hm i) (a i),by |
| apply (memℓp_gen_iff (by norm_num : 0<(2:ℝ≥0∞).toReal)).mpr |
| simpa only [ENNReal.toReal_ofNat,Real.rpow_two,axialRateLp_norm_sq] using ha.div_const 2⟩ |
|
|
| def cylinderRateField (m : TorusModes → ℝ) (hm : ∀ i,0 ≤ m i) |
| (a : TorusModes → ℂ) (ha : Summable (fun i => m i*‖a i‖^2)) : |
| CylinderL2 EndAxisMeasure := |
| cylinderFourier EndAxisMeasure (axialRateSequence m hm a ha) |
|
|
| lemma cylinderRateField_norm_sq (m : TorusModes → ℝ) (hm : ∀ i,0 ≤ m i) |
| (a : TorusModes → ℂ) (ha : Summable (fun i => m i*‖a i‖^2)) : |
| ‖cylinderRateField m hm a ha‖^2=(∑' i,m i*‖a i‖^2)/2 := by |
| unfold cylinderRateField |
| rw [cylinderFourier_norm_sq] |
| change (∑' i,‖axialRateLp (m i) (hm i) (a i)‖^2)=_ |
| simp only [axialRateLp_norm_sq,tsum_div_const] |
|
|
| lemma cylinderRateField_hasSum (m : TorusModes → ℝ) (hm : ∀ i,0 ≤ m i) |
| (a : TorusModes → ℂ) (ha : Summable (fun i => m i*‖a i‖^2)) : |
| HasSum (fun i => cylinderMode EndAxisMeasure i (axialRateLp (m i) (hm i) (a i))) |
| (cylinderRateField m hm a ha) := |
| cylinderFourier_hasSum EndAxisMeasure (axialRateSequence m hm a ha) |
|
|
| abbrev FiniteAxisMeasure (R : ℝ) : Measure ℝ := volume.restrict (Ioc 0 R) |
| abbrev FiniteAxisL2 (R : ℝ) := Lp ℂ 2 (FiniteAxisMeasure R) |
|
|
| def axialDecayField (m : ℝ) (a : ℂ) (t : ℝ) : ℂ := (Real.exp (-m*t):ℂ)*a |
|
|
| lemma axialDecayField_continuous (m : ℝ) (a : ℂ) : |
| Continuous (axialDecayField m a) := by unfold axialDecayField; fun_prop |
|
|
| lemma axialDecayField_memLp (m R : ℝ) (a : ℂ) : |
| MemLp (axialDecayField m a) 2 (FiniteAxisMeasure R) := by |
| apply (memLp_two_iff_integrable_sq_norm (axialDecayField_continuous m a).aestronglyMeasurable).mpr |
| exact (((axialDecayField_continuous m a).norm.pow 2).integrableOn_Icc).mono_set Ioc_subset_Icc_self |
|
|
| def finiteDecayLp (m R : ℝ) (a : ℂ) : FiniteAxisL2 R := |
| (axialDecayField_memLp m R a).toLp _ |
|
|
| lemma finiteDecayLp_ae (m R : ℝ) (a : ℂ) : |
| finiteDecayLp m R a =ᵐ[FiniteAxisMeasure R] axialDecayField m a := |
| (axialDecayField_memLp m R a).coeFn_toLp |
|
|
| lemma finiteDecayLp_norm_sq {m R : ℝ} (hm : 0 ≤ m) (hR : 0≤R) (a : ℂ) : |
| ‖finiteDecayLp m R a‖^2≤R*‖a‖^2 := by |
| rw [complexLp_norm_sq] |
| have he : (∫ t,‖finiteDecayLp m R a t‖^2 ∂FiniteAxisMeasure R)= |
| ∫ t in Ioc (0:ℝ) R,‖axialDecayField m a t‖^2 := by |
| apply integral_congr_ae |
| filter_upwards [finiteDecayLp_ae m R a] with t ht |
| rw [ht] |
| rw [he] |
| calc |
| (∫ t in Ioc (0:ℝ) R,‖axialDecayField m a t‖^2) ≤ |
| ∫ t in Ioc (0:ℝ) R,‖a‖^2 := by |
| apply setIntegral_mono_on |
| ((axialDecayField_memLp m R a).integrable_norm_pow (by decide : (2:ℕ)≠0)) |
| (integrable_const _) measurableSet_Ioc |
| intro t ht |
| have hb : Real.exp (-m*t)≤1 := Real.exp_le_one_iff.mpr |
| (mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr hm) ht.1.le) |
| have hn : ‖axialDecayField m a t‖≤‖a‖ := by |
| unfold axialDecayField |
| rw [norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos (Real.exp_pos _)] |
| exact (mul_le_mul_of_nonneg_right hb (norm_nonneg a)).trans_eq (one_mul _) |
| exact pow_le_pow_left₀ (norm_nonneg _) hn 2 |
| _=R*‖a‖^2 := by |
| rw [integral_const] |
| change ((volume.restrict (Ioc (0:ℝ) R)) univ).toReal * ‖a‖^2=_ |
| rw [Measure.restrict_apply MeasurableSet.univ,univ_inter,Real.volume_Ioc, |
| sub_zero,ENNReal.toReal_ofReal hR] |
|
|
| lemma finiteDecayLp_add (m R : ℝ) (a b : ℂ) : |
| finiteDecayLp m R (a+b)=finiteDecayLp m R a+finiteDecayLp m R b := by |
| apply Lp.ext |
| filter_upwards [finiteDecayLp_ae m R (a+b),finiteDecayLp_ae m R a,finiteDecayLp_ae m R b, |
| Lp.coeFn_add (finiteDecayLp m R a) (finiteDecayLp m R b)] with t h h1 h2 h3 |
| rw [h,h3] |
| simp only [Pi.add_apply] |
| rw [h1,h2] |
| exact mul_add _ _ _ |
|
|
| lemma finiteDecayLp_smul (m R : ℝ) (c a : ℂ) : |
| finiteDecayLp m R (c*a)=c•finiteDecayLp m R a := by |
| apply Lp.ext |
| filter_upwards [finiteDecayLp_ae m R (c*a),finiteDecayLp_ae m R a, |
| Lp.coeFn_smul c (finiteDecayLp m R a)] with t h h1 h2 |
| rw [h,h2] |
| simp only [Pi.smul_apply,smul_eq_mul] |
| rw [h1] |
| unfold axialDecayField |
| ring |
|
|
| lemma finiteDecayLp_zero (m R : ℝ) : finiteDecayLp m R 0=0 := by |
| simpa only [zero_mul,zero_smul] using finiteDecayLp_smul m R 0 0 |
|
|
| lemma finiteAxisMeasure_le (R : ℝ) : FiniteAxisMeasure R≤EndAxisMeasure := |
| Measure.restrict_mono (fun _ h => h.1) le_rfl |
|
|
| def endAxisRestrict (R : ℝ) : EndAxisL2 →L[ℝ] FiniteAxisL2 R := |
| Lp.LpToLpOfMeasureLeSMul (c:=1) (by simp) (by simpa only [one_smul] using finiteAxisMeasure_le R) |
|
|
| lemma endAxisRestrict_ae (R : ℝ) (f : EndAxisL2) : |
| endAxisRestrict R f =ᵐ[FiniteAxisMeasure R] f := |
| Lp.coeFn_LpToLpOfMeasureLeSMul (c:=1) (by simp) |
| (by simpa only [one_smul] using finiteAxisMeasure_le R) f |
|
|
| lemma endAxisRestrict_norm (R : ℝ) (f : EndAxisL2) : ‖endAxisRestrict R f‖≤‖f‖ := by |
| have hn : ‖endAxisRestrict R‖≤1 := by |
| simpa only [endAxisRestrict,ENNReal.toReal_one,Real.one_rpow] using |
| Lp.norm_LpToLpOfMeasureLeSMul_le (E:=ℂ) (p:=2) (c:=1) (by simp) |
| (by simpa only [one_smul] using finiteAxisMeasure_le R) |
| exact ((endAxisRestrict R).le_opNorm f).trans |
| ((mul_le_mul_of_nonneg_right hn (norm_nonneg f)).trans_eq (one_mul _)) |
|
|
| lemma finiteDecayLp_rate (m R : ℝ) (hm : 0 ≤ m) (a : ℂ) : |
| finiteDecayLp m R ((m:ℂ)*a)=endAxisRestrict R (axialRateLp m hm a) := by |
| apply Lp.ext |
| filter_upwards [finiteDecayLp_ae m R ((m:ℂ)*a), |
| endAxisRestrict_ae R (axialRateLp m hm a), |
| (Measure.absolutelyContinuous_of_le (finiteAxisMeasure_le R)).ae_eq (axialRateLp_ae m hm a)] |
| with t h h1 h2 |
| rw [h,h1,h2] |
| simp only [axialDecayField,axialRateField,Complex.ofReal_mul] |
| ring |
|
|
| lemma finiteDecayLp_rate_norm_sq (m R : ℝ) (hm : 0 ≤ m) (a : ℂ) : |
| ‖finiteDecayLp m R ((m:ℂ)*a)‖^2≤(m*‖a‖^2)/2 := by |
| rw [finiteDecayLp_rate m R hm a,←axialRateLp_norm_sq m hm a] |
| exact pow_le_pow_left₀ (norm_nonneg _) (endAxisRestrict_norm R _) 2 |
|
|
| def finiteDecaySequence {ι : Type*} (m : ι → ℝ) (hm : ∀ i,0 ≤ m i) |
| (R : ℝ) (hR : 0≤R) (a : SpectralL2 ι) : lp (fun _ : ι => FiniteAxisL2 R) 2 := |
| ⟨fun i => finiteDecayLp (m i) R (a i),by |
| apply (memℓp_gen_iff (by norm_num : 0<(2:ℝ≥0∞).toReal)).mpr |
| simp only [ENNReal.toReal_ofNat,Real.rpow_two] |
| have ha := (lp.memℓp a).summable (by norm_num : 0<(2:ℝ≥0∞).toReal) |
| simp only [ENNReal.toReal_ofNat,Real.rpow_two] at ha |
| exact (ha.mul_left R).of_nonneg_of_le (fun i => sq_nonneg _) |
| (fun i => finiteDecayLp_norm_sq (hm i) hR (a i))⟩ |
|
|
| end ScalarConductivity |
|
|
| end |
|
|
| end OAI |
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|
