| import OAI.Analysis.HenonEmden.WeightedEnergy |
|
|
| namespace OAI |
|
|
| section IntegratedRecovery |
|
|
| open MeasureTheory Set Metric |
| open scoped ENNReal |
| noncomputable section |
| namespace HenonLaneEmden.SpatialComponents |
| open Cutoff AdaptiveCap LayerTonelli |
|
|
| lemma lintegral_weight_layer {n : ℕ} {u : Space n → ℝ} {q B : ℝ} |
| (hu : Measurable u) (hu0 : ∀ x, 0 ≤ u x) (hf : Measurable (source u q B)) |
| {χ : Space n → ℝ} (hχ : Measurable χ) (_hχ0 : ∀ x, 0 ≤ χ x) : |
| (∫⁻ z, ENNReal.ofReal (χ z.1*u z.1) ∂layerMeasure volume (source u q B)) = |
| weightedEnergyN χ u q B := by |
| rw [lintegral_layerMeasure_fst volume hf (F := fun x => ENNReal.ofReal (χ x*u x)) (by fun_prop)] |
| apply lintegral_congr |
| intro x |
| rw [← ENNReal.ofReal_mul (source_nonneg hu0 q B x)] |
| congr 1 |
| ring |
|
|
| lemma lintegral_pressure_root {n : ℕ} {u : Space n → ℝ} {q B : ℝ} |
| (hu : Measurable u) (hup : ∀ x, 0 < u x) (hq : 0 < q) (hf : Measurable (source u q B)) |
| {χ : Space n → ℝ} (hχ : Measurable χ) (hχ0 : ∀ x, 0 ≤ χ x) : |
| (∫⁻ z, ENNReal.ofReal (χ z.1*(u z.1-sourceRoot q B z)) ∂layerMeasure volume (source u q B)) = |
| ENNReal.ofReal (1/(q+1))*weightedEnergyN χ u q B := by |
| rw [lintegral_layerMeasure volume hf (by fun_prop)] |
| simp_rw [ENNReal.ofReal_mul (hχ0 _)] |
| have he (x : Space n) : (∫⁻ t in Ioo 0 (source u q B x), ENNReal.ofReal (χ x)* |
| ENNReal.ofReal (u x-sourceRoot q B (x,t))) = |
| ENNReal.ofReal (1/(q+1))*ENNReal.ofReal (χ x*source u q B x*u x) := by |
| rw [lintegral_const_mul _ (by fun_prop),lintegral_source_residual hup hq, |
| ← ENNReal.ofReal_mul (hχ0 x),← ENNReal.ofReal_mul (one_div_nonneg.mpr (by linarith : 0 ≤ q+1))] |
| congr 1 |
| ring |
| simp_rw [he] |
| exact lintegral_const_mul _ (by fun_prop) |
|
|
| lemma IsSolution_endpointIntegral_bound {n : ℕ} (hn : 2 ≤ n) {p q A B κ δ : ℝ} |
| {u v : Space n → ℝ} (h : IsSolution n p q A B u v) (hq : 0 < q) |
| (hκ : 0 < κ) (hκ1 : κ ≤ 1) (hδ : 0 ≤ δ) (e : sphere (0:Space n) 1) : |
| endpointIntegral u (source u q B) κ δ e ≤ |
| ENNReal.ofReal (1+endpointConstant q B*δ)*weightedEnergyN (weight volume κ e) u q B := by |
| have : Nontrivial (Space n) := Module.nontrivial_of_finrank_pos (R := ℝ) (by simp; omega) |
| have huM := h.continuous_u.measurable |
| have hfM := measurable_source h.continuous_u q B |
| have hm := (continuous_weight volume hκ hκ1 e).measurable |
| have hd : Measurable (d : Space n → ℝ) := continuous_d.measurable |
| have hc := endpointConstant_nonneg q B |
| have hpoint (z : Space n × ℝ) (hz : z ∈ sourceSubgraph (source u q B)) : |
| lineWeight volume (source u q B) κ δ (e,z)*endpointAverage u (source u q B) (e,z) ≤ |
| (1+endpointConstant q B*δ)*(weight volume κ e z.1*u z.1) := by |
| have hχ := lineWeight_nonneg volume (source u q B) κ δ (e,z) |
| have hh := weight_nonneg volume κ e z.1 |
| have hu0 := (h.positive_u z.1).le |
| by_cases hpos : 0 < lineWeight volume (source u q B) κ δ (e,z) |
| · have he := (abs_le.mp (source_endpoint_error (by omega) h.regular_u h.positive_u hq hκ hκ1 hδ hpos)).2 |
| have hr := (sourceRoot_bounds h.positive_u hq hz).2 |
| change sourceRoot q B z ≤ u z.1 at hr |
| have hε : endpointConstant q B*δ*d z.1*u z.1 ≤ endpointConstant q B*δ*u z.1 := by |
| exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left (d_le_one z.1) (mul_nonneg hc hδ)) hu0 |>.trans_eq (by ring) |
| have hle : endpointAverage u (source u q B) (e,z) ≤ (1+endpointConstant q B*δ)*u z.1 := by |
| change endpointAverage u (source u q B) (e,z)-sourceRoot q B z ≤ _ at he |
| nlinarith |
| calc |
| _ ≤ lineWeight volume (source u q B) κ δ (e,z)*((1+endpointConstant q B*δ)*u z.1) := |
| mul_le_mul_of_nonneg_left hle hχ |
| _ ≤ weight volume κ e z.1*((1+endpointConstant q B*δ)*u z.1) := |
| mul_le_mul_of_nonneg_right (lineWeight_le volume _ _ _ _) (by positivity) |
| _ = _ := by ring |
| · rw [le_antisymm (not_lt.mp hpos) hχ,zero_mul] |
| positivity |
| calc |
| _ ≤ ∫⁻ z, ENNReal.ofReal ((1+endpointConstant q B*δ)*(weight volume κ e z.1*u z.1)) |
| ∂layerMeasure volume (source u q B) := by |
| apply lintegral_mono_ae |
| filter_upwards [ae_restrict_mem (measurableSet_sourceSubgraph hfM)] with z hz |
| exact ENNReal.ofReal_le_ofReal (hpoint z hz) |
| _ = _ := by |
| simp_rw [ENNReal.ofReal_mul (show 0 ≤ 1+endpointConstant q B*δ by positivity)] |
| rw [lintegral_const_mul _ (by fun_prop),lintegral_weight_layer h.continuous_u.measurable |
| (fun x => (h.positive_u x).le) hfM hm (weight_nonneg volume κ e)] |
|
|
| lemma IsSolution_endpointIntegral_finite {n : ℕ} (hn : 2 ≤ n) {p q A B κ δ : ℝ} |
| {u v : Space n → ℝ} (h : IsSolution n p q A B u v) (hq : 0 < q) |
| (hκ : 0 < κ) (hκ1 : κ ≤ 1) (hδ : 0 ≤ δ) (e : sphere (0:Space n) 1) : |
| endpointIntegral u (source u q B) κ δ e ≠ ⊤ := by |
| have : Nontrivial (Space n) := Module.nontrivial_of_finrank_pos (R := ℝ) (by simp; omega) |
| exact ne_top_of_le_ne_top (ENNReal.mul_ne_top ENNReal.ofReal_ne_top |
| (h.weightedEnergyN_finite hn (continuous_weight volume hκ hκ1 e) (compactSupport_weight κ e))) |
| (IsSolution_endpointIntegral_bound hn h hq hκ hκ1 hδ e) |
|
|
| lemma IsSolution_integrated_pressure_recovery {n : ℕ} (hn : 2 ≤ n) {p q A B κ δ : ℝ} |
| {u v : Space n → ℝ} (h : IsSolution n p q A B u v) (hq : 0 < q) |
| (hκ : 0 < κ) (hκ1 : κ ≤ 1) (hδ : 0 ≤ δ) (e : sphere (0:Space n) 1) : |
| ENNReal.ofReal (1/(q+1))*weightedEnergyN (weight volume κ e) u q B+ |
| endpointIntegral u (source u q B) κ δ e ≤ |
| positionIntegral u (source u q B) κ δ e+lossIntegral u (source u q B) κ δ e+ |
| ENNReal.ofReal (endpointConstant q B*δ)*weightedEnergyN (fun x => d x*weight volume κ e x) u q B := by |
| have : Nontrivial (Space n) := Module.nontrivial_of_finrank_pos (R := ℝ) (by simp; omega) |
| have huM := h.continuous_u.measurable |
| have hfM := measurable_source h.continuous_u q B |
| have hf := isOpen_source_subgraph h.regular_u h.positive_u q B |
| have hm := (continuous_weight volume hκ hκ1 e).measurable |
| have hmχ : Measurable (fun z : Space n × ℝ => lineWeight volume (source u q B) κ δ (e,z)) := |
| (measurable_lineWeight volume hf hκ hκ1 δ).comp (measurable_const.prodMk measurable_id) |
| have hlo := measurable_lowerPoint hf |
| have hhi := measurable_upperPoint hf |
| have hd : Measurable (d : Space n → ℝ) := continuous_d.measurable |
| have hc := endpointConstant_nonneg q B |
| have hrec := lintegral_mono_ae (μ := layerMeasure volume (source u q B)) (show ∀ᵐ z ∂layerMeasure volume (source u q B), |
| ENNReal.ofReal (weight volume κ e z.1*(u z.1-sourceRoot q B z))+ |
| ENNReal.ofReal (lineWeight volume (source u q B) κ δ (e,z)*endpointAverage u (source u q B) (e,z)) ≤ |
| ENNReal.ofReal (lineWeight volume (source u q B) κ δ (e,z)*u z.1)+ |
| ENNReal.ofReal ((weight volume κ e z.1-lineWeight volume (source u q B) κ δ (e,z))*u z.1)+ |
| ENNReal.ofReal (endpointConstant q B*δ*d z.1*weight volume κ e z.1*u z.1) from by |
| filter_upwards [ae_restrict_mem (measurableSet_sourceSubgraph hfM)] with z hz |
| exact source_pressure_recovery_pointwise (by omega) h.regular_u h.positive_u hq hκ hκ1 hδ e hz) |
| rw [lintegral_add_left (by fun_prop),lintegral_add_left (by fun_prop),lintegral_add_left (by fun_prop), |
| lintegral_pressure_root huM h.positive_u hq hfM hm (weight_nonneg volume κ e)] at hrec |
| change _ ≤ positionIntegral u (source u q B) κ δ e+lossIntegral u (source u q B) κ δ e+_ at hrec |
| convert hrec using 1 |
| congr 1 |
| have he (z : Space n × ℝ) : ENNReal.ofReal (endpointConstant q B*δ*d z.1*weight volume κ e z.1*u z.1) = |
| ENNReal.ofReal (endpointConstant q B*δ)*ENNReal.ofReal ((d z.1*weight volume κ e z.1)*u z.1) := by |
| rw [← ENNReal.ofReal_mul (mul_nonneg hc hδ)]; congr 1; ring |
| simp_rw [he] |
| rw [lintegral_const_mul _ (by fun_prop), |
| lintegral_weight_layer huM (fun x => (h.positive_u x).le) hfM (by fun_prop) |
| (fun x => mul_nonneg (d_nonneg x) (weight_nonneg volume κ e x))] |
|
|
| end HenonLaneEmden.SpatialComponents |
| end |
| end IntegratedRecovery |
|
|
| end OAI |
|
|
