| import OAI.MathematicalPhysics.AlternatingFlow.Analytic |
|
|
| namespace OAI |
|
|
| open scoped BigOperators ENNReal NNReal Topology ContDiff |
| open MeasureTheory |
| namespace AlternatingNS |
| namespace Analytic |
| noncomputable section |
|
|
| lemma spatial_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (F : Field E) : spatial (fun _ => 0) F = F := by simp [spatial] |
|
|
| lemma spatial_single {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (F : Field E) (i : Fin 3) (r : ℕ) : |
| spatial (fun j => if j = i then r else 0) F = (dx i)^[r] F := by |
| fin_cases i <;> simp [spatial] |
|
|
| lemma h_memLp {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (F : Field E) {k : ℕ} {T t : ℝ} (hF : ContinuousInH k F T) |
| (ht : t ∈ Set.Icc 0 T) : MemLp (F t) 2 (volume : Measure Space) := by |
| obtain ⟨g, _, hg⟩ := hF (fun _ => 0) (by simp) |
| have hm := (Lp.memLp (g t)).ae_eq (hg t ht) |
| simpa only [spatial_zero] using hm |
|
|
| lemma h_dx_memLp {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (F : Field E) {k : ℕ} {T t : ℝ} (hF : ContinuousInH k F T) (hk : 1 ≤ k) |
| (ht : t ∈ Set.Icc 0 T) (i : Fin 3) : MemLp (dx i F t) 2 (volume : Measure Space) := by |
| obtain ⟨g, _, hg⟩ := hF (fun j => if j = i then 1 else 0) (by simpa using hk) |
| have hm := (Lp.memLp (g t)).ae_eq (hg t ht) |
| simpa only [spatial_single, Function.iterate_one, Function.comp_apply] using hm |
|
|
| lemma h_dxx_memLp {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (F : Field E) {T t : ℝ} (hF : ContinuousInH 2 F T) |
| (ht : t ∈ Set.Icc 0 T) (i : Fin 3) : MemLp (dx i (dx i F) t) 2 (volume : Measure Space) := by |
| obtain ⟨g, _, hg⟩ := hF (fun j => if j = i then 2 else 0) (by simp) |
| have hm := (Lp.memLp (g t)).ae_eq (hg t ht) |
| simpa only [spatial_single, Function.iterate_succ_apply, Function.iterate_zero_apply] using hm |
|
|
| lemma L2_inner_integrable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| {f g : Space → E} (hf : MemLp f 2 (volume : Measure Space)) (hg : MemLp g 2 volume) : |
| Integrable (fun x => inner ℝ (f x) (g x)) volume := by |
| apply (hf.norm.integrable_mul hg.norm).mono' |
| (hf.aestronglyMeasurable.inner hg.aestronglyMeasurable) |
| exact Filter.Eventually.of_forall (fun x => norm_inner_le_norm (f x) (g x)) |
| |
| lemma L2_bounded_smul {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {f : Space → E} {a : Space → ℝ} (hf : MemLp f 2 (volume : Measure Space)) |
| (ha : AEStronglyMeasurable a volume) (C : ℝ) (hC : ∀ x, ‖a x‖ ≤ C) : |
| MemLp (fun x => a x • f x) 2 volume := by |
| apply hf.of_le_mul (ha.smul hf.aestronglyMeasurable) (c := C) |
| filter_upwards [] with x |
| change ‖a x • f x‖ ≤ C * ‖f x‖ |
| rw [norm_smul] |
| exact mul_le_mul_of_nonneg_right (hC x) (norm_nonneg _) |
| |
| lemma L2_coord {f : Space → Space} (hf : MemLp f 2 (volume : Measure Space)) (i : Fin 3) : |
| MemLp (fun x => f x i) 2 volume := |
| hf.continuousLinearMap_comp (PiLp.proj 2 (𝕜 := ℝ) (fun _ : Fin 3 => ℝ) i) |
| |
| lemma d_coord (f : Space → Space) (hf : Differentiable ℝ f) (i j : Fin 3) (x : Space) : |
| Spatial.d i (fun x => f x j) x = Spatial.d i f x j := by |
| have hd := (Spatial.coord_hasFDeriv j (f x)).comp x (hf x).hasFDerivAt |
| change fderiv ℝ (fun x => f x j) x (e i) = _ |
| change fderiv ℝ ((fun x : Space => x j) ∘ f) x (e i) = _ |
| rw [hd.fderiv] |
| rfl |
| |
| lemma sum_basis (x : Space) : ∑ i : Fin 3, x i • e i = x := by |
| ext j |
| simp [WithLp.ofLp_sum, Spatial.e_apply] |
| |
| lemma directional_sum (f : Space → Space) (x v : Space) : |
| fderiv ℝ f x v = ∑ i : Fin 3, v i • Spatial.d i f x := by |
| conv_lhs => rw [← sum_basis v] |
| simp [Spatial.d, map_sum] |
| |
| lemma L2_directional {f v : Space → Space} |
| (hd : ∀ i, MemLp (Spatial.d i f) 2 (volume : Measure Space)) |
| (hv : AEStronglyMeasurable v volume) (C : ℝ) (hC : ∀ x, ‖v x‖ ≤ C) : |
| MemLp (fun x => fderiv ℝ f x (v x)) 2 volume := by |
| simp_rw [directional_sum] |
| apply memLp_finsetSum |
| intro i _ |
| apply L2_bounded_smul (hd i) |
| (((Spatial.coord_smooth i).continuous.comp_aestronglyMeasurable hv)) C |
| intro x |
| exact (PiLp.norm_apply_le (v x) i).trans (hC x) |
| |
| lemma diffusion_identity (w : Space → Space) (i : Fin 3) |
| (hw : Differentiable ℝ w) (hdw : Differentiable ℝ (Spatial.d i w)) |
| (hm : MemLp w 2 (volume : Measure Space)) (hdm : MemLp (Spatial.d i w) 2 volume) |
| (hddm : MemLp (Spatial.d i (Spatial.d i w)) 2 volume) : |
| (∫ x : Space, inner ℝ (w x) (Spatial.d i (Spatial.d i w) x)) = |
| -(∫ x : Space, ‖Spatial.d i w x‖ ^ 2) := by |
| have h := integral_bilinear_fderiv_right_eq_neg_left_of_integrable |
| (μ := (volume : Measure Space)) (B := innerSL ℝ) (v := e i) |
| (f := w) (g := Spatial.d i w) (L2_inner_integrable hdm hdm) |
| (L2_inner_integrable hm hddm) (L2_inner_integrable hm hdm) |
| (fun x _ => hw x) (fun x _ => hdw x) |
| change (∫ x : Space, inner ℝ (w x) (Spatial.d i (Spatial.d i w) x)) = |
| -(∫ x : Space, inner ℝ (Spatial.d i w x) (Spatial.d i w x)) at h |
| simpa only [real_inner_self_eq_norm_sq] using h |
| |
| lemma norm_sq_d (w : Space → Space) (hw : Differentiable ℝ w) (i : Fin 3) (x : Space) : |
| Spatial.d i (fun x => ‖w x‖ ^ 2) x = 2 * inner ℝ (w x) (Spatial.d i w x) := by |
| unfold Spatial.d |
| rw [(hw x).hasFDerivAt.norm_sq.fderiv] |
| simp |
| |
| lemma bounded_L2_sq {w : Space → Space} (hm : MemLp w 2 (volume : Measure Space)) |
| (W : ℝ) (hW : ∀ x, ‖w x‖ ≤ W) : MemLp (fun x => ‖w x‖ ^ 2) 2 volume := by |
| apply hm.of_le_mul (hm.aestronglyMeasurable.norm.pow 2) (c := W) |
| filter_upwards [] with x |
| rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _), pow_two] |
| exact mul_le_mul_of_nonneg_right (hW x) (norm_nonneg _) |
| |
| lemma pressure_identity (w : Space → Space) (p : Space → ℝ) |
| (hw : Differentiable ℝ w) (hp : Differentiable ℝ p) |
| (hm : MemLp w 2 (volume : Measure Space)) (hdm : ∀ i, MemLp (Spatial.d i w) 2 volume) |
| (pm : MemLp p 2 volume) (pdm : ∀ i, MemLp (Spatial.d i p) 2 volume) |
| (hdiv : ∀ x, ∑ i : Fin 3, Spatial.d i w x i = 0) : |
| (∫ x : Space, ∑ i : Fin 3, Spatial.d i p x * w x i) = 0 := by |
| have hc (i : Fin 3) (x : Space) : fderiv ℝ (fun x => w x i) x (e i) = |
| Spatial.d i w x i := d_coord w hw i i x |
| have hi (i : Fin 3) : (∫ x : Space, Spatial.d i p x * w x i) = |
| -(∫ x : Space, p x * Spatial.d i w x i) := by |
| have h := integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable |
| (μ := (volume : Measure Space)) (v := e i) (f := p) (g := fun x => w x i) |
| ((pdm i).integrable_mul (L2_coord hm i)) |
| (by |
| simp only [hc] |
| exact pm.integrable_mul (L2_coord (hdm i) i)) |
| (pm.integrable_mul (L2_coord hm i)) (fun x _ => hp x) |
| (fun x _ => (Spatial.coord_smooth i).differentiable (by simp) (w x) |>.comp x (hw x)) |
| have h' : (∫ x : Space, p x * Spatial.d i w x i) = |
| -(∫ x : Space, Spatial.d i p x * w x i) := by |
| simpa only [hc, Spatial.d, Pi.mul_apply] using h |
| linarith |
| rw [integral_finsetSum Finset.univ (f := fun i x => Spatial.d i p x * w x i) |
| (fun i _ => (pdm i).integrable_mul (L2_coord hm i))] |
| simp_rw [hi] |
| rw [Finset.sum_neg_distrib, ← integral_finsetSum Finset.univ (f := fun i x => p x * Spatial.d i w x i) |
| (fun i _ => pm.integrable_mul (L2_coord (hdm i) i))] |
| simp_rw [← Finset.mul_sum, hdiv, mul_zero] |
| simp |
|
|
| lemma transport_identity (w v : Space → Space) |
| (hw : Differentiable ℝ w) (hv : Differentiable ℝ v) |
| (hm : MemLp w 2 (volume : Measure Space)) (hdm : ∀ i, MemLp (Spatial.d i w) 2 volume) |
| (vdm : ∀ i, MemLp (Spatial.d i v) 2 volume) |
| (W V : ℝ) (hW : ∀ x, ‖w x‖ ≤ W) (hV : ∀ x, ‖v x‖ ≤ V) |
| (hdiv : ∀ x, ∑ i : Fin 3, Spatial.d i v x i = 0) : |
| (∫ x : Space, inner ℝ (w x) (fderiv ℝ w x (v x))) = 0 := by |
| have hvc (i : Fin 3) : ∀ x, ‖v x i‖ ≤ V := |
| fun x => (PiLp.norm_apply_le (v x) i).trans (hV x) |
| have hvm (i : Fin 3) : AEStronglyMeasurable (fun x => v x i) volume := |
| ((Spatial.coord_smooth i).continuous.comp hv.continuous).aestronglyMeasurable |
| have hq := bounded_L2_sq hm W hW |
| have hA (i : Fin 3) : Integrable |
| (fun x : Space => v x i * inner ℝ (w x) (Spatial.d i w x)) volume := |
| (L2_inner_integrable hm (hdm i)).bdd_mul (hvm i) (Filter.Eventually.of_forall (hvc i)) |
| have hB (i : Fin 3) : Integrable |
| (fun x : Space => Spatial.d i v x i * ‖w x‖ ^ 2) volume := |
| (L2_coord (vdm i) i).integrable_mul hq |
| have hi (i : Fin 3) : |
| 2 * (∫ x : Space, v x i * inner ℝ (w x) (Spatial.d i w x)) = |
| -(∫ x : Space, Spatial.d i v x i * ‖w x‖ ^ 2) := by |
| have hfg' : Integrable |
| (fun x : Space => v x i * fderiv ℝ (fun x => ‖w x‖ ^ 2) x (e i)) volume := by |
| have he : (fun x : Space => v x i * fderiv ℝ (fun x => ‖w x‖ ^ 2) x (e i)) = |
| fun x => 2 * (v x i * inner ℝ (w x) (Spatial.d i w x)) := by |
| funext x |
| rw [← Spatial.d, norm_sq_d w hw] |
| ring |
| rw [he] |
| exact (hA i).const_mul 2 |
| have hc (x : Space) : fderiv ℝ (fun x => v x i) x (e i) = Spatial.d i v x i := |
| d_coord v hv i i x |
| have hq (x : Space) : fderiv ℝ (fun x => ‖w x‖ ^ 2) x (e i) = |
| 2 * inner ℝ (w x) (Spatial.d i w x) := norm_sq_d w hw i x |
| have h := integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable |
| (μ := (volume : Measure Space)) (v := e i) (f := fun x => v x i) |
| (g := fun x => ‖w x‖ ^ 2) |
| (by simpa only [hc] using hB i) hfg' |
| ((hm.integrable_norm_pow (by norm_num)).bdd_mul (hvm i) (Filter.Eventually.of_forall (hvc i))) |
| (fun x _ => ((Spatial.coord_smooth i).differentiable (by simp) (v x)).comp x (hv x)) |
| (fun x _ => (hw x).hasFDerivAt.norm_sq.differentiableAt) |
| simp only [hc, hq] at h |
| convert h using 1 |
| rw [← integral_const_mul] |
| congr 1 |
| funext x |
| ring |
| have hsum : 2 * (∫ x : Space, inner ℝ (w x) (fderiv ℝ w x (v x))) = 0 := by |
| simp_rw [directional_sum, inner_sum, real_inner_smul_right] |
| rw [integral_finsetSum _ (fun i _ => hA i), Finset.mul_sum] |
| simp_rw [hi] |
| rw [Finset.sum_neg_distrib, ← integral_finsetSum _ (fun i _ => hB i)] |
| simp_rw [← Finset.sum_mul, hdiv, zero_mul] |
| simp |
| linarith |
|
|
| lemma spatial_energy_inequality (ν : ℝ) (hν : 0 ≤ ν) (w v U a : Space → Space) |
| (p : Space → ℝ) |
| (hw : Differentiable ℝ w) (hdw : ∀ i, Differentiable ℝ (Spatial.d i w)) |
| (hv : Differentiable ℝ v) (hp : Differentiable ℝ p) |
| (hm : MemLp w 2 (volume : Measure Space)) |
| (hdm : ∀ i, MemLp (Spatial.d i w) 2 volume) |
| (hddm : ∀ i, MemLp (Spatial.d i (Spatial.d i w)) 2 volume) |
| (vdm : ∀ i, MemLp (Spatial.d i v) 2 volume) |
| (Udm : ∀ i, MemLp (Spatial.d i U) 2 volume) |
| (pm : MemLp p 2 volume) (pdm : ∀ i, MemLp (Spatial.d i p) 2 volume) |
| (W V C : ℝ) (hW : ∀ x, ‖w x‖ ≤ W) (hV : ∀ x, ‖v x‖ ≤ V) |
| (hC : ∀ x, ‖fderiv ℝ U x‖ ≤ C) |
| (hdivw : ∀ x, ∑ i : Fin 3, Spatial.d i w x i = 0) |
| (hdivv : ∀ x, ∑ i : Fin 3, Spatial.d i v x i = 0) |
| (heq : ∀ x, a x = ν • (∑ i : Fin 3, Spatial.d i (Spatial.d i w) x) - |
| fderiv ℝ w x (v x) - fderiv ℝ U x (w x) - |
| (EuclideanSpace.equiv (Fin 3) ℝ).symm (fun i => Spatial.d i p x)) : |
| (∫ x : Space, inner ℝ (w x) (a x)) ≤ C * (∫ x : Space, ‖w x‖ ^ 2) := by |
| have hD (i : Fin 3) := L2_inner_integrable hm (hddm i) |
| have hDsum : Integrable (fun x => ∑ i : Fin 3, |
| inner ℝ (w x) (Spatial.d i (Spatial.d i w) x)) volume := |
| integrable_finsetSum _ (fun i _ => hD i) |
| have hT := L2_inner_integrable hm (L2_directional hdm hv.continuous.aestronglyMeasurable V hV) |
| have hS := L2_inner_integrable hm (L2_directional Udm hw.continuous.aestronglyMeasurable W hW) |
| have hP : Integrable (fun x => ∑ i : Fin 3, Spatial.d i p x * w x i) volume := |
| integrable_finsetSum _ (fun i _ => (pdm i).integrable_mul (L2_coord hm i)) |
| have hDI : (∫ x : Space, ∑ i : Fin 3, inner ℝ (w x) (Spatial.d i (Spatial.d i w) x)) = |
| -(∑ i : Fin 3, ∫ x : Space, ‖Spatial.d i w x‖ ^ 2) := by |
| rw [integral_finsetSum _ (fun i _ => hD i)] |
| simp_rw [diffusion_identity w _ hw (hdw _) hm (hdm _) (hddm _)] |
| rw [Finset.sum_neg_distrib] |
| have hDN : 0 ≤ ∑ i : Fin 3, ∫ x : Space, ‖Spatial.d i w x‖ ^ 2 := by |
| apply Finset.sum_nonneg |
| intro i _ |
| exact integral_nonneg (fun x => sq_nonneg _) |
| have hTI := transport_identity w v hw hv hm hdm vdm W V hW hV hdivv |
| have hPI := pressure_identity w p hw hp hm hdm pm pdm hdivw |
| have hSI : -(∫ x : Space, inner ℝ (w x) (fderiv ℝ U x (w x))) ≤ |
| C * (∫ x : Space, ‖w x‖ ^ 2) := by |
| rw [← integral_neg, ← integral_const_mul] |
| apply integral_mono hS.neg ((hm.integrable_norm_pow (by norm_num)).const_mul C) |
| intro x |
| calc |
| -inner ℝ (w x) (fderiv ℝ U x (w x)) ≤ ‖w x‖ * ‖fderiv ℝ U x (w x)‖ := |
| (neg_le_abs _).trans (by |
| simpa only [Real.norm_eq_abs] using norm_inner_le_norm (𝕜 := ℝ) (w x) (fderiv ℝ U x (w x))) |
| _ ≤ ‖w x‖ * (C * ‖w x‖) := by |
| gcongr |
| exact ((fderiv ℝ U x).le_opNorm _).trans |
| (mul_le_mul_of_nonneg_right (hC x) (norm_nonneg _)) |
| _ = C * ‖w x‖ ^ 2 := by ring |
| have he (x : Space) : inner ℝ (w x) (a x) = |
| ν * (∑ i : Fin 3, inner ℝ (w x) (Spatial.d i (Spatial.d i w) x)) - |
| inner ℝ (w x) (fderiv ℝ w x (v x)) - inner ℝ (w x) (fderiv ℝ U x (w x)) - |
| ∑ i : Fin 3, Spatial.d i p x * w x i := by |
| rw [heq] |
| simp only [inner_sub_right, real_inner_smul_right, inner_sum] |
| congr 1 |
| have hDT : Integrable (fun x => ν * (∑ i : Fin 3, |
| inner ℝ (w x) (Spatial.d i (Spatial.d i w) x)) - |
| inner ℝ (w x) (fderiv ℝ w x (v x))) volume := (hDsum.const_mul ν).sub hT |
| have hDTS : Integrable (fun x => ν * (∑ i : Fin 3, |
| inner ℝ (w x) (Spatial.d i (Spatial.d i w) x)) - |
| inner ℝ (w x) (fderiv ℝ w x (v x)) - |
| inner ℝ (w x) (fderiv ℝ U x (w x))) volume := hDT.sub hS |
| simp_rw [he] |
| rw [integral_sub hDTS hP, integral_sub hDT hS, |
| integral_sub (hDsum.const_mul ν) hT, integral_const_mul, hDI, hTI, hPI] |
| nlinarith [mul_nonneg hν hDN] |
|
|
| lemma d_sub {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (f g : Space → E) (hf : Differentiable ℝ f) (hg : Differentiable ℝ g) (i : Fin 3) : |
| Spatial.d i (fun x => f x - g x) = fun x => Spatial.d i f x - Spatial.d i g x := by |
| funext x |
| simp [Spatial.d, fderiv_fun_sub (hf x) (hg x)] |
|
|
| lemma grad_zero (t : ℝ) (x : Space) : grad (fun _ _ => 0) t x = 0 := by |
| ext i |
| simp [grad, dx, EuclideanSpace.equiv] |
|
|
| lemma navierStokes_energy_inequality (ν : ℝ) (hν : 0 ≤ ν) (f U v : Velocity) (p : Pressure) |
| (hU : NavierStokes ν f U (fun _ _ => 0)) (hv : NavierStokes ν f v p) |
| {T t : ℝ} (ht : t ∈ Set.Icc 0 T) |
| (hUH : ContinuousInH 2 U T) (hvH : ContinuousInH 2 v T) (hpH : ContinuousInH 1 p T) |
| (CU Cv C : ℝ) (hCU : ∀ x, ‖U t x‖ ≤ CU) (hCv : ∀ x, ‖v t x‖ ≤ Cv) |
| (hC : ∀ x, ‖fderiv ℝ (U t) x‖ ≤ C) : |
| (∫ x : Space, inner ℝ (v t x - U t x) (dt v t x - dt U t x)) ≤ |
| C * (∫ x : Space, ‖v t x - U t x‖ ^ 2) := by |
| let w : Space → Space := fun x => v t x - U t x |
| have hUD := hU.1.2.1 t ht.1 |
| have hvD := hv.1.2.1 t ht.1 |
| have hwD : Differentiable ℝ w := hvD.sub hUD |
| have hd (i : Fin 3) : Spatial.d i w = fun x => dx i v t x - dx i U t x := |
| d_sub (v t) (U t) hvD hUD i |
| have hdd (i : Fin 3) : Spatial.d i (Spatial.d i w) = |
| fun x => dx i (dx i v) t x - dx i (dx i U) t x := by |
| rw [hd] |
| exact d_sub _ _ (hv.1.2.2.1 t ht.1 i) (hU.1.2.2.1 t ht.1 i) i |
| have hdD (i : Fin 3) : Differentiable ℝ (Spatial.d i w) := by |
| rw [hd] |
| exact (hv.1.2.2.1 t ht.1 i).sub (hU.1.2.2.1 t ht.1 i) |
| have hdM (i : Fin 3) : MemLp (Spatial.d i w) 2 (volume : Measure Space) := by |
| rw [hd] |
| exact (h_dx_memLp v hvH (by norm_num) ht i).sub (h_dx_memLp U hUH (by norm_num) ht i) |
| have hddM (i : Fin 3) : MemLp (Spatial.d i (Spatial.d i w)) 2 (volume : Measure Space) := by |
| rw [hdd] |
| exact (h_dxx_memLp v hvH ht i).sub (h_dxx_memLp U hUH ht i) |
| apply spatial_energy_inequality ν hν w (v t) (U t) _ (p t) hwD hdD hvD |
| (hv.1.2.2.2.1 t ht.1) ((h_memLp v hvH ht).sub (h_memLp U hUH ht)) hdM hddM |
| (h_dx_memLp v hvH (by norm_num) ht) (h_dx_memLp U hUH (by norm_num) ht) |
| (h_memLp p hpH ht) (h_dx_memLp p hpH (by norm_num) ht) |
| (Cv + CU) Cv C (fun x => (norm_sub_le _ _).trans (add_le_add (hCv x) (hCU x))) hCv hC |
| · intro x |
| simp only [hd, PiLp.sub_apply, Finset.sum_sub_distrib] |
| change div v t x - div U t x = 0 |
| rw [(hv.2.2 t ht.1 x).1, (hU.2.2 t ht.1 x).1, sub_self] |
| · exact fun x => (hv.2.2 t ht.1 x).1 |
| · intro x |
| have h1 := eq_sub_of_add_eq (hv.2.2 t ht.1 x).2 |
| have h2 := eq_sub_of_add_eq (hU.2.2 t ht.1 x).2 |
| rw [grad_zero, neg_zero, zero_add] at h2 |
| simp only [hdd, Finset.sum_sub_distrib, smul_sub] |
| rw [h1, h2] |
| simp only [w, fderiv_fun_sub (hvD x) (hUD x), sub_apply, map_sub, |
| laplacian, advection] |
| change (-grad p t x + ν • (∑ i, dx i (dx i v) t x) + f t x - |
| fderiv ℝ (v t) x (v t x)) - |
| (ν • (∑ i, dx i (dx i U) t x) + f t x - fderiv ℝ (U t) x (U t x)) = _ |
| unfold grad dx Spatial.d |
| abel |
|
|
| lemma l2_inner_representatives (g h : Lp Space 2 (volume : Measure Space)) |
| (w a : Space → Space) (hw : ∀ᵐ x ∂volume, g x = w x) (ha : ∀ᵐ x ∂volume, h x = a x) : |
| inner ℝ g h = ∫ x : Space, inner ℝ (w x) (a x) := by |
| rw [L2.inner_def] |
| apply integral_congr_ae |
| filter_upwards [hw, ha] with x hx hy |
| rw [hx, hy] |
|
|
| lemma l2_norm_sq_representative (g : Lp Space 2 (volume : Measure Space)) |
| (w : Space → Space) (hw : ∀ᵐ x ∂volume, g x = w x) : |
| ‖g‖ ^ 2 = ∫ x : Space, ‖w x‖ ^ 2 := by |
| rw [← real_inner_self_eq_norm_sq, l2_inner_representatives g g w w hw hw] |
| simp only [real_inner_self_eq_norm_sq] |
|
|
| theorem energy_velocity_unique (ν : ℝ) (hν : 0 ≤ ν) (f U v : Velocity) (p : Pressure) |
| (hU : NavierStokes ν f U (fun _ _ => 0)) (hv : NavierStokes ν f v p) |
| (hUE : EnergyClass U (fun _ _ => 0)) (hvE : CompetitorEnergyClass v p) : |
| ∀ t, 0 ≤ t → ∀ x, v t x = U t x := by |
| intro T hT |
| obtain ⟨hUH, hUL, _, CU, hCU⟩ := hUE.1 T hT |
| obtain ⟨hvH, hvL, hpH, Cv, hCv⟩ := hvE T hT |
| obtain ⟨C, hC⟩ := hUE.2 T hT |
| obtain ⟨gU, gU', hgUc, _, hgUd, hgU, hgU'⟩ := hUL |
| obtain ⟨gv, gv', hgvc, _, hgvd, hgv, hgv'⟩ := hvL |
| let g := fun t => gv t - gU t |
| let g' := fun t => gv' t - gU' t |
| have hgc : ContinuousOn g (Set.Icc 0 T) := hgvc.sub hgUc |
| have hgd : ∀ t ∈ Set.Icc 0 T, HasDerivWithinAt g (g' t) (Set.Icc 0 T) t := |
| fun t ht => (hgvd t ht).sub (hgUd t ht) |
| have hgw (t : ℝ) (ht : t ∈ Set.Icc 0 T) : |
| ∀ᵐ x ∂(volume : Measure Space), (g t) x = v t x - U t x := by |
| filter_upwards [Lp.coeFn_sub (gv t) (gU t), hgv t ht, hgU t ht] with x hx hvx hUx |
| change (gv t - gU t) x = _ |
| rw [hx] |
| change gv t x - gU t x = _ |
| rw [hvx, hUx] |
| have hga (t : ℝ) (ht : t ∈ Set.Icc 0 T) : |
| ∀ᵐ x ∂(volume : Measure Space), (g' t) x = dt v t x - dt U t x := by |
| filter_upwards [Lp.coeFn_sub (gv' t) (gU' t), hgv' t ht, hgU' t ht] with x hx hvx hUx |
| change (gv' t - gU' t) x = _ |
| rw [hx] |
| change gv' t x - gU' t x = _ |
| rw [hvx, hUx] |
| have hg0 : g 0 = 0 := by |
| apply Lp.ext |
| filter_upwards [hgw 0 ⟨le_rfl, hT⟩, Lp.coeFn_zero Space 2 (volume : Measure Space)] with x hx h0x |
| rw [hx, h0x] |
| change v 0 x - U 0 x = 0 |
| rw [hU.2.1, hv.2.1, sub_self] |
| have hgi (t : ℝ) (ht : t ∈ Set.Icc 0 T) : inner ℝ (g t) (g' t) ≤ C * ‖g t‖ ^ 2 := by |
| rw [l2_inner_representatives (g t) (g' t) _ _ (hgw t ht) (hga t ht), |
| l2_norm_sq_representative (g t) _ (hgw t ht)] |
| exact navierStokes_energy_inequality ν hν f U v p hU hv ht hUH hvH hpH CU Cv C |
| (hCU t ht) (hCv t ht) (fun x => (le_add_of_nonneg_left (norm_nonneg _)).trans (hC t ht x)) |
| have hrd (t : ℝ) (ht : t ∈ Set.Ico 0 T) : |
| HasDerivWithinAt (fun t => ‖g t‖ ^ 2) (2 * inner ℝ (g t) (g' t)) (Set.Ici t) t := by |
| apply (hgd t ⟨ht.1, ht.2.le⟩).norm_sq.mono_of_mem_nhdsWithin |
| filter_upwards [self_mem_nhdsWithin, mem_nhdsWithin_of_mem_nhds (Iio_mem_nhds ht.2)] with s hs hs' |
| exact ⟨ht.1.trans hs, hs'.le⟩ |
| have hbound := le_gronwallBound_of_liminf_deriv_right_le |
| (f := fun t => ‖g t‖ ^ 2) (f' := fun t => 2 * inner ℝ (g t) (g' t)) |
| (δ := 0) (K := 2 * C) (ε := 0) (hgc.norm.pow 2) |
| (fun t ht r hr => (hrd t ht).liminf_right_slope_le hr) |
| (by simp [hg0]) (fun t ht => by nlinarith [hgi t ⟨ht.1, ht.2.le⟩]) |
| T ⟨hT, le_rfl⟩ |
| rw [gronwallBound_ε0_δ0] at hbound |
| have hgT : g T = 0 := by |
| apply norm_eq_zero.mp |
| nlinarith [norm_nonneg (g T)] |
| have he : v T = U T := by |
| apply ((hv.1.2.1 T hT).continuous.ae_eq_iff_eq volume (hU.1.2.1 T hT).continuous).mp |
| filter_upwards [hgw T ⟨hT, le_rfl⟩, Lp.coeFn_zero Space 2 (volume : Measure Space)] with x hx h0x |
| rw [hgT, h0x] at hx |
| exact sub_eq_zero.mp hx.symm |
| exact congrFun he |
|
|
| lemma pressure_zero_of_gradient_zero (p : Space → ℝ) (hp : Differentiable ℝ p) |
| (hm : MemLp p 2 (volume : Measure Space)) |
| (hd : ∀ x i, Spatial.d i p x = 0) : ∀ x, p x = 0 := by |
| have hfd (x : Space) : fderiv ℝ p x = 0 := by |
| ext y |
| rw [← sum_basis y, map_sum] |
| have hde : ∀ i, fderiv ℝ p x (e i) = 0 := hd x |
| simp only [map_smul, hde, smul_zero, Finset.sum_const_zero, zero_apply] |
| have hc (x : Space) : p x = p 0 := is_const_of_fderiv_eq_zero hp hfd x 0 |
| have hpc : p = fun _ => p 0 := funext hc |
| rw [hpc] at hm |
| have hzero : p 0 = 0 := by |
| simpa [memLp_const_iff (by norm_num : (2 : ℝ≥0∞) ≠ 0) (by simp)] using hm |
| intro x |
| rw [hc, hzero] |
|
|
| theorem energy_unique (ν : ℝ) (hν : 0 ≤ ν) (f U v : Velocity) (p : Pressure) |
| (hU : NavierStokes ν f U (fun _ _ => 0)) (hv : NavierStokes ν f v p) |
| (hUE : EnergyClass U (fun _ _ => 0)) (hvE : CompetitorEnergyClass v p) : |
| ∀ t, 0 ≤ t → ∀ x, v t x = U t x ∧ p t x = 0 := by |
| have he := energy_velocity_unique ν hν f U v p hU hv hUE hvE |
| have hef (t : ℝ) (ht : 0 ≤ t) : v t = U t := funext (he t ht) |
| have hdt (t : ℝ) (ht : 0 ≤ t) (x : Space) : dt v t x = dt U t x := by |
| unfold dt |
| rw [fderivWithin_congr' (fun s (hs : s ∈ Set.Ici 0) => he s hs x) ht] |
| intro t ht x |
| refine ⟨he t ht x, ?_⟩ |
| apply pressure_zero_of_gradient_zero (p t) (hv.1.2.2.2.1 t ht) |
| (h_memLp p (hvE t ht).2.2.1 ⟨ht, le_rfl⟩) |
| intro y i |
| have h1 := (hv.2.2 t ht y).2 |
| have h2 := (hU.2.2 t ht y).2 |
| rw [grad_zero, neg_zero, zero_add] at h2 |
| have ha : advection v t y = advection U t y := by simp only [advection, hef t ht] |
| have hdx (j : Fin 3) : dx j v t = dx j U t := by |
| funext z |
| simp only [dx, hef t ht] |
| have hl : laplacian v t y = laplacian U t y := by simp only [laplacian, dx, hdx] |
| rw [hdt t ht y, ha, hl, h2] at h1 |
| have hg : grad p t y = 0 := by |
| have hn : -grad p t y = 0 := by |
| apply add_right_cancel (b := ν • laplacian U t y + f t y) |
| simpa only [zero_add, add_assoc] using h1.symm |
| exact neg_eq_zero.mp hn |
| have hcoord := congrArg (fun z : Space => z i) hg |
| exact hcoord |
| |
| end |
| end Analytic |
| |
| namespace Construction |
| |
| theorem velocity_unique (ν : ℝ) (hν : 0 < ν) (M : Machine) (w : List ℕ) |
| (v : Velocity) (p : Pressure) (hv : NavierStokes ν (force ν M w) v p) |
| (hvE : CompetitorEnergyClass v p) : |
| ∀ t, 0 ≤ t → ∀ x, v t x = velocity M w t x ∧ p t x = 0 := by |
| apply Analytic.energy_unique ν hν.le (force ν M w) (velocity M w) v p |
| (Analytic.residual_navierStokes ν _ (velocity_smooth M w) (velocity_rest M w) (velocity_div M w)) |
| hv (velocity_energyClass M w) hvE |
| |
| end Construction |
| end AlternatingNS |
| |
| end OAI |
| |
