| import OAI.MathematicalPhysics.RapidForcing.ClassicalSolution |
| import OAI.MathematicalPhysics.RapidForcing.Energy |
| import OAI.MathematicalPhysics.RapidForcing.Uniqueness |
|
|
| namespace OAI |
|
|
| section |
| open scoped BigOperators ENNReal Topology NNReal InnerProductSpace |
| open Set MeasureTheory |
| namespace RapidForcing |
|
|
| lemma L2ContinuousOn.memLp {E : Type} [NormedAddCommGroup E] |
| {f : Field E} {T t : ℝ} (hf : L2ContinuousOn f T) (ht : t ∈ Icc 0 T) : |
| MemLp (f t) 2 volume := by |
| obtain ⟨v, _, hv⟩ := hf |
| exact (Lp.memLp (v t)).ae_eq (hv t ht) |
|
|
| @[simp] lemma spatialMulti_zero {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (f : Field E) : spatialMulti 0 f = f := by simp [spatialMulti] |
|
|
| lemma spatialMulti_single {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (f : Field E) (i : Fin 3) (n : ℕ) : |
| spatialMulti (Pi.single i n) f = (spatialD i)^[n] f := by |
| fin_cases i <;> simp [spatialMulti] |
|
|
| lemma order_single (i : Fin 3) (n : ℕ) : order (Pi.single i n) = n := by |
| simp [order] |
|
|
| lemma EnergyRegularity.memLp_velocity {u : Field Space} {p : Field ℝ} |
| (h : EnergyRegularity u p) {t : ℝ} (ht : 0 ≤ t) : MemLp (u t) 2 volume := by |
| have he := (h t ht).1 0 (by simp [order]) |
| simpa using he.memLp (right_mem_Icc.mpr ht) |
|
|
| lemma EnergyRegularity.memLp_pressure {u : Field Space} {p : Field ℝ} |
| (h : EnergyRegularity u p) {t : ℝ} (ht : 0 ≤ t) : MemLp (p t) 2 volume := by |
| have he := (h t ht).2.2 0 (by simp [order]) |
| simpa using he.memLp (right_mem_Icc.mpr ht) |
|
|
| lemma EnergyRegularity.memLp_velocityD {u : Field Space} {p : Field ℝ} |
| (h : EnergyRegularity u p) {t : ℝ} (ht : 0 ≤ t) (i : Fin 3) : |
| MemLp (spaceD i (u t)) 2 volume := by |
| have he := (h t ht).1 (Pi.single i 1) (by rw [order_single]; norm_num) |
| have this := he.memLp (right_mem_Icc.mpr ht) |
| rw [spatialMulti_single] at this |
| exact this |
|
|
| lemma EnergyRegularity.memLp_velocityDD {u : Field Space} {p : Field ℝ} |
| (h : EnergyRegularity u p) {t : ℝ} (ht : 0 ≤ t) (i : Fin 3) : |
| MemLp (spaceD i (spaceD i (u t))) 2 volume := by |
| have he := (h t ht).1 (Pi.single i 2) (by rw [order_single]) |
| have this := he.memLp (right_mem_Icc.mpr ht) |
| rw [spatialMulti_single] at this |
| exact this |
|
|
| lemma EnergyRegularity.memLp_pressureD {u : Field Space} {p : Field ℝ} |
| (h : EnergyRegularity u p) {t : ℝ} (ht : 0 ≤ t) (i : Fin 3) : |
| MemLp (spaceD i (p t)) 2 volume := by |
| have he := (h t ht).2.2 (Pi.single i 1) (by rw [order_single]) |
| have this := he.memLp (right_mem_Icc.mpr ht) |
| rw [spatialMulti_single] at this |
| exact this |
|
|
| lemma timeD_sub {u v : Field Space} (hu : ∀ x, DifferentiableOn ℝ (fun t => u t x) (Ici 0)) |
| (hv : ∀ x, DifferentiableOn ℝ (fun t => v t x) (Ici 0)) {t : ℝ} (ht : 0 ≤ t) (x : Space) : |
| timeD (fun t x => u t x - v t x) t x = timeD u t x - timeD v t x := |
| derivWithin_sub (hu x t ht) (hv x t ht) |
|
|
| lemma spaceD_sub_twice {u v : Space → Space} |
| (hu : ContDiff ℝ 2 u) (hv : ContDiff ℝ 2 v) (i : Fin 3) : |
| spaceD i (spaceD i (fun x => u x - v x)) = |
| fun x => spaceD i (spaceD i u) x - spaceD i (spaceD i v) x := by |
| rw [spaceD_sub (hu.differentiable (by norm_num)) (hv.differentiable (by norm_num))] |
| exact spaceD_sub ((spaceD_contDiff hu i).differentiable (by norm_num)) |
| ((spaceD_contDiff hv i).differentiable (by norm_num)) i |
|
|
| lemma advection_difference {u v : Space → Space} |
| (hu : Differentiable ℝ u) (hv : Differentiable ℝ v) (x : Space) : |
| (∑ i, v x i • spaceD i v x) - (∑ i, u x i • spaceD i u x) = |
| (∑ i, v x i • spaceD i (fun z => v z - u z) x) + |
| fderiv ℝ u x (v x - u x) := by |
| simp only [spaceD_sub hv hu, smul_sub, Finset.sum_sub_distrib, |
| spaceD_sum, map_sub] |
| abel |
|
|
| lemma navierStokes_difference_energy {ν : ℝ} (hν : 0 ≤ ν) |
| {f u v : Field Space} {p q : Field ℝ} |
| (hu : Classical u p) (hv : Classical v q) |
| (heu : EnergyRegularity u p) (hev : EnergyRegularity v q) |
| (hnsu : NavierStokes ν f u p) (hnsv : NavierStokes ν f v q) |
| {t C B A : ℝ} (ht : 0 ≤ t) |
| (huC : ∀ x, ‖fderiv ℝ (u t) x‖ ≤ C) |
| (hvB : ∀ x, ‖v t x‖ ≤ B) (huA : ∀ x, ‖u t x‖ ≤ A) : |
| (∫ x, ⟪v t x - u t x, timeD v t x - timeD u t x⟫_ℝ) ≤ |
| C * ∫ x, ‖v t x - u t x‖ ^ 2 := by |
| let w : Space → Space := fun x => v t x - u t x |
| let r : Space → ℝ := fun x => q t x - p t x |
| have hu₂ := (hu.1 t ht).1 |
| have hv₂ := (hv.1 t ht).1 |
| have hp₁ := (hu.1 t ht).2 |
| have hq₁ := (hv.1 t ht).2 |
| have hu' := hu₂.differentiable (by norm_num) |
| have hv' := hv₂.differentiable (by norm_num) |
| have hp' := hp₁.differentiable (by norm_num) |
| have hq' := hq₁.differentiable (by norm_num) |
| have hw1 (i : Fin 3) : MemLp (spaceD i w) 2 volume := by |
| dsimp [w]; rw [spaceD_sub hv' hu'] |
| exact (hev.memLp_velocityD ht i).sub (heu.memLp_velocityD ht i) |
| have hw2 (i : Fin 3) : MemLp (spaceD i (spaceD i w)) 2 volume := by |
| dsimp [w]; rw [spaceD_sub_twice hv₂ hu₂] |
| exact (hev.memLp_velocityDD ht i).sub (heu.memLp_velocityDD ht i) |
| have hr1 (i : Fin 3) : MemLp (spaceD i r) 2 volume := by |
| dsimp [r]; rw [spaceD_sub hq' hp'] |
| exact (hev.memLp_pressureD ht i).sub (heu.memLp_pressureD ht i) |
| have hdv (x : Space) : ∑ i, spaceD i (v t) x i = 0 := hnsv.2.1 t ht x |
| have hdw (x : Space) : ∑ i, spaceD i w x i = 0 := by |
| dsimp [w] |
| simp_rw [spaceD_sub hv' hu', PiLp.sub_apply, Finset.sum_sub_distrib] |
| change divergence v t x - divergence u t x = 0 |
| rw [hnsv.2.1 t ht x, hnsu.2.1 t ht x, sub_self] |
| have hbound := spatial_energy_bound hu₂ hv' (hv₂.sub hu₂) (hq'.sub hp') |
| ((hev.memLp_velocity ht).sub (heu.memLp_velocity ht)) |
| ((hev.memLp_pressure ht).sub (heu.memLp_pressure ht)) |
| (hev.memLp_velocityD ht) hw1 hw2 hr1 hν huC hvB |
| (fun x => (norm_sub_le (v t x) (u t x)).trans (add_le_add (hvB x) (huA x))) hdv hdw |
| have heq (x : Space) : timeD v t x - timeD u t x = |
| -(∑ i, spaceD i r x • basis i) + ν • (∑ i, spaceD i (spaceD i w) x) - |
| (∑ i, v t x i • spaceD i w x) - fderiv ℝ (u t) x (w x) := by |
| have ha := advection_difference hu' hv' x |
| change advection v t x - advection u t x = |
| (∑ i, v t x i • spaceD i w x) + fderiv ℝ (u t) x (w x) at ha |
| have hl : (∑ i, spaceD i (spaceD i w) x) = laplace v t x - laplace u t x := by |
| simp only [w, spaceD_sub_twice hv₂ hu₂, Finset.sum_sub_distrib] |
| rfl |
| have hg : (∑ i, spaceD i r x • basis i) = gradient q t x - gradient p t x := by |
| simp only [r, spaceD_sub hq' hp', sub_smul, Finset.sum_sub_distrib] |
| rfl |
| rw [hl, hg, smul_sub] |
| have hvns := hnsv.1 t ht x |
| have huns := hnsu.1 t ht x |
| linear_combination (norm := module) hvns - huns - ha |
| change (∫ x, ⟪w x, -(∑ i, spaceD i r x • basis i) + |
| ν • (∑ i, spaceD i (spaceD i w) x) - (∑ i, v t x i • spaceD i w x) - |
| fderiv ℝ (u t) x (w x)⟫_ℝ) ≤ C * ∫ x, ‖w x‖ ^ 2 at hbound |
| simpa only [← heq] using hbound |
|
|
| lemma energy_zero {E : Type} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| {v v' : ℝ → E} {T C : ℝ} (hT : 0 ≤ T) |
| (hv : ContinuousOn v (Icc 0 T)) |
| (hd : ∀ t ∈ Icc 0 T, HasDerivWithinAt v (v' t) (Icc 0 T) t) |
| (h0 : v 0 = 0) |
| (hb : ∀ t ∈ Icc 0 T, ⟪v t, v' t⟫_ℝ ≤ C * ‖v t‖ ^ 2) : |
| ∀ t ∈ Icc 0 T, v t = 0 := by |
| let e : ℝ → ℝ := fun s => Real.exp (-2 * C * s) * ‖v s‖ ^ 2 |
| let e' : ℝ → ℝ := fun s => Real.exp (-2 * C * s) * |
| (2 * ⟪v s, v' s⟫_ℝ - 2 * C * ‖v s‖ ^ 2) |
| have he : ContinuousOn e (Icc 0 T) := |
| ((Real.continuous_exp.comp (continuous_const.mul continuous_id)).continuousOn).mul (hv.norm.pow 2) |
| have he' (s : ℝ) (hs : s ∈ Icc 0 T) : HasDerivWithinAt e (e' s) (Icc 0 T) s := by |
| have h := (((hasDerivAt_id s).const_mul (-2 * C)).exp.hasDerivWithinAt).mul (hd s hs).norm_sq |
| convert h using 1 <;> first | rfl | (dsimp only [e, e', id_eq]; ring) |
| have hn (s : ℝ) (hs : s ∈ Icc 0 T) : e' s ≤ 0 := by |
| apply mul_nonpos_of_nonneg_of_nonpos (Real.exp_nonneg _) |
| have := hb s hs |
| linarith |
| have hanti : AntitoneOn e (Icc 0 T) := |
| antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc 0 T) he |
| (fun s hs => (he' s (interior_subset hs)).mono interior_subset) |
| (fun s hs => hn s (interior_subset hs)) |
| intro t ht |
| have h := hanti (left_mem_Icc.mpr hT) ht ht.1 |
| have hle : Real.exp (-2 * C * t) * ‖v t‖ ^ 2 ≤ 0 := by simpa [e, h0] using h |
| have hz : ‖v t‖ ^ 2 = 0 := le_antisymm (nonpos_of_mul_nonpos_right hle (Real.exp_pos _)) |
| (sq_nonneg _) |
| exact norm_eq_zero.mp (eq_zero_of_pow_eq_zero hz) |
|
|
| theorem energy_velocity_unique {ν : ℝ} (hν : 0 ≤ ν) |
| {f u v : Field Space} {p q : Field ℝ} |
| (hu : Classical u p) (hv : Classical v q) |
| (heu : EnergyClass u p) (hev : EnergyCompetitor v q) |
| (hnsu : NavierStokes ν f u p) (hnsv : NavierStokes ν f v q) : |
| ∀ t : ℝ, 0 ≤ t → v t = u t := by |
| intro t ht |
| obtain ⟨U, U', hU, hU', huCoe, hu'Coe, hdU⟩ := (heu.1 t ht).2.1 |
| obtain ⟨V, V', hV, hV', hvCoe, hv'Coe, hdV⟩ := (hev.1 t ht).2.1 |
| obtain ⟨C, hC⟩ := heu.2 t ht |
| obtain ⟨B, hB⟩ := hev.2 t ht |
| let W := fun s => V s - U s |
| let W' := fun s => V' s - U' s |
| have hCoe (s : ℝ) (hs : s ∈ Icc 0 t) : |
| ((W s) : Space → Space) =ᵐ[volume] (fun x => v s x - u s x) := by |
| filter_upwards [Lp.coeFn_sub (V s) (U s), hvCoe s hs, huCoe s hs] with x hw hv hu |
| change (V s - U s) x = _ |
| rw [hw, Pi.sub_apply, hv, hu] |
| have h'Coe (s : ℝ) (hs : s ∈ Icc 0 t) : |
| ((W' s) : Space → Space) =ᵐ[volume] (fun x => timeD v s x - timeD u s x) := by |
| filter_upwards [Lp.coeFn_sub (V' s) (U' s), hv'Coe s hs, hu'Coe s hs] with x hw hv hu |
| change (V' s - U' s) x = _ |
| rw [hw, Pi.sub_apply, hv, hu] |
| have h0 : W 0 = 0 := by |
| apply Lp.ext |
| filter_upwards [hCoe 0 (left_mem_Icc.mpr ht), Lp.coeFn_zero (E := Space) (p := 2) |
| (μ := (volume : Measure Space))] with x hw hz |
| rw [hw, hnsv.2.2 x, hnsu.2.2 x, sub_self, hz] |
| rfl |
| have henergy (s : ℝ) (hs : s ∈ Icc 0 t) : ⟪W s, W' s⟫_ℝ ≤ C * ‖W s‖ ^ 2 := by |
| have hi : ⟪W s, W' s⟫_ℝ = ∫ x, ⟪v s x - u s x, timeD v s x - timeD u s x⟫_ℝ := by |
| rw [L2.inner_def] |
| apply integral_congr_ae |
| filter_upwards [hCoe s hs, h'Coe s hs] with x hw hw' |
| rw [hw, hw'] |
| have hn : ‖W s‖ ^ 2 = ∫ x, ‖v s x - u s x‖ ^ 2 := by |
| rw [← real_inner_self_eq_norm_sq, L2.inner_def] |
| apply integral_congr_ae |
| filter_upwards [hCoe s hs] with x hx |
| rw [hx, real_inner_self_eq_norm_sq] |
| rw [hi, hn] |
| apply navierStokes_difference_energy hν hu hv heu.1 hev.1 hnsu hnsv hs.1 |
| · intro x |
| exact (le_add_of_nonneg_left (norm_nonneg (u s x))).trans (hC s hs x) |
| · exact hB s hs |
| · intro x |
| exact (le_add_of_nonneg_right (norm_nonneg (fderiv ℝ (u s) x))).trans (hC s hs x) |
| have heq : W t = 0 := energy_zero ht (hV.sub hU) |
| (fun s hs => (hdV s hs).sub (hdU s hs)) h0 henergy t (right_mem_Icc.mpr ht) |
| apply ((hv.1 t ht).1.continuous.ae_eq_iff_eq volume (hu.1 t ht).1.continuous).mp |
| have h := hCoe t (right_mem_Icc.mpr ht) |
| rw [heq] at h |
| filter_upwards [h, Lp.coeFn_zero (E := Space) (p := 2) (μ := (volume : Measure Space))] |
| with x hx hz |
| rw [hz] at hx |
| exact sub_eq_zero.mp hx.symm |
|
|
| lemma volume_space_univ : volume (univ : Set Space) = ∞ := by |
| have h := (PiLp.volume_preserving_ofLp (Fin 3)).measure_preimage MeasurableSet.univ |
| change volume (univ : Set Space) = volume (univ : Set (Fin 3 → ℝ)) at h |
| rw [h, ← pi_univ (univ : Set (Fin 3)), volume_pi_pi] |
| simp |
|
|
| lemma gradient_apply (p : Field ℝ) (t : ℝ) (x : Space) (i : Fin 3) : |
| gradient p t x i = spatialD i p t x := by |
| simp [gradient, basis, Pi.single_apply, apply_ite] |
|
|
| theorem energy_pressure_unique {ν : ℝ} {f u v : Field Space} {p q : Field ℝ} |
| (hu : Classical u p) (hv : Classical v q) |
| (heu : EnergyRegularity u p) (hev : EnergyRegularity v q) |
| (hnsu : NavierStokes ν f u p) (hnsv : NavierStokes ν f v q) |
| (heq : ∀ t : ℝ, 0 ≤ t → v t = u t) : |
| ∀ t : ℝ, 0 ≤ t → ∀ x : Space, q t x = p t x := by |
| intro t ht |
| have htime (x : Space) : timeD v t x = timeD u t x := by |
| apply derivWithin_congr (fun s hs => congrFun (heq s hs) x) (congrFun (heq t ht) x) |
| have hgrad (x : Space) : gradient q t x = gradient p t x := by |
| have h := hnsv.1 t ht x |
| have h' := hnsu.1 t ht x |
| have ha : advection v t x = advection u t x := by |
| simp only [advection, spatialD, heq t ht] |
| have hl : laplace v t x = laplace u t x := by |
| unfold laplace spatialD |
| rw [heq t ht] |
| rw [htime, ha, hl] at h |
| linear_combination (norm := module) h - h' |
| let r : Space → ℝ := fun x => q t x - p t x |
| have hp := (hu.1 t ht).2.differentiable (by norm_num) |
| have hq := (hv.1 t ht).2.differentiable (by norm_num) |
| have hdr (x : Space) : fderiv ℝ r x = 0 := by |
| ext z |
| have hd (i : Fin 3) : fderiv ℝ r x (basis i) = 0 := by |
| have h := congrArg (fun z : Space => z i) (hgrad x) |
| rw [gradient_apply, gradient_apply] at h |
| simpa [r, fderiv_fun_sub (hq x) (hp x), spatialD] using sub_eq_zero.mpr h |
| calc |
| (fderiv ℝ r x) z = (fderiv ℝ r x) (∑ i, z i • basis i) := by rw [sum_basis] |
| _ = 0 := by simp [map_sum, hd] |
| have hconst : r = fun _ => r 0 := funext fun x => |
| is_const_of_fderiv_eq_zero (hq.sub hp) hdr x 0 |
| have hr : MemLp r 2 volume := (hev.memLp_pressure ht).sub (heu.memLp_pressure ht) |
| rw [hconst] at hr |
| have hz := (memLp_const_iff (by norm_num : (2 : ℝ≥0∞) ≠ 0) (by simp)).mp hr |
| have hz : r 0 = 0 := hz.resolve_right (by simp) |
| intro x |
| exact sub_eq_zero.mp ((congrFun hconst x).trans hz) |
|
|
| theorem energy_solution_unique {ν : ℝ} (hν : 0 ≤ ν) |
| {f u : Field Space} {p : Field ℝ} (hu : Classical u p) |
| (heu : EnergyClass u p) (hnsu : NavierStokes ν f u p) : |
| UniqueEnergySolution ν f u p := by |
| refine ⟨hu, heu, hnsu, ?_⟩ |
| intro v q hv hev hnsv |
| have heq := energy_velocity_unique hν hu hv heu hev hnsu hnsv |
| have hp := energy_pressure_unique hu hv heu.1 hev.1 hnsu hnsv heq |
| exact fun t ht x => ⟨congrFun (heq t ht) x, hp t ht x⟩ |
|
|
| theorem addressed_uniqueEnergySolution {ν : ℝ} (hν : 0 ≤ ν) (M : Machine) (w : M.Input) : |
| UniqueEnergySolution ν (addressedForce ν M w) (addressedVelocity M w) (fun _ _ => 0) := by |
| exact energy_solution_unique hν (addressed_classical_solution ν M w).2.2.2.2.1 |
| (addressed_energyClass M w) (addressed_pointwise_equations ν M w) |
|
|
| end RapidForcing |
|
|
| end |
|
|
| end OAI |
|
|
