openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Programs /FiniteTimeUniqueness.lean
| import OAI.MathematicalPhysics.NavierStokes.ShearFlows.ClassicalUniqueness | |
| namespace OAI | |
| /-! The energy uniqueness argument only needs a derivative bound on each | |
| finite interval. This covers the increasing-speed torus detector. -/ | |
| noncomputable section | |
| namespace ForcedComputation.VelocityDetector | |
| open ShearFlows Set MeasureTheory | |
| open scoped ContDiff | |
| theorem finite_interval_classical_velocity_unique {L ν : ℝ} (hL : 0 < L) (hν : 0 ≤ ν) | |
| {f u v : Velocity} {p : Pressure} | |
| (hu : IsClassicalSolution L ν f u p) (hv : IsClassicalSolution L ν f v (fun _ => 0)) | |
| (hb : ∀ T, 0 ≤ T → ∃ B : ℝ, ∀ t ∈ Icc 0 T, ∀ x, | |
| ‖fderiv ℝ (fun y => v (t,y)) x‖ ≤ B) : | |
| ∀ t, 0 ≤ t → ∀ x, u (t,x) = v (t,x) := by | |
| let W : Velocity := fun y => u y - v y | |
| let G : Velocity := fun y => initialTimeDerivative u y.1 y.2 - initialTimeDerivative v y.1 y.2 | |
| let E : ℝ → ℝ := fun t => ∫ x in fundamentalCube L, dot (W (t,x)) (W (t,x)) | |
| have hWs (t : ℝ) (ht : 0 ≤ t) : ContDiff ℝ 2 (fun x => W (t,x)) := | |
| (hu.regularity.spatial_u t ht).sub (hv.regularity.spatial_u t ht) | |
| have hWp (t : ℝ) (ht : 0 ≤ t) : CubePeriodic L (fun x => W (t,x)) := by | |
| intro x n | |
| dsimp [W] | |
| rw [hu.periodic_u t ht, hv.periodic_u t ht] | |
| have hWc (T : ℝ) (hT : 0 ≤ T) : ContinuousOn W (timeCylinder T) := | |
| (hu.regularity.continuous_u T hT).sub (hv.regularity.continuous_u T hT) | |
| have hGc (T : ℝ) (hT : 0 ≤ T) : ContinuousOn G (timeCylinder T) := | |
| (hu.regularity.continuous_ut T hT).sub (hv.regularity.continuous_ut T hT) | |
| have hGd (t : ℝ) (ht : 0 < t) (x : Space) : HasDerivAt (fun s => W (s,x)) (G (t,x)) t := | |
| ((hu.regularity.temporal_u t ht.le x).hasDerivWithinAt.hasDerivAt (Ici_mem_nhds ht)).sub | |
| ((hv.regularity.temporal_u t ht.le x).hasDerivWithinAt.hasDerivAt (Ici_mem_nhds ht)) | |
| have hE0 : E 0 = 0 := by simp [E,W,hu.initial,hv.initial] | |
| have hEpos (t : ℝ) : 0 ≤ E t := integral_nonneg (fun _ => dot_self_nonneg _) | |
| have hEc (T : ℝ) (hT : 0 ≤ T) : ContinuousOn E (Icc 0 T) := | |
| continuousOn_cube_integral (dot_continuousOn (hWc T hT) (hWc T hT)) | |
| have hEd (T : ℝ) (hT : 0 ≤ T) (t : ℝ) (ht : t ∈ Ioo 0 T) : | |
| HasDerivAt E (2 * ∫ x in fundamentalCube L, dot (W (t,x)) (G (t,x))) t := by | |
| have hh := hasDerivAt_cube_integral (L := L) | |
| (dot_continuousOn (hWc T hT) (hWc T hT)) | |
| (continuousOn_const.mul (dot_continuousOn (hWc T hT) (hGc T hT))) | |
| (fun s hs x => hasDerivAt_dot_self (hGd s hs.1 x)) ht | |
| simpa only [Pi.mul_apply, integral_const_mul] using hh | |
| intro T hT x | |
| obtain ⟨B, hB⟩ := hb T hT | |
| have hineq (t : ℝ) (htT : t ∈ Icc 0 T) : | |
| (∫ x in fundamentalCube L, dot (W (t,x)) (G (t,x))) ≤ 3 * B * E t := by | |
| have ht : 0 ≤ t := htT.1 | |
| apply integrated_comparison_inequality hL.le hν (hWs t ht) | |
| ((hu.regularity.spatial_u t ht).of_le (by norm_num)) | |
| ((hv.regularity.spatial_u t ht).of_le (by norm_num)) | |
| (hu.regularity.spatial_p t ht) (hWp t ht) (hu.periodic_u t ht) (hu.periodic_p t ht) | |
| · intro x | |
| rw [divergence_sub ((hu.regularity.spatial_u t ht).differentiable (by norm_num)) | |
| ((hv.regularity.spatial_u t ht).differentiable (by norm_num)), | |
| hu.divergence_zero t ht, hv.divergence_zero t ht, sub_self] | |
| · exact hu.divergence_zero t ht | |
| · exact hB t htT | |
| · intro x | |
| dsimp only [G, W] | |
| rw [← initialTimeDerivative_sub (hu.regularity.temporal_u t ht x) | |
| (hv.regularity.temporal_u t ht x)] | |
| exact classical_difference_equation hu hv ht x | |
| have hEt : E T = 0 := by | |
| apply zero_of_energy_inequality (C := 6 * B) hT (hEc T hT) hE0 (fun s _ => hEpos s) | |
| intro s hs | |
| refine ⟨_, hEd T hT s hs, ?_⟩ | |
| have hh := hineq s ⟨hs.1.le, hs.2.le⟩ | |
| nlinarith | |
| exact sub_eq_zero.mp (periodic_field_zero_of_energy_zero hL (hWs T hT).continuous | |
| (hWp T hT) hEt x) | |
| private theorem compact_scalar_bound {L T : ℝ} {g : SpaceTime → ℝ} | |
| (hg : ContinuousOn g (Icc 0 T ×ˢ fundamentalCube L)) : | |
| ∃ B : ℝ, ∀ t ∈ Icc 0 T, ∀ x ∈ fundamentalCube L, g (t, x) ≤ B := by | |
| have hk : IsCompact (Icc (0 : ℝ) T ×ˢ fundamentalCube L) := | |
| isCompact_Icc.prod isCompact_Icc | |
| obtain ⟨B, hB⟩ := hk.bddAbove_image hg | |
| refine ⟨B, ?_⟩ | |
| intro t ht x hx | |
| exact hB ⟨(t, x), ⟨ht, hx⟩, rfl⟩ | |
| theorem classical_compact_derivative_bound {L ν : ℝ} | |
| {f v : Velocity} {p : Pressure} (hv : IsClassicalSolution L ν f v p) | |
| (T : ℝ) (hT : 0 ≤ T) : | |
| ∃ B : ℝ, ∀ t ∈ Icc 0 T, ∀ x ∈ fundamentalCube L, | |
| ‖fderiv ℝ (fun y => v (t,y)) x‖ ≤ B := by | |
| let D : SpaceTime → Space →L[ℝ] Space := fun y => fderiv ℝ (fun x => v (y.1, x)) y.2 | |
| have hc : ContinuousOn D (Icc 0 T ×ˢ fundamentalCube L) := | |
| (hv.regularity.continuous_du T hT).mono (fun _ hy => ⟨hy.1, mem_univ _⟩) | |
| exact compact_scalar_bound (g := fun y => ‖D y‖) hc.norm | |
| theorem classical_spatial_derivative_periodic {L ν : ℝ} | |
| {f v : Velocity} {p : Pressure} (hv : IsClassicalSolution L ν f v p) | |
| {t : ℝ} (ht : 0 ≤ t) : CubePeriodic L (fderiv ℝ (fun x => v (t, x))) := by | |
| have hd : Differentiable ℝ (fun x => v (t, x)) := | |
| (hv.regularity.spatial_u t ht).differentiable (by norm_num) | |
| exact CubePeriodic.fderiv (hv.periodic_u t ht) hd | |
| theorem classical_finite_derivative_bound {L ν : ℝ} (hL : 0 < L) | |
| {f v : Velocity} {p : Pressure} (hv : IsClassicalSolution L ν f v p) | |
| (T : ℝ) (hT : 0 ≤ T) : | |
| ∃ B : ℝ, ∀ t ∈ Icc 0 T, ∀ x, ‖fderiv ℝ (fun y => v (t,y)) x‖ ≤ B := by | |
| obtain ⟨B, hB⟩ := classical_compact_derivative_bound hv T hT | |
| refine ⟨B, ?_⟩ | |
| intro t ht x | |
| obtain ⟨y, hy, n, rfl⟩ := fundamentalCube_representative hL x | |
| rw [classical_spatial_derivative_periodic hv ht.1 y n] | |
| exact hB t ht y hy | |
| theorem finite_time_classical_unique {L ν : ℝ} (hL : 0 < L) (hν : 0 ≤ ν) | |
| {f v : Velocity} (hv : IsClassicalSolution L ν f v (fun _ => 0)) : | |
| ∀ u p, IsClassicalSolution L ν f u p → | |
| ∀ t, 0 ≤ t → ∀ x, u (t,x) = v (t,x) ∧ p (t,x) = 0 := by | |
| intro u p hu | |
| have he := finite_interval_classical_velocity_unique hL hν hu hv | |
| (classical_finite_derivative_bound hL hv) | |
| have hp := classical_pressure_unique hL hu hv he | |
| exact fun t ht x => ⟨he t ht x, hp t ht x⟩ | |
| end ForcedComputation.VelocityDetector | |
| end | |
| end OAI | |