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37bcf52 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 | 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
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