A global bound on positive-time slabs #
This file proves lem:pv-positive-bound: under the critical hypothesis
u ∈ L^∞_t L³_x of thm:ess-global, a Leray–Hopf solution on ℝ³ × (0, T) is
essentially bounded on every slab ℝ³ × (δ, T), δ > 0. The local regularity
theorem thm:ess-local is assumed in the form of hE1.
The proof fixes the scale R = √δ and, around each point z of the slab, the
backward cylinder of radius R / 4 ending at min (z.2 + R² / 32, T), which
may end at the terminal time. Far from the origin the tails of |u|³ and of
|p|^{3/2} for the canonical pressure of thm:assoc-pressure make the
normalized energy small, and pv_epsilon_bound gives the bound C / R. On the
bounded part thm:ess-local gives local bounds, and a finite subcover of the
compact set closure B × [δ, T] makes them uniform.
theorem
ESS.pvPositiveBound
(hE1 :
∀ (u : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3)
(Du : CKN.Foundation.Parabolic.ParabolicPoint → Fin 3 → CKN.Foundation.Parabolic.Vec3)
(p : CKN.Foundation.Parabolic.ParabolicPoint → ℝ),
MeasureTheory.AEStronglyMeasurable u
(MeasureTheory.volume.restrict (CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball 0 1) (Set.Ioo (-1) 0))) →
MeasureTheory.AEStronglyMeasurable Du
(MeasureTheory.volume.restrict (CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball 0 1) (Set.Ioo (-1) 0))) →
MeasureTheory.AEStronglyMeasurable p
(MeasureTheory.volume.restrict
(CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball 0 1) (Set.Ioo (-1) 0))) →
essSup
(fun (t : ℝ) =>
∫⁻ (x : CKN.Foundation.Parabolic.Vec3) in CKN.Foundation.Parabolic.vec3Ball 0 1, ‖u (x, t)‖ₑ ^ 2)
(MeasureTheory.volume.restrict (Set.Ioo (-1) 0)) < ⊤ →
∫⁻ (z : CKN.Foundation.Parabolic.ParabolicPoint) in CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball 0 1) (Set.Ioo (-1) 0), ‖u z‖ₑ ^ 2 + ‖Du z‖ₑ ^ 2 < ⊤ →
MeasureTheory.MemLp p (ENNReal.ofReal (3 / 2))
(MeasureTheory.volume.restrict
(CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball 0 1) (Set.Ioo (-1) 0))) →
essSup
(fun (t : ℝ) =>
∫⁻ (x : CKN.Foundation.Parabolic.Vec3) in CKN.Foundation.Parabolic.vec3Ball 0 1, ENNReal.ofReal (CKN.Foundation.Parabolic.vec3EuclideanNorm (u (x, t))) ^ 3)
(MeasureTheory.volume.restrict (Set.Ioo (-1) 0)) < ⊤ →
(∀ᵐ (t : ℝ) ∂MeasureTheory.volume.restrict (Set.Ioo (-1) 0), ∀ (i : Fin 3),
CKN.HasWeakGradientOn (CKN.Foundation.Parabolic.vec3Ball 0 1)
(fun (x : CKN.Vec 3) => u (x, t) i) fun (x : CKN.Vec 3) => Du (x, t) i) →
(∀ ψ ∈ CKN.spaceTimeTestFunction (CKN.Foundation.Parabolic.vec3Ball 0 1) (Set.Ioo (-1) 0),
∫ (z : CKN.Foundation.Parabolic.ParabolicPoint) in CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball 0 1) (Set.Ioo (-1) 0), ∑ i : Fin 3, u z i * CKN.spatialPartial ψ i z = 0) →
(∀ φ ∈ CKN.spaceTimeTestFunction (CKN.Foundation.Parabolic.vec3Ball 0 1) (Set.Ioo (-1) 0),
∫ (z : CKN.Foundation.Parabolic.ParabolicPoint) in CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball 0 1) (Set.Ioo (-1) 0), -∑ i : Fin 3,
u z i * CKN.timePartial (fun (y : CKN.Foundation.Parabolic.ParabolicPoint) => φ y i)
z - ∑ i : Fin 3,
∑ j : Fin 3,
u z i * u z j * CKN.spatialPartial
(fun (y : CKN.Foundation.Parabolic.ParabolicPoint) => φ y i) j z + ∑ i : Fin 3,
∑ j : Fin 3,
Du z i j * CKN.spatialPartial
(fun (y : CKN.Foundation.Parabolic.ParabolicPoint) => φ y i) j z - p z * ∑ i : Fin 3,
CKN.spatialPartial (fun (y : CKN.Foundation.Parabolic.ParabolicPoint) => φ y i) i
z - ∑ i : Fin 3, 0 z i * φ z i = 0) →
∃ (γ : ℝ),
0 < γ ∧ γ ≤ 1 ∧ ∃ (w : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3),
w =ᵐ[MeasureTheory.volume.restrict
(CKN.Foundation.Parabolic.parabolicCylinder 0 0 (1 / 2))]
u ∧ CKN.ParabolicHolderVecOn
(closure (CKN.Foundation.Parabolic.parabolicCylinder 0 0 (1 / 2))) w γ)
{T : ℝ}
{a : CKN.Foundation.Parabolic.Vec3 → CKN.Foundation.Parabolic.Vec3}
{u : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3}
{Du : CKN.Foundation.Parabolic.ParabolicPoint → Fin 3 → CKN.Foundation.Parabolic.Vec3}
(hLH : CKN.IsLerayHopfSolution T a u Du)
(hL3 :
essSup
(fun (t : ℝ) =>
∫⁻ (x : CKN.Foundation.Parabolic.Vec3), ENNReal.ofReal (CKN.Foundation.Parabolic.vec3EuclideanNorm (u (x, t))) ^ 3)
(MeasureTheory.volume.restrict (Set.Ioo 0 T)) < ⊤)
{δ : ℝ}
(hδ : 0 < δ)
:
∃ (B : ℝ),
∀ᵐ (z :
CKN.Foundation.Parabolic.ParabolicPoint) ∂MeasureTheory.volume.restrict (CKN.spaceTimeSet Set.univ (Set.Ioo δ T)), CKN.Foundation.Parabolic.vec3EuclideanNorm (u z) ≤ B
A global bound on positive-time slabs, lem:pv-positive-bound.