The local weakly continuous L³ representative #
This module upgrades the time-slice L² representative to L³ and extends
its smooth-test continuity to all L^{3/2} pairings.
The spatial ball of radius 3/4 used for the weak continuity L3 conclusion.
Equations
Instances For
theorem
ESS.weakContL3
{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}
(hu :
MeasureTheory.AEStronglyMeasurable u
(MeasureTheory.volume.restrict (CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball 0 1) (Set.Ioo (-1) 0))))
(hDu :
MeasureTheory.AEStronglyMeasurable Du
(MeasureTheory.volume.restrict (CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball 0 1) (Set.Ioo (-1) 0))))
(henergy :
∫⁻ (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 < ⊤)
(hp :
MeasureTheory.MemLp p (ENNReal.ofReal (3 / 2))
(MeasureTheory.volume.restrict (CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball 0 1) (Set.Ioo (-1) 0))))
(hL3 :
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)) < ⊤)
(hgrad :
∀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)
(hS3 :
∀ 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)
:
∃ (v :
↑(Set.Icc (-(3 / 4) ^ 2) 0) →
↥(MeasureTheory.Lp CKN.Foundation.Parabolic.L2Vec3 3
(MeasureTheory.volume.restrict (CKN.Foundation.Parabolic.vec3Ball 0 (3 / 4))))),
(∀ (t : ↑(Set.Icc (-(3 / 4) ^ 2) 0)), ‖v t‖ ≤ weakContL3MomentBound.toReal) ∧ (∀
(w :
↥(MeasureTheory.Lp CKN.Foundation.Parabolic.L2Vec3 (ENNReal.ofReal (3 / 2))
(MeasureTheory.volume.restrict (CKN.Foundation.Parabolic.vec3Ball 0 (3 / 4))))),
Continuous fun (t : ↑(Set.Icc (-(3 / 4) ^ 2) 0)) =>
∫ (x : CKN.Foundation.Parabolic.Vec3) in CKN.Foundation.Parabolic.vec3Ball 0 (3 / 4), inner ℝ (↑↑(v t) x) (↑↑w x)) ∧ ∀ᵐ (t : ℝ) ∂MeasureTheory.volume.restrict (Set.Ioo (-(3 / 4) ^ 2) 0), ∃ (ht : t ∈ Set.Icc (-(3 / 4) ^ 2) 0) (huSlice :
MeasureTheory.MemLp (fun (x : CKN.Foundation.Parabolic.Vec3) => WithLp.toLp 2 (u (x, t))) 3
(MeasureTheory.volume.restrict (CKN.Foundation.Parabolic.vec3Ball 0 (3 / 4)))),
v ⟨t, ht⟩ = MeasureTheory.MemLp.toLp (fun (x : CKN.Foundation.Parabolic.Vec3) => WithLp.toLp 2 (u (x, t))) huSlice
The local velocity has the all-time weakly continuous L³ representative
from lem:weak-cont-L3.