The compact-weight step of lem:reg-tails

For a compact, nonnegative smooth weight q with |∇q| ≤ M, the localized energy of a regularized solution grows on (s, t) by at most 18 M B² t^{1/2} + Q M B³ t^{1/4}, where B is the Euclidean L² norm of the initial datum (eq:reg-tails, lem:reg-tails).

noncomputable def CKN.Leray.regTailsFluxConstant : ℝ

The constant multiplying B³ t^{1/4} in the flux estimate of lem:reg-tails.

Equations

• One or more equations did not get rendered due to their size.

theorem CKN.Leray.regTailsFluxConstant_nonneg : 0 ≤ regTailsFluxConstant

The flux constant of lem:reg-tails is nonnegative.

theorem CKN.Leray.regTailsFinal_euclideanNorm_memLp {f : Foundation.Parabolic.Vec3 → Foundation.Parabolic.Vec3} (hf : MeasureTheory.MemLp f 2 MeasureTheory.volume) : MeasureTheory.MemLp (fun (x : Foundation.Parabolic.Vec3) => Foundation.Parabolic.vec3EuclideanNorm (f x)) 2 MeasureTheory.volume

A square-integrable field has square-integrable Euclidean length, as used for the exterior energies in lem:reg-tails.

theorem CKN.Leray.regTailsFinal_compact_weight_bound (ρ : RegMollifierProfile) (uε : (a : Foundation.Parabolic.Vec3 → Foundation.Parabolic.Vec3) → IsInJ a → ℝ → Foundation.Parabolic.ParabolicPoint → Foundation.Parabolic.Vec3) (pε : (a : Foundation.Parabolic.Vec3 → Foundation.Parabolic.Vec3) → IsInJ a → ℝ → Foundation.Parabolic.ParabolicPoint → ℝ) (hregularised : ∀ (a : Foundation.Parabolic.Vec3 → Foundation.Parabolic.Vec3) (ha : IsInJ a) (ε : ℝ) (hε : 0 < ε), have u := uε a ha ε; have p := pε a ha ε; have D := fun (z : Foundation.Parabolic.ParabolicPoint) (i j : Fin 3) => spatialPartial (fun (y : Foundation.Parabolic.ParabolicPoint) => u y i) j z; have DD := fun (z : Foundation.Parabolic.ParabolicPoint) (i j k : Fin 3) => spatialPartial (fun (y : Foundation.Parabolic.ParabolicPoint) => spatialPartial (fun (x : Foundation.Parabolic.ParabolicPoint) => u x i) j y) k z; have Dt := fun (z : Foundation.Parabolic.ParabolicPoint) (i : Fin 3) => timePartial (fun (y : Foundation.Parabolic.ParabolicPoint) => u y i) z; have Dp := fun (z : Foundation.Parabolic.ParabolicPoint) (i : Fin 3) => spatialPartial (fun (y : Foundation.Parabolic.ParabolicPoint) => p y) i z; (∃ (hSlice : ∀ (t : ℝ), 0 ≤ t → MeasureTheory.MemLp (fun (x : Foundation.Parabolic.Vec3) => u (x, t)) 2 MeasureTheory.volume), (Continuous fun (t : ↑(Set.Ici 0)) => realVectorL2OfCoordinateFunction (fun (x : Foundation.Parabolic.Vec3) => u (x, ↑t)) ⋯) ∧ (fun (x : Foundation.Parabolic.Vec3) => u (x, 0)) =ᵐ[MeasureTheory.volume] regUniformMollifiedInitial ρ ε hε a ∧ ∀ (t : ℝ), 0 ≤ t → IsWeakDivFreeL2 fun (x : Foundation.Parabolic.Vec3) => u (x, t)) ∧ (∀ (i : Fin 3), ContinuousOn (fun (z : Foundation.Parabolic.ParabolicPoint) => u z i) (spaceTimeSet Set.univ (Set.Ioi 0))) ∧ (∀ (i j : Fin 3), ContinuousOn (fun (z : Foundation.Parabolic.ParabolicPoint) => D z i j) (spaceTimeSet Set.univ (Set.Ioi 0))) ∧ (∀ (i j k : Fin 3), ContinuousOn (fun (z : Foundation.Parabolic.ParabolicPoint) => DD z i j k) (spaceTimeSet Set.univ (Set.Ioi 0))) ∧ (∀ (i : Fin 3), ContinuousOn (fun (z : Foundation.Parabolic.ParabolicPoint) => Dt z i) (spaceTimeSet Set.univ (Set.Ioi 0))) ∧ ContinuousOn p (spaceTimeSet Set.univ (Set.Ioi 0)) ∧ (∀ (i : Fin 3), ContinuousOn (fun (z : Foundation.Parabolic.ParabolicPoint) => Dp z i) (spaceTimeSet Set.univ (Set.Ioi 0))) ∧ (∀ (i : Fin 3), ContDiffOn ℝ 1 (fun (z : Foundation.Parabolic.ParabolicPoint) => u z i) (spaceTimeSet Set.univ (Set.Ioi 0))) ∧ (∀ z ∈ spaceTimeSet Set.univ (Set.Ioi 0), ∀ (i j : Fin 3), DifferentiableAt ℝ (fun (x : Foundation.Parabolic.Vec3) => D (x, z.2) i j) z.1) ∧ (∀ z ∈ spaceTimeSet Set.univ (Set.Ioi 0), DifferentiableAt ℝ (fun (x : Foundation.Parabolic.Vec3) => p (x, z.2)) z.1) ∧ (∀ (δ T : ℝ), 0 < δ → δ < T → ∃ (C : ℝ), 0 ≤ C ∧ ∀ z ∈ spaceTimeSet Set.univ (Set.Icc δ T), Foundation.Parabolic.vec3EuclideanNorm (u z) ≤ C ∧ |p z| ≤ C ∧ (∀ (i j : Fin 3), |D z i j| ≤ C) ∧ (∀ (i j k : Fin 3), |DD z i j k| ≤ C) ∧ (∀ (i : Fin 3), |Dt z i| ≤ C) ∧ ∀ (i : Fin 3), |Dp z i| ≤ C) ∧ (∀ (δ T : ℝ), 0 < δ → δ < T → (∀ (i : Fin 3), MeasureTheory.MemLp (fun (z : Foundation.Parabolic.ParabolicPoint) => u z i) 2 (MeasureTheory.volume.restrict (spaceTimeSet Set.univ (Set.Ioo δ T)))) ∧ (∀ (i j : Fin 3), MeasureTheory.MemLp (fun (z : Foundation.Parabolic.ParabolicPoint) => D z i j) 2 (MeasureTheory.volume.restrict (spaceTimeSet Set.univ (Set.Ioo δ T)))) ∧ (∀ (i j k : Fin 3), MeasureTheory.MemLp (fun (z : Foundation.Parabolic.ParabolicPoint) => DD z i j k) 2 (MeasureTheory.volume.restrict (spaceTimeSet Set.univ (Set.Ioo δ T)))) ∧ (∀ (i : Fin 3), MeasureTheory.MemLp (fun (z : Foundation.Parabolic.ParabolicPoint) => Dt z i) 2 (MeasureTheory.volume.restrict (spaceTimeSet Set.univ (Set.Ioo δ T)))) ∧ MeasureTheory.MemLp p 2 (MeasureTheory.volume.restrict (spaceTimeSet Set.univ (Set.Ioo δ T)))) ∧ (∀ (z : Foundation.Parabolic.ParabolicPoint), 0 < z.2 → ∀ (i : Fin 3), Dt z i - ∑ j : Fin 3, DD z i j j + ∑ j : Fin 3, regUniformMollifiedVelocity ρ ε hε u z j * D z i j + Dp z i = 0) ∧ (∀ (t : ℝ), 0 < t → ∃ (hF : ∀ (i j : Fin 3), MeasureTheory.MemLp (fun (x : Foundation.Parabolic.Vec3) => regUniformMollifiedVelocity ρ ε hε u (x, t) i * u (x, t) j) (ENNReal.ofReal 2) MeasureTheory.volume), ((fun (x : Foundation.Parabolic.Vec3) => p (x, t)) =ᵐ[MeasureTheory.volume] rieszPressureSliceRepresentative 2 regTailsFluxConstant._proof_1 fun (i j : Fin 3) => MeasureTheory.MemLp.toLp (fun (x : Foundation.Parabolic.Vec3) => regUniformMollifiedVelocity ρ ε hε u (x, t) i * u (x, t) j) ⋯) ∧ ∀ (ψ : Foundation.Parabolic.Vec3 → ℝ), ContDiff ℝ (↑⊤) ψ → HasCompactSupport ψ → ∫ (x : Foundation.Parabolic.Vec3), p (x, t) * spatialLaplacian ψ x = -∑ i : Fin 3, ∑ j : Fin 3, ∫ (x : Foundation.Parabolic.Vec3), regUniformMollifiedVelocity ρ ε hε u (x, t) i * u (x, t) j * mixedSecond ψ i j x) ∧ ∀ (t : ℝ), 0 ≤ t → MeasureTheory.eLpNorm (regUniformVelocitySlice u t) 2 MeasureTheory.volume ^ 2 + 2 * regUniformDissipation u D t = MeasureTheory.eLpNorm (regMollifyVector ρ ε hε (regUniformSpatialField a)) 2 MeasureTheory.volume ^ 2) (a : Foundation.Parabolic.Vec3 → Foundation.Parabolic.Vec3) (ha : IsInJ a) (ε : ℝ) (hε : 0 < ε) (q : Foundation.Parabolic.Vec3 → ℝ) (hq : ContDiff ℝ (↑⊤) q) (hqc : HasCompactSupport q) (hq0 : ∀ (x : Foundation.Parabolic.Vec3), 0 ≤ q x) {M : ℝ} (hM : 0 ≤ M) (hDq : ∀ (x : Foundation.Parabolic.Vec3) (j : Fin 3), |spatialDeriv q j x| ≤ M) {s t : ℝ} (hs : 0 < s) (hst : s < t) : regTailsEnergyWeight (uε a ha ε) q t ≤ regTailsEnergyWeight (uε a ha ε) q s + (18 * M * (MeasureTheory.eLpNorm (regUniformSpatialField a) 2 MeasureTheory.volume).toReal ^ 2 * t ^ (1 / 2) + regTailsFluxConstant * M * (MeasureTheory.eLpNorm (regUniformSpatialField a) 2 MeasureTheory.volume).toReal ^ 3 * t ^ (1 / 4))

For a compact nonnegative smooth weight q with |∂_j q| ≤ M, the localized energy of the regularized solution increases on (s, t) by at most 18 M B² t^{1/2} + Q M B³ t^{1/4} (lem:reg-tails).