thm:leray by the direct route

The regularized solutions of thm:regularised are the pointwise velocity of the global regularized mild curve and its canonical Riesz pressure. With the global Bessel lift of the curve, the pressure regularity, the classical time derivative and joint continuous differentiability, they satisfy the full contract (R1)–(R5). Together with lem:reg-tails and prop:leray-limit, the direct limiting argument then gives thm:leray.

theorem CKN.Leray.regR12_regularised_unconditional (ρ : RegMollifierProfile) (a : Foundation.Parabolic.Vec3 → Foundation.Parabolic.Vec3) (ha : IsInJ a) (ε : ℝ) (hε : 0 < ε) : have u := regR12Uε ρ a ha ε; have p := regR12Pε ρ 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 regR12_regularised_unconditional._proof_55 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

thm:regularised, (R1)–(R5), for the regularized velocities and pressures, with every input discharged.

theorem CKN.Leray.leray_existence_direct_route (ρ : RegMollifierProfile) (a : Foundation.Parabolic.Vec3 → Foundation.Parabolic.Vec3) : IsInJ a → ∃ (u : Foundation.Parabolic.ParabolicPoint → Foundation.Parabolic.Vec3) (Du : Foundation.Parabolic.ParabolicPoint → Fin 3 → Foundation.Parabolic.Vec3) (p : Foundation.Parabolic.ParabolicPoint → ℝ), IsGlobalLerayHopfSolution a u Du ∧ ∀ (q : ℝ), 5 / 2 < q → IsSuitableWeakSolution Set.univ (Set.Ioi 0) q u Du p 0

thm:leray by the direct route, for the regularized solutions with the mollifier profile ρ.