Uniform one-sided velocity and pressure Morrey bounds #
Small-cylinder estimates follow from decay. Large-cylinder estimates use
the original unit-cylinder small-data hypothesis, so the constants are
independent of the solution. Both estimates concern extension by zero from
the intermediate backward cylinder of radius 5/8.
theorem
CKN.Core.Endgame.closure_small_cylinder_subset_unit
{z : Foundation.Parabolic.ParabolicPoint}
{r : ℝ}
(hz : z ∈ Foundation.Parabolic.parabolicCylinder 0 0 (3 / 4))
(hr : 0 ≤ r)
(hrquarter : r ≤ 1 / 4)
:
closure (Foundation.Parabolic.parabolicCylinder z.1 z.2 r) ⊆ closure (Foundation.Parabolic.parabolicCylinder 0 0 1)
Quarter-scale cylinders centered in the three-quarter cylinder have closure in the closed unit cylinder.
The explicit uniform velocity Morrey constant.
Equations
- CKN.Core.Endgame.oneSidedVelocityMorreyBound M r₀ ε₀ = CKN.Core.Endgame.oneSidedMorreyBound 3 (25 / 3) r₀ (ENNReal.ofReal ((2 * CKN.gagliardoConstant * M) ^ 3)) (ENNReal.ofReal ε₀)
Instances For
The explicit uniform pressure Morrey constant.
Equations
- CKN.Core.Endgame.oneSidedPressureMorreyBound M r₀ ε₀ = CKN.Core.Endgame.oneSidedMorreyBound (3 / 2) (25 / 8) r₀ (ENNReal.ofReal (M ^ (3 / 2))) (ENNReal.ofReal ε₀)
Instances For
The velocity constant is finite.
The pressure constant is finite.
theorem
CKN.Core.Endgame.oneSided_velocity_morrey_of_decay
(M r₀ ε₀ : ℝ)
(hM : 0 ≤ M)
(hr₀ : 0 < r₀)
(hrquarter : r₀ ≤ 1 / 4)
{Ω : Set Foundation.Parabolic.Vec3}
{I : Set ℝ}
{q : ℝ}
{u : Foundation.Parabolic.ParabolicPoint → Foundation.Parabolic.Vec3}
{Du : Foundation.Parabolic.ParabolicPoint → Fin 3 → Foundation.Parabolic.Vec3}
{p : Foundation.Parabolic.ParabolicPoint → ℝ}
{f : Foundation.Parabolic.ParabolicPoint → Foundation.Parabolic.Vec3}
(hsol : IsSuitableWeakSolutionIntegrable Ω I q u Du p f)
(hQ₁ : closure (Foundation.Parabolic.parabolicCylinder 0 0 1) ⊆ spaceTimeSet Ω I)
(hdata :
∫⁻ (z : Foundation.Parabolic.ParabolicPoint) in Foundation.Parabolic.parabolicCylinder 0 0 1, ENNReal.ofReal (Foundation.Parabolic.vec3EuclideanNorm (u z)) ^ 3 + ENNReal.ofReal |p z| ^ (3 / 2) + ENNReal.ofReal (Foundation.Parabolic.vec3EuclideanNorm (f z)) ^ q ≤ ENNReal.ofReal ε₀)
(hdec :
∀ z ∈ Foundation.Parabolic.parabolicCylinder 0 0 (3 / 4),
∀ (r : ℝ), 0 < r → r ≤ r₀ → max (max (alpha u z r) (beta u Du z r)) (delta p z r ^ 2) ≤ M * r ^ (2 / 5))
(i : Fin 3)
:
Foundation.Parabolic.Morrey.morreyNorm 3 (25 / 3)
((Foundation.Parabolic.parabolicCylinder 0 0 (5 / 8)).indicator fun (z : Foundation.Parabolic.ParabolicPoint) =>
u z i) ≤ oneSidedVelocityMorreyBound M r₀ ε₀
Uniform scalar velocity Morrey control follows from one-sided decay and the original unit-cylinder smallness hypothesis.
theorem
CKN.Core.Endgame.oneSided_pressure_morrey_of_decay
(M r₀ ε₀ : ℝ)
(hM : 0 ≤ M)
(hr₀ : 0 < r₀)
(hrquarter : r₀ ≤ 1 / 4)
{Ω : Set Foundation.Parabolic.Vec3}
{I : Set ℝ}
{q : ℝ}
{u : Foundation.Parabolic.ParabolicPoint → Foundation.Parabolic.Vec3}
{Du : Foundation.Parabolic.ParabolicPoint → Fin 3 → Foundation.Parabolic.Vec3}
{p : Foundation.Parabolic.ParabolicPoint → ℝ}
{f : Foundation.Parabolic.ParabolicPoint → Foundation.Parabolic.Vec3}
(hsol : IsSuitableWeakSolutionIntegrable Ω I q u Du p f)
(hQ₁ : closure (Foundation.Parabolic.parabolicCylinder 0 0 1) ⊆ spaceTimeSet Ω I)
(hdata :
∫⁻ (z : Foundation.Parabolic.ParabolicPoint) in Foundation.Parabolic.parabolicCylinder 0 0 1, ENNReal.ofReal (Foundation.Parabolic.vec3EuclideanNorm (u z)) ^ 3 + ENNReal.ofReal |p z| ^ (3 / 2) + ENNReal.ofReal (Foundation.Parabolic.vec3EuclideanNorm (f z)) ^ q ≤ ENNReal.ofReal ε₀)
(hdec :
∀ z ∈ Foundation.Parabolic.parabolicCylinder 0 0 (3 / 4),
∀ (r : ℝ), 0 < r → r ≤ r₀ → max (max (alpha u z r) (beta u Du z r)) (delta p z r ^ 2) ≤ M * r ^ (2 / 5))
:
Foundation.Parabolic.Morrey.morreyNorm (3 / 2) (25 / 8)
((Foundation.Parabolic.parabolicCylinder 0 0 (5 / 8)).indicator p) ≤ oneSidedPressureMorreyBound M r₀ ε₀
Uniform pressure Morrey control follows from one-sided decay and the original unit-cylinder smallness hypothesis.