Lin34 Centred Potential Source #
Part of the Caffarelli–Kohn–Nirenberg partial regularity proof.
Local L^{3/2} membership and linear growth of the centred potentials p₂–p₆ #
The uniqueness half of the Newtonian representation ext:newtonian applies Liouville's
theorem to a function that is L^{3/2} on every round ball about the origin, with local
norm growing at most like 1 + R. This file produces that pair of statements for the
group p₂ + p₃ + p₄ + p₅ + p₆ of the pressure decomposition
prop:pressure-decomposition of paper/ckn.tex, run with the mollified cut-off of
B_ρ(x₀) and the doubly centred tensor eq:Uhat of prop:lin34.
The velocity potentials p₂, p₃, p₄ have sources that are products of a second- or
first-order derivative of the cut-off with the centred tensor, so they inherit the
L^{3/2} bound of eq:Chat from the cubic integrability of the mean-free velocity on
B_ρ(x₀). The pressure potentials p₅, p₆ have sources that are products of a
derivative of the cut-off with the pressure slice, so they inherit the L^{3/2} bound
of the slice itself. All five sources vanish off a closed ball about the origin, which
is what lets the single-potential engines of
CKN.Foundation.Euclidean.PotentialLocalLpGrowth be applied.
The square of the mean-free velocity is almost everywhere strongly measurable on
the ambient space after multiplication by the indicator of its ball. This is the
measurability input of the L^{3/2} source bound eq:Chat, where the velocity enters
only through its slice on B_ρ(x₀).
A function vanishing off B(x₀,ρ) and bounded in absolute value by M times the
square of a velocity field there is of class L^{3/2}, whenever the velocity cube is
integrable on that ball. The constant M is allowed to be any nonnegative real.
Measurability of the factors of the five sources #
The five sources of the centred potentials #
The source g₂ of the centred potential p₂: the entry i j of the centred
tensor eq:Uhat multiplied by the mixed second derivative ∂_i ∂_j η of the cut-off.
Equations
- CKN.lin34CentredTensorHessian u x₀ hρ s i j y = CKN.mixedSecond (CKN.mollifiedBallCutoff x₀ hρ) i j y * CKN.pressureUTensor (CKN.lin34CentredVelocity u x₀ ρ) 0 (y, s) i j
Instances For
The source of the centred potential p₃: the entry i j of the centred tensor
eq:Uhat multiplied by the first derivative ∂_i η of the cut-off.
Equations
- CKN.lin34CentredTensorGradientI u x₀ hρ s i j y = CKN.pressureUTensor (CKN.lin34CentredVelocity u x₀ ρ) 0 (y, s) i j * CKN.spatialDeriv (CKN.mollifiedBallCutoff x₀ hρ) i y
Instances For
The source of the centred potential p₄: the entry i j of the centred tensor
eq:Uhat multiplied by the first derivative ∂_j η of the cut-off.
Equations
- CKN.lin34CentredTensorGradientJ u x₀ hρ s i j y = CKN.pressureUTensor (CKN.lin34CentredVelocity u x₀ ρ) 0 (y, s) i j * CKN.spatialDeriv (CKN.mollifiedBallCutoff x₀ hρ) j y
Instances For
The source of the centred potential p₅: the pressure slice multiplied by the
spatial Laplacian Δη of the cut-off.
Equations
- CKN.lin34CentredPressureLaplacian p x₀ hρ s y = p (y, s) * CKN.spatialLaplacian (CKN.mollifiedBallCutoff x₀ hρ) y
Instances For
The source of the centred potential p₆: the first derivative ∂_j η of the
cut-off multiplied by the pressure slice.
Equations
- CKN.lin34CentredPressureGradient p x₀ hρ s j y = CKN.spatialDeriv (CKN.mollifiedBallCutoff x₀ hρ) j y * p (y, s)
Instances For
Step B: the five sources are L^{3/2} and vanish off a closed ball #
The source g₂ lies in L^{3/2}(ℝ³) and vanishes off the closed ball of radius
‖x₀‖ + ρ about the origin.
The source of p₃ lies in L^{3/2}(ℝ³) and vanishes off the closed ball of
radius ‖x₀‖ + ρ about the origin.
The source of p₄ lies in L^{3/2}(ℝ³) and vanishes off the closed ball of
radius ‖x₀‖ + ρ about the origin.
The source of p₅ lies in L^{3/2}(ℝ³) and vanishes off the closed ball of
radius ‖x₀‖ + ρ about the origin.
The source of p₆ lies in L^{3/2}(ℝ³) and vanishes off the closed ball of
radius ‖x₀‖ + ρ about the origin.