Gradient Slot Duhamel Tested #
Part of the Caffarelli–Kohn–Nirenberg partial regularity proof.
Gradient Slot Duhamel Transfers #
Part of the Caffarelli–Kohn–Nirenberg partial regularity proof.
Scalar and pressure forms of the localized gradient transfers #
The paper label lem:local-equation records the localized form of the
Caffarelli–Kohn–Nirenberg equation tested against a product cutoff φ · ψ.
The two statements below extract from it the two ingredients used when the
cutoff is frozen in the time variable and only the spatial slot structure
matters:
gradientSlot_diffusion_transfer_of_swsis the scalar-coordinate form of the vector diffusion transferlocalized_diffusion_transfer_of_sws; it is the version in which every component of the vector test field is the same scalar fieldψin the selected sloti.gradientSlot_pressure_transfertransfers the pressure pairing against the product cutoff to a weak pressure gradient, with no solution hypothesis at all, since the pressure enters the localized equation only through its weak gradient.
Scalar form of the diffusion transfer of lem:local-equation: testing the
vector localized equation with the vector field whose i-th component is a
scalar cutoff ψ and whose other components vanish collapses the transfer
identity to the scalar identity in the i-th coordinate. No divergence
information is used beyond what IsSuitableWeakSolutionIntegrable already provides.
Pressure transfer of lem:local-equation against the product cutoff: the
pairing of the pressure with the spatial derivative of φ · ψ equals the
pairing of the weak pressure gradient with φ · ψ. The pressure enters the
localized equation only through its weak gradient, so this transfer carries no
solution hypothesis: all it needs is the weak-gradient pairing rule on the
support box of φ.
Gradient Slot Duhamel Split #
Part of the Caffarelli–Kohn–Nirenberg partial regularity proof.
Regrouped integrands for the local equation #
The tested form of the localized equation (lem:local-equation) pairs the
divergence-form sources localizedDivergenceG, localizedDivergenceH with a
test field and its first spatial derivative. The paper writes the same
expression after moving the derivative of the product φ * ψ onto the cutoff
factor, which is what the identities below record.
gradientSlot_divergence_integrand is the divergence-form regrouping: the
source integrand together with the derivative slot equals the time, force,
convection, derivative and pressure contributions collected on the right.
gradientSlot_gradient_integrand is the corresponding regrouping for the
pressure-gradient slot of localizedGradientSourceG, where the second spatial
derivatives of the cutoff appear explicitly; it needs no smoothness hypotheses
because the product rule is not used.
The divergence-form tested integrand of the local equation, regrouped onto
the cutoff product φ * ψ (lem:local-equation). The only analytic input is
the product rule spatialPartial_mul_full for the smooth cutoff factors.
The pressure-gradient tested integrand of the local equation, regrouped
onto the cutoff product φ * ψ (lem:local-equation). No derivative of the
test field is moved, so no smoothness hypothesis is needed.
Trading the divergence-form pressure slot for the gradient slot #
Paper label lem:local-equation. The cutoff-tested identity for the localized
velocity is first obtained with the pressure sitting in the divergence-form
slot, as p ∂ᵢφ together with the diagonal entry δᵢⱼ p φ. The estimates of
Step 4 instead need the pressure as φ Dp in the heat slot, the convection in
the form φ (u · ∇) u, and the second cutoff derivative Δφ u explicit. The
theorem below performs that exchange once and for all at the level of the
tested identity: the divergence-form right-hand side and the gradient-slot
right-hand side agree for every space-time test function. The three inputs are
the convection transfer (∑ⱼ ∫ uᵢuⱼ ∂ⱼ(φψ) = −∫ φψ (u · ∇)uᵢ), the diffusion
transfer (one integration by parts in ∂ⱼφ ψ) and the weak pressure gradient
tested against the product cutoff φψ.
The divergence-form and gradient-slot right-hand sides of the tested local
equation agree, for every space-time test function ψ. Paper label
lem:local-equation: this is the passage from the raw tested identity to the
displayed equation eq:local-equation, in which the pressure enters through
its weak gradient Dp.