Pressure Gradient Origin KPAffine Slot #
Part of the Caffarelli–Kohn–Nirenberg partial regularity proof.
A homogeneous small-cell budget for the origin pressure-gradient majorant #
The small-cell budget A of oneSidedMorreyBound (6/5) κ ρ₀ A B bounds a
time integral of the 6/5 power of a spatial L^{6/5} norm by A · r^θ.
It is therefore homogeneous of degree 6/5 in any norm of the pressure
gradient. The budget of oneSidedPressureGradientKP is linear in the
source bound X = 3·KU·KD + forceSourceMorreyBound q ε, so it has one fifth
of a power too little: the zero-velocity datum with linear pressure a·x₀
and balancing force a·e₀ has cell integrals equal to X^{6/5} r⁵ at the
sharp data size, which exceeds any fixed multiple of X once a is large.
The budget originKPAffineASlot keeps the old linear term, adds its 6/5
power (the source term, also used by oneSidedPressureGradientKP'), and
adds a pressure-mass term 128 · ε^{4/5}. The last term is the 6/5 power
of the L^{3/2} size ε^{2/3} of the pressure; it is needed for pressure
gradients driven by a fast time oscillation of a spatially constant velocity,
which carry no force and no velocity-gradient source.
lem:pressure-gradient-morrey asserts only that the bound is a finite
function of the incoming Morrey norms, the local energy, pressure and force
norms, the exponents and the radii. oneSidedPressureGradientKPAffine is
one such function, written out. Its two summands are the two regimes of the
one-sided transfer: originKPAffineASlot controls every small cell through
its growth in the radius, and the second summand is the mass of the gradient
over the whole carrier, which pays for the remaining cells.
The small-cell coefficient is not linear in the source size. With
X = 3·K_U·K_D plus the force contribution, the datum with zero velocity,
pressure a·x₀ and balancing force a·e₀ has cell integrals of size
X^{6/5} r⁵, which no fixed multiple of X dominates as a grows. The
coefficient therefore carries the 6/5 power as well, together with a term
of size ε^{4/5}, the 6/5 power of the L^{3/2} size of the pressure,
which is needed for gradients driven by a fast time oscillation of a
spatially constant velocity: such data carry no force and no
velocity-gradient source.
Main results #
oneSidedPressureGradientKP'_le_oneSidedPressureGradientKPAffine: the majorant with the6/5-power budget is dominated by the new one.oneSidedPressureGradientKP_le_oneSidedPressureGradientKPAffine: the original majorant is dominated by the new one, with no parameter assumptions.source_time_power_le_originKPAffineASlot: the source time-power coefficientX^{6/5}lies below the new budget.clipped_clause_le_oneSidedPressureGradientKPAffine,restricted_clause_le_oneSidedPressureGradientKPAffine: the clipped and restricted scale conversions land below the new majorant for0 < R₁ < 1.oneSidedPressureGradientQuantitative_to_KPAffine: the explicit AE binder with the original majorant yields the same binder with the new one.
The enlarged small-cell budget for the origin pressure-gradient
estimate. With c = |C_CZ| + 1 and X = 3·KU·KD + forceSourceMorreyBound q ε
it is c·3X + (c·3X)^{6/5} + c·128·ε^{4/5}.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The enlarged origin pressure-gradient majorant: the small-cell budget
originKPAffineASlot together with the whole-carrier budget of
oneSidedPressureGradientKP', which carries the clipped-scale inflation.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The coefficient |C_CZ| + 1 is at least one.
The budget of oneSidedPressureGradientKP' lies below the enlarged
budget.
Any quantity whose 6/5 power controls the budget: every Y ≤ c·3X
has Y^{6/5} below the enlarged budget.
The source time-power coefficient (3·KU·KD + forceSourceMorreyBound q ε)^{6/5}
lies below the enlarged budget.
The majorant with the 6/5-power budget is dominated by the enlarged
majorant, with no parameter assumptions.
The original majorant is dominated by the enlarged majorant, with no parameter assumptions.
Clipped route. Small-cell budget below originKPAffineASlot and a
whole-carrier budget below c·(|R₀| + |R₁| + |ε| + 1) at the clipped scale
(1 - R₁)/2 give a constant below the enlarged majorant, for every
0 < R₁ < 1; the scale inflation is absorbed into the whole-carrier budget.
Restricted route. Small-cell budget below originKPAffineASlot and a
whole-carrier budget inflated by the doubled-margin ratio
(R₁ / (2·((1 - R₁)/4)))^θ give a constant below the enlarged majorant, for
every 0 < R₁ < 1.
The explicit AE binder with the original majorant yields the same binder
with the enlarged majorant: each output estimate is carried upward by
oneSidedPressureGradientKP_le_oneSidedPressureGradientKPAffine.