The circulation bound eq:aniso:circulation:bound #
The last step of lem:aniso:annulus. On the regular annulus {R_- < |x| < R_+} × (t_0, 0)
the velocity is bounded, so for R_* ∈ (R_-, R_+) the circulation Γ = r u_θ is bounded on
the lateral boundary ∂B(R_*) × (t_0, 0) by 2 R_* M; and u is smooth at time t_0, so
Γ is bounded on B(R_*) there. Between those two data sets the axis maximum
principle CIV.axisMaximumPrinciple (lem:aniso:axis) fills in the interior: Γ obeys
eq:aniso:axis:pde with drift (u_r, u_z), exponent k = −1, no zeroth-order term and
source r f_θ, and it vanishes on the axis, so it is bounded on the closed half-disc by its
parabolic-boundary values plus the accumulated source R_* M_f (t − t_0).
Two remarks on the shape of the argument. First, the maximum principle is applied with its
final time s set to the time at which the bound is read, so the constant it produces never
mentions an auxiliary s; the manuscript's s ↑ 0 is the monotone step
t − t_0 ≤ −t_0 = |t_0|. Second, Γ is invariant under the rotations about the axis
(circulation_rotZ_invariant) and every point of Vec3 is such a rotation of a meridional
point with r ≥ 0 (exists_rotZ_meridional), so the half-disc bound transfers to the whole
ball B(R_*) — axis and inner ball included, which is what
eq:aniso:circulation:bound asserts.
The data of the circulation instantiation of lem:aniso:axis #
The radial drift u_r, read on the meridional plane.
Equations
- CIV.circulationDriftR u p = u (CIV.meridional p.1.1 p.1.2, p.2) 0
Instances For
The vertical drift u_z, read on the meridional plane.
Equations
- CIV.circulationDriftZ u p = u (CIV.meridional p.1.1 p.1.2, p.2) 2
Instances For
The source r f_θ of the circulation equation, read on the meridional plane.
Equations
- CIV.circulationSource f p = p.1.1 * f (CIV.meridional p.1.1 p.1.2, p.2) 1
Instances For
The source of the circulation equation obeys |r f_θ| ≤ R_* M_f on the half-disc of
radius R_*, the uniform source bound CIV.axisMaximumPrinciple asks for.
The circulation bound on the meridional half-disc #
The circulation bound on the closed meridional half-disc r ≥ 0, by one application of
the axis maximum principle CIV.axisMaximumPrinciple to the circulation profile with
drift (u_r, u_z), exponent k = −1, zeroth-order coefficient γ = 0 and source r f_θ.
The bound is read at the final time s := t, so the constant max (M_b, M_i) + R_* M_f (t − t_η) does not depend on any auxiliary final time: this is the manuscript's
"neither depends on s".
From the half-disc to the whole ball #
The circulation bound on the whole ball B(R_*), axis and inner ball included: the
circulation is invariant under the rotations about the axis (circulation_rotZ_invariant),
and every point of Vec3 is such a rotation of a meridional point with r ≥ 0
(exists_rotZ_meridional), so the half-disc bound transfers verbatim. The final time enters
only through t − t_η ≤ −t_η, which is the manuscript's M_f |t_0| and its s ↑ 0.
The initial data at time t_η #
At any time of (−1, 0) the circulation is bounded on B(R_*) for R_* < 1: u is
smooth there, so Γ is continuous on the compact closure of the ball. This is the
manuscript's "the former are finite since u is smooth at time t_0".
The azimuthal force bound of eq:interior:force:c-two #
The order-0 case of CIV.ForceC2Bounded (eq:interior:force:c-two): one constant bounds the
azimuthal component of the force on the whole cylinder.
eq:aniso:circulation:bound #
The circulation bound eq:aniso:circulation:bound on B(R_*) × (t_0, 0), given a bound
M_b for Γ on the lateral boundary ∂B(R_*) × (t_0, 0). The initial constant is produced
here, from the smoothness of u at time t_0.
The circulation bound eq:aniso:circulation:bound of lem:aniso:annulus: from the
regular annulus chosen in its proof — a bound M for |u| on {R_- < |x| < R_+} × (t_0, 0) — the
circulation obeys eq:aniso:circulation:bound on B(R_*) × (t_0, 0) for every
R_* ∈ (R_-, R_+), axis and inner ball included. The lateral data is |Γ| ≤ 2 R_* M on
∂B(R_*) (circulation_hbdry_of_annulus_bound); the initial data comes from the smoothness
of u at time t_0; the interior is filled in by the maximum principle
CIV.axisMaximumPrinciple applied to Γ with k = −1, γ = 0 and source r f_θ.
lem:aniso:annulus's circulation clause in full: for 0 ≤ R₁ < R₀ < 1 there are radii
R₁ < R_- < R_+ < R₀ and a time t_0 ∈ (−1, 0) such that, for every R_* ∈ (R_-, R_+),
the circulation is bounded on the whole cylinder B(R_*) × (t_0, 0).
The circulation bound of lem:aniso:annulus #
The circulation bound eq:aniso:circulation:bound in the proof of lem:aniso:annulus.
Fix R_* ∈ (R_-, R_+). For s ∈ (t_0, 0) the set closure B(R_*) × [t_0, s] is a compact
subset of Q on which u is smooth, so Γ = r u_θ is continuous there
(continuousOn_circulationProfile), smooth off the axis (hreg_circulationProfile), and
vanishes on the axis since |Γ| ≤ r |u| (circulationProfile_axis_eq_zero). The circulation
equation eq:aniso:circulation:pde is of the form eq:aniso:axis:pde with k = −1,
γ = 0 and source r f_θ with |r f_θ| ≤ R_* M_f (hpde_circulationProfile,
abs_circulationSource_le, from eq:interior:force:c-two). lem:aniso:axis, in the form
CIV.axisMaximumPrinciple, bounds |Γ| on closure B(R_*) × [t_0, s] by its values at time
t_0 and on ∂B(R_*) × [t_0, s] plus M_f |t_0|; the former are finite since u is smooth
at time t_0 (exists_bound_circulation_slice), the latter at most 2 R_* times the bound
on u on the annulus (circulation_hbdry_of_annulus_bound); neither depends on s, and
s ↑ 0 gives eq:aniso:circulation:bound.