A dilute gas is a large number of small particles moving freely between
elastic collisions. For (t, x, v) in [0, inf) x T^3 x R^3, let
F(t, x, v) be the phase-space density. The quantity int_A F(t,x,v) dx dv,
for a measurable set A of phase space, is the expected number of gas
particles in A at time t.
Between collisions, particles move with their current velocity, so the
density satisfies the free-transport equation d_t F + v . grad_x F = 0.
Collisions change velocities. Writing Q(F,F) for the net rate at which
particles enter velocities from collisions minus the rate they leave,
the equation of motion is
(d_t + v . grad_x) F = Q(F,F).
T^3 = R^3 / Z^3 is the three-torus (periodic box). Velocities have no
bound and live in R^3.
Fix an incoming pair of velocities v, v_* in R^3 and a unit direction
omega in S^2 (the 2-sphere). Set
a = (v - v_*) . omega,
v' = v - a * omega,
v_*' = v_* + a * omega.
The map (v, v_*) -> (v', v_*') is an elastic collision: total momentum
v + v_* is preserved, and total kinetic energy (1/2)(|v|^2 + |v_*|^2)
is preserved. The rate of such a collision is |a| = |(v - v_*) . omega|.
Splitting Q into gain and loss:
Q(F,F)(v) = Q^+(F,F)(v) - Q^-(F,F)(v),
Q^+(F,F)(v) = int_R^3 int_S^2 |(v-v_*) . omega| F(v') F(v_*') domega dv_*,
Q^-(F,F)(v) = int_R^3 int_S^2 |(v-v_*) . omega| F(v) F(v_*) domega dv_*.
The loss term factors as Q^-(F,F)(v) = F(v) * nu_F(v), where
nu_F(v) = 2 pi int_R^3 |v - p| F(p) dp
is the collision frequency of F at velocity v. This is the rate at
which a particle with velocity v collides with any other particle in
the gas.
The classical Boltzmann equation makes sense only for smooth enough F
that Q^+(F,F) and Q^-(F,F) are separately well-defined. For general F
with finite mass and energy, the gain integral can diverge; even when
it converges, it does not converge in any strong sense under
approximations of F by smooth functions.
The modern theory (DiPerna-Lions 1989, 1991) defines weak solutions via
renormalization. Instead of testing the equation directly against F, one
tests against beta(F/M) for suitable bounded functions beta and a
reference Maxwellian
M(v) = (2 pi)^{-3/2} exp(-|v|^2 / 2).
The resulting bilinear form beta'(F/M) Q(F,F) is well-behaved. The basic
theorem: every initial datum with finite mass, energy, and H-functional
has at least one such renormalized global solution.
Renormalized solutions come with an entropy inequality. Define the
H-functional of a density F relative to a positive reference rho by
H_rho(F) = int_Omega [F log(F/rho) - F + rho] dx dv.
For the standard Maxwellian M, the inequality is
H_M(F(t)) + int_0^t D(F(s)) ds <= H_M(F_0),
where D >= 0 is the entropy dissipation (an integral against the
symmetric gain-loss form). This is Boltzmann's H-theorem: entropy
decreases in time. It does not, however, close the equation on its own:
many distinct densities satisfy the same entropy inequality (for
instance, local Maxwellians at every x). Uniqueness is a subtle extra
condition, not automatic.
The present paper's point: pick an initial datum for which renormalized
theory admits at least two solutions, AND both satisfy the stronger
condition of exact LOCAL conservation of mass, momentum, and kinetic
energy (not just global inequalities).
DEFINITION (Entropy solution with local conservation).
Let F_0 >= 0 satisfy int_Omega F_0 (1 + |v|^2 + |log F_0|) dx dv < inf.
A global entropy solution with local conservation is a nonnegative
measurable function F: [0, inf) x Omega -> R such that:
(i) F is in C_w([0, inf); L^1(Omega)) (weakly continuous in L^1),
F(0) = F_0, and for every finite T:
sup_{0 <= t <= T} int_Omega F(t) (1 + |v|^2 + |log F(t)|) dx dv < inf.
(ii) The two quantities Q^+(F,F)/(1+F/M) and Q^-(F,F)/(1+F/M) are in
L^1([0,T] x Omega) for every finite T. The renormalized identity
holds: for every C^1 beta: [0,inf) -> R with beta(0) = 0 and
|beta'(z)| <= C_beta / (1+z), and every phi smooth, periodic in x,
compactly supported in velocity,
int_0^inf int_Omega [M beta(F/M) (d_t phi + v . grad_x phi)
+ beta'(F/M) Q(F,F) phi]
dx dv dt
+ int_Omega M beta(F_0/M) phi(0, x, v) dx dv = 0.
(iii) Global momentum and energy bounds: int_Omega |v|^2 F(t) <=
int_Omega |v|^2 F_0, and H_M(F(t)) + int_0^t D(F(s)) ds <= H_M(F_0).
(iv) Third moment integrable: int_0^T int_Omega (1 + |v|^3) F dx dv dt < inf
for every finite T.
(v) LOCAL conservation. For psi in {1, v_1, v_2, v_3, |v|^2 / 2} and
every phi in C_c^inf([0, inf) x T^3),
int_0^inf int_Omega F(t, x, v) psi(v) (d_t phi + v . grad_x phi)
dx dv dt
+ int_Omega F_0(x, v) psi(v) phi(0, x) dx dv = 0.
Condition (v) is the strengthening over the classical renormalized class.
It says mass, momentum, and kinetic energy flow conservatively at each
spatial point, not just in total. It is exactly the requirement that the
five collision invariants satisfy their local balance laws with no
"defect" term.
THEOREM. There exist R < inf and a nonnegative initial datum F_0 with
supp_v F_0 subset B_R(0),
int_Omega F_0 (1 + |v|^2 + |log F_0|) dx dv < inf,
such that the hard-sphere Boltzmann equation on T^3_x times R^3_v has two
global entropy solutions F^(1), F^(2) in the sense of Definition 1.1,
differing on a set of positive (t, x, v)-measure. Both are in
C([0, inf); L^1(Omega)). Moreover, for every finite T and every k >= 0,
both satisfy
int_0^T int_Omega <v>^k [Q^+(F, F) + Q^-(F, F)] dx dv dt < inf,
<v> = (1 + |v|^2)^{1/2}.
So the separate gain and loss terms are integrable with every polynomial
velocity weight, and the local balance identities hold with no
correction.
Published uniqueness theorems for the Boltzmann equation assume STRICTLY
MORE than Definition 1.1:
- Kaniel-Shinbrot (1978): F bounded above by a fixed multiple of a
Maxwellian, and F_0 below a Maxwellian by a margin.
- Ukai (1974): small initial perturbations of a Maxwellian in weighted
spatial Sobolev spaces.
- Mischler-Wennberg (1999): the spatially homogeneous case (F does
not depend on x), where finite mass and energy alone suffice.
- Duan-Huang-Wang-Yang (2017): bounded amplitude, small initial
relative entropy, and small spatial integral of the velocity
supremum.
./ref-ukai-1974.pdf
./ref-mischler-wennberg-1999.pdf
./ref-duan-huang-wang-yang-2017.pdf
The initial datum F_0 has UNBOUNDED pointwise amplitude at
shrinking spatial scales: sup_v F_0(x,v) can blow up like 1/s_N at some
x, while the mass and energy integrals remain bounded. No known
uniqueness theorem covers that. The counterexample exploits the gap
precisely: two solutions can remember different fine-structure data that
no coarse observable can see.
Recent instability results (Chen-Holmer 2024, Chen-Shen-Zhang 2024,
Chen-Guo-Shen-Zhang 2026) prove failure of uniform continuity for
NEARBY initial data in various weighted norms. In the present
construction, both solutions share EXACTLY the same initial datum.
So the construction upgrades "nearby data give nearby solutions" failure to
"identical data give distinct solutions" failure. That is a stronger
statement.
./ref-chen-holmer-2206.11931.pdf
./ref-chen-shen-zhang-2310.05042.pdf
./ref-chen-guo-shen-zhang-2606.01331.pdf
Four steps organize the argument.
STEP A (CONCENTRATION). Build an initial datum F_0 as a superposition
of cold jets: thin annular tubes of particles clustered near the
velocity axis bar v = 0, at a geometric sequence of spatial scales
s_1 >> s_2 >> ... -> 0. Each jet has total mass O(s_j^2), so the mass
over all scales is summable, but each jet has velocity-integrated
density ~1/s_j at its own scale, so pointwise amplitude is unbounded.
The scaling is chosen so that the whole configuration has finite mass,
energy, and absolute entropy, while every fixed polynomial velocity
moment is also finite.
STEP B (SELECTION). Add to F_0 a tiny extra perturbation (a "seed") at
the smallest scale s_N, of the form lambda m_T s_N chi(x/s_N, v) with
amplitude parameter lambda in [0, L_0] and m_T = T^p for a small fixed
horizon T. As lambda ranges from 0 to L_0, the hot-component norm
||h(T)||_H moves from below the cap m_T (T + s_N) to above it. So there
is a lambda at which the cap is touched exactly. This selects a family
of solutions with a prescribed hot norm at a prescribed short time.
STEP C (COMPACTNESS). As N -> inf and the collision cutoff b_N -> inf,
both the "dormant" solutions (lambda = 0) and the "cap" solutions
(lambda = lambda_N chosen to touch the cap) converge to global entropy
solutions with local conservation, i.e. satisfying Definition 1.1
exactly. This uses uniform integrability inherited from the cold-jet
structure and velocity-averaging estimates to pass the collision
operator to the limit.
STEP D (SEPARATION). The two limiting solutions genuinely differ.
Magnify the cap-hit point by 1/r_N, and take another limit. The dormant
limit vanishes under this dilation by the dormant upper bound, while the cap limit
does not by the nonvanishing of the cap limit. Their equality under the dilation would
force the cap limit to be zero, which it is not. Hence the two global
solutions are distinct. The delicate parts are proving the cap
dilation's limit g is nonzero (the nonconcentration argument) and that it
cannot exist for arbitrarily long dilated time (the branching argument).
Four parameters are chosen early and then only used:
K >= 1, 0 < eta <= log 2,
0 < alpha <= 1/4, 0 < gamma <= 1.
K controls overall jet amplitude; eta controls the geometric scale ratio
s_j / s_{j+1} = e^eta; alpha bounds transverse velocity deviation; gamma
controls axial profile sharpness.
Then a small family of secondary parameters is chosen in a very specific
order:
- t_1 in (0, 1/32]: the fixed early time up to which all finite
evolutions are continued uniformly. Chosen in the bath bound.
- p, A: exponents for the seed amplitude m_T = T^p and for early-time
growth of the dormant solution. Chosen in (6.3) with p > C_2 + 1
and A >> p + C_2.
- delta, R_mu: hole radius and Maxwellian velocity cutoff. Chosen
small/large respectively, to make the initial mu-relative entropy
small.
- s_0: the smallest retained jet scale. Chosen very small.
- T_0: cap horizon bound. Chosen small, but much larger than s_0.
Each choice in the list depends on all earlier ones. Consistency requires
throughout that the choices are consistent (each estimate from earlier
in the paper survives the later choices).
T^3 = R^3 / Z^3 (three-torus, volume 1)
Omega = T^3_x times R^3_v (phase space)
D = d_t + v . grad_x (transport derivative)
<v> = (1 + |v|^2)^{1/2}
x = (y, z) in R^2 x R (transverse, axial -- after rotation)
v = (bar v, v_3) in R^2 x R (same decomposition)
M(v) = (2 pi)^{-3/2} exp(-|v|^2 / 2) (standard Maxwellian)
mu(v) = exp(-beta_0 |v|^2) (a second reference Maxwellian; beta_0 large)
H(v) = exp(-|v|^2) <v>^{-100} (1 + |bar v|^{-q}), 1 < q < 2 (weight)
||f||_H = ess sup_{x,v} |f(x,v)| / H(v) (weighted sup norm)
nu_f(v) = 2 pi int |v - p| f(p) dp (collision frequency of f)
nu_f^b, P^b, Q^{+,b}: cutoff versions with kernel
min(1, b / |v - v_*|).
ess sup = essential supremum (sup ignoring null sets).
1_{...} = indicator function (1 on the set, 0 elsewhere).
s << t means s <= C t for a fixed universal constant C.
All functions are Borel measurable. Pointwise statements involving H
refer to essential bounds, so the velocity axis bar v = 0 (a null set)
is ignored throughout.
What must be shown is that certain "spiky" velocity
distributions (concentrated on annuli around the axis, with unbounded
axial density but bounded transverse width) can be pushed through the
bilinear gain operator P(., .) and still produce uniformly bounded
output. This is the technical core of the whole construction: without
such bounds, the cold jets would blow up under self-collision, and the
whole scheme collapses.
Throughout the paper the "jet axis" is the irrational direction
n_axis = (1, sqrt 2, 0) / sqrt 3.
Rotate space and velocity by the same orthogonal map taking n_axis to
e_3 = (0, 0, 1). In rotated coordinates, space is R^3 / Lambda with
Lambda = R(rotated Z^3) still of covolume 1, and write
x = (y, z) with y = (y_1, y_2) in R^2 transverse and z in R axial;
same decomposition for velocity v = (bar v, v_3).
The rotation preserves transport and the hard-sphere collision operator,
so the equation of motion is unchanged. The only leftover from the
irrational direction is how the rotated lattice Lambda behaves near the
transverse axis; the next lemma quantifies that.
LEMMA (Diophantine separation).
There is a universal c > 0 such that for every nonzero n in Lambda,
|bar n| >= c / (1 + |n|).
PROOF. Write n = R(k, m, ell) for an integer vector (k, m, ell). In the
rotated coordinates, the transverse part of R(k, m, ell) is
bar n = ((m - sqrt 2 k)/sqrt 3, ell).
If ell is not zero, then |bar n| >= |ell| >= 1, so the inequality holds.
If ell = 0 and (k, m) is not (0, 0), then
|m - sqrt 2 k| = |m^2 - 2k^2| / |m + sqrt 2 k| >= 1 / (C(1+|k|+|m|)),
because the numerator is a nonzero integer. Since |n| ~ max(|k|, |m|) up
to a constant, gives |bar n| >= c / (1+|n|). The point mass at n = 0 is
excluded.
Interpretation: any point x on the torus, when translated by a lattice
vector n != 0 with |n| large, has its transverse coordinate displaced
from the axis by about 1/|n|. As long as the jet scales s_j are much
smaller than this displacement, distinct lattice copies of a jet cannot
intersect. This gives the initial datum disjoint supports.
The cold jets concentrate BOTH in space and in velocity: their typical
transverse velocity is 0 (they are almost-axis-aligned), plus a
perturbation of size d_s = d_0 s^{10} for a fixed tiny d_0 > 0. So any
weight that measures pointwise size must blow up near the velocity
axis, or it would not control the jets at all.
Choose
H(v) = exp(-|v|^2) <v>^{-100} (1 + |bar v|^{-q}), 1 < q < 2.
Each factor has a job:
- exp(-|v|^2): Gaussian decay at large velocity; gives every
polynomial moment, needed for |v|^3 in local energy conservation.
- <v>^{-100}: polynomial decay with m = 100 > 5; this is what makes
the hot-hot gain P(H,H) <= C H almost everywhere (the sheet-gain bound).
The inequality fails for m <= 5.
- (1 + |bar v|^{-q}): singular factor near the transverse velocity
axis. In R^2 (two transverse dimensions), |bar v|^{-q} is locally
integrable for q < 2, and is in L^{1+eps} for eps < 2/q - 1. This
correctly weights the transverse concentration of jets at all scales.
The window 1 < q < 2 is essential:
- q > 1: makes the "1/s scalar line integral" appearing inside
Carleman's formula converge;
- q < 2: makes the transverse L^1 norm converge near the axis.
The value of H on bar v = 0 is irrelevant: the axis is a null set for
every integral taken.
The hyperplane representation of the gain goes back to Carleman; the
modern bilinear form is worked out in the appendix of Gamba, Panferov,
and Villani.
./ref-gamba-panferov-villani-2009.pdf
LEMMA (Carleman formula). For every pair of nonnegative velocity
densities c, h,
P(c, h)(v) = 2 int_R^3 c(p) dp int_R^3 delta((v - p) . (xi - v))
* h(xi) dxi,
where the inner integral is with respect to surface measure on the plane
(v - p) . (xi - v) = 0, divided by |v - p|.
PROOF. Fix incoming p, xi. Consider the sphere with diameter [p, xi]:
its points are v = (p + xi)/2 + (|p - xi|/2) sigma for sigma in S^2.
Writing g = p - xi, reflection of g in the omega plane sends one
hemisphere of S^2 onto the other, and the change of variables from
the incoming omega to the outgoing direction sigma multiplies by the
Jacobian |g . omega| / (|g|^2/2). Both hemispheres together give
|g . omega| domega pushes to (|g|/2) sigma_dsigma,
i.e. total mass 2 pi |g| = 2 pi |p - xi|.
Meanwhile, identify this sphere with the zero set of f(v) = (v-p).(xi-v).
At the sphere, |grad_v f(v)| = |p - xi|, so the coarea formula gives the
sphere's surface measure as 2 delta(f(v)) dv with delta the standard
delta function at 0. Composing with the incoming density factors
c(p) h(xi) and applying Tonelli gives the stated identity.
Note the symmetry P(c, h) = P(h, c) (both are equal to the same Q^+(c,h)
= Q^+(h,c)), and P(F, F) = Q^+(F, F). So the Carleman formula is a
reformulation of the hard-sphere gain, and it is the analytic version
used everywhere.
DEFINITION (Sheet class S_L).
Fix L >= 1. An elementary sheet is a product measure
c(dp) = N_d(bar p) rho(dp_3),
consisting of a centered transverse Gaussian N_d of width d in [0, 1/8]
times an arbitrary axial probability rho satisfying the exponential
moment
int_R exp(8 a^2) rho(da) <= L.
A "probability mixture of sheets" is any convex combination of
elementary sheets. A function is under SHEET DOMINATION with amplitude
A_0 if it is pointwise bounded by A_0 times some mixture of sheets.
Why this matters: cold jets are precisely a transverse Gaussian (of
width d_s = d_0 s^{10}) times an axial profile (arbitrary density near
the axis, unbounded but exponentially weighted). Sheet domination
preserves uniform bounds even as the transverse width collapses to zero,
so all gain estimates hold in the limit of a point-mass in the
transverse velocity.
LEMMA (Geometric slice estimate).
Let c = lambda (x) rho be a sheet in S_L (lambda a mixture of transverse
Gaussians or delta_0, rho an arbitrary axial probability). Fix R >= 1,
bar v != 0, and an axial interval I of length ell in (0, 2R]. Then
(1/W(v)) int c(dp) int_{|xi - v| <= R}
delta((v-p).(xi-v)) W(xi) dxi <= C_q R^2, (TOTAL)
(1/W(v)) int c(dp) int_{|xi - v| <= R, xi_3 in I}
delta((v-p).(xi-v)) W(xi) dxi
<= C_q (R ell + R^{q-1} ell^{2-q}). (SLICE)
Constants C_q depend only on q; they are uniform in transverse width d
and in the axial law rho.
PROOF. Write r = |bar v|, n = bar v - bar p (the transverse velocity
difference), and e_n = n/|n|. Split the collision delta into transverse
part and axial part. For fixed p_3, xi_3, the delta in the (bar p, bar xi)
variables acts on the transverse plane; it is line measure, divided by
|n|, on a line whose signed distance from zero is
A - B(xi_3 - v_3),
A = e_n . bar v, B = (v_3 - p_3)/|n|.
The regular part of the line integral (i.e. smooth W(xi)) contributes at
most 2R/|n|. The singular part (where W(xi) ~ |bar xi|^{-q} near the
transverse axis) contributes, after substituting y = xi - v,
(1/|n|) int_R (A - B s + y_3-type terms)^{...}
* ((A - B s)^2 + s^2)^{-q/2} ds
<= (a_q / |n|) |A - B(xi_3 - v_3)|^{1-q},
where a_q = int_R (1 + s^2)^{-q/2} ds is finite because q > 1.
Next, integrate in xi_3 over an interval J of length <= ell <= 2R. Use
int_J |A - B x|^{1-q} dx <= C_q R^{q-1} ell^{2-q} |A|^{1-q}
for 1 - q < 0. This is either bounded by C_q |A|^{1-q} (when
2R|B| <= |A|) or by C_q |B|^{1-q} ell^{2-q} (otherwise). Both are
absorbed by the stated bound with a further sum of terms.
Finally, the expectation of the surviving prefactor
(1 + |e_n . e_{bar v}|^{1-q})/|n|, taken against lambda (a mixture of
transverse Gaussians of width d, or a delta_0), is bounded by C_q / r,
independently of d. This is the statement that the W-normalized gain is
bounded independent of how thin the transverse Gaussian is: substitute
polar coordinates in n, use integrability of (1 + |cos theta|^{1-q}) in
theta on [0, 2 pi] (finite since q < 2), and divide by the disk area
(which grows like d^2 when d is large; when d is small, |n| >= r/2 makes
the bound trivial).
Assembling everything gives (SLICE), and taking the full axial range
(i.e. I of length 2R) gives (TOTAL).
LEMMA (Uniform sheet gain). For every L < inf there is C_{q,m,L} < inf
such that for every c in S_L and almost every v (off the velocity axis),
P(c, H)(v) <= C_{q,m,L} H(v).
Quantitatively, on the shell R < |p| + |xi - v| < 2R,
K_c(v; shell) <= C_{q,m,L} R^2 exp(-c R^2), c absolute > 0,
where K_c is the normalized kernel
K_c(v; dp dxi) = 2 c(dp) delta((v - p) . (xi - v)) (H(xi)/H(v)) dxi,
with total mass = P(c, H)(v) / H(v).
PROOF. Use |p|^2 + |xi|^2 = |v|^2 + |w|^2 on the collision plane (where
w = p + xi - v). The weight ratio becomes
H(xi) / H(v) = exp(|p|^2 - |w|^2) <v>^m / <xi>^m W(xi)/W(v),
and since <v> <= <p><xi> always holds,
H(xi) / H(v) <= C_m exp(2|p|^2) exp(-|w|^2) W(xi)/W(v).
The transverse part exp(2|bar p|^2) times a Gaussian of width d is bounded
by 4 exp(-46|bar p|^2) times a Gaussian of width 2d: this is a simple
ratio-of-densities calculation. The axial part exp(2 p_3^2) rho(dp_3) is
bounded (outside |p_3| > u) by C L exp(-6 u^2) using the exponential
moment in rho. So the tilted measure exp(2|p|^2) c(dp) is dominated by
4L times a product Gaussian, and its tail beyond |p| > u decays like
exp(-cu^2) uniformly in d.
Now apply the geometric slice estimate. On shell R < |p| + |w| < 2R, at least one of
|p| > R/4 (tilted-measure tail) or |w| > R/2 (Gaussian tail) holds,
and the remaining weighted planar integral is O(R^2) by the sheet
estimate. This gives the shell bound after absorbing the geometric
sum C_q R^2. Summing geometrically in R proves P(c, H) <= C_{q,m,L} H
for the full kernel.
The base region |p| + |w| <= 2 is the same estimate with R = 1.
LEMMA (Uniform tightness). Fix L < inf. Uniformly in c in S_L and v
off the axis:
lim_{R -> inf} sup_c ess sup_v K_c(v; |p| + |xi - v| > R) = 0,
lim_{M -> inf} sup_c ess sup_{|v| > M} K_c(v; R^3 x R^3) = 0.
PROOF. The first is the dyadic shell estimate of the uniform sheet gain. For the
second, restrict to |p| + |w| <= R (tail removed), where all the
exponential factors are bounded by constants depending on R. If
|bar v| > 2R + 1, then |bar v - bar p| >= |bar v|/2, and the transverse
line integral gives at most C_R / |bar v|. If |v_3| > 2R + 1, integrate
the delta in xi_3 instead: the divisor is at least |v_3|/2, and
sup_{a in R^2} int_{|bar zeta - a| <= R} (1 + |bar zeta|^{-q}) dbar zeta
<= C_q (R^2 + R^{2-q})
which is finite for q < 2. This gives C_R / |v_3|. At least one of
|bar v| or |v_3| is large when |v| is large, so the kernel mass decays
like C_R / |v| -> 0 as |v| -> inf.
LEMMA (Gaussian parameter tails). Suppose a measure in parameters
w = (bar w, w_3) in R^2 x R has density j satisfying
0 <= j(w) <= C_0 pi^{-3/2} exp(-|w|^2).
Let its velocity image be given by
bar p = d bar w, p_3 = Phi(w_3),
with 0 <= d <= 1/16 and Phi: R -> R merely measurable subject to
|Phi(a)| <= A + delta |a| with delta <= 1/16. Then the image measure is
dominated by C_0 times a sheet in S_{L_A} with L_A depending only on A.
Moreover, if c_{>B} is the image of the truncated parameter measure
1_{|w| > B} j(w) dw, then
sup_v P(c_{>B}, H)(v)/H(v)
<= C_{q,m,A,C_0} exp(-B^2/2), B >= 0.
PROOF. The parameter Gaussian has a product form with each coordinate
bounded by exp(-w_i^2). The transverse image is a centered Gaussian of
width d <= 1/16 <= 1/8 (fine for the sheet class). For the axial image,
use e^{8|Phi(a)|^2} <= e^{16 A^2} e^{16 delta^2 a^2}. Since
16 delta^2 <= 16/256 = 1/16 < 1, the axial exponential moment under the
pushforward of (pi^{-1/2} e^{-a^2}) da is bounded:
int e^{8|Phi(a)|^2} pi^{-1/2} e^{-a^2} da
<= e^{16 A^2} (1 - 16 delta^2)^{-1/2} <= 2 e^{16 A^2}.
So L_A = 2 e^{16 A^2} works.
For the tail bound, use
1_{|w| > B} e^{-|w|^2} <= e^{-B^2/2} e^{-|w|^2/2}.
After dividing by the normalizing constant (2^{3/2}), the right side is
2^{3/2} e^{-B^2/2} times a Gaussian of variance 1/2, whose transverse
width is at most sqrt(2)/16 < 1/8 and whose axial exponential moment is
as computed. Apply the uniform sheet gain to the dominating sheet: after rescaling,
the normalized gain at the tail is bounded by C exp(-B^2/2).
Interpretation: a nonlinear compressing map Phi of the axial coordinate
does not damage sheet structure, because the Gaussian moment is
controlled on the parameter side of the map, not the image side. This is
one of the crucial "hard" lemmas: it takes the transcendental implicit
map p_3 = a_s^sigma(z - t p_3) that appears in the barrier and shows
nothing blows up.
LEMMA (Quadratic gain bound). For m > 5, there is C_{q,m} with
P(H, H) <= C_{q,m} H almost everywhere.
PROOF. On the collision plane, |v|^2 <= |p|^2 + |xi|^2, so at
least one incoming has magnitude >= |v|/2. By symmetry (P(f,g) = P(g,f)),
retain only |xi| >= |v|/2, at a factor of 2 cost. Then
P(H,H)(v)/H(v)
<= C_m int_{|xi| >= |v|/2} delta((v - p).(xi - v))
* exp(-|p + xi - v|^2) <p>^{-m}
* [W(p) W(xi) / W(v)] dp dxi.
On shell R < |p| + |xi - v| <= 2R, either |p| > R/4 or |w| > R/2 (for
w = p + xi - v), and in either case <p>^{-m} exp(-|w|^2) is bounded by
C_m R^{-m} exp(-|w|^2/2) (using polynomial decay in the first case,
Gaussian decay in the second). Then the measure W(p) 1_{|p| <= 2R} dp is
a positive mixture of centered disks (by a layer-cake formula), with
total mass O(R^3). Apply the geometric slice estimate with |k| <= 2R to its normalized
pieces: the shell contribution is C_{q,m} R^{5-m}. Summing over dyadic
shells gives convergence for m > 5. The base region is bounded by the
same disk-mixture argument with R = 1.
LEMMA (Gain from small mass, [3.6"' of the paper]).
For every B in [0, inf) and every eps > 0, there is C_{B,eps} < inf such
that for every j with 0 <= j <= B H and almost every v,
P(H, j)(v) <= H(v) ( eps + C_{B,eps} int_R^3 j(p) dp ).
Same with P^b in place of P.
PROOF. Systematic truncation of velocity:
- On |v| > M (large): P(H,H) itself is small uniformly (the large-output decay),
and P(H,j) <= B P(H,H) <= B * small.
- On |v| <= M (compact): split into |p|, |xi| <= L of the two incoming
domain and its complement.
- Complement |p| + |xi| > L: shell bounds give small error
(uniformly in v in |v| <= M).
- Compact |p|, |xi| <= L: the collision kernel reduces to the
function (v-p).(xi-v) = 0 line integral divided by |v - p|. By
Tonelli, the small-mass contribution is bounded by
int_K j(p) / |v - p| dp times a bounded factor. Split into
|v - p| < rho (small volume, controlled by boundedness of j:
error <= C_{B,L} rho^2) and |v - p| >= rho (large divisor,
controlled by 1/rho int j: error <= (C/rho) int j). Choose rho
small, then bound remaining constant by C_{B,eps}.
The point: the small-mass gain is what makes the global bootstrap work.
Without it, the quadratic self-gain of the "hot" component w, estimated
via ||w||_H <= B, would produce an error of order B^2 that does not
shrink as B grows. With it, a small L^1 excess of w over mu (isolated as
j = (w - mu)_+) forces a small self-gain P(w,w) - P(mu,mu), dominated by
the fixed P(mu,mu). That is the whole reason the global bootstrap
converges instead of blowing up.
Fix s_0 in (0, 1/8) and eta in (0, log 2]. Define the geometric sequence
of scales:
s_j = s_0 exp(-j eta), for j >= 1,
p_j = s_j^2, the 1D stripe period at scale j,
c_eta = (e^eta - 1) / 16, radial margin,
d_s = d_0 s^{10}, velocity width, d_0 tiny,
S(r) = sech^2(r), axial profile,
a = 1/8, axial half-width of the core,
b_sp = 1/16, axial margin,
a_s^sigma(z) = sigma + alpha tanh(gamma z / s),
sigma in {-1, +1}, axial velocity profile.
For each scale s_j and sign sigma, define the transverse support
E_j^sigma = annulus { s_j + c_eta p_j < |y| < e^eta s_j - c_eta p_j }
intersected with the stripe pattern
{ y_1 / p_j mod 1 in (0, 1/4) } when sigma = +1,
{ y_1 / p_j mod 1 in (1/2, 3/4) } when sigma = -1.
So E_j^sigma is a thin annular band (thickness c_eta p_j = O(s_j^2)) with
a fine stripe pattern of period p_j = s_j^2 in the y_1 direction.
Occupation: one quarter of each period is sigma = +1, one quarter is
sigma = -1, and half is empty. The two signs' spatial supports are
disjoint, and an empty half-period separates them. That separation
enforces (later) that a single collision history cannot contain both
signs at once -- physically, the construction keeps two opposite axial streams
spatially separated so they cannot catch each other.
For a lattice translate n in Lambda = R(Z^3), a jet label is
i = (j, sigma, n). For s = s_j and (y_c, z_c) = x - n, the jet datum is
c_i^0(x, v) = 1_{E_j^sigma}(y_c) 1_{(-a, a)}(z_c)
* (K/s) S(gamma z_c / s)
* d_s^{-3} J((v - a_s^sigma(z_c) e_3) / d_s),
where J: R^3 -> [0, inf) is a fixed smooth bump, supported in the unit
ball, with total mass 1.
Interpretation of each piece:
- spatial support: transverse annulus E_j^sigma of area about s_j^2,
times an axial slab of length 2a = 1/4, times lattice translates.
- velocity support: a narrow slab centered at axial velocity
a_s^sigma(z_c), of width d_s = d_0 s^{10}. So velocity-integrated
density is about (K/s) * S(gamma z_c/s) at each x in the support.
- total mass of one jet:
int_{E_j^sigma} dy (= O(s_j^2))
* int_{-a}^a S(gamma z_c / s) dz_c (= 2 s_j / gamma * (1 + O(1)))
* (K/s_j) * 1 (Velocity part integrates to 1)
= O(K s_j^2).
Summable in j.
- entropy of one jet: O(s_j^2 (1 + |log s_j|)). Also summable in j.
- fixed polynomial velocity moments: bounded by C_k s_j^2. Also
summable.
So the sum over (j, sigma, n) of c_i^0 is a well-defined density on Omega
with finite mass, finite energy, finite absolute entropy, finite every
velocity moment.
Fix beta_0 large, mu(v) = exp(-beta_0 |v|^2). Fix delta in (0, 1/32),
and let D_hole be the open (2 delta)-neighborhood in the torus of the
axial segment [-1/3, 1/3] centered at 0. So D_hole is a thin tube
around the rotated z-axis, radius ~2 delta, length 2/3.
The bath is
u^0(x, v) = 1_{D_hole^c}(x) mu(v) 1_{|v| <= R_mu}.
No bath particles inside the tube. That is crucial for the small-time
dynamics: the jets (which live inside D_hole) don't interact with the
bath until later. The bath provides the collision background only after
a fixed time t_1 > 0, once the jets have settled into their cold shape.
The full initial datum is
F_0(x, v) = u^0(x, v) + sum_{i} c_i^0(x, v).
The sum is over all j >= 1, sigma in {+1, -1}, and n in Lambda.
The irrational axis (1, sqrt 2, 0)/sqrt 3 has been rotated to e_3, so
the "z-axis" here is (1, sqrt 2, 0)/sqrt 3 in original coordinates. That
is why the jet supports (thin tubes around this rotated axis) are
disjoint from their lattice translates by the Diophantine separation.
For the family selecting a specific solution, add a small
"seed" aimed at the smallest retained scale s_N:
h^0_{N, T, lambda}(x, v) =
lambda m_T s_N sum_{n in Lambda} chi((x - n)/s_N, v),
where 0 <= lambda <= L_0, m_T = T^p for a fixed exponent p > 0 (fixed later), chi is a fixed nonzero nonnegative smooth bump in x and v,
and L_0 is chosen with L_0 ||chi||_H > 1.
Dormant family (lambda = 0): no seed.
Cap family (lambda = lambda_N): the seed is chosen so the solution
touches a specific weighted cap at a specific short time T .
The seed is tiny: its contribution to entropy is O(s_N^4 (1 + |log s_N|)),
vanishing as N -> inf. So the initial datum is essentially "dormant" for
the seed, but a cleverly chosen tiny seed can still nudge the solution
into a different long-time behaviour.
LEMMA (Datum bounds and approximation). F_0 >= 0, has bounded
velocity support, and satisfies
int_Omega F_0 (1 + |v|^2 + |log F_0|) dx dv < inf.
The truncations with j <= N, with or without seed, converge to F_0 in
weighted L^1 (every polynomial velocity weight) and in relative entropy
with respect to M or mu. The convergence is uniform for 0 < T <= t_1
and 0 <= lambda <= L_0. Moreover, their mu-relative entropy can be made
arbitrarily small, uniformly in N, T, lambda, by first decreasing delta,
then increasing R_mu, then decreasing s_0.
PROOF. Integrability of the cold jets: mass is sum over labels of
O(K s_j^2), summable. Same for entropy, moments (see 4.1).
Integrability of bath: bounded above by mu * 1_{|v| <= R_mu}. Its
mass and entropy are finite.
Integrability of seed: writing b = h^0_{N, T, lambda}, scaling
x |-> s_N x gives
int b <v>^k dx dv <= C_k s_N^4
(since b is supported on a set of volume s_N^3 in x, of amplitude
lambda m_T s_N |chi|, and the v-part integrates to 1). Also |log b| is
bounded by C(1 + |log s_N| + |log(lambda m_T chi)|) on its support, and
|log(lambda m_T)| is bounded on the box [0, L_0] x (0, t_1]. Hence
int b |log b| dx dv <= C s_N^4 (1 + |log s_N|),
vanishing as N -> inf.
For small mu-relative entropy: removing the hole costs H_mu-difference
equal to the mass of mu inside D_hole, which is O(delta) in H_mu-cost.
Removing the velocity tail |v| > R_mu costs H_mu-difference O(exp(-beta_0
R_mu^2)). And inserting the cold jets costs, in H_mu, sum over j of
O(s_j^2 (1 + |log s_j|)) -> 0 as s_0 -> 0. So the initial mu-relative
entropy can be made arbitrarily small in the order
delta -> 0, R_mu -> inf, s_0 -> 0.
For fixed (N, b, T, lambda) and time in [0, t_1], write
F = c + u + h with c = sum_{j <= N, sigma, n} c_i, u = bath, h = hot
(containing any seed and the retired colors). The colored system is
D c_i + nu_F^b c_i = P^b(c_i, c_i), for each retained color i,
D u + nu_F^b u = P^b(u, u), for the bath,
D h + nu_F^b h = P^b(h, h) + 2 P^b(c + u, h) + S,
S = 2 P^b(c, u) + sum_{i != i'} P^b(c_i, c_{i'}). (source term)
nu_F^b and P^b are the cutoff versions. The cutoff kernel is
min(1, b / |v - v_*|): it smooths the hard-sphere kernel near zero
relative speed, and preserves all particle symmetries.
Key observation: summing the equations gives D F = Q^b(F, F). So this
is just a formal reorganization of the single Boltzmann equation, with
individual cold jets tracked separately. This lets the argument bound each
jet independently of the others.
At "retirement times" L_s = s^{-1/2}, the construction stops tracking scale-s colors
separately and merge them into h (and later into w). S is continuous
across this merge, because the total density has no jump. Retirements
are the only events that require bookkeeping; between them, the
evolution is smooth in the weighted H-norm.
PROPOSITION (Finite evolution and its identities).
For fixed (N, b, T, lambda), the colored equations have a unique
nonnegative local mild solution with bounded total H-norm. It continues
while that H-norm stays bounded, depends continuously in the H-norm on
its initial data on every common bounded interval of existence, with
prescribed transfers at retirement times. Each component's norm is
continuous between transfers. On every such interval:
- total mass, momentum, and kinetic energy are conserved;
- the exact local collision-invariant balances hold, with all
polynomial velocity fluxes integrable;
- for rho = M or rho = mu, the entropy dissipation inequality
H_rho(F(t)) + int_0^t D^b(F(s)) ds <= H_rho(F(0))
holds.
PROOF. Weighted-norm Lipschitz. The gain estimate P(H,H) <= C H
(the quadratic gain bound) and the frequency bound
||nu_f^b||_inf <= 2 pi b ||f||_H int H(p) dp
combine to make both the gain bilinear form and the loss operator
locally Lipschitz in the weighted H-norm. Picard iteration in free
coordinates (integrating along free flights) gives a local mild
solution. Nonnegativity: use the loss integrating factor together with
nonnegative gain iteration. Repetition on short subintervals gives
continuation and the difference estimate. Transfers only regroup
finitely many scales and preserve the total density.
Conservation and local balances. The weight H has every polynomial
moment and is in L^{1+eps}(R^3) for some eps > 0. So the cutoff equation
has integrable gain and loss with all polynomial weights on every
bounded interval. The path formula
d/dt int_Omega F(t, x + tv, v) psi(v) dx dv = 0 in D',
when integrated against a smooth spatial test phi, gives the local
balance for every collision invariant psi. Integrability of the flux
comes from (1 + |v|^3) F in L^1.
Entropy. Regularize by F_eps = F + eps g with g(v) = exp(-2|v|^2).
Differentiate F_eps log(F_eps / rho) - F_eps + rho along flights. Its
derivative is Q^b(F,F) log(F_eps / rho). Replacing Q^b(F,F) by
Q^b(F_eps, F_eps) changes the integral by O(eps (1 + |log eps|)) (which
vanishes as eps -> 0), since
|log(F_eps / rho)| <= C(1 + |log eps| + |v|^2 + log^+(1/|bar v|))
is integrable against H. For the shifted density, particle interchange
and the elastic pre/post-collisional substitution turn the collision sum
into the symmetric dissipation. The log of either Maxwellian reference
is a collision invariant; symmetrization therefore gives the negative
dissipation. Passing to the limit gives the entropy inequality.
For later use, define
D_F(t, x) = int_R^3 <v> |F(t, x, v) - mu(v)| dv,
D_-(t, x) = int_R^3 <v> (mu(v) - F(t, x, v))_+ dv.
D_F is the local L^1 distance of F(t,x,.) from the Maxwellian mu; D_- is
its "deficit" (only the negative part). Both are integrable in x (by
Cauchy-Schwarz against mass/energy). the entropy-distance bound makes their L^1_x
norm small by taking the initial mu-relative entropy small.
LEMMA (Entropy-distance from mu). For every prescribed sufficiently
small eps_E > 0, the parameters of the datum construction can be chosen so that
throughout every finite approximation's interval of existence,
||D_F(t)||_{L^1_x} <= eps_E,
0 <= D_-(t, x) <= int_R^3 <v> mu(v) dv for every (t, x).
PROOF. Use (sqrt z - 1)^2 <= z log z - z + 1 and Cauchy-Schwarz:
||D_F(t)||_{L^1_x} <= H_mu(F(t))^{1/2}
* (2 int_Omega <v>^2 (F(t) + mu) dx dv)^{1/2}.
The second factor is uniformly bounded by mass/energy conservation
(and initial moment bounds). The first factor is uniformly small by
the entropy inequality (Prop 4.2) plus the small initial entropy
(the datum construction). The bound on D_- is immediate from positivity of mu.
Only an L^1_x bound is asserted here. Getting POINTWISE control
of D_F (via averaging along free flights) is the crucial job of the averaging step.
COLD JETS: BARRIER FUNCTIONS AND POINTWISE MAJORANTS
A jet at scale s begins as a thin annulus of area about s^2 whose
velocity-integrated mass K/s concentrates in a slab of width
d_s = d_0 s^{10}. Free transport spreads the jet along its axis at O(1)
velocity while leaving the transverse geometry nearly frozen. Self-
collisions are controlled by an explicit Gaussian supersolution, and
without such control the cold jets would blow up under their own gain
and the whole construction would collapse.
For all s > 0, signs sigma in {-1, +1}, and t >= 0, define the axial
profile U_{s,sigma}(t,z) by the implicit equation
U_{s,sigma}(t,z) = a_s^sigma(z - t U_{s,sigma}(t,z)),
where a_s^sigma(z) = sigma + alpha tanh(gamma z / s). The map
zeta -> zeta + t a_s^sigma(zeta) is onto and has derivative at least
one, so the profile exists, is unique, and is smooth. The bound
|U_{s,sigma}| <= 1 + alpha holds everywhere.
For a fixed label, abbreviate s = s_j, L = L_s = s^{-1/2},
U = U_{s,sigma}(t,z_c), and put
tau = z_c - t U, xi = v - U e_3,
D_y = dist(y_c, E_j^sigma), D_z = dist(tau, [-a, a]).
Free transport of the jet is f_i(t,x,v) = c_i^0(x - tv, v). The barrier
is the Gaussian upper bound
m_i(t,x,v) = (C_m K / s) d_s^{-3} e^{t/L}
exp( -(2 - t/L) |xi|^2 / d_s^2
-(D_y^2 + D_z^2) / (8 d_s^2 L t) ),
0 < t <= L.
The constant C_m depends only on the profile norm |J|_infty.
LEMMA (Pointwise barrier comparison).
For every label and every 0 < t <= L_s, the jet satisfies
0 <= f_i(t,x,v) <= m_i(t,x,v) almost everywhere.
PROOF. Take a flight that begins at time zero inside the core: the
start x^0 lies in the spatial support, and v = a_s^sigma(z_c^0) e_3 +
d_s w for some |w| <= 1. At the point x = x^0 + t v, the function
p -> p - a_s^sigma(z_c - t p)
is strictly increasing. Its derivative is at least one, its value at
p = v_3 is d_s w_3, and its value at p = U is zero. Monotonicity gives
|xi| <= d_s,
D_y <= t d_s |bar w|,
D_z <= t d_s |w_3|.
The distance exponent in the barrier is therefore at most t/(8L), and
the velocity exponent is at most two. A fixed C_m absorbs both, giving
f_i <= m_i. Because L_s d_s = d_0 s^{19/2} is far below s^2, the
transverse displacement is smaller than the enlargement radius. The
axial speed is at most two, so the flight also stays inside the axial
interval of the enlargement. End of proof.
At fixed (t,x), the barrier is a scalar multiple of a Maxwellian centred
at U e_3. Detailed balance gives P(m_i, m_i) = m_i nu_{m_i}. Its
velocity mass is at most CK/s, and its first moment about U e_3 is at
most CK d_s / s, uniformly for t <= L_s. The inequality
|v - p| <= |xi| + |p - U e_3| yields
P(m_i, m_i)(t,x,v) <= (CK/s + CK d_s/s) m_i(t,x,v).
For 0 <= t <= L_s, integrating the barrier in velocity,
int_R^3 m_i(t,x,v) dv <= CK / (t + s),
int_{|v| > R} m_i(t,x,v) dv <= CK e^{-c R^2} / (t + s) (R >= 3),
with constants independent of position, sign, and lattice copy.
LEMMA (Pointwise control of a cold jet).
Let 0 < t_1 <= L_s. Suppose m_i <= C_s H throughout [0, t_1] for one
fixed constant C_s. Then on t_1 <= t <= L_s every solution of the
colored equation satisfies
c_i(t,x,v) <= U_i(t,x,v),
where U_i is the Gaussian barrier centred at U_{s,sigma}(t,z_c) with
initial data C_s m_i(t_1). This comparison is valid in the measurable
mild-solution class: at a fixed approximation with m_i <= C_s H
throughout the interval, the difference z_i = (c_i - U_i)_+ has a
pathwise integral bound from zero with source 2 P^b(U_i, z_i) +
P^b(z_i, z_i). The bilinear H-norm estimate plus Gronwall's inequality
gives ||z_i||_H = 0, first on a positive-time interval and then by
continuation to the whole [t_1, L_s]. End of proof.
Consequences. For every fixed k >= 0 and every fixed s and label,
int <p>^k c_i(t,x,p) dp <= C_k / (t + s),
int_{|p| > R} <p>^k c_i(t,x,p) dp <= C_k e^{-c_k R^2} / (t + s) (R >= 3),
uniform over positions on the torus, lattice copies, and signs. If a
strictly positive external frequency kappa satisfies nu_F^b >= kappa
throughout [t_1, L_s], then the barrier U_i gains a factor
c_i(t,x,v) <= e^{-kappa (t - t_1)} U_i(t,x,v),
because e^{-kappa (t - t_1)} U_i is a supersolution of D + kappa: its
linear derivative carries the factor e^{-kappa (t-t_1)}, its quadratic
gain carries e^{-2 kappa (t-t_1)}, and the initial value at t_1 still
dominates c_i(t_1). The comparison c_i <= U_i absorbed into this weaker
form is the exponential decay used in the global bootstrap, and the
mass bound int <p>^k c_i <= C_k/(t+s) follows by dropping the
exponential and using the barrier mass estimate above.
The mass at fixed (t,x) is the source of the sheet amplitudes used later.
For every fixed k >= 0 and every 0 < t <= L_s, summing <p>^k c_i over
lattice points inside a ball of radius R at a fixed height gives at
worst a fixed polynomial factor in 1/s, absorbed by the exponential
tail.
CORE FORMULA AND SHEET AMPLITUDES
At fixed (t,x) the velocity density c_i(t,x,p) of an active jet is the
pushforward of a parameter measure on R^3_w. Write x = (y,z) and let
(y_c, z_c) be the center of the appropriate lifted core: transverse
projection to a shoulder and axial projection to z. The parameter
measure is
j_i(t,x; w) dw
= [K S(gamma(z_c - t p_3) / s)
/ (s + t alpha gamma S(gamma(z_c - t p_3) / s))]
J(w) dw,
where the velocity coordinate of x - t v is (d_s bar w, p_3) and
p_3 is the unique solution of
p_3 = a_s^sigma(z_c - t p_3) + d_s w_3.
LEMMA (Core formula and Jacobian).
At fixed (t,x), the velocity density satisfies
c_i(t,x,p) dp
= [K S(gamma(z_c - t p_3) / s)
/ (s + t alpha gamma S(gamma(z_c - t p_3) / s))]
* (d^3 w / d^3 p) * J(w) dw,
where the parameter-to-velocity Jacobian is
d^3 p / d^3 w = d_s^3 / (1 + t (a_s^sigma)'(z_c - t p_3)).
PROOF. The transverse map w -> d_s bar w is linear on R^2 with
Jacobian d_s^2. The axial map w_3 -> p_3 is strictly increasing, because
the defining equation has left side p_3 + t a_s^sigma(z_c - t p_3)
whose derivative in p_3 equals 1 + t (a_s^sigma)'(z_c - t p_3) >= 1.
Differentiating implicitly,
dp_3 / dw_3 = d_s / (1 + t (a_s^sigma)'(z_c - t p_3)).
Multiplying the transverse and axial factors yields the displayed
inverse. End of proof.
The axial factor S(s) / (s + t alpha gamma S) is the only place a
possible singularity of the axial map enters, and the positivity
(t alpha gamma S >= 0) keeps the coefficient bounded above by S/s and
below by zero. No differentiability of the axial profile beyond the
implicit equation is needed.
DEFINITION (Sheet class). Fix L >= 1. An elementary sheet is a product
measure
c(dp) = N_d(bar p) rho(dp_3),
consisting of a centered transverse Gaussian N_d of width d in [0, 1/8]
times an axial probability rho satisfying the exponential moment
int_R exp(8 a^2) rho(da) <= L.
A "probability mixture of sheets" is any convex combination of
elementary sheets, and a "sheet at amplitude A_0" is any pointwise
multiple of such a mixture by A_0.
LEMMA (Sheet domination of a main color).
Suppose m_i <= C_s H on [0, t_1] with t_1 > 0 fixed, and let
t_1 <= t <= L_s. Then at fixed (t,x) the full upper bound U_i(t) is a
sheet at amplitude C/(t+s), with transverse width d_s and axial
probability rho_s whose exponential moment depends only on
K, alpha, gamma, |J|_infty:
U_i(t,x,p) dp <= (C/(t+s)) N_{d_s}(bar p) rho_s(dp_3),
int_R exp(8 a^2) rho_s(da) <= 2 exp(16(1 + alpha)^2)
exp(16 (2 d_s + 1)^2).
PROOF. The transverse part of c_i dp is the linear image of J(w) dw
under w_bar -> d_s w_bar, which is a centered transverse Gaussian of
width d_s times a bounded transverse amplitude. The axial pushforward
is the law of p_3 = a_s^sigma(z_c - t p_3) + d_s w_3. Its exponential
moment is bounded using |a_s^sigma| <= 1 + alpha and the Gaussian
parameter control. The amplitude follows from the Jacobian and the S/s
factor. End of proof.
The remainder that separates the barrier from the sheet is epsilon_s m_i.
After replacing the barrier's transverse Gaussian by its amplitude at
scale epsilon_s, the sheet amplitude drops to
C epsilon_s / s = C L_s d_s / s^2 <= C s^{15/2},
which is below 1/(t+s) for all sufficiently small s and for all
t <= L_s. So U_i coincides with the sheet at amplitude C/(t+s) up to
a factor that is far below the main amplitude.
Consequences for the sheet amplitudes used in the global bootstrap:
int <p>^k c_i(t,x,p) dp <= C_k / (t + s),
int_{|p| > R} <p>^k c_i(t,x,p) dp <= C_k e^{-c_k R^2} / (t + s) (R >= 3),
uniform over positions on the torus, lattice copies, and signs, with
C_k depending only on (q, m, K, alpha, gamma, |J|_infty).
NONMAIN COLORS, RETIREMENT, AND TUBE GEOMETRY
For each label i = (j, sigma, n) with 1 <= j <= N, define the enlarged
active tube O_i(t) in phase space:
- transverse: the closed s_0-neighborhood of the current lifted shoulder,
- axial: |z_c - t U| <= a + 2t + b_sp.
The projection to the torus is an open set in x. Set O_j(t) as the union
over all sigma and all lattice points n at scale s_j, and O(t) as the
union over all j.
LEMMA (Disjointness of tubes and uniqueness of main color).
For 0 <= t <= L_{s_0} and s_0 sufficiently small:
(i) the sets O_j(t) at distinct scales are pairwise disjoint in phase
space;
(ii) within each scale, distinct lattice copies are disjoint;
(iii) at every (t,x) at most one active main color sits in O(t).
PROOF. The axial half-length obeys a + 2t + b_sp <= 1/8 + 1/16 + 1/16 = 1/4
for t <= 1/32; extending to t up to s_0^{-1/2} keeps it below 1/2 once
s_0 is small. Distinct lattice points have axial distance at least 1/2
in the rotated lattice by the Diophantine separation, so distinct copies
at a fixed scale have disjoint axial support. Between scales, the
ratio s_j / s_{j+1} = e^eta so the tubal envelopes at adjacent scales
are separated by at least s_0 - s_0 e^eta > 0 for small s_0. Uniqueness
of the main color at each (t,x) follows. End of proof.
LEMMA (Sum of nonmain colors).
Fix M > 0 and a compact velocity region |p| <= R. Uniformly over
(t,x), active labels, and lattice points,
sum over nonmain labels of c_i(t,x,p) <= C_{M,R} s^{-M} e^{-4|p|^2},
where s denotes the smallest active scale and C_{M,R} depends only on
(K, alpha, gamma, |J|_infty, M, R, s_0), not on the approximation
index, the cutoff b, or the cap horizon.
PROOF. A nonmain color is a jet whose (t,x) lies outside the interior
of its O_i(t) tube, so either D_y >= s_0 or D_z >= a, or the barrier
picks up the |xi| >= R Gaussian decay. In every case the barrier m_i
carries a factor exp(-D^2 / (8 d_s^2 L t)) for the relevant distance
D. With d_s = d_0 s^{10} and L = s^{-1/2}, the denominator at t = L is
8 d_0^2 s^{19}. Multiplied by the fixed D^2 (which is at least a^2 or
s_0^2), the exponent yields doubly-exponential decay in s, so a fixed
polynomial 1/s loss from the lattice sum over the s^{-M}-many nonmain
labels is absorbed by the exponential. End of proof.
LEMMA (Size at retirement and volume of active tubes).
There are fixed constants C, C' > 0 such that
|| sum_{i : s_i = s} U_i(L_s) ||_H <= C s^{-C'},
and uniformly over active label sets and all times t >= 0,
|O(t)| <= sum_j C s_j^2 (1 + L_{s_j}) <= C s_0^{3/2}.
PROOF. The retirement bound uses f_i <= m_i and the Gaussian lattice
sum without extracting a small distance factor. The prefactor and the
number of central copies are polynomial in 1/s. Completing the velocity
square bounds the remaining factor by C e^{-4|v|^2} <= CH. For the tube
volume, the transverse area at scale s_j is O(s_j^2), the axial length
while active is O(1 + L_{s_j}) = O(s_j^{-1/2}), and projection to the
torus does not increase volume. Summation gives C sum_j (s_j^2 + s_j^{3/2})
<= C s_0^{3/2}. End of proof.
Convention. Every pointwise assertion is an essential
bound. It holds at the time sections defined by the finite path
equations: free coordinates preserve measure, and Fubini's theorem
supplies a full-measure set of paths on which the inequalities
integrate. For each fixed terminal time this gives a full-measure
endpoint set. Countably many approximation indices, lattice copies,
and interval choices are handled simultaneously.
Concluding remark. With the pointwise control c_i <= U_i, the sheet
domination, the nonmain smallness, and the retirement size, the cold
jets are uniformly controlled throughout their lifetimes, and their
combined contribution to any weighted supremum estimate is summable
over the geometric scales. This closes the analysis of the cold
component and prepares the early continuation.
EARLY SELECTION AND CONTINUATION
The bath starts close to the Maxwellian mu and remains confined to a
fixed region, away from the jets throughout the initial time interval.
The cold jets, in turn, occupy only a thin tubular neighborhood of their
initial positions because they move at bounded axial speed and their
transverse positions are frozen. Together these imply that on a fixed
short interval [0, t_1] the bath stays below a Maxwellian barrier away
from a hole around the origin, and inside that hole the jets dominate.
BATH BARRIER
LEMMA (Bath barrier). There are t_1 in (0, 1/32], beta' > 4, and
constants C, c > 0 independent of delta, R_mu, and s_0, such that every
finite evolution of the colored equations satisfies
0 <= u(t,x,v) <= C e^{-beta' |v|^2} exp(-c D_0(x)^2 / t),
0 < t <= t_1,
where D_0(x) = dist(x, D_hole^c) is the distance from x to the
complement of the initial spatial hole D_hole.
PROOF. The bath density starts below the Maxwellian mu. In the colored
system the bath obeys
D u + nu_F^b u = P^b(u, u).
Dropping the nonnegative loss from the left side and using P^b <= P
(since b >= 1), the bath satisfies
D u <= P(u, u) <= C_bath H(v)
for a fixed Maxwellian constant C_bath, from the quadratic gain bound
P(H,H) <= C_{q,m} H. The free transport of the initial bath decays as
a Gaussian away from its support, and its spatial support is bounded
by a distance function D_0 that controls the distance to the jets.
The comparison proceeds along free flights. Consider the path x - tv
at time zero. For a flight starting outside the hole (D_0(x) = 0), the
free transport of the bath is dominated by mu. For a flight starting
inside the hole (D_0(x) > 0), the trace is zero, as is the initial bath,
and the comparison is trivial. Integrate the differential inequality
from a small positive time; the Gaussian source contributes a factor
C exp(-c D_0^2 / t) by the free heat kernel bound. The trace of the
bath along such a path dominates u^0 because D_0(x + tv) tends to one.
End of proof.
LEMMA (Hole distance along tubes). For s_0 small enough depending on
the hole radius delta, and 0 <= t <= t_1, every point of O(t) is at
distance at least delta from the complement of the hole:
D_0(x) >= delta for x in O(t).
PROOF. The axial interval of O_i(t) has half-length at most
a + 2t + b_sp, which for t <= 1/32 is at most 1/8 + 1/16 + 1/16 = 1/4,
still inside 1/3. The transverse distance from the lifted axis is at
most s_0. So every point of O(t) lies within distance s_0 of the
segment defining the hole in the lifted picture. Since the hole is the
open 2 delta neighborhood of this segment, the triangle inequality
gives D_0 >= 2 delta - s_0 >= delta once s_0 <= delta. End of proof.
Combining the two bounds. Inside O(t), the bath is at most
C e^{-c delta^2 / t} H(v), which is exponentially small in 1/t and
contributes a tiny multiple of H (t + s_N)^{-1} to the cold-bath
source. Away from O(t), the cold jets are at most a doubly-exponential
small multiple of H by the nonmain bound. The total cold-bath cross
term is a small multiple of H times (t + s_N)^{-1} + rho(t).
COLD-BATH AND COLD-COLD SOURCE INTEGRAL
LEMMA (Source integral). Fix a small time t_b in (0, t_1). For every
epsilon > 0 there is a reduction of s_0 after which the source term
S = 2 P^b(c, u) + sum_{i != i'} P^b(c_i, c_{i'}) satisfies
|| int_{t_b}^t S(s, x - (t-s)v, v) ds ||_H
<= C_{t_b} epsilon(s_0) + t_1 omega_{t_b}(R) + C_{t_b, q'} t_1 r^{q-q'}
+ C_{t_b, R} s_0 / r,
with the terms on the right tending to zero in the specified order:
s_0 -> 0 first, then R -> infinity, then r -> 0.
PROOF. Split S into two pieces:
(a) the cold-bath cross term 2 P^b(c, u);
(b) the cold-cold cross term sum_{i != i'} P^b(c_i, c_{i'}).
For (a): at a point in O_i(t) the bath is at most C e^{-c delta^2 / t}
H(v). Outside O_i(t) the cold jet is at most a doubly-exponential small
multiple of H. The upper bound P(c, H) <= C H (t + s_N)^{-1} then
gives P(c, u) <= C e^{-c delta^2 / t} H / (t + s_N).
For the H-norm of the integral of S along paths, use the path-formula
estimate. The Gaussian spreading of the initial bath over a distance
delta/2 dominates the direct source by an exponential e^{-c delta^2 / t}
with a constant uniform in (t,x,v).
For (b): the two colors are on disjoint spatial tubes (via the
disjointness of O_i(t)), so P^b(c_i, c_{i'}) at an output point either
involves nonmain colors or one input sits at a main color of the
opposite sign. Both fall under the nonmain bound or the opposite-sign
disjointness. Summation over 1 <= i != i' <= N gives at most
rho(t)^2 H.
The path-integral form: writing the H-norm as an essential supremum
over (t,x,v) of |S|/H, use the Gaussian envelope to convert the path
integral into the stated expression. The q' < q term comes from the
Gaussian tail decay in the nonmain colors, and the r^{q-q'} term from
the low transverse-velocity weight. The s_0/r term comes from the
cylinder-crossing argument: a free flight of duration t_1 crosses only
O(1) transverse cylinders of radius s_0. End of proof.
SMALL HOT NORM THROUGH t_1
The hot norm Z(t) = ||h(t)||_H satisfies, on [0, t_1] and while Z <= 1,
Z(t) <= Z(r) + int_r^t (C_2 / (s + s_N)) Z(s) ds
+ int_r^t C_{A, delta} s^{A - 1} ds, 0 <= r <= t <= t_1.
The constant C_2 includes the quadratic hot gain, because Z^2 <= Z and
s + s_N <= 2. It is independent of delta, R_mu, s_0, and N.
CAP SELECTION
The cap is a prescribed upper bound on the hot norm at some time before
a horizon T. It selects specific finite solutions whose hot component
reaches a target size, and dilation at the attaining time will turn a
finite mass into a nonzero limit.
Fix T, N, and the cutoff. Let I be the set of lambda in [0, L_0] whose
solutions exist on [0, T] and remain strictly below
h(t,x,v) <= lambda m_T s_N chi_bounded H(v)
at every such time. The norm is the weighted supremum || h ||_H.
LEMMA (Continuous dependence on lambda). For each lambda in [0, L_0],
the map lambda -> h_lambda obtained by the finite evolution depends
continuously in Z(t) = || h_lambda(t) ||_H on every bounded interval
of existence, uniformly in T <= t_1 and in the cutoff.
PROOF. The colored system is locally Lipschitz in the H-norm with
constants uniform in lambda and cutoff. Picard iteration on free flight
coordinates gives existence with a uniform lifespan. The difference
between h_lambda and h_{lambda'} obeys a Duhamel formula whose dominant
factor is bounded by the fixed quadratic estimates. Taking the H-norm
gives a Gronwall bound with || lambda chi - lambda' chi' ||_H on the
right. End of proof.
The set I is relatively open in [0, L_0], contains 0, and excludes L_0
because L_0 || chi ||_H > 1. Let lambda_* be the right endpoint of the
connected component of I containing zero. Every hot component of
h_{lambda_k} for lambda_k up to lambda_* is uniformly bounded by the
cap in the limit, and the cold and bath estimates bound the remaining
colors. So the lambda_k solutions are Cauchy in the fixed-approximation
norm, and their limit is the solution at lambda_*.
LEMMA (Cap attained). At lambda = lambda_* the solution exists on
[0, T] and satisfies
sup_{0 <= t <= T} (|| h (t) ||_H) / ((T - t)^p (T - t + s_N)) = 1.
PROOF. The norm Z is continuous in time, the cap denominator
(T - t)^p (T - t + s_N) is positive on [0, T), so the ratio has a
finite maximum. If that maximum were less than one, continuous dependence
and continuation would place a neighbourhood of lambda_* in I, contrary
to the right-endpoint property. Its maximum equals one. End of proof.
UNIFORM EARLY CONTINUATION
Fix delta > 0 and zeta in (0, 1). There are s_0_bar > 0 and
T_0 in (0, t_1), depending only on (delta, zeta) and on the fixed jet
parameters, such that for every 0 < s_0 <= s_0_bar, every finite
approximation, and every 0 < T <= T_0, the dormant solution and every
solution selected above continue to t_1 with
sup_{0 <= t <= t_1} || h(t) ||_H <= zeta.
The choices are uniform in N, the collision cutoff, R_mu, and T.
PROOF. Choose an intermediate time t_b in (0, t_1) and then require
T_0 < t_b. At the cap horizon both selected and dormant solutions obey
Z(T) <= T^p (T + s_N) <= 2 T^p,
using T, s_N <= 1; no relation between T and s_N is needed. For
T <= t <= t_b, the scalar comparison and (s + s_N)^{-1} <= s^{-1} give,
while Z <= 1,
Z(t) <= 2 T^{p - C_2} t^{C_2} + C'_A t^A.
Multiplying the scalar comparison by t^{-C_2} integrates the source to
at most C''_A t^{A - C_2} / (A - C_2). On [t_b, t_1], the source
integral is at most C_{t_b} epsilon(s_0) by the source-integral lemma,
and the Z term contributes an exponential e^{C_2 t_b^{-1} t_1} that is
absorbed into the final constant. Choose T_0 and s_0 to make the
right side smaller than zeta. End of proof.
For dormant solutions, the same computation starting from zero source
gives an initial-interval bound with exponent A > p + 1,
Z(t) <= C t^A,
on an initial interval independent of (N, b, T, R_mu). This is the
growth that the cap selection uses.
GLOBAL CONTINUATION
The estimates on the cold jets and the early interval give every finite
evolution a common starting point. What remains is to prove
that all such evolutions exist for all time, with a uniform bound on the
weighted H-norm of the portion containing the bath, the hot component,
and the retired jets. The additional control after t_1 comes from the
bath: it makes F close to mu in entropy, which supplies a positive
collision frequency and thereby a uniform bound on the hot nonlinear
gain after averaging along free flights.
Record the two quantities measuring the entropy distance of F from
mu. Write
D_F(t,x) = int <v> |F(t,x,v) - mu(v)| dv,
D_-(t,x) = int <v> (mu(v) - F(t,x,v))_+ dv.
The second is bounded pointwise by int <v> mu(v) dv, independent of any
bootstrap. The first has an L^1_x bound uniform over time, from the
entropy inequality and the small initial mu-relative entropy. Both
statements are uniform over the approximation index.
POSITIVE FREQUENCY
LEMMA (Positive collision frequency). There is a constant kappa > 0,
depending only on the fixed Maxwellian mu and t_1, with the following
property. For each fixed B > B_0 + 2, if eps_E is small enough
depending on B, then every finite evolution satisfying || w(t) ||_H
<= B for t >= t_1 obeys
nu_F^b(t,x,v) >= kappa, t >= t_1.
The constant is uniform in b >= 1 and in the approximation index.
PROOF. The bath starts near mu and the excess mass of F over mu is a
small multiple of eps_E in L^1_x. Split F = mu + (F - mu), and lower
bound nu_F^b against nu_mu^b. The difference is bounded by the small
mass of |F - mu| plus the fixed mass of F outside a large velocity
truncation. Choosing the entropy tolerance small enough and the
truncation large enough produces a strictly positive lower bound, since
nu_mu(v) is uniformly bounded below by nu_mu(v) >= c_m (1 + |v|). End
of proof.
COLD DECAY
Lemma (Exponential decay of active jets). Whatever the bootstrap size B,
every active jet at scale s = s_j >= s_N satisfies
c_i(t,x,v) <= e^{-kappa (t - t_1)} U_i(t,x,v), t_1 <= t <= L_s.
This is the barrier gain of the previous chapter. At a point in the
active tube union O(t) there is at most one main color, whose sheet
amplitude is at most C/t_1. Outside that union all colors are nonmain.
Combining the sheet domination with the nonmain smallness, and enlarging
the small coefficient if necessary, gives constants C_3, C_4 independent
of B such that
P(c, H) <= C_3 e^{-kappa (t - t_1)} H (ind_{O(t)} + theta_0),
S_c = sum_{i != i'} P^b(c_i, c_{i'}) <= C_4 theta_0 H,
where theta_0 = theta_0(s_0) -> 0 as s_0 -> 0. For the second estimate,
if c_* is the main color and r the sum of the remaining colors, the
ordered cross sum is bounded by 2 P(c_*, r) + P(r, r). Here
r <= theta_0 e^{-4|v|^2} and the sheet bound controls P(c_*, H); if no
main color is present, then c <= theta_0 e^{-4|v|^2} directly.
RETIREMENT INCREMENTS
At a retirement time L_s, the color at scale s joins w. The increment
is bounded by J_s H, where
J_s = C s^{-C} e^{-kappa (L_s - t_1)}, sum_s J_s <= J_0(s_0),
with J_0(s_0) -> 0 as s_0 -> 0. To see the last limit, the exponential
e^{-kappa s^{-1/2}} dominates every power of s: for sufficiently small s,
C s^{-C} e^{-kappa (s^{-1/2} - t_1)} <= s^2,
and the remaining geometric sum is at most s_0^2 / (e^{2 eta} - 1). So
the sum of all retirement increments is small uniformly in N.
UNIFORM NONLINEAR GAIN
The remaining obstruction is the quadratic term P^b(w, w). Estimating
it directly from w <= B H would produce a constant of order B^2. Instead
compare w with the fixed Maxwellian. Entropy bounds the spatial integral
of the positive excess, and an averaging argument along free flights
turns this into a small excess mass at every position. The small-mass
gain estimate then controls the quadratic term by a constant chosen
before B.
LEMMA (Uniform nonlinear gain). Let B > B_0 + 2 be fixed and suppose
|| w(t) ||_H <= B on an interval beginning at t_1. After taking eps_E,
zeta, and s_0 small enough depending on B, one has on that interval
P^b(w, w) <= C_5 H, C_5 := || P(mu, mu) ||_H + 1.
The constant C_5 is independent of B, the approximation, and the cap
horizon, and the estimate holds also at the appropriate one-sided
sections of retirement times.
PROOF. Choose the entropy tolerance small enough for the positive
frequency lemma, so that kappa and the cold decay estimates are
available. Define
j = (w - mu)_+, m_j(t,x) = int j(t,x,v) dv.
Since w <= F and j <= B H,
0 <= m_j <= B M_H, || m_j(t) ||_1 <= eps_E,
where M_H = int H dv.
Write C_mu = || mu ||_H < infinity and A_B = 2 C_mu + B. The
inequality w <= mu + j, bilinearity, and positivity give
P^b(w, w) - P^b(mu, mu) <= 2 P(mu, j) + P(j, j) <= A_B P(H, j).
The frequency deficit has the bound
(nu_mu^b - nu_F^b) mu <= 2 pi mu(v) int |v - p| (mu(p) - F(p))_+ dp
<= C H(v) D_-(t,x),
using mu(v) <v> <= C H(v).
FLIGHT-AVERAGE EXCESS AND POINTWISE SMALL MASS
The L^1 smallness of m_j alone does not directly bound P(H, j) at a
point, because the small-mass gain estimate absorbs a pointwise m_j.
Fix rho in (0, t_1/2). For t >= t_1 and R > 0, the flight-average of
m_j obeys
int_{|v| <= R} H(v) m_j(t,x - tv, v) dv
<= C_{rho, R} (eps_E + e^{-c rho} M_H),
where H is the Maxwellian weight and the constant does not depend on
the bootstrap size B.
PROOF. Free transport of int m_j = || m_j ||_1 is the almost-everywhere
mass of excess over mu. Bounding the contribution at a point reduces
to the average of || m_j ||_1 over a transverse slab of width R rho,
by the change of variable (x, v) -> (x - tv, v). Since || m_j ||_1 <= eps_E
uniformly, the total mass in the slab is eps_E plus the polynomial tail
over |v| > R. End of proof.
Choosing rho small enough and R large enough, and then eps_E and s_0
small enough, gives
m_j(t,x) <= eps, for all (t,x), t >= t_1,
for an arbitrarily small prescribed eps. Applying the small-mass gain
estimate from the kernel chapter,
P(H, j)(v) <= H(v) (eps' + C_{B, eps'} m_j(t,x)) <= H(v) eps''
for arbitrarily small eps''. Combining with the bilinear comparison
P^b(w, w) <= P^b(mu, mu) + A_B P(H, j) and the fixed bound
|| P(mu, mu) ||_H = C_mu < infinity gives the claimed
P^b(w, w) <= C_5 H, C_5 = C_mu + 1,
once eps'' is at most 1/A_B. End of proof of the uniform gain lemma.
GLOBAL CONTINUATION PROPOSITION
LEMMA (Uniform global continuation). The construction parameters can
be fixed so that every dormant finite evolution and every cap solution
with 0 < T <= T_0 continues globally. There are constants B < infinity
and kappa > 0, independent of N, b_N >= 1, and T, such that
|| w(t) ||_H <= B, nu_F^{b_N}(t,x,v) >= kappa, t >= t_1.
Every active jet satisfies the exponential-decay estimate through its
retirement, the total retirement increment is bounded by J_0(s_0), and
the hot component satisfies the small bound of the earlier continuation
on [0, t_1].
PROOF. On a bootstrap interval let Z_w(t) = || w(t) ||_H. The Duhamel
formula for the w-equation, with all retirement increments included,
and the frequency bound, the cold decay estimates, the cold source
bound, and the uniform nonlinear gain give
Z_w(t) <= e^{-kappa (t - t_1)} B_0
+ (C_5 + C_4 theta_0) / kappa
+ J_0
+ 2 C_3 (1 + theta_0) int_{t_1}^t
e^{-kappa (t - s)} e^{-kappa (s - t_1)} Z_w(s) ds.
Taking s_0 small enough that theta_0, J_0 <= 1 and discarding the
first exponential in the last integrand,
Z_w(t) <= B_0 + (C_4 + C_5) / kappa + 1
+ 4 C_3 int_{t_1}^t e^{-kappa (s - t_1)} Z_w(s) ds.
The integral version of Gronwall's inequality yields
Z_w(t) <= (B_0 + (C_4 + C_5) / kappa + 1) e^{4 C_3 / kappa} =: B_*.
The constants defining B_* were fixed before B, so taking B > B_* +
something closes the bootstrap: the bound at each time is strictly
smaller than B, a strict improvement.
CHOOSING THE PARAMETERS IN ORDER
The parameters are chosen one after another, each small or large
depending only on those fixed before it:
(a) The bath parameters c, beta', t_1 from the bath barrier lemma.
(b) The early horizon p, A from the exponent comparison A > p + C_2 + 1.
(c) The hole radius delta and the bath cutoff R_mu, to make the initial
mu-relative entropy small.
(d) The cold source coefficient zeta, and s_0 small enough for the
sheet amplitudes, the nonmain bound, the retirement sum, the source
integral, and the cold decay.
(e) The constant B and the entropy tolerance eps_E once the previous
choices make theta_0, J_0 small.
(f) The cap horizon T_0 finally, chosen after all the above so the
early continuation applies.
None of these choices depends on N, on the chosen cap solution, on its
horizon T in (0, T_0], or on the cutoff b_N. End of proof of the global
continuation proposition.
GLOBAL LIMITS AND LOCAL CONSERVATION
The uniform bounds of the global continuation allow the cutoff and the
initial truncation to be removed. The needed compactness is stronger
than mass and entropy: the same-position collision products must be
uniformly integrable with every polynomial velocity weight. This will
preserve the local energy flux and rule out defects in all five
collision-invariant balances.
GLOBAL LIMITS
Fix the construction parameters, and let b_N -> infinity. Either the
dormant family or a family of cap solutions at one fixed horizon
T in (0, T_0] has a subsequence of total densities F_N converging
weakly in L^1([0, tau] x Omega) for every finite tau to a nonnegative
density F. This subsequence can be chosen so that
F_N(t) weakly converges to F(t) in L^1(Omega) at every t >= 0,
with uniformly small polynomially weighted velocity tails on each
bounded time interval.
WEIGHTED MOMENTS AND ANNULAR SINGULARITY
Let A_{s_j}(t) be the union, in one periodic cell, of the enlarged
sets O_i(t) of the two signs at scale s_j, including all lattice
copies, when t <= L_{s_j}; set A_{s_j}(t) = empty for t > L_{s_j}.
The active enlarged sets are disjoint across scales and within a fixed
scale's lattice copies.
On [0, tau], the axial lengths of the active tubes are bounded by
C_tau, so
|A_{s_j}(t)| <= C_tau s_j^2, 0 <= t <= tau.
The free-core amplitude estimate, the cold remainder bound, and the
bounds on bath and hot component give the moment envelope
R_{k,N}(t,x) = int <v>^k F_N(t,x,v) dv
<= C_{k, tau} (1 + sum_j ind_{A_{s_j}(t)}(x) / (t + s_j)).
The square of this envelope controls the collision products.
Disjointness ensures no product of two distinct singular terms appears,
and one scale contributes at most
int_0^tau C_tau s_j^2 / (t + s_j)^2 dt <= C_tau s_j.
Set A_{< eps}(t) = union_{s_j < eps} A_{s_j}(t). Since the scales form
a geometric sequence,
sup_N int_0^tau int_{A_{<eps}(t)} R_{k,N}(t,x)^2 dx dt
<= C_{k,tau} sum_{s_j<eps} (s_j + s_j^2) -> 0 as eps -> 0.
The first power of the envelope gives uniformly at individual times
sup_{N, 0 <= t <= tau} int_{A_{<eps}(t)} R_{k,N}(t,x) dx
<= C_{k, tau} sum_{s_j<eps} (s_j + s_j^2) -> 0.
Outside A_{<eps}(t), a main color can have only one of the finitely
many scales s_j >= eps. Their thermal widths are positive and fixed.
The Gaussian velocity bounds plus summation over lattice copies give
F_N(t,x,v) <= Psi_{eps,tau}(v) for x not in A_{<eps}(t),
0 <= Psi_{eps,tau} <= C_{eps,tau} H.
The same majorant includes bounded hot, bath, and retired parts, plus
the Gaussian nonmain tails, and it has every polynomial velocity moment.
The remaining singularity on the velocity axis is integrable.
These bounds prove uniform integrability in the full spacetime and
two-velocity variables of
<v>^k <v_*>^k F_N(t,x,v) F_N(t,x,v_*), k >= 0.
Their integrals on the removed spatial sets are bounded by the s_j-sum
estimate above; on the complement they are dominated by the fixed
integrable function <v>^k Psi_{eps,tau}(v) <v_*>^k Psi_{eps,tau}(v_*).
First make the removed contribution small, then use absolute continuity
of this fixed integral on arbitrary small measurable sets. The same
decomposition gives uniformly small weighted velocity tails. Using the
first power instead of the square proves the corresponding one-particle
assertions, both in spacetime and uniformly at all individual times.
LOCAL COMPACTNESS
Averaging in velocity converts the transport equation into an equation
where spatial compactness follows from the convolution structure.
Because the weighted velocity bounds are uniform, the resulting local
L^1 compactness can be transferred back to the full distribution by an
exhausting sequence in velocity.
Let phi in C_c^infinity(R^3) be a nonnegative test with int phi = 1, and
put phi_L(v) = L^3 phi(L v). For a smooth velocity test psi compactly
supported, the average
A_L[F_N](t,x) = int psi(v) phi_L * F_N(t,x,v) dv
is bounded and satisfies a transport equation with a source equal to
the L^1-slope of a velocity average of Q_N. The latter is controlled by
the pair-product bounds above.
By Cauchy-Schwarz, for a Fourier transform in (t, x) with Fourier
series in the torus case,
| int psi(v) hat(u_n)(sigma, k, v) dv |^2
<= I_psi(sigma, k) int | hat(r_n)^L(sigma, k, v) |^2 dv,
where I_psi(sigma, k) = int |psi(v)|^2 / (1 + (sigma + k.v)^2) dv. On a
compact velocity support, I_psi vanishes uniformly as |sigma| + |k|
grows. So Plancherel's theorem gives uniformly small high-frequency
L^2 tails. Convolution with a smooth approximate identity therefore
approximates the averages in L^2 uniformly. On compact time-space sets
the convolutions are bounded and equicontinuous, hence relatively
compact. So local L^2, and a fortiori local L^1, compactness holds at
fixed L. The uniform L^1 error as L -> infinity completes the argument.
The same reasoning applies unchanged to the dual lattice of the rotated
torus.
IDENTIFICATION OF COLLISION TERMS
The uniform integrability and velocity tightness give a subsequence
with F_N weakly converging to F in L^1([0, tau] x Omega). Since
D F_N = Q_N^+ - Q_N^-, the averaging lemma applies. After extraction
for a countable dense family of compact velocity tests, every smooth
compact velocity average converges strongly in local L^1_{t,x} to the
corresponding average of F. The extension from the countable family
follows from the uniform integrated mass bound.
Set W_N(t,x,v,v_*) = F_N(t,x,v) F_N(t,x,v_*). By the weighted pair-
product bound, a further subsequence has a weak L^1 limit W. For
smooth compactly supported psi, chi,
int psi(v) chi(v_*) W_N(t,x,v,v_*) dv dv_*
= ( int psi(v) F_N(t,x,v) dv ) ( int chi(v_*) F_N(t,x,v_*) dv_* ).
The right-hand side converges in measure locally in (t,x). It is
uniformly integrable there: its absolute value is bounded by
int |psi(v) chi(v_*)| W_N, and the full pair-product bounds control
this on small time-space sets. Vitali's theorem therefore gives
convergence in local L^1. It follows that W and the product
F(t,x,v) F(t,x,v_*) have identical integrals against separated velocity
tests for almost every (t,x). A countable determining family gives a
common full-measure set; finite sums of separated tests are dense among
continuous functions on compact two-velocity sets. Hence
W_N weakly converges to F(t,x,v) F(t,x,v_*),
and the gain and loss of the cutoff Boltzmann operator converge to the
gain and loss of the full one.
ENTROPY LOWER SEMICONTINUITY
Let Phi(A, B) = (A - B) log(A / B) with its nonnegative lower-
semicontinuous extended value at zero. It is jointly convex: it is the
sum of the two convex perspectives A log(A / B) and B log(B / A). On
any bounded collision domain the pair-limit and its primed version give
joint weak convergence of the two arguments of Phi. Restriction to
[0, t] preserves this for each fixed t. Weak lower semicontinuity
applies to the integral of |a| Phi; the weight absorbs into the
arguments by Phi(cA, cB) = c Phi(A, B) for c >= 0. On |v|, |v_*| <= R
the cutoff disappears once b_N >= 2R. Exhausting velocity space yields
int_0^t D(F(s)) ds <= lim inf_N int_0^t D^{b_N}(F_N(s)) ds.
The same convex lower-semicontinuity argument applies to H_M(F_N(t))
under the weak L^1 convergence at each time. Combining the two
nonnegative terms with the finite-approximation entropy inequality and
initial entropy convergence gives
H_M(F(t)) + int_0^t D(F(s)) ds <= H_M(F_0), 0 <= t <= tau.
The extraction is independent of t; no exceptional set of time
endpoints is introduced by the lower-semicontinuity argument.
SINGLE TIME REPRESENTATIVE
For a smooth periodic phi(x, v) compactly supported in velocity, the
cutoff path equations give
|<F_N(t) - F_N(s), phi>|
<= int_s^t int_Omega
( F_N |v . grad_x phi| + (Q_N^+ + Q_N^-) |phi| )
dx dv du.
The first term is bounded by C_phi |t - s|. The second has a modulus
tending to zero with |t - s|, uniformly in N, by the short-time
integral conclusion. For a countable family dense in the uniform norm
on each compact velocity set, Arzela-Ascoli supplies uniform convergence
of the pairings on [0, tau] after one extraction.
At any individual time, the fixed-time smallness and fixed-majorant
bounds make {F_N(t)} relatively weakly compact in L^1(Omega). Every
cluster point has the smooth pairings just obtained, and hence is the
same density. So this one subsequence converges weakly in L^1 at every
time, defining a representative F(t) whose smooth pairings are
continuous and which agrees with the spacetime limit. Weighted fixed-
time tightness gives the same conclusion for moments, so the cutoff
mass, momentum, and energy pass at every time. In particular the initial
trace of this representative is F_0.
RENORMALIZED EQUATION
For a smooth test phi, first pass to velocity-averaged form:
D<F_N, phi> = <Q_N^+, phi> - <Q_N^-, phi>.
The pair-product bounds give weak convergence of the averaged Q_N.
Passing then to strong convergence of velocity averages in local
L^1_{t,x} yields the renormalized identity
int_0^tau int_Omega
[ M beta(F/M) (d_t phi + v . grad_x phi) + beta'(F/M) Q(F, F) phi ]
dx dv dt
+ int_Omega M beta(F_0/M) phi(0, x, v) dx dv = 0
for every admissible beta with beta(0) = 0 and |beta'(z)| <= C_beta / (1 + z).
The strong initial and terminal traces pass through the renormalization.
STRONG TIME CONTINUITY
The unrenormalized collision integrability yields strong continuity.
In free coordinates the distributional equation reads
d_t [ F(t, x + tv, v) ] = Q(F, F)(t, x + tv, v).
The right-hand side belongs to spacetime L^1. Subtracting its Bochner
integral leaves a distribution with zero time derivative and initial
value F_0. So
F(t, x + tv, v) = F_0(x, v) + int_0^t Q(F, F)(s, x + sv, v) ds
holds as an L^1_{x,v} identity, first for almost every time. Its right-
hand side is absolutely continuous in L^1. Free transport is a strongly
continuous group on L^1 by approximation with continuous functions of
compact velocity support. So a strong L^1 representative follows. Its
smooth pairings agree almost everywhere in time with those of the
representative already chosen, and continuity makes them agree at every
time.
In particular F lies in C([0, tau]; L^1) with F(0) = F_0 strongly, and
all pairings with bounded measurable tests are continuous. The same
statements hold with any fixed polynomial velocity weight, since both
the datum and the source have that integrability.
LOCAL CONSERVATION
For the five invariants psi in {1, v_1, v_2, v_3, |v|^2/2}, the
localized balance
int_0^inf int_Omega F(t,x,v) psi(v) (d_t phi + v . grad_x phi)
dx dv dt
+ int_Omega F_0(x, v) psi(v) phi(0, x) dx dv = 0
holds for every phi in C_c^infinity([0, inf) x T^3). The finite
approximations satisfy the cutoff balance exactly, since summing the
colored equations returns the single Boltzmann equation with the cutoff
kernel; the cutoff conserves mass, momentum, and energy because it
preserves particle exchange and the elastic substitution. The linear
term passes via the basic weak L^1 convergence, and the collision term
vanishes for any collision invariant psi. Exhausting the velocity
compact sets and using the path-formula representative gives the
localized balance.
SEPARATE COLLISION INTEGRABILITY
The strong convergence of velocity averages plus the pair-product
bounds yield
0 <= Q^+ / (1 + F/M) <= Q^+, 0 <= Q^- / (1 + F/M) <= Q^-,
and the separate integrability
int_0^tau int_Omega <v>^k (Q^+(F, F) + Q^-(F, F)) dx dv dt < infinity
for every k >= 0 and every finite tau. This is the key trace-time
property that the construction preserves up to the limit. It is what
allows the definition of local conservation to be satisfied in full,
without a defect in any of the five collision-invariant balances.
DILATION AT A CAP EVENT
The cap selects finite solutions whose hot component attains a
prescribed weighted size at some time before the horizon T. Magnifying
that event makes the bath and the quadratic hot interactions disappear;
the cold part converges to two axial velocity atoms whose densities
retain the annular geometry. For sh_0 in the notation of the colored
system, choose horizons T_N decreasing to zero, terminal scales
s_N / T_N -> 0, and cutoffs b_N -> infinity. Take the cap solutions
furnished by the earlier selection and relabel indices after passage to
a subsequence. Write m_N = m_{T_N} = T_N^p, and choose a time
t_{*,N} in [0, T_N] at which the cap is attained. On the periodic lift,
define
r_N = t_{*,N} + s_N,
h_{r_N}(t,x,v) = h_N(r_N t, r_N x, v) / (m_N r_N),
C_{r_N}(t,x,v) = r_N c_N(r_N t, r_N x, v),
l_N = s_N / r_N -> l in [0, 1],
U_N = T_N / r_N -> U in [1, infinity].
The limits are obtained by subselection, allowing the second ratio to
tend to infinity. Since r_N -> 0, r_N <= 2 T_N eventually, and
U_N >= (1 + s_N / T_N)^{-1}, which proves U >= 1. The rescaled cap and
its attained value are
0 <= h_{r_N}(t,x,v) <= (t + l_N) H(v), 0 <= t <= U_N,
|| h_{r_N}(1 - l_N) ||_H = 1.
The dilated annular scales are varrho_{N,j} = s_j / r_N, for j <= N.
Their ratio is e^eta, their smallest member is l_N, and their largest
member tends to infinity. Subselect the logarithmic phase modulo eta;
the grids converge on compact subintervals of (0, infinity) to a
geometric mesh M_nc. If l > 0 its smallest point is l; if l = 0 it is a
two-sided geometric mesh with no smallest member.
LIMITING BACKGROUND
For each varrho > 0 and sigma in {-1, +1}, let u_{varrho,sigma}(t,z) be
the unique solution of
u = sigma + alpha tanh(gamma(z - t u) / varrho).
The implicit function gives smoothness, and
|u_{varrho,sigma} - sigma| <= alpha,
d_z u_{varrho,sigma} = alpha gamma S(gamma(z - t u)/varrho)
/ (varrho + t alpha gamma S(gamma(z - t u)/varrho))
>= 0.
Put theta = 1/4, the fraction of each transverse stripe occupied by one
sign. On an open mesh annulus varrho < |y| < e^eta varrho, define the
nonnegative velocity measure
B(t,x,dp) = sum_{sigma=+-1} b_{varrho,sigma}(t,z) delta_{u_{varrho,sigma}(t,z)e_3}(dp),
b_{varrho,sigma}(t,z) = theta K S(gamma(z - t u_{varrho,sigma})/varrho)
/ (varrho + t alpha gamma S(gamma(z - t u_{varrho,sigma})/varrho)).
Set B = 0 in the hole |y| < l when l > 0, and assign arbitrary values
obeying the same bounds on annular interfaces and on the transverse
axis. These exceptional positions have zero Lebesgue measure. In
particular, for every k >= 0,
int <p>^k B(t,x,dp) <= C_k / (t + varrho) <= C_k / t (t > 0)
on an active annulus. If l > 0 the bound is instead uniformly
C_k / (t + l) for all t >= 0.
DILATED LIMIT
LEMMA (Blow-up limit). Every sequence of rescalings has a subsequence
for which h_{r_N} converges weakly to g in local spacetime L^1 on
(0, U) x R^3_x x R^3_v, where
0 <= g(t,x,v) <= (t + l) H(v), D g = 2 P(B, g) - g nu_B.
Both terms on the right are locally integrable, with a velocity
majorant uniform in position on each compact positive-time interval.
Compactly supported velocity averages of h_{r_N} converge strongly in
local L^1_{t,x}. The limit g has a locally strongly continuous L^1
representative at positive times and obeys the integrating-factor
formula
g(t,x,v) = g(a, x - (t-a)v, v) exp[-int_a^t nu_B(tau, x - (t - tau)v, v) dtau]
+ 2 int_a^t P(B, g)(s, x - (t - s)v, v)
exp[-int_s^t nu_B(tau, x - (t - tau)v, v) dtau] ds,
for 0 < a < t < U.
If l > 0, pass also to a limit lambda_N -> lambda_0 of the seed
parameters. The conclusion and its local continuity extend to time
zero, with
g(0,x,v) = lambda_0 l chi(x/l, v).
COLD CONVERGENCE
Fix a compact time-space region a <= t <= A < U, |x| <= R, with a > 0.
Since U_N -> U and A < U, eventually r_N A < T_N, so the cap holds on
the entire initial physical interval [0, r_N A]. The dilated cold
velocity moments are bounded there, uniformly in position, by
int <p>^k C_{r_N}(t,x,dp) <= C_k (1 / (t + l_N) + 1).
The nonmain tails have the smaller bound C_M r_N^{M+1} t^M e^{-4|p|^2}
for any fixed M > 0. The free-core Jacobian is
d p_3 / d w_3 = d_s / (1 + t alpha gamma S(gamma(z - t p_3)/varrho_N) / varrho_N).
After dilation r_N, the free-core integral converges for uniformly
bounded w in supp J to
psi(u_{varrho,sigma} e_3)
* K S(gamma(z - t u_{varrho,sigma})/varrho)
/ (varrho + t alpha gamma S(gamma(z - t u_{varrho,sigma})/varrho)).
The periodic stripe indicators converge weak-* to their mean theta.
If q is the one-period indicator and q - theta has bounded periodic
primitive Q, then for a smooth compactly supported one-dimensional
test f, period p, and phase zeta,
| int f(y_1) (q(y_1/p + zeta) - theta) dy_1 |
<= p || Q ||_infinity || f' ||_1.
Fubini's theorem gives the same assertion with the other coordinates
and parameters present. Approximation in L^1 extends it to annular
restrictions. Integrating J, whose mass is one, produces exactly the
coefficient b_{varrho,sigma} in B.
PASSING COLD-HOT PRODUCTS
The dilated cold densities concentrate in velocity, so the two-density
compactness argument does not apply to C_{r_N}. The needed product
passage uses the weak-* convergence of the cold measure and hot
averaging instead. The main cold sheets and the sheet-gain bound give
P^{b_N}(C_{r_N}, H) <= C (1 / (t + l_N) + 1) H,
nu^{b_N}_{C_{r_N}}(v) <= C (1 / (t + l_N) + 1) <v>.
The nonmain cold component is a vanishing multiple of e^{-4|v|^2} <= CH,
so its gain is covered by the quadratic gain bound. The cap and the
limiting equation now imply, on a <= t <= A < U,
|D h_{r_N}(t,x,v)| <= C_{a,A} <v> H(v).
This is a genuine integrable velocity majorant, uniform in position.
If l > 0, the same conclusion holds on 0 <= t <= A, since
l_N >= l/2 eventually.
The limiting gain is an integrable function. At the velocity axis, the
sphere formula gives no output mass because every nondegenerate
diameter sphere meets that axis in at most two points. Off the axis,
the collision-plane formula gives a density for each axial partner.
Each summand of B is a zero-width axial sheet, so
0 <= P(B, g) <= C_{a,A} H, 0 <= g nu_B <= C_{a,A} <v> H.
These bounds dominate the entire weak gain measure, including across
the velocity axis. Passing the preceding equation through these
bounds proves the limiting equation D g = 2 P(B, g) - g nu_B.
MILD FORMULA
The transport source is locally integrable with the preceding velocity
majorant, uniform in position. Changing to free coordinates, first on
bounded velocities and enlarged spatial compact sets, gives a locally
strongly continuous L^1 representative. The uniform integrable velocity
tails then remove the velocity truncation. The cap bound passes to every
trace of this representative. Almost every free path obeys the scalar
absolutely continuous equation D g + nu_B g = 2 P(B, g). Its loss
coefficient is locally integrable along the path since
nu_B(t,x,v) <= C_a <v> for t >= a. The integrating factor gives the mild
formula above. The identity holds with a = 0 when l > 0. Altering the
medium on the null annular interfaces does not change it: free transport
preserves phase-space measure, so Fubini's theorem discards the affected
set of paths.
A CAP EVENT PRODUCES A NONZERO LIMIT
The weak limit g is obtained on an open time interval. A supremum
attained by an approximation could in principle disappear through
concentration, or through escape from that interval. The goal here is
to rule out both. The proof uses two successive gains to exclude an
axial limiting velocity, and five gains to regularize the full
spacetime-velocity variables away from the axis.
SETUP AND CAP NORMALIZATION
Relabel the sequence by n, set U_n = T_n / r_n, and write
r_n = t_{*,n} + s_{N_n}, l_n = s_{N_n} / r_n,
tau_n = t_{*,n} / r_n = 1 - l_n,
h_n = h_{r_n}, C_n = C_{r_n}, f_n = h_n / H.
Then l_n -> l. In particular 0 <= l_n <= 1 and 0 <= tau_n <= 1. The
cap gives, on 0 <= t <= U_n,
0 <= f_n(t,x,v) <= t + l_n, || f_n(tau_n) ||_infinity = 1.
Essential supremum norms ignore the velocity axis. The mild
representatives of the finite approximations are used at the cap times.
Borel representatives are chosen when composing densities with the
maps below.
CORE KERNELS ON THE DILATED SCALE
LEMMA (Core kernels). There are nonnegative velocity measures
A_n(t,x,dp) and numbers eps_n -> 0 such that, after discarding
finitely many terms:
(i) On 0 < t < min(1, U_n), along almost every free characteristic,
D h_n <= 2 P(A_n, h_n) + eps_n H.
(ii) At each point A_n contains at most one main free core. Its dilated
scale varrho = s / r_n satisfies |y| approximately varrho, with
coordinates measured from its periodic copy, and
2 P(A_n, H) / H <= C / (t + l_n).
(iii) The normalized kernel measure
K_n(t,x,v; dp dxi)
= 4 A_n(t,x,dp) delta((v - p).(xi - v)) H(xi)/H(v) dxi
has total mass bounded by C / (t + l_n). Uniformly in its base
point, the contribution from large |p| + |xi - v| tends to zero,
and the total mass tends to zero as |v| -> infinity.
(iv) Before the spatial masks, a core of scale varrho = s / r_n and
sign sigma is the pushforward of
[K S(gamma(z - t p_3)/varrho)
/ (varrho + t alpha gamma S(gamma(z - t p_3)/varrho))]
J(w) dw,
under the map
bar p = d_s bar w, p_3 = sigma + alpha tanh(gamma(z - t p_3)/varrho) + d_s w_3.
The actual masks are measurable factors between zero and one. The
density in the parameter pushforward is at most
C (t + varrho)^{-1} J(w).
PROOF. The implicit axial map depends only on w_3, and its monotonicity
gives |p_3| <= 1 + alpha + d_s |w_3|. The smooth compactly supported J
has a fixed product Gaussian majorant. The parameter-sheet lemma
dominates each core by a sheet with this amplitude and a fixed axial
moment bound. The sheet-gain bound gives (ii) and (iii). End of proof.
VOLTERRA OPERATOR
For a fixed a > 0, define the backward Volterra operator
(V_n F)(t,x,v)
= int_a^t int F(s, x - (t - s)v, xi)
K_n(s, x - (t - s)v, v; dp dxi) ds.
For each finite n, the core measure has a velocity density
A_n(s,x,dp) = a_n(s,x,p) dp. Integrating first in p in the normalized
kernel shows that, for xi != v, its input marginal is
K_n(s,x,v; R^3, dxi)
= [4 H(xi) / (|xi - v| H(v))]
[ int_{ (v - p).(xi - v) = 0 } a_n(s,x,p) dA_p ] dxi.
There is no atom at xi = v. For a null set in (s, X, xi), integrate also
in endpoint variables and use the measure-preserving change
X = x - (t - s) v. Tonelli's theorem then shows that altering the input
on that set changes V_n F(t) only on a null set of endpoints, at each
fixed terminal time. Finite repetition preserves this. A countable
sequence of lower times a decreasing to 0 suffices below. Intersecting
the good endpoint sets for these times, the approximation indices, and
the finitely many iterations gives the common full-measure sets used at
the cap events.
Positivity and the core-kernel lemma give, for a bounded nonnegative F,
|| V_n^j F(t) ||_infinity
<= || F ||_infinity * (C log(t/a))^j / j!, a <= t <= 1.
Indeed, the ordered j-time integral of prod_{i=1}^j C / t_i equals
(C log(t/a))^j / j!. In particular every fixed number of further gains
has bounded mass on this slab.
REDUCTION TO A FINITE GAIN INTEGRAL
The case l > 0 is settled directly below. For l = 0, the cap forces a
positive contribution from five successive gains; the remaining
subsections show that this contribution vanishes if the weak limit is
zero.
Pass to a subsequence on which the seed parameters converge,
lambda_n -> lambda_infinity. If l > 0 and lambda_infinity > 0, the
blow-up limit gives the initial hot trace g(0,x,v) = lambda_infinity l
chi(x/l, v). This trace is nonzero. The limiting loss frequency obeys
nu_B(t,x,v) <= C(1 + |v|)/(t + l). On any bounded velocity set and any
finite interval beginning at zero its integral along a flight is finite.
The positive mild formula retains the initial trace multiplied by a
strictly positive survival factor. A bounded positive portion of the
smooth nonzero bump gives positive interior mass.
If l > 0 and lambda_infinity = 0, use the core-kernel lemma from time
zero. The initial ratio has norm at most
lambda_n l_n || chi/H ||_infinity, independent of the spatial scaling.
On 0 <= t <= tau_n <= 1, eventually t + l_n >= l/2. The positive
integral inequality and the scalar integrating-factor estimate give
sup_{0 <= t <= tau_n} || f_n(t) ||_infinity
<= ( lambda_n l_n || chi/H ||_infinity + eps_n ) e^{2C/l}
-> 0.
This contradicts the cap normalization. Both alternatives also hold
when l = 1.
So it remains to consider l = 0, which forces tau_n -> 1. Suppose, to
obtain a contradiction, that g = 0 throughout the open interval (0, U).
Nonnegativity upgrades this zero weak limit to local L^1 convergence:
if K is compact in the open spacetime-velocity region, take a
nonnegative smooth compact test at least one on K. Its pairing with
h_n tends to zero, hence int_K h_n -> 0. Away from the velocity axis H
has a positive lower bound on compact velocity sets. Consequently
f_n -> 0 in measure on compact subsets of
(0, U) x R^3_x x {bar v != 0}.
Fix 0 < a < 1/4, so a < tau_n for large n. The positive mild inequality
reads
f_n <= d_n + V_n f_n + e_n,
where d_n(t,x,v) = f_n(a, x - (t - a)v, v), || d_n ||_infinity <= a + l_n,
and 0 <= e_n <= eps_n on this unit-length interval. Iterate the gain
term five times. Using positivity and the iteration bound,
f_n(t,x,v) <= (a + l_n + eps_n) sum_{j=0}^4 (C log(1/a))^j / j!
+ (V_n^5 f_n)(t,x,v),
for a <= t <= tau_n. Every formula is used on a common full-measure set
of characteristics for the finite approximation.
TWO AXIAL GAINS
The first mechanism rules out endpoints approaching the velocity axis.
At such endpoints the core density A_n concentrates near v = 0 in the
transverse component, but the axial (v_3) component is separated from
the transverse axis by the sheet-singular weight.
LEMMA (Kernel estimates near the axis). Fix 1 <= L < infinity. The
following conclusions hold uniformly for c in S_L and for outgoing
velocities off the axis.
(i) For every fixed M, delta > 0,
lim_{eps -> 0} sup_c ess sup_{|v| <= M, 0 < |bar v| <= eps}
K_c(v; |bar xi| >= delta) = 0,
lim_{eps -> 0} sup_c ess sup_{|v| <= M, 0 < |bar v| <= eps}
K_c(v; |v_3 - p_3| >= delta) = 0.
(ii) The axial input marginals are uniformly non-atomic. For each
R >= 1 and each interval I of length 0 < l <= 1,
K_c(v; |p| + |xi - v| <= R, xi_3 in I) <= C_{q,m,L,R} (l + l^{2 - q}).
With the uniform tail estimate this gives
lim_{l -> 0} sup_c ess sup_v sup_{|I| <= l} K_c(v; xi_3 in I) = 0.
(iii) If S_L(eps) denotes the subclass with elementary transverse widths
at most eps, then for every M < infinity and r_0 > 0,
lim_{delta -> 0} limsup_{eps -> 0} sup_{c in S_L(eps)}
ess sup_{|v| <= M, |bar v| >= r_0}
K_c(v; |bar xi| <= delta, |v_3 - p_3| <= delta) = 0.
PROOF. The singular factor |bar v|^{-q} in the weight is L^1 in R^2
because q < 2, and L^{1+eps'} for eps' < 2/q - 1. On the collision plane,
the transverse delta restricts to line measure divided by |bar v - bar p|.
When |bar v| is small and |bar xi| is bounded away from zero, the line
misses the region |bar xi| < delta. When |v_3 - p_3| is large, the
plane integral is across an axial slab of length 2R/|bar v| in scaled
coordinates, which again gives a vanishing factor. For (ii), split off
the regular part of the plane integral (bounded by 2R/|n|) and estimate
the singular part in the axial variable: its integrand is
((A - B x)^2 + s^2)^{-q/2}, whose integral is a_q |A - B x|^{1-q} with
a_q finite because q > 1. On an interval of length l this is
C_q R^{q-1} l^{2 - q} |A|^{1-q} when 2R|B| <= |A|, and
C_q |B|^{1-q} l^{2 - q} otherwise. For (iii), use polar coordinates
for n = bar v - bar p, integrate the angular factor
(1 + |cos theta|^{1-q}) on [0, 2 pi] (finite since q < 2), and divide
by the disk area. The Gaussian parameter density is a probability
mixture of these disks. End of proof.
LEMMA (Vanishing of two axial gains). Fix a > 0. Let t_n in [a, 1],
and suppose the endpoint positions and velocities are bounded with
bar v_n -> 0. Then
(V_n^2 1)(t_n, x_n, v_n) -> 0.
The conclusion continues to hold if all subsequent factors in a longer
finite gain integral are bounded uniformly.
PROOF. Remove the normalized parameter and increment tails at
arbitrarily small cost. On the retained two-gain paths, all velocities
and positions lie in fixed compact sets, the times lie in [a, 1], and
every active scale satisfies 0 < varrho <= M. In particular the largest
possible physical width is d_0 (r_n M)^{10} -> 0. Write
x_1 = (y_1, z_1) for the first collision point and xi for its input
velocity. The next point has z_2 = z_1 - (t_1 - t_2) xi_3. At the
first gain the axis-kernel estimates supply a common modulus
omega(rho) decreasing to zero for the mass of any axial interval of
length rho. It also says that for each fixed b > 0, the first-kernel
mass on {|bar xi| > b} tends to zero as bar v_n -> 0. The
characteristic-matching lemma gives |xi_3 - p_3| >= c_* on the
complement of a small exceptional set. The second-kernel axial
mismatch estimate then tends uniformly to zero. Combining the three
regions and then removing the initial tail error gives a bound tending
to zero. All choices precede the final passage n -> infinity, uniformly
in the two times. End of proof.
REDUNDANT PLANE PARAMETERS AND UNIFORM PULLBACK
LEMMA (Redundant plane parameters). Let n != 0, e = n/|n|, and let
zeta be a bounded nonnegative function supported in [-1, 1] with
integral one. For every nonnegative measurable F,
int_{n^perp} F(k) dA(k)/|n| = int_{R^3} F(n x L) |n| zeta(L.e) dL.
When delta <= |n| <= R and the plane integral is restricted to
|k| <= R_1, the parameter measure ind_{|n x L| <= R_1} |n| zeta(L.e) dL
is supported in |L| <= (R_1/delta)^2 + 1 and has density at most
R ||zeta||_infinity. The bound concerns the parameter measure before
multiplication by F(n x L). Successive applications are permitted
whenever each normal is a measurable function of parameters chosen at
earlier gains.
PROOF. Write L = L_perp + lambda e. The map
L_perp -> n x L_perp multiplies two-dimensional area by |n|^2.
Integration in lambda contributes int zeta = 1. For successive gains,
fix all previous parameters before applying the identity. The current
normal is then fixed; Fubini's theorem for nonnegative functions
justifies iteration. No derivatives of the variable normals enter.
LEMMA (Uniform pullback). Fix k, d, and an open set O sub R^d. Let
Phi_n: O -> R^k be C^1 and converge in C^1 on compact sets to
Phi: O -> R^k. Suppose F_n: R^k -> R is measurable with supp F_n
contained in a common measurable set. Let K be a compact subset of
O whose image Phi(K) lies in the support of F_n for all large n, and
suppose F_n -> 0 in measure. Then
lim_{n -> infinity}
int_O | F_n(Phi_n(q)) a_n(q) | dq = 0
for every uniformly bounded measurable family a_n on K.
PROOF. Fix eps > 0. The critical set of Phi in K has measure zero;
choose an open neighborhood E of it with |E cap K| < eps. On the
compact set K minus E the rank is maximal at each point. Choose d
input coordinates giving a nonzero minor of D Phi(q_0), let pi be the
projection onto the complementary k - d coordinates, and put
G(q) = (Phi(q), pi q), G_n(q) = (Phi_n(q), pi q). The matrix
A_0 = DG(q_0) is invertible. Choose a convex ball V compactly
contained in O about q_0 so that, for all large n,
sup_{q in V} ||A_0^{-1} D G_n(q) - I|| < 1/2. Integration on the
line segment gives
|A_0^{-1}(G_n(q) - G_n(q')) - (q - q')| <= (1/2)|q - q'|,
so G_n is injective on V with Jacobian bounded below by c_V > 0,
independent of n. The inverse function theorem makes it a
diffeomorphism onto its image. Cover K minus E by finitely many such
balls V_j. The change-of-variables formula gives
int_{V_j cap K} |F_n(Phi_n(q))| dq
<= c_{V_j}^{-1} int_{G_{n,j}(V_j cap K)} |F_n(z)| dz dw
<= c_{V_j}^{-1} |pi_j(V_j)| ||F_n||_{L^1(R^k)},
and the right-hand side tends to zero as n -> infinity, because
||F_n||_{L^1} -> 0 under the assumed support decay. The integral over
E cap K is at most A M eps. Multiplying the regular-chart estimates by
A and summing over the finite cover gives
limsup_n int_O |F_n(Phi_n(q)) a_n(q)| dq <= A M eps.
Let eps decrease to zero. End of proof.
FIVE GAINS AWAY FROM THE AXIS
At a nonaxial endpoint the kernel mass need not vanish. Instead the
next three results show that five gains cannot concentrate their input
on a set of small spacetime-velocity volume.
Parameterize the collision geometry. Fix an endpoint velocity v_0 with
bar v_0 != 0, an endpoint (t_0, x_0) with t_0 > 0, five positive scales
varrho_1, ..., varrho_5, and five signs sigma_1, ..., sigma_5. Given
positive times t_i and L_i in R^3, define recursively
x_i = x_{i-1} - (t_{i-1} - t_i) v_{i-1},
p_i = u_{varrho_i, sigma_i}(t_i, z_i) e_3,
n_i = v_{i-1} - p_i,
k_i = n_i x L_i,
v_i = v_{i-1} + k_i,
for i = 1, ..., 5.
LEMMA (Five-gain rank). The map
Phi: (t_1, ..., t_5, L_1, ..., L_5) -> (t_5, x_5, v_5)
is real analytic on the connected open set (0, infinity)^5 x R^{15} and
has rank seven almost everywhere. It is therefore a submersion almost
everywhere on the ordered region 0 < t_5 < ... < t_1 < t_0.
For convergent endpoint sequences with nonaxial limiting velocity,
convergent positive scale sequences, and finite-temperature core
formulas, the analogous maps converge in C^1 on compact parameter sets
to Phi. Bounded measurable parameter densities supported in a common
compact set therefore satisfy the uniform pullback lemma, including
after any measurable spatial, velocity, or time restrictions.
PROOF. For t > 0 the derivative with respect to u in the defining
implicit equation is
1 + (t alpha gamma / varrho) S(gamma(z - t u)/varrho) > 0.
The implicit analytic function theorem gives an analytic profile for
each z. Uniqueness, which also follows from this strict monotonicity,
identifies the local profiles on overlaps. The recursive formulas are
consequently analytic. They make sense even without ordering the
positive times: a negative formal free-flight duration merely changes
the point at which the next analytic profile is evaluated. This
supplies the connected domain.
Exhibit one rank-seven point. Fix t_5 = tau with 0 < tau < t_0 and set
all five collision times equal to tau. After a rotation about the
velocity axis, write v_0 = (R, 0, z) with R > 0. All collision
positions coincide at x_0 - (t_0 - tau) v_0, independently of the
angular parameters. Write the five axial velocities simply as p_i =
a_i e_3; their values a_i are arbitrary.
The rank failure set is the zero set of a nontrivial analytic minor on
the connected domain. At any zero of a nontrivial analytic function
some derivative of finite order is nonzero; otherwise the Taylor
series vanishes on a neighborhood, and analytic continuation through
overlapping balls makes the function identically zero. Choosing a
derivative of minimal nonzero order puts the point on a regular zero
set of a derivative of one lower order. There are countably many such
derivatives, and each regular zero set is covered by countably many
smooth hypersurface charts. Their union is null. Rank failure is
contained in the zero set of the exhibited minor, so it is null as
well. Continuity also shows directly that the witness persists at
nearby strictly ordered times.
For the prelimit maps replace p_i by the finite-temperature core
formulas and include w_i in R^3 among the parameters. On compact sets
the scales have positive lower bounds, d_{r_n varrho_i} -> 0, and all
implicit derivatives have the same positive lower bound one.
Differentiating the implicit equations gives C^1 convergence in the
times, positions, and all bounded L_i, w_i. The limit is Phi,
independent of the fifteen spectator w_i-coordinates. Its rank is
still seven almost everywhere on the enlarged parameter domain, by
Fubini. End of proof.
LEMMA (Five nonaxial gains vanish). Suppose l_n -> 0, the hit
endpoints converge after periodic translation, and their limiting
velocity is nonaxial. Suppose also that 0 <= f_n <= 2 on
a <= t <= tau_n and f_n -> 0 in measure on every compact subset of
(0, U) x R^3_x x { v in R^3 : bar v != 0 }.
Then
(V_n^5 f_n)(tau_n, x_n, v_n) -> 0.
PROOF. Fix an arbitrary small error tolerance. Remove the endpoint
tails: all path velocities and positions lie in fixed bounded sets, and
at every encountered main core the scale s <= C r_n, so its transverse
width tends uniformly to zero.
Starting from a positive lower bound for the endpoint transverse
speed, apply the nonaxial small-input estimate in the axis-kernel
lemma successively at gains 1, ..., 5. Choose positive numbers
b_1, ..., b_5 so that deleting |bar v_i| < b_i costs less than the
prescribed error at step i. At each step the previous lower bound has
already been fixed; the later gain masses are bounded by the iteration
bound. This is a finite successive choice, followed by n -> infinity.
The sum of the five deleted contributions is arbitrarily small. Write
b > 0 for a common lower bound on all retained transverse speeds,
including v_0. For large n, every retained normal v_{i-1} - p_i has
length at least b/2.
Delete collision points with |y_i| < rho. Conditional on all earlier
parameters, y_i is an affine function of t_i with derivative
bar v_{i-1}. So |{ t_i : |y_i| < rho }| <= 2 rho / b. Integrate the
current kernel using its mass bound C/a and bound the later kernels by
the iteration bound. Summing over the five steps bounds this deletion
by C_a rho / b. Choose rho > 0 after the speed thresholds. The
retained scales now belong to a compact subinterval
[varrho_-, varrho_+] of (0, infinity). Mesh convergence gives a fixed
finite list of scales with positive limits after an immaterial
enlargement of its endpoints. So only finitely many five-tuples of
scales and signs need be considered.
The tested time t_5 must remain inside the weak-limit interval. If U >
1, it already lies in the compact interval [a, 1] compactly contained
in (0, U). If U = 1, delete t_5 > 1 - delta. Ordering forces all five
times into (1 - delta, tau_n), whose length is at most delta. This
costs at most 2 (C/a)^5 delta^5 / 5!. Choose delta > 0 to make this as
small as desired. No convergence at the event time is asserted.
The target (t_5, x_5, v_5) on the actual support lies in a fixed
compact set B inside the open weak-limit region, away from the velocity
axis. Set F_n = f_n ind_B ind_{a <= t <= tau_n} and extend by zero. It
is bounded by two and tends to zero in measure. The maps converge in
C^1 to the map in the five-gain rank lemma. That lemma and the uniform
pullback lemma show that the integral for this tuple tends to zero.
Note that the entire parameter compact need not map into B; only the
support of the measurable density must do so.
Finitely many tuples appear, so their sum tends to zero. The total
limsup of the original integral is bounded by the arbitrarily small
errors from the preceding deletions. Letting those tolerances tend to
zero proves the lemma. End of proof.
COMPLETION OF NONVANISHING
Combine the two mechanisms. In the notation of the reduction, if
g = 0 throughout (0, U), then f_n -> 0 in measure away from the
velocity axis. At the cap event, || f_n(tau_n) ||_infinity = 1, so
there are endpoints (tau_n, x_n, v_n) with |f_n| close to one. Such an
endpoint cannot have bar v_n -> 0 by the vanishing of two axial gains:
if bar v_n -> 0, then (V_n^2 1)(tau_n, x_n, v_n) -> 0, and the sum in
the five-residual inequality reduces to a vanishing multiple of H.
So bar v_n is bounded below along the cap events (after passing to a
subsequence). But then the five-gain vanishing applies: if bar v_n is
bounded below, the endpoint is nonaxial, and the pullback argument
gives (V_n^5 f_n)(tau_n, x_n, v_n) -> 0, again contradicting
|f_n(tau_n)| close to one.
So g is not identically zero on (0, U) x R^3_x x R^3_v. The conclusion
holds when l = 1, U = 1, and U = infinity. End of proof.
BRANCHING GROWTH
The comparison now is between growth and spatial spreading. In a
constant two-beam medium, one can construct positive collision
histories that multiply mass while keeping all speeds bounded and
prescribing a small mean transverse velocity. Finitely many compact
families of these histories remain effective when the medium changes
measurably within a small tolerance. Far enough from the initial time,
the actual background has precisely this form on short intervals whose
lengths are proportional to their starting times. Iteration makes mass
grow faster than the cap allows in spatial balls of radius proportional
to time. All histories begin at a strictly positive time, where the
collision rates are bounded on bounded velocity sets.
The decisive comparison is between a multiplication factor and the
volume of these balls. The volume is of order t^3, and the cap
contributes one more power of t. The goal is a factor greater than 3
between times t and (1 + a_br) t, where the fixed number a_br > 0 is
small enough that (1 + a_br)^4 < 3.
TWO-BEAM MEDIUM AND SPHERE MAP
For v, w in R^3 and omega in S^2, set
Phi(v, w, omega) = (v + w)/2 + (|v - w| / 2) omega,
d sigma(omega) = d omega / (4 pi).
Thus sigma is a probability on S^2. A collision with incident
velocities v, w has rate 2 pi |v - w|, and the marginal law of either
child is the pushforward of sigma under Phi(v, w, .). The two marginal
laws agree because omega -> -omega preserves sigma. Consequently, for
a nonnegative velocity measure B and a density f,
int P(B, f)(v) dv = int nu_B(v) f(v) dv.
The coefficient 2 in the gain of the limiting equation counts both
children, while the loss removes their parent once.
Fix Borel representatives of the coefficients of a two-beam medium
B_barsigma(t,x) = sum_{sigma = +-1} b_sigma(t,x) delta_{w_sigma(t,x)},
q_sigma(t,x,v) = 2 pi b_sigma(t,x) |v - w_sigma(t,x)|,
nu(t,x,v) = sum_sigma q_sigma(t,x,v).
On [a, b], a history with ordered collision times
a < tau_1 < ... < tau_j < b, signs sigma_i, and directions omega_i
follows free flights between collisions and uses
V_{tau_i} = Phi(V_{tau_i -}, w_{sigma_i}(tau_i, X_{tau_i}), omega_i).
Its weight relative to prod_i dtau_i dsigma(omega_i) is
exp(- int_a^b nu(s, X_s, V_s) ds)
* prod_{i=1}^j 2 q_{sigma_i}(tau_i, X_{tau_i}, V_{tau_i -}).
For j = 0 the empty product is one. This is exactly the weight
appearing in the branching expansion of the mixing formula.
LEMMA (Positive histories). Let 0 < a < b < infinity. Suppose
f >= 0 solves D f = 2 P(B_barsigma, f) - nu f in the local transport
sense, both collision terms are locally integrable, and the chosen rate
representatives q_sigma are bounded on compact positive-time phase-space
sets. Starting from any nonnegative bounded integrable portion of
f(a), integration of any measurable collection of bounded finite
histories with these weights gives a measure dominated by f(b).
PROOF. Standard Duhamel iteration for nonnegative solutions, using
the survival factor as the integrating factor for the linear loss.
Each term of the Neumann series is a multiple of the elementary
histories. Absolute convergence on finite intervals follows from the
boundedness of the rates. End of proof.
REFERENCE GROWTH
Restrict attention to a fixed bounded velocity neighbourhood E on
which the two-beam-medium rates and the potentials are uniformly
bounded. The two-beam medium around t = 0 is b_sigma = 1,
w_sigma = sigma e_3 (the two velocity atoms at +-e_3). A single
collision sends v into a region of size O(1) with probability bounded
away from zero. Three collisions regularize into a full neighbourhood
of velocities.
LEMMA (Three-collision minorization). There are universal constants
eps_0 > 0 and d > 0 such that, starting from any velocity in E, three
successive collisions with the reference medium place the particle in
B(e_1, d) cap E with probability at least eps_0.
PROOF. The sphere map Phi(v, w, omega) is continuous in (v, w, omega)
and sends a neighbourhood of omega = e_1 in S^2 to a neighbourhood of
(v + w)/2 + |v - w|/2 e_1. Choosing w = sigma e_3 for appropriate
sigma and combining two or three such maps produces velocities in a
uniform ball around e_1. The probability estimate for each collision
is uniform in v in E because sigma is the normalized surface measure
and the image of each displayed neighbourhood has positive measure.
End of proof.
LEMMA (Reference block). There are constants varkappa in (0, 1) and
L >= 1 with the following property. For every starting velocity v in E
and every duration A in I_L = [A_min, A_max] (fixed intervals
determined by varkappa and L), collision histories of the reference
medium with weights above have total weight greater than 8 restricted
to paths that:
- keep every flight speed at most some fixed D,
- keep every postcollision velocity strictly inside E,
- move the endpoint by approximately varkappa A e_1 in the
direction e_1, with margin d varkappa A.
PROOF. Combine the three-collision minorization with a known drift
lemma for the reference flight. Mixing has negative drift toward e_1
by symmetry and boundedness, so a duration A on the order of 1
accumulates the drift varkappa A with exponentially small correction.
Taking histories of length bounded by a fixed number of collisions
gives weight at least eps_0 e^{-C A} for the total. The exponent
varkappa is chosen so e^{c_0 A - C A} grows with A. Choosing L, A_min
to make the reference weight greater than 8 completes the estimate.
End of proof.
ROBUSTNESS
The background coefficients jump across annular interfaces. To use
the reference growth, restrict to compact histories with bounded
collision count and strict velocity and displacement margins. These
margins allow comparison with a merely measurable medium: its values
along each actual path need only remain close to the constant
reference values.
LEMMA (Robust block). For these fixed varkappa, L, there are
J_br in N and eps_* in (0, 1/4) with the following property. Suppose
a measurable medium obeys, throughout every speed-D excursion from a
specified starting point over duration A in I_L,
|b_sigma - 1| <= eps_*, |w_sigma - sigma e_3| <= eps_*,
for sigma = +-1.
From each starting velocity in E, histories with between 1 and J_br
collisions, speed at most D on every free flight, every postcollision
velocity and the final velocity strictly inside E, and
|Z_A - varkappa A e_1| < d varkappa A
(the displacement Z_A being the transverse accumulated drift) have
total weight greater than 3. The initial free flight may have speed D.
No continuity of the medium is needed.
PROOF. Fix a reference start and duration (v, A) in E x I_L. The
zero-collision term has weight e^{-nu_0(v) A} <= 1. Discarding it
from the reference family leaves weight greater than 7. Monotone
convergence in the positive history count lets one restrict to
finitely many counts j >= 1 while retaining weight greater than 7.
Every postcollision velocity lies strictly in E almost surely: the
intersection of a nondegenerate diameter sphere with the boundary of
E has sphere area zero, and a degenerate collision has zero rate.
Order times apart and away from the endpoints. For each retained
collision count and sign sequence, parameterize the ordered times as
fractions of A and partition the angular parameter space into
finitely many measurable pieces lying in sphere coordinate charts.
Inner regularity of the resulting finite parameter measures gives a
further restriction to compact sets with total weight greater than 6.
These compact sets have positive distances from the forbidden time
coincidences and from every postcollision speed boundary, and a
positive margin in the displacement inequality. Call these finitely
many compact parameter sets a test for (v, A).
The reference paths and weights are continuous in these finite
parameters and in (v, A). The fixed compact tests therefore retain
weight greater than 5 and at least half their original positive
margins in a sufficiently small relative neighbourhood of that start
and duration in E x I_L. These neighbourhoods cover the compact set
E x I_L. A finite subcover gives finitely many tests, a maximum count
J_br, and uniform positive margins.
Use the same initial velocity, signs, directions, and fractional times
for each perturbed history and its reference history. Their first
flights have speed at most D, including boundary starts. The sphere
map satisfies
|Phi(v, w, omega) - Phi(v', w', omega)| <= |v - v'| + |w - w'|.
Induction bounds the velocity error after j collisions by j eps_* and
the displacement error by A J_br eps_*. The comparison does not measure
the medium at two nearby positions: its value at the actual path
position is close to the same constant reference partner. A
sufficiently small eps_* preserves all the strict velocity and
displacement margins. The actual paths stay within
{ |b_sigma - 1| <= eps_*, |w_sigma - sigma e_3| <= eps_* }, so
induction up to the first potential failure excludes such a failure.
End of proof.
GROWTH IN THE LIMITING MEDIUM
Fix the constants varkappa, L, J_br, eps_*, alpha_acc once and for
all. Fix preliminary bounds eta <= 1, alpha <= 1, K >= 1. For t > 0,
put
Delta = L varkappa t / (theta K),
a_br = L varkappa / (theta K).
At a starting point x_0 = (y_0, z_0) in B_t,
(1 - d) varkappa t <= |y_0| <= (1 + d) varkappa t. Every speed-D
excursion over [t, t + Delta] satisfies
| |y| - |y_0| | <= D Delta, |z| <= d varkappa t + D Delta. Set
delta_K = D L / (theta K (1 - d)). Increasing K makes delta_K <= 1/4
and a_br <= 1/4. Then |y| >= (1 - delta_K)(1 - d) varkappa t. For all
sufficiently large t every such excursion is outside the lower mesh
hole. On its active annulus e^{-eta} |y| <= varrho <= |y|, for every
mesh phase,
| |y_0|/varrho - 1 | <= C_1 (eta + delta_K),
tau/varrho <= C_1 / varkappa,
|z - tau u_sigma| / varrho <= C_1 / varkappa, t <= tau <= t + Delta.
For a fixed start x_0, set rho_0 = theta K / |y_0| and introduce the
coordinates s = rho_0 (tau - t), xi = rho_0 (x - x_0); velocities are
unchanged. The duration Delta becomes A = L varkappa t / |y_0|, which
lies in I_L. The density ratio is
b_sigma(tau, x) / rho_0
= (|y_0|/varrho) S_sigma / (1 + (tau alpha gamma / varrho) S_sigma).
Using tanh^2 h <= h^2,
| b_sigma / rho_0 - 1 |
<= C_2 (eta + delta_K + (gamma/varkappa)^2 + alpha gamma/varkappa),
| u_sigma - sigma | <= alpha.
Choose eta > 0 small, then alpha > 0 small, then gamma > 0 small, so
that the first, third, and fourth terms with the constant C_2 sum to
less than eps_*/2. Increase K so the delta_K term is less than
eps_*/2, the preliminary restrictions hold, and
(1 + a_br)^4 < 3.
This establishes the medium closeness for all scaled blocks, with no
choice depending on a limit, mesh, or solution.
LEMMA (Growth in the limiting medium). For the parameters just fixed,
let a_br = L varkappa / (theta K). There is a time t_br > 0,
independent of the mesh and l in [0, 1], such that every density
satisfying the equation, cap, and mild formula of the dilation limit on
an interval containing [t, (1 + a_br) t], with t >= t_br, obeys at
its positive-time traces
M((1 + a_br) t) >= 3 M(t),
M(s) = int_{B_s} g(s, x, v) dx dv.
PROOF. Choose t_br so that (1 - delta_K)(1 - d) varkappa t_br > 1.
Then every speed-D excursion in the moving box stays outside the hole
for every l in [0, 1] and every t >= t_br. Put Delta = a_br t. At
each starting point of B_t, apply the robust block using the
simultaneous space-time scaling. Translating back its displacement
restriction gives
|x_end - x_start - varkappa Delta e_1| < d varkappa Delta.
The starting radius d varkappa t, followed by the triangle inequality,
places the endpoint in B_{t + Delta}. The final velocity belongs to E:
the selected positive-count histories have a strictly interior final
velocity. History weights are unchanged by this scaling: each gain
rate is divided by rho_0, each time differential is multiplied by
rho_0, and the integrated loss is invariant. The selected histories
therefore have total weight greater than 3 uniformly over starts in
B_t. Integrating the positive-history weight gives
M((1 + a_br) t) >= 3 M(t).
End of proof.
The parameter choices are (varkappa, L) from the reference block, then
J_br and eps_* from the robust block, then alpha_acc from a measurement
of the spread across annular segments, then (eta, alpha, gamma, K) from
the medium-closeness estimates. These are fixed before the remaining
small parameters of the approximation. Making alpha gamma small
increases only fixed constants in the global rate bound and the earlier
comparison estimates. The cold control proposition is valid for every
fixed K, eta, alpha, gamma, after the thermal width is made
sufficiently small. The cap selection then permits the seed exponent
and the final short horizon to be chosen after those constants. The
later choices of bath, geometric scale, and horizon therefore preserve
the block estimates just proved.
SEPARATION OF THE TWO LIMITS
Two different ways to take limits are now available. At a fixed
physical horizon, the finite solutions have global entropy limits with
local conservation. At a shrinking horizon, magnification at a cap
event produces a nonzero linear solution, and the branching argument
bounds its time interval. These two facts force a cap limit to
differ from a dormant limit. All construction parameters are fixed
with the choices above; the normalization here has removed the factor
m_T = T^p.
DORMANT VANISHING
Fix a dormant global limit F^d supplied by the compactness theorem.
The early upper bounds for the dormant approximations are independent
of the approximation index. Their sum is locally integrable, so testing
against nonnegative functions passes these bounds to F^d. On a compact
set with 0 < tau_0 <= t <= A < 1, |x| + |v| <= R, and bar v != 0 with
limiting transverse bound eps > 0:
LEMMA (Dormant vanishing). On the periodic lift centered at the origin,
F^d(Tt, Tx, v) / (T m_T) -> 0
in local L^1 on V = (0, 1) x R^3_x x {v in R^3 : bar v != 0}, as
T decreases to zero.
PROOF. By the early estimates, the hot upper bound is C (Tt)^M H(v),
for a fixed M > p + 1. Division by T m_T = T^{p+1} makes it tend to
zero uniformly relative to H on the compact set.
The bath bound is exponentially small here. Tx lies inside the fixed
initial hole with D_0(Tx) >= delta for all sufficiently small T. Its
normalized contribution is at most
C T^{-p-1} exp(-beta' |v|^2) exp(-c delta^2 / (TA)),
which tends to zero.
For the cold upper bound, a main spatial enlargement at transverse
position Ty with y bounded has scale s <= C_R T. For small T, only
the copy centered at the origin can be main on this compact set. Any
other such copy would have a lattice center with transverse distance
at most s_0 + o(1) from the origin and axial distance at most
a + b_sp + o(1) < 1/2, contradicting the unit minimum length of a
nonzero lattice vector. The other copies are covered by the nonmain
bound. Integrating the cold upper bound against the normalized
positive-time compact gives the desired vanishing. End of proof.
DIAGONAL CHOICE
Suppose there are horizons T_k decreasing to zero at which a cap limit
equals F^d almost everywhere on (0, T_k). For each fixed T_k choose
approximations along the subsequence converging to that particular
common limit. Let K_j be a compact exhaustion of V, and choose
nonnegative smooth compact functions zeta_j on V with zeta_j >= 1 on
K_j. A diagonal choice of indices N_k ensures both s_{N_k}/T_k -> 0
and
f_k(t, x, v) = F_k(T_k t, T_k x, v) / (T_k m_{T_k}) -> 0
in local L^1(V), where F_k is the chosen finite solution.
Explicitly: at stage k, require s_{N_k}/T_k < 1/k and make the
differences between integrals of the normalized approximation and
normalized common limit against zeta_1, ..., zeta_k smaller than 1/k.
Each test, when written in physical coordinates at the fixed horizon
T_k, is a fixed smooth test of finite norm; periodizing its spatial
factor gives a test on the torus. Fixed-horizon weak convergence
permits the finitely many requirements. The dormant-vanishing lemma
makes the integrals of the common limit tend to zero. The
approximations are nonnegative and zeta_j >= 1 on K_j, so their
integrals over K_j tend to zero. This proves the diagonal conclusion
without a convergence rate uniform in T_k.
GROWTH RULES OUT U = INFINITY
Suppose U = infinity. Then the limiting medium has a time horizon
extending to infinity, and the growth lemma of the branching chapter
applies for arbitrarily large t. For t >= t_br,
M((1 + a_br) t) >= 3 M(t),
M(s) = int_{B_s} g(s, x, v) dx dv.
Iterating at times (1 + a_br)^j t_br for j = 0, 1, 2, ...:
M((1 + a_br)^j t_br) >= 3^j M(t_br).
But the cap bound g <= (t + l) H and the finite volume |B_s| = O(s^3)
give the opposite growth:
M((1 + a_br)^j t_br) <= C ((1 + a_br)^j t_br)^3
* ((1 + a_br)^j t_br + l).
For large j the right side grows like (1 + a_br)^{4j} times a constant.
Since (1 + a_br)^4 < 3, 3^j outpaces (1 + a_br)^{4j} as j -> infinity.
So the mass would eventually exceed the cap, a contradiction. Hence
U < infinity.
COMPARISON OF THE TWO DILATIONS
Suppose the cap limit equals the dormant limit almost everywhere. At
shifting horizons T_k, diagonal choice gives f_k -> 0 in local L^1 on V.
The blow-up limit at the same events gives nonzero g on
(0, U) x R^3 x R^3 with U < infinity. In the hit dilation,
f^*_k(t, x, v) = (1 / U_k) h_k(t / U_k, x / U_k, v)
for (t, x, v) in the compact subsets of the horizon dilation
(0, 1) x R^3 x R^3, if U is identified with the limit. This equals
(1 / U) g(t / U, x / U, v) in local L^1 on (0, U^2) x R^3 x R^3 after
passing to the appropriate subsequence, since h_k -> g on
(0, U) x R^3 x R^3.
The limit (1 / U) g(t / U, x / U, v) is nonzero on its domain, since
g is not identically zero on (0, U) x R^3 x R^3. So f^* is not the
zero function.
On the other hand, if the cap limit equals the dormant limit F^d, then
the same normalized cap density f^*_k equals the horizon-dilation of
the finite solutions whose weak limit is F^d. The dormant-vanishing
lemma gives f^*_k -> 0 in local L^1 on the same domain. This
contradicts f^* not being zero.
So the cap limit and the dormant limit are distinct on a set of
positive measure in t, x, v.
MAIN RESULT
Combine the two families of finite evolutions and their two global
limits. The initial datum F_0 has compact velocity support, finite
mass, energy, and absolute entropy. Both limits lie in
C([0, infinity); L^1(Omega)) and satisfy the entropy inequality. Both
satisfy local conservation of mass, momentum, and kinetic energy. The
separate gain and loss terms are integrable with every polynomial
velocity weight on every bounded time interval. The two limits differ
on a set of positive measure. So the hard-sphere Boltzmann equation on
T^3_x times R^3_v admits two global entropy solutions with local
conservation for one and the same initial datum, and the two solutions
are not equal almost everywhere.
The mechanism is spatial concentration that preserves energy locally.
Thin jets are aimed along a quadratic-irrational direction, are kept
apart until their lifetime ends, and are damped by a nearly
Maxwellian bath. A finite cap over a short horizon selects one family;
the same mechanism with seed set to zero produces the other. Dilation
at the cap event magnifies the jets into a nonzero linear limit that
does not exist in the dormant family, and branching alone bounds the
magnified horizon. Finiteness of that horizon then transfers the
comparison back to the original physical variables.
Every step of the argument is deterministic. The use of Maxwellian
weights, entropy comparison, and the Duhamel formula is classical, and
the analytic inputs (Carleman's formula, the sheet formalism, the
geometric slice estimate) go back to Boltzmann, Carleman, Grad, and
DiPerna-Lions. The passage to limits preserving local conservation is
the technical work of the compactness chapter. The growth mechanism at
dilations borrows from Doeblin-type branching representations of
linear Boltzmann flows. The construction is completed by combining
these ingredients with an explicit initial datum whose scales are
chosen geometrically and whose amplitudes decrease geometrically with
the same rate.
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