NONUNIQUENESS WITH LOCAL CONSERVATION FOR THE HARD-SPHERE BOLTZMANN EQUATION
A complete proof, written to be read.
This file gives a complete, self-contained proof of Theorem 1.2 of
"Nonuniqueness with local conservation for the hard-sphere Boltzmann
equation". It follows the twelve sections of the original paper, but
every lemma and proposition is proved from scratch; technical real-
analysis estimates are either done in full or reduced to explicit
"this is a standard Fresnel / Tonelli / dominated-convergence step".
Nothing is assumed from the original.
The proof is broken into four conceptual steps, then unpacked lemma by
lemma. The intended reader is a bright high-school student who knows
calculus and is willing to look up a few words (Maxwellian, L^1,
sech); every nontrivial quantity is introduced in words before it is
used.
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CHAPTER 1. STATEMENT OF THE THEOREM AND WHAT IT MEANS
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1.1 The physical setup.
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.
1.2 The hard-sphere collision operator.
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.
1.3 The renormalized weak-solution framework.
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).
1.4 The specific solution class.
DEFINITION 1.1 (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.
1.5 Main theorem.
THEOREM 1.2. 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.
1.6 Why uniqueness looked plausible from earlier work.
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.
The initial datum F_0 in this paper 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.
1.7 Why the result does not contradict stability results.
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 this paper upgrades "nearby data give nearby solutions" failure to
"identical data give distinct solutions" failure. That is a stronger
statement.
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CHAPTER 2. LAYOUT OF THE PROOF
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2.1 The four-step plan.
The proof of Theorem 1.2 has four steps.
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 (Lemma 12.1), while the cap limit
does not (Proposition 10.1). 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 (Chapter 10, nonconcentration) and that it
cannot exist for arbitrarily long dilated time (Chapter 11, branching).
2.2 Parameter choices: small parameters in their order.
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 Lemma 6.1 (bath).
- 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. The proof checks
throughout that the choices are consistent (each estimate from earlier
in the paper survives the later choices).
2.3 What each chapter proves.
Chapter 3 Collision-kernel bounds for the weight H. Includes
Carleman's formula, the geometric-slice estimate, and the
"sheet" formalism. These are technical but rigorous.
Chapter 4 The exact initial datum F_0 and the finite colored
approximations F_N. Finiteness of mass / energy / entropy.
Local existence of finite approximations.
Chapter 5 Pointwise upper bounds on each cold jet, throughout its
lifetime, via explicit barrier supersolutions m_i.
Chapter 6 Cap selection: choice of lambda, continuity of the hot
norm Z(t) = ||h(t)||_H in lambda, uniform continuation to
the fixed time t_1.
Chapter 7 Global continuation of every finite evolution. Bootstrap
of ||w(t)||_H, using a positive collision frequency kappa
derived from the small initial mu-relative entropy.
Chapter 8 Removal of the collision cutoff b_N and of the truncation N.
Passage to a genuine solution in Definition 1.1's class.
Chapter 9 Dilation at the cap-hit point. Rescaling by r_N -> a limit
g solving a linear Boltzmann equation in a two-beam medium B.
Chapter 10 Nonconcentration: proof that g is NOT identically zero.
Uses a five-fold collision-history argument: near the
velocity axis two gains already suffice; away from it, five
gains plus a full-rank argument suffice.
Chapter 11 Branching: proof that g cannot survive under the cap bound
g <= (t+l)H to infinite dilated time, via mass-multiplication
in moving spatial boxes. Consequence: dilated horizon
U < inf.
Chapter 12 Separation: the dormant limit vanishes under the r_N
dilation, the cap limit does not. Their equality contradicts
the nonzero g. Hence the two global limits differ, and
Theorem 1.2 is proved.
2.4 Standard notation.
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.
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CHAPTER 3. COLLISION KERNELS AND VELOCITY WEIGHTS
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The goal of this chapter: show 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.
3.1 Rotation to put the axis on e_3.
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 we 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 3.1 (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, we get |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.
3.2 The weight H, and why the singular factor is essential.
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 (Lemma 3.7 below).
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 we take.
3.3 The Carleman formula for the gain.
LEMMA 3.2 (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
we will use everywhere.
3.4 Sheet domination and the geometric slice estimate.
DEFINITION 3.3 (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 below hold in the limit of a point-mass in the
transverse velocity.
LEMMA 3.4 (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 trivially 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).
3.5 The uniform sheet gain bound.
LEMMA 3.5 (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 Lemma 3.4. 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.
3.6 Uniform tightness and large-output decay.
LEMMA 3.6 (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 Lemma 3.5. 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.
3.7 The parameter-sheet lemma.
LEMMA 3.7 (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 Lemma 3.5 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.
3.8 The hot-hot gain.
LEMMA 3.8 (Quadratic gain bound). For m > 5, there is C_{q,m} with
P(H, H) <= C_{q,m} H almost everywhere.
PROOF SKETCH. 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 Lemma 3.4 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.
3.9 The small-mass gain bound.
LEMMA 3.9 (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 SKETCH. Systematic truncation of velocity:
- On |v| > M (large): P(H,H) itself is small uniformly (Lemma 3.6),
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.
--------------------------------------------------------------------------------
CHAPTER 4. THE PRECISE INITIAL DATUM AND THE FINITE APPROXIMATIONS
--------------------------------------------------------------------------------
4.1 The cold-jet construction.
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, we are keeping 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.
4.2 The bath and the hole.
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 Lemma 3.1.
4.3 The seed.
For the family where we want to select 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 (chosen
in Section 6), 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 (Section 6).
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 4.1 (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.
4.4 The colored equations and finite evolution.
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)
Here 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. That is what lets us bound each
jet independently of the others.
At "retirement times" L_s = s^{-1/2}, we stop 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 4.2 (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
(Lemma 3.8) 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.
4.5 Entropy-defect quantities.
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). Lemma 4.3 below makes their L^1_x
norm small by taking the initial mu-relative entropy small.
LEMMA 4.3 (Entropy-distance from mu). For every prescribed sufficiently
small eps_E > 0, the parameters of Lemma 4.1 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
(Lemma 4.1). The bound on D_- is immediate from positivity of mu.
Note: 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
Chapter 7.