Uniqueness and related constructions

Classical uniqueness theorems impose additional control, such as Maxwellian upper bounds (Kaniel and Shinbrot 1978). For periodic cutoff hard potentials, Ukai’s perturbative theory gives global existence and uniqueness near a Maxwellian in a velocity-weighted spatial Sobolev class, with mass, momentum, and energy matched to that Maxwellian (Ukai 1974, Theorem 1.2).

The distinction between spatially homogeneous and inhomogeneous evolution is also important. For the homogeneous hard-sphere equation, Mischler and Wennberg prove uniqueness in a conservative class from finite mass and energy alone (Mischler and Wennberg 1999, Theorem 1.1). Wennberg’s homogeneous nonuniqueness examples for hard potentials permit increasing energy (Wennberg 1999). Theorem 1 instead concerns a spatially inhomogeneous datum, with the energy and entropy-dissipation inequalities retained.

Recent instability constructions give another relevant comparison. Chen and Holmer prove failure of uniform continuity in weighted Sobolev spaces for the constant collision kernel (Chen and Holmer 2024, Theorem 1.3); Chen, Guo, Shen, and Zhang establish failure of uniform continuity for the hard-sphere solution map on \(\mathbb R^3_x\times\mathbb R^3_v\), in Gaussian-weighted \(L^2_vH^s_x\) spaces for \(0\le s<1\) (Chen et al. 2026, Theorem 1.1). Those results compare solutions with arbitrarily close initial data. Here the two initial data agree exactly, on periodic space, and both resulting solutions satisfy the admissibility conditions above. The instability in (Chen et al. 2026) is driven by collision loss and dispersion. In the construction below, collision gain amplifies a vanishing hot seed and produces a nonzero weak limit near the initial spatial singularity.

For the related Krieger–Strain model, Gismondi, Golding, and Novack construct rough stationary solutions in a mollifier-confluent sense: the collision operators of mollified densities tend to zero in distributions for every nonnegative Schwartz mollifier of mass one (Gismondi et al. 2026, Definition 1.3 and Theorem 1.5). They also conjecture nonuniqueness of renormalized inhomogeneous Landau and Boltzmann solutions for a dense class of initial data. Theorem 1 concerns one periodic hard-sphere datum in the specified admissibility class.

One datum and two approximation families

Write \(x=(y,z)\in\mathbb R^2\times\mathbb R\) in a lift of the torus, and \(v=(\bar v,v_3)\in\mathbb R^2\times\mathbb R\). The datum consists of narrow velocity distributions, called cold jets, near the axial directions \(\pm e_3\). They occupy transverse annuli of geometrically decreasing radii \(s_j\downarrow0\). Within each annulus the two velocity signs occupy disjoint, finely alternating stripes. Their thermal widths are much smaller than the stripe gaps. An expanding axial velocity profile bounds the spatial density of each freely transported jet at scale \(s\) by \(C/(t+s)\). Annular mass is \(O(s^2)\), and annular absolute entropy is \(O(s^2(1+|\log s|))\), so all scales together define a single finite-entropy datum.

Assign a color to each cold jet. A collision within one color retains that color; collisions between different colors, or with particles already declared hot, produce hot particles. This is an exact decomposition: the sum of the colored equations is the Boltzmann equation. The narrow velocity distributions suppress self-collisions, and the spatial gaps suppress collisions between distinct cold jets at early times.

Retain the first \(N\) annuli and impose a collision cutoff that is removed as \(N\to\infty\). Let \(h_N\) denote the hot component. The estimates use an integrable velocity weight \(H(v)\), Gaussian at infinity and locally comparable to \(|\bar v|^{-q}\) near the velocity axis, with \(1<q<2\). A cap is an almost-everywhere upper bound on \(h_N/H\). The two approximation families have the following properties:

1. With no hot particles initially, the hot density remains bounded by \(t^2H(v)\). We call this the dormant family.

2. For every sufficiently short observation horizon \(T>0\), a smooth hot seed of vanishing mass, energy, and entropy can be selected so that \(h_N\le(t+s_N)H\) on \([0,T]\), with equality in the weighted essential supremum at some time. Here \(s_N\) is the smallest retained annular radius.

At each fixed horizon, uniform integrability of both the densities and their two-particle products gives admissible limits with exactly the same initial datum. A vanishing seed and a cap hit alone do not imply that these limits differ: the hit might carry arbitrarily little mass. Excluding that loss of information is the central step.

From a cap hit to separation

Along a sequence of observation horizons shrinking to zero, rescale time and space at a cap time \(t_{*,N}\) by \(r_N=t_{*,N}+s_N\), and divide the hot density by \(r_N\). The resulting cap is of order one. The cold density has a different normalization: multiplying it by \(r_N\) produces a locally bounded velocity measure at positive times. Its two axial velocity signs coexist in the limit because the increasingly fine spatial stripes average to their occupied fractions. Strong convergence of hot velocity averages identifies the cold–hot product. This use of velocity averaging follows the uniformly integrable formulation of (Golse et al. 1988, Proposition 3). The limit obeys a linear Boltzmann equation in that two-sign background. This dilation is distinct from the fixed-horizon compactness used to construct admissible solutions.

To show that the dilated hot limit is nonzero, we analyze normalized Carleman gain kernels. Carleman’s representation expresses a collision gain as an integral over velocity planes; its normalization here is derived in Section 3. Iterated-gain regularization has an earlier role in the homogeneous theory (Mischler and Wennberg 1999, Lemma 2.1). In the present construction the estimates must be uniform as the cold velocity distributions collapse onto the axis, despite the singularity of \(H\). Two complementary arguments provide this control. Near the velocity axis, two successive gains have vanishing normalized mass: the first has a uniformly non-atomic axial input, while a second large gain would force that input to match a small collection of characteristic cold speeds. Away from the axis, five gains suffice. Their times and angular parameters map to final time, position, and velocity with rank seven almost everywhere. A compact submersion argument then turns vanishing local hot mass into a vanishing gain contribution. Together these facts prevent the cap hit from disappearing in the weak limit.

The limiting hot equation also has a branching interpretation. A hot particle colliding with a cold jet creates two hot outputs. We retain finite collision histories with bounded velocities and show that their mass grows by a fixed factor inside an expanding spatial box. If the dilated observation horizon were infinite, this growth would exceed the polynomial mass bound supplied by the cap. The horizon is therefore finite, so the cap scale remains comparable to the physical observation horizon along the extracted sequence.

Finally, the dormant solution vanishes off the velocity axis under the corresponding shrinking-horizon dilation. If cap limits agreed with it at arbitrarily small fixed horizons, a diagonal choice of approximants would force the nonzero dilated hot limit to vanish as well. This contradiction yields two distinct early solutions of the original equation with the same datum. The periodic existence theorem then continues each solution globally while preserving admissibility. Figure 1 separates the two compactness procedures and their roles in this final comparison.

Organization

Section 2 gives the full admissibility definition and the colored equations. Section 3 establishes uniform collision-kernel estimates, and Section 4 constructs the datum and controls its cold components. Section 5 selects the approximation families; Section 6 constructs their fixed-horizon limits and verifies admissibility. Section 7 identifies the dilated linear equation, and Section 8 proves that its hot limit is nonzero. Section 9 excludes an infinite dilated horizon. Section 10 completes the same-datum comparison and global continuation.

A velocity weight and colored components

Fix \(1<q<2\) and \(m=100\). The weight used throughout is \[ H(v)=e^{-|v|^2}\langle v\rangle^{-m} \bigl(1+|\bar v|^{-q}\bigr),\qquad \langle v\rangle=(1+|v|^2)^{1/2}. \tag{3}\] Its values on the velocity axis can be assigned arbitrarily; all weighted supremum estimates are essential estimates. It is integrable with every polynomial velocity weight and lies in \(L^p\) for some \(p>1\). Write \(\|f\|_H=\mathop{\mathrm{ess\,sup}}_{x,v}|f(x,v)|/H(v)\).

The cold components are \(c_i\), their sum is \(c\), and \(h\) denotes the hot component. Particles retain a cold label only in collisions within that label. All other collisions create hot particles. With \(F=c+h\), this gives \[ \begin{split} (D_t)c_i&=P(c_i,c_i)-c_i\nu_F,\\ (D_t)h&=2P(c,h)+P(h,h)+\sum_{i\ne j}P(c_i,c_j)-h\nu_F. \end{split} \tag{4}\] The sum over \(i\ne j\) is ordered. Summing these equations gives \((D_t)F=Q(F,F)\). This identity also holds with a common collision cutoff. When labels include periodic copies, the individual functions are defined in the lift and are covariant under translations; their sums are periodic.

Constants denoted \(C\), with or without subscripts, may depend on the fixed construction parameters. They never depend on the approximation index or the shrinking time horizon unless explicitly stated. The parameters needed for branching growth will be fixed before the final small geometric and time parameters. Section 9 records the order of choices.

The exact Carleman formula

The hyperplane representation used here is a Carleman formula; see also (Gamba et al. 2009, Appendix C). We derive its coefficient for the normalization of this paper.

The cutoff gain obtained by inserting \(\mathbf 1_{|p-\xi|<b}\) is bounded by the uncut gain. Thus all upper estimates in this section also hold for the collision cutoffs used later.

A geometric slice estimate

For \(d>0\), let \(\mathcal G_d\) be the centered transverse Gaussian probability measure with density \[\frac{1}{\pi d^2}e^{-|\bar p|^2/d^2},\] and let \(\mathcal G_0=\delta_0\). We shall also use the uniform probability measure \(\mathcal U_d\) on \(\{\bar p:|\bar p|<d\}\).

Sheets and normalized kernel measures

Every axial probability supported in \([-A,A]\) satisfies this definition with \(L=e^{8A^2}\). In particular, axial atoms and limits of arbitrarily narrow axial densities are permitted. The exponential moment is used only to remove large axial velocities, and imposes no axial regularity.

For such a sheet and \(\bar v\ne0\), define its normalized kernel by \[ \mathcal K_c(v;\,\mathrm dp\,\,\mathrm d\xi) =2\,c(\,\mathrm dp)\, \delta\bigl((v-p)\cdot(\xi-v)\bigr) \frac{H(\xi)}{H(v)}\,\,\mathrm d\xi. \tag{13}\] The total mass is \(P(c,H)(v)/H(v)\). Kernel restrictions always refer to this complete measure, including the ratio \(H(\xi)/H(v)\).

The sheet gain bound controls total normalized mass. We next control its velocity tails and its behavior as the outgoing velocity approaches the axis. The key distinction is that the hot input can concentrate transversely on the axis while its axial marginal remains uniformly non-atomic.

Gaussian parameter laws and compressed axial maps

The next formulation records the uniformity needed when a cold profile is parameterized before its axial change of variables.

For example, suppose an axial velocity is determined by \(p_3=a(z-tp_3)+d w_3\), where \(a\) is nondecreasing, and \(u\) solves \(u=a(z-tu)\). The function \(p\mapsto p-a(z-tp)\) is increasing with inverse Lipschitz constant at most one. Hence \[ |p_3-u|\leq d|w_3|. \tag{30}\] If \(|u|\leq A\), Lemma 8 applies. A large Jacobian compression of the axial law does not affect its constants. Gaussian majorants with any fixed positive exponent in place of one are covered by a fixed rescaling of the parameters and a corresponding decrease of the upper bounds on the physical widths.

The hot–hot gain

The geometry and the datum

Put \[s_j=s_0e^{-j\eta},\qquad p_j=s_j^2,\qquad c_\eta=\frac{e^\eta-1}{16},\qquad \delta_j=\frac{c_\eta}{4}s_j^2,\qquad j\ge1.\] For \(s=s_j\), define the annular core and the two stripe patterns by \[\begin{align*} A_j&=\{y\in\mathbb R^2: s_j+c_\eta p_j<|y|<e^\eta s_j-c_\eta p_j\},\\ P_j^+&=\bigcup_{k\in\mathbb Z} \{y:kp_j<y_1<(k+\tfrac14)p_j\},\\ P_j^-&=\bigcup_{k\in\mathbb Z} \{y:(k+\tfrac12)p_j<y_1<(k+\tfrac34)p_j\},\\ E_j^\sigma&=A_j\cap P_j^\sigma,\qquad \sigma\in\{-1,+1\}. \end{align*}\] Each stripe pattern has period \(p_j\) and fraction \(\theta=1/4\). This is a statement about the periodic pattern: intersecting it with the annulus need not preserve the fraction exactly. We retain the original sets \(E_j^\sigma\) in the datum.

Let \[\widehat E_j^\sigma =\{y:\mathop{\mathrm{dist}}(y,E_j^\sigma)\le\delta_j\}, \qquad a=\tfrac18,\qquad b_{\mathrm{sp}}=\tfrac1{16}.\] For a label \(i=(j,\sigma,n)\), \(n=(\bar n,n_3)\in\mathbb Z^3\), set \[ \mathcal O_i(t)= (\widehat E_j^\sigma+\bar n) \times[n_3-a-2t-b_{\mathrm{sp}}, n_3+a+2t+b_{\mathrm{sp}}]. \tag{36}\] These enlargements will only localize upper bounds; they will not replace the stripe masks when their average is computed.

Fix a nonnegative smooth probability density \(J\), supported in the unit ball of \(\mathbb R^3\). Write \[S(\ell)=\operatorname{sech}^2\ell,\qquad a_s^\sigma(z)=\sigma+\alpha\tanh(\gamma z/s), \qquad d_s=d_0s^{10}.\] The positive parameter \(d_0\le1\) will be decreased below finitely many times. In particular, we require that \[ d_0\overline s^{10}\le\tfrac1{64},\qquad \overline T d_0\overline s^8\le c_\eta/8 . \tag{37}\] The width \(1/64\), including the fixed Gaussian widenings used below, lies within the range of Lemma 6. For \(i=(j,\sigma,n)\), with \(s=s_j\), define on the lift \[c_i^0(x,v)= \mathbf 1_{E_j^\sigma}(y-\bar n)\mathbf 1_{(-a,a)}(z-n_3) \frac K s S\!\left(\frac{\gamma(z-n_3)}s\right)d_s^{-3} J\!\left(\frac{v-a_s^\sigma(z-n_3)e_3}{d_s}\right).\] The periodic datum is \[ F_0(x,v)=\sum_{j\ge1}\sum_{\sigma=\pm1} \sum_{n\in\mathbb Z^3}c_{(j,\sigma,n)}^0(x,v). \tag{38}\] At a given spatial point at most one summand is nonzero. Integer translations permute the labels, so the sum is periodic.

A transported Maxwellian barrier

For a fixed scale and sign, the equation \[ u_s^\sigma(t,z)=a_s^\sigma(z-tu_s^\sigma(t,z)) \tag{39}\] defines \(u_s^\sigma\) for all \(t\ge0,z\in\mathbb R\). Indeed \(\tau\mapsto\tau+t a_s^\sigma(\tau)\) is strictly increasing, has derivative at least one, and maps \(\mathbb R\) onto \(\mathbb R\). In particular, \(|u_s^\sigma|\le1+\alpha\). For label \(i=(j,\sigma,n)\), abbreviate \[u_i(t,z)=u_{s_j}^\sigma(t,z-n_3),\quad \tau_i=z-n_3-tu_i(t,z),\quad w_i=v-u_i(t,z)e_3,\] and set \[D_{y,i}=\mathop{\mathrm{dist}}(y-\bar n,E_j^\sigma),\qquad D_{z,i}=\mathop{\mathrm{dist}}(\tau_i,[-a,a]).\] Choose \(c_{\mathrm{sp}}=1/8\) and a fixed sufficiently large \(C_m\), depending only on \(J\), and define, for \(t>0\), \[ m_i(t,x,v)= C_m\frac K{s_j}d_{s_j}^{-3} e^t \exp\left[-\frac{(2-t)|w_i|^2}{d_{s_j}^2} -\frac{c_{\mathrm{sp}}(D_{y,i}^2+D_{z,i}^2)} {d_{s_j}^2t}\right]. \tag{40}\] Its trace at \(t=0\) is defined by the limit along free characteristics. Boundary values of the spatial masks are irrelevant. Denote the free transport of the \(i\)-th initial color by \[f_i(t,x,v)=c_i^0(x-tv,v).\]

Cold control and the density Jacobian

The labels for the finite approximation are \[\mathcal I_N=\{(j,\sigma,n):1\le j\le N,\ \sigma=\pm1,\ n\in\mathbb Z^3\}.\] For \(i=(j,\sigma,n)\) we also write \(s_i=s_j\). The infinitely many integer-copy labels merely implement periodicity: translating \(x\) permutes them. We consider nonnegative colored mild solutions with these initial colors and any nonnegative hot component, with a common finite collision cutoff \(b>0\). The statement below applies throughout every interval on which the finite-approximation solution exists with bounded weighted sup norm. Its constants do not use that norm or any bound on the hot component.

It is useful to record the spatial form of the moment bound. For a fixed \(k\ge0\), Proposition 14 implies \[ \int_{\mathbb R^3}\langle v\rangle^k c(t,x,v)\,\,\mathrm dv \le C_k\sum_{i\in\mathcal I_N} \frac{\mathbf 1_{\mathcal O_i(t)}(x)}{t+s_i} +C_{A,k}t^A. \tag{52}\] At most one term of the displayed sum is present at any point. In one periodic cell, the spatial volume of the enlargement belonging to a fixed annular sign, with its periodic copies included, is at most \(Cs_i^2\). The same bound with a factor tending to zero as the velocity cutoff increases holds for the integrated velocity tails.

Quantitative survival on a free core

The upper estimates did not use a hot bound. The following consequence, used only for the dilation, adds the cap assumption.

Conservation and entropy before passage to the limit

At a fixed approximation the bounds already proved imply \(F_N\leq A_N H\) on the whole solution interval, with a finite constant allowed to depend on \(N,b_N,T\). In particular there are all polynomial velocity moments, and there is an \(L^p\) majorant for some \(1<p<2/q\). The equation holds absolutely continuously on almost every free path. Its cutoff gain and loss are integrable, including after multiplication by any fixed polynomial in velocity.

The collision map preserves volume, kinetic energy, and \(|v-v_*|\), so it also preserves the cutoff. For an integrable velocity test \(\psi\) the collision symmetry gives \[\int_{\mathbb R^3}\psi(v)Q^{b_N}(F_N,F_N)(v)\,\mathrm dv =\frac12\int_{\mathbb R^3\times\mathbb R^3\times S^2} |a|\mathbf 1_{\{|v-v_*|<b_N\}}F_NF_{N,*} (\psi'+\psi_*'-\psi-\psi_*)\,\mathrm dv\,\mathrm dv_*\,\mathrm d\omega.\] The right side vanishes for \(\psi=1,v_j,|v|^2\). Integrating the path equation and using periodicity therefore proves mass, momentum, and energy conservation at every time of the finite approximation. The same calculation with a spatially dependent mass test gives its local continuity equation.

To justify entropy at zeros, put \(E(v)=e^{-2|v|^2}\), take \(0<\delta\leq1\), and set \(F_{N,\delta}=F_N+\delta E\). Define \[\mathcal A_\delta(f,v) =M(v)+\int_0^f\log\frac{u+\delta E(v)}{M(v)}\,\mathrm du.\] Its derivative with respect to \(f\) is \(\log(F_{N,\delta}/M)\). At the fixed approximation, \[ \left|\log\frac{F_{N,\delta}}M\right| \leq C_N\bigl(1+|\log\delta|+|v|^2 +\log^+|\bar v|^{-1}\bigr). \tag{58}\] Indeed \(F_{N,\delta}\geq\delta e^{-2|v|^2}\) gives the lower logarithmic bound, and \(F_N\leq A_NH\) gives the upper bound. Every factor on the right of (58) is integrable against \(H\), apart from the harmless constant \(|\log\delta|\): in the transverse plane this uses \(\int_0^1 r^{1-q}|\log r|\,\mathrm dr<\infty\).

Since \(E\leq C H\), the fixed-cutoff bilinear bounds imply \[|Q^{b_N}(F_{N,\delta},F_{N,\delta}) -Q^{b_N}(F_N,F_N)|\leq C_{N,b_N}\delta H.\] Consequently replacing the latter operator by the former in its integral against \(\log(F_{N,\delta}/M)\) costs at most \(C_{N,b_N,T}\delta(1+|\log\delta|)\). The path chain rule, followed by spatial integration, now gives \[ \int_\Omega\mathcal A_\delta(F_N(t),v) +\int_0^t\mathcal D_{b_N}(F_{N,\delta}(s))\,\mathrm ds \leq\int_\Omega\mathcal A_\delta(F_N(0),v) +C_{N,b_N,T}\delta(1+|\log\delta|). \tag{59}\] Here \(\mathcal D_b\) denotes the dissipation with the invariant cutoff inserted. The collision identity used in this step is \[\int_\Omega Q^{b_N}(F_{N,\delta},F_{N,\delta}) \log(F_{N,\delta}/M) =-\mathcal D_{b_N}(F_{N,\delta}).\] It follows by averaging over interchange of the two particles and over the elastic substitution. The \(\log M\) term cancels because it is a linear combination of mass and energy. All integrals for this calculation are finite at \(\delta>0\) by (58).

As \(\delta\downarrow0\), the primitive converges to \(f\log(f/M)-f+M\). Its integrals converge as well. For example its lower bound is that limiting primitive, while its upper bound follows by replacing \(\delta\) by one in the increasing integrand. The negative part of \(f\log f\) is integrable under a finite second moment, and the upper bound is integrable by (58) without the \(|\log\delta|\) term. Fatou’s lemma applied to the nonnegative dissipation integrand in (59) therefore yields, for every \(t\in[0,T]\), \[ \mathcal H_M(F_N(t))+\int_0^t\mathcal D_{b_N}(F_N(s))\,\mathrm ds \leq\mathcal H_M(F_N(0)). \tag{60}\]

Lemma 12 gives convergence of the initial mass, energy, and relative entropy to those of (38). Its estimates include the hot seed, whose mass and energy are \(O(s_N^4)\) and whose absolute entropy is \(O(s_N^4(1+|\log s_N|))\), as well as the seed’s overlap with the cold density in the relative entropy. The convergence is uniform for the allowed seed parameters \(0\leq\lambda\leq L_0\).

Uniform integrability of densities and collision products

For \(s=s_j\), let \(A_s\) be the annulus \(\{y\in\mathbb R^2:s<|y|<e^\eta s\}\), viewed as a subset of \(\mathbb T^2\). By Lemma 11, these annuli are pairwise disjoint and contain the enlarged stripes of both signs at their respective scales. Their areas satisfy \(|A_s|\leq Cs^2\). Proposition 14, the cap, and the dormant bound imply, for every fixed \(k\geq0\), \[ R_{k,N}(t,x):=\int_{\mathbb R^3}\langle v\rangle^kF_N(t,x,v)\,\mathrm dv \leq C_k\left(1+\sum_{j\leq N} \frac{\mathbf 1_{A_{s_j}}(y)}{t+s_j}\right). \tag{61}\] The bounded term includes the hot density and escaped tails. The cold remainder has density coefficient \(O(d_s/s^2)=O(s^8)\), and is also covered by this estimate. All constants are independent of \(N\).

The square of the envelope is integrable uniformly in \(N\), including at the initial surface. In fact the contribution of one annulus is \[ \int_0^T\frac{Cs^2}{(t+s)^2}\,\mathrm dt =Cs^2\left(\frac1s-\frac1{T+s}\right)\leq Cs. \tag{62}\] Since the scales form a geometric sequence, \(\sum_{s_j<\varepsilon}s_j\leq C_\eta\varepsilon\). If \(A_{<\varepsilon}=\bigcup_{s_j<\varepsilon}A_{s_j}\), disjointness of the annuli thus gives \[ \sup_N\int_0^T\int_{A_{<\varepsilon}\times\mathbb T} R_{k,N}(t,x)^2\,\mathrm dx\,\mathrm dt\longrightarrow0 \qquad(\varepsilon\downarrow0). \tag{63}\] The contribution of the bounded term is at most \(C T|A_{<\varepsilon}|\).

Outside \(A_{<\varepsilon}\), the main color has one of finitely many scales \(s_j\geq\varepsilon\). Those velocity widths are positive and fixed. Their Gaussian bounds, with axial centers in a fixed bounded interval, give a function \(\Psi_\varepsilon(v)\), independent of \(N,t,x\), such that \[ F_N(t,x,v)\leq\Psi_\varepsilon(v) \quad\hbox{if }y\notin A_{<\varepsilon},\qquad \int\langle v\rangle^k\Psi_\varepsilon(v)\,\mathrm dv<\infty \quad(k\geq0). \tag{64}\] One may take a finite sum of broadened Gaussians about the bounded axial interval, together with a multiple of \(H\) for the hot component and escaped tails. There is no need for this majorant to be bounded on the velocity axis.

Equations (63)–(64) prove uniform integrability of \[\langle v\rangle^k\langle w\rangle^k F_N(t,x,v)F_N(t,x,w)\] in the full spacetime and two-velocity variables. Indeed its integral over the removed annuli is at most the left side of (63); elsewhere it has the fixed integrable majorant \(\langle v\rangle^k\Psi_\varepsilon(v) \langle w\rangle^k\Psi_\varepsilon(w)\). This also proves uniformly small velocity tails. Taking first powers instead proves uniform integrability of the one-particle densities in spacetime. At individual times the estimate is \[ \sup_{N,\,0\leq t\leq T} \int_{A_{<\varepsilon}\times\mathbb T}R_{k,N}(t,x)\,\mathrm dx \leq C_k\sum_{s_j<C\varepsilon}\frac{s_j^2}{s_j} +C_k|A_{<\varepsilon}|\longrightarrow0. \tag{65}\] Together with (64), this proves weighted one-particle uniform integrability and velocity tightness uniformly at all individual times.

The gains and losses enjoy the same spacetime conclusion. To see this directly, collision symmetry and energy conservation imply \[\int\langle v\rangle^k Q_N^\pm(t,x,v)\,\mathrm dv\leq C_k R_{k+1,N}(t,x)^2, \qquad Q_N^\pm=Q^{\pm,b_N}(F_N,F_N).\] Their weighted integrals on the small annuli therefore tend uniformly to zero. On the complement, positivity and (64) give the fixed bounds \[Q_N^+(t,x,v)\leq P(\Psi_\varepsilon,\Psi_\varepsilon)(v),\qquad Q_N^-(t,x,v)\leq\Psi_\varepsilon(v)\nu_{\Psi_\varepsilon}(v).\] Both right sides are integrable with every fixed polynomial velocity weight: integrate first in the output velocity and use the preceding collision-moment inequality for the fixed function \(\Psi_\varepsilon\). Thus \(\{Q_N^\pm\}\) is uniformly integrable in spacetime, with uniformly small weighted velocity tails. In particular its integrals over time intervals of length tending to zero vanish uniformly in \(N\). This last assertion follows by first discarding the small annuli and then integrating the fixed majorants over the time interval.

An elementary averaging argument

We use a uniformly integrable version of velocity averaging (Golse et al. 1988, Proposition 3). The following proof gives the precise local statement needed here.

Identification of the nonlinear terms

The uniform integrability just proved supplies a subsequence with \(F_N\rightharpoonup F\) in spacetime \(L^1\); velocity tightness allows the whole velocity space. Since \(D_tF_N=Q_N^+-Q_N^-\), Lemma 19 applies. After a diagonal extraction for a countable dense family of compact velocity tests, all such velocity averages converge strongly in local \(L^1_{t,x}\) to the corresponding averages of \(F\). Extension from that family to every smooth compact velocity test uses the uniform mass bound.

Set \(W_N(t,x,v,w)=F_N(t,x,v)F_N(t,x,w)\). The full pair-product uniform integrability supplies a weak \(L^1\) limit \(W\), after another extraction. For smooth compact velocity tests \(\psi,\chi\), \[\int\psi(v)\chi(w)W_N\,\mathrm dv\,\mathrm dw =\left(\int\psi F_N\,\mathrm dv\right) \left(\int\chi F_N\,\mathrm dw\right).\] Both factors converge in local \(L^1\), so their product converges in measure. The products are uniformly integrable in \((t,x)\), by (62) and the fixed majorants after removal of small annuli. Their convergence is consequently in \(L^1\): to verify this last implication, discard a set of arbitrarily small measure on which convergence is not uniform, and use uniform integrability on the discarded set. We obtain \[\int\psi(v)\chi(w)W\,\mathrm dv\,\mathrm dw =\left(\int\psi F\,\mathrm dv\right) \left(\int\chi F\,\mathrm dw\right).\] Finite sums of separated velocity tests are uniformly dense among continuous functions on compact two-velocity sets. Testing also in \((t,x)\) identifies \[ W_N\rightharpoonup F(t,x,v)F(t,x,w) \quad\hbox{in }L^1([0,T]\times\mathbb T^3\times\mathbb R^3\times\mathbb R^3). \tag{66}\] The same conclusion holds with any fixed polynomial weight for which it is needed. Interior time compactness suffices: the uniform integrability already proved removes intervals near zero and \(T\).

Adjoining the finite angular measure preserves weak convergence. The elastic map on \((v,w,\omega)\) is measure preserving and involutive, so (66) also identifies the primed pair product. On a bounded two-velocity set the factor \(|a|\) is bounded, and the cutoff is eventually absent. The weighted tail estimates allow these sets to exhaust all velocities. Since \(Q_N^\pm\) themselves are uniformly integrable, testing their weak limits and using these identifications proves \[ Q_N^\pm\rightharpoonup Q^\pm(F,F) \quad\hbox{in }L^1([0,T]\times\Omega). \tag{67}\] Passing the equation, including its initial boundary term, is now legitimate. The initial densities converge in \(L^1\), and the limit satisfies the unrenormalized distributional equation with datum \(F_0\).

Every-time traces, entropy, and the chain rule

We first specify representatives at every time. For a smooth periodic test \(\varphi(x,v)\) compactly supported in velocity, the finite path equations give \[|\langle F_N(t)-F_N(s),\varphi\rangle| \leq\int_s^t\int_\Omega \bigl(F_N|v\cdot\nabla_x\varphi| +(Q_N^++Q_N^-)|\varphi|\bigr)\,\mathrm dx\,\mathrm dv\,\mathrm du.\] The first term is bounded by \(C_\varphi|t-s|\); the second has a uniform modulus tending to zero with \(|t-s|\), by the time-integral conclusion above. For a countable uniformly dense collection of such tests, extract uniform convergence of their pairings on \([0,T]\). At each individual time (65) and (64) make \(\{F_N(t)\}\) relatively weakly compact in \(L^1\). Every cluster point has the prescribed smooth pairings, hence is the same density. Thus this single subsequence converges weakly in \(L^1\) at every time. It defines a weakly continuous representative against smooth tests, agreeing with the spacetime limit. The weighted fixed-time tightness also passes total mass, momentum, and energy from the finite approximations. In particular energy is conserved on this early interval, although only its inequality will be needed.

Let \(\Phi(A,B)=(A-B)\log(A/B)\) with its nonnegative extended values. It is jointly convex and lower semicontinuous: it is the sum of the two convex perspectives \(A\log(A/B)\) and \(B\log(B/A)\). On every bounded collision set, (66) and its primed version therefore give lower semicontinuity of the integral of \(|a|\Phi\). One can absorb the weight into the arguments because \(\Phi(cA,cB)=c\Phi(A,B)\) for \(c\geq0\). The lower semicontinuity follows, for example, by expressing a nonnegative convex lower-semicontinuous function as the supremum of its affine supporting functions and using weak convergence on a finite partition. Exhausting bounded velocity sets, and using that the cutoff is eventually absent on each of them, gives for every \(t\in[0,T]\) \[\int_0^t\mathcal D(F(s))\,\mathrm ds \leq\liminf_N\int_0^t\mathcal D_{b_N}(F_N(s))\,\mathrm ds.\] The same convex lower-semicontinuity argument applies to \(\mathcal H_M(F_N(t))\) under weak \(L^1\) convergence at that time. Combining the two nonnegative terms, using (60) and initial entropy convergence, proves \[ \mathcal H_M(F(t))+\int_0^t\mathcal D(F(s))\,\mathrm ds \leq\mathcal H_M(F_0)\qquad(0\leq t\leq T). \tag{68}\] There is no time-dependent subsequence in this conclusion.

In fact the time representative is strongly continuous. In free coordinates the distributional equation and its initial trace read \[\partial_t\bigl[F(t,x+tv,v)\bigr]=Q(F,F)(t,x+tv,v).\] The right side is integrable in \((t,x,v)\). Subtracting its Bochner integral leaves a distribution with zero time derivative, whose initial value is \(F_0\). Therefore \[ F(t,x+tv,v)=F_0(x,v) +\int_0^t Q(F,F)(s,x+sv,v)\,\mathrm ds \tag{69}\] as an \(L^1_{x,v}\) identity, initially for almost every time and then for every time by the representative just constructed. The right side is absolutely continuous in \(L^1\). Free transport is a strongly continuous group on \(L^1(\mathbb T^3\times\mathbb R^3)\), as follows by approximating an \(L^1\) function by continuous functions with compact velocity support. Thus \(F\in C([0,T];L^1)\). This proves in particular continuity against every bounded measurable test.

Equation (69) also holds on almost every individual free path. Since \(M(v)>0\) is constant on such a path, the ordinary absolutely continuous chain rule gives, for every \(\beta\) in Definition 2, \[D_t\bigl[M\beta(F/M)\bigr]=\beta'(F/M)Q(F,F).\] The source is integrable because \(|\beta'|\leq C_\beta\) and \(Q\in L^1\). Moreover \[|M\beta(f/M)-M\beta(g/M)|\leq C_\beta|f-g|, \qquad |M\beta(f/M)|\leq C_\beta f.\] Hence the renormalization is strongly continuous at the initial and terminal traces. Integrating its path equation proves exactly the renormalized identity and trace in Definition 2. The separate integrability requirement follows from \(0\leq Q^\pm/(1+F/M)\leq Q^\pm\).

Integrating the unrenormalized equation in velocity is justified by absolute collision integrability. The collision integral has zero mass, and the flux is integrable, so it gives the required local continuity equation with its initial density. Momentum has already passed at every time. Finally, the negative part of \(F\log F\) is bounded uniformly by mass and energy. For example split \(\{F<1\}\) at \(F=e^{-1-|v|^2}\): above that threshold use \(-F\log F\leq(1+|v|^2)F\), and below it use a fixed multiple of \(e^{-(1+|v|^2)/2}\). Equation (68) then bounds the positive part as well. This proves the uniform absolute logarithmic integral required in the definition.

Continuation and inherited bounds

At time \(T\), the density just constructed has finite mass, energy, and absolute logarithmic integral. We use the global existence theorem of DiPerna and Lions (DiPerna and Lions 1989, 1991), in the precise periodic formulation of (Levermore and Masmoudi 2010, sec. 4.1, Theorem 4.1). We verify its interface with Definition 2 before concatenating.

For our kernel \(b_{\mathrm{phys}}(\omega,v-w)=|(v-w)\cdot\omega|\), local integrability holds and, for every compact \(K\subset\mathbb R^3\), \[\frac1{1+|v|^2}\int_{K\times S^2} b_{\mathrm{phys}}(\omega,v-w)\,\mathrm d\omega\,\mathrm dw =\frac{2\pi}{1+|v|^2}\int_K|v-w|\,\mathrm dw\longrightarrow0 \quad(|v|\to\infty).\] This is the hypothesis of that theorem. To match its collision and time normalizations, put \[C_b=\int_{\mathbb R^3\times\mathbb R^3\times S^2} b_{\mathrm{phys}}(\omega,v-w)M(v)M(w)\,\mathrm d\omega\,\mathrm dv\,\mathrm dw, \qquad \epsilon=C_b^{-1}.\] The constant \(C_b\) is finite and positive. Use the source kernel \(b_{\mathrm{phys}}/C_b\) and its equation \(\epsilon\partial_sG+v\cdot\nabla_xG =\epsilon^{-1}Q_{b_{\mathrm{phys}}/C_b}(G,G)\), with initial density \(F(T)/M\). Here the relative-density operator satisfies \(M Q_{b_{\mathrm{phys}}/C_b}(G,G)=Q(MG,MG)/C_b\). Then \(\widetilde F(t,x,v)=M(v)G(\epsilon t,x,v)\) has precisely our collision normalization. The same time substitution converts the source dissipation coefficient \(\epsilon^{-2}\) into the coefficient one of \(\mathcal D\).

The cited theorem allows arbitrary finite relative-entropy initial data. It gives weak continuity of \(G\) in \(L^1(M\,\mathrm dx\,\mathrm dv)\), which is exactly weak continuity of \(\widetilde F\) in \(L^1(\,\mathrm dx\,\mathrm dv)\), and the renormalized identities with the initial trace for continuously differentiable functions satisfying \(|\beta'(z)|\le C/\sqrt{1+z}\). Our renormalizations are included. Its local mass identity is the one required here. Integrating its momentum identity over the torus removes the spatial divergence of the momentum defect, giving total momentum conservation. This extends to every time by weak continuity and the uniform second-moment bound: the first-moment tail outside \(|v|\le R\) is at most \(R^{-1}\) times the second moment. Dropping the nonnegative defect contributions in the theorem’s energy and relative-entropy inequalities gives the two inequalities in Definition 2 at every time. The negative-entropy estimate above then supplies the uniform absolute logarithmic integral.

We also need separate global integrability of the renormalized gain and loss, rather than only their difference. For any density \(f\ge0\), write \(g=f/M\). Since \(f/(1+g)\le M\), \[\int_\Omega\frac{Q^-(f,f)}{1+g} \le\int_\Omega M\nu_f \le C\int_\Omega(1+|v|)f.\] For \(A,B\ge0\), with the same extended-value convention as for the dissipation, \[A\le2B+\frac2{\log2}(A-B)\log(A/B).\] Indeed the first term suffices when \(A\le2B\); otherwise \(A-B>A/2\) and \(\log(A/B)>\log2\). Apply this with \(A=f'f_*'\), \(B=ff_*\), divide by \(1+g(v)\), and integrate with the collision kernel. Nonnegativity of the dissipation gives \[\int_\Omega\frac{Q^+(f,f)}{1+g} \le2\int_\Omega\frac{Q^-(f,f)}{1+g} +\frac8{\log2}\mathcal D(f).\] The moment and entropy-dissipation bounds for \(\widetilde F\) make both terms integrable over every finite time interval. Thus the continuation satisfies all of Definition 2.

Concatenate \(\widetilde F\) with \(F\) at time \(T\). The two weak \(L^1\) traces agree, so the concatenation is continuous against every bounded measurable test. Its renormalized identity follows by adding the two identities: the early terminal term \(M\beta(F(T)/M)\) cancels the continuation’s prescribed initial term. The Lipschitz estimate above supplies that early terminal trace. The mass continuity identities concatenate in the same way.

For \(t\geq T\), add the continuation inequality \[\mathcal H_M(\widetilde F(t-T)) +\int_0^{t-T}\mathcal D(\widetilde F(s))\,\mathrm ds \leq\mathcal H_M(F(T))\] to (68) at \(T\). This gives the global entropy inequality, with no entropy equality at the splice assumed. Energy inequalities and momentum conservation concatenate using the common terminal moments. Collision integrability in the renormalized sense holds on both pieces, hence on every bounded interval of the concatenation. The uniform moment and absolute entropy bounds on such an interval follow from those of the two pieces.

For a dormant sequence, sum the cold upper bounds of Proposition 14 over all labels and add \(t^2H\). This is a common, locally integrable pointwise upper bound for the finite total densities. Testing against nonnegative functions and passing to the weak limit preserves it. All assertions of Proposition 18 are now proved.

The limiting mesh and background

The dilated annular scales are \(\mu_{N,j}=s_j/r_N\), \(j\leq N\). Their ratio is \(e^\eta\), their smallest member is \(l_N\), and their largest member tends to infinity. Subselect their logarithmic phase modulo \(\eta\). The grids then converge on compact subintervals of \((0,\infty)\) to a geometric mesh \(\mathcal M\) unbounded above. If \(l>0\), its members are \(l,e^\eta l,e^{2\eta}l,\ldots\). If \(l=0\), it is a two-sided geometric mesh with no smallest member. To verify this description, choose a grid point in \((e^{-\eta},1]\), pass to its limit, and multiply it by the fixed factors \(e^{j\eta}\). When the smallest point stays positive its integer distance in the grid from that chosen point is bounded, so can also be fixed by subselection. Coincident phase endpoints give the same mesh.

For each \(\mu>0\) and \(\sigma\in\{-1,1\}\), let \(u_{\mu,\sigma}(t,z)\) be the unique solution of \[u=\sigma+\alpha\tanh\frac{\gamma(z-tu)}\mu.\] The left side minus the right side has derivative in \(u\) equal to \(1+t\alpha\gamma S(\gamma(z-tu)/\mu)/\mu>0\); its values have opposite signs for sufficiently large positive and negative \(u\). This proves existence and uniqueness. The implicit function theorem gives smoothness, and \[|u_{\mu,\sigma}-\sigma|\leq\alpha, \qquad \partial_z u_{\mu,\sigma} =\frac{\alpha\gamma S(\gamma(z-tu)/\mu)} {\mu+t\alpha\gamma S(\gamma(z-tu)/\mu)}\geq0.\] On an open mesh annulus \(\mu<|y|<e^\eta\mu\), define the nonnegative velocity measure \[ \begin{split} B(t,x,\,\mathrm dp) &=\sum_{\sigma=\pm1}b_{\mu,\sigma}(t,z) \delta_{u_{\mu,\sigma}(t,z)e_3}(\,\mathrm dp),\\ b_{\mu,\sigma}(t,z) &=\frac{\theta K S(\gamma(z-tu_{\mu,\sigma})/\mu)} {\mu+t\alpha\gamma S(\gamma(z-tu_{\mu,\sigma})/\mu)}. \end{split} \tag{72}\] Set \(B=0\) in the hole \(|y|<l\) if \(l>0\), and assign arbitrary values obeying the same bounds on annular interfaces and the transverse axis. These exceptional positions have zero Lebesgue measure. In particular, for every \(k\geq0\), \[ \int\langle p\rangle^kB(t,x,\,\mathrm dp) \leq\frac{C_k}{t+\mu}\leq\frac{C_k}{t} \quad(t>0) \tag{73}\] on an active annulus. If \(l>0\), the bound is instead uniformly \(C_k/(t+l)\) for all \(t\geq0\).

Cold convergence: survival, Jacobian, and stripe averaging

We first identify the cold measure independently of the hot limit. Fix a compact time-space region \[a\leq t\leq A<U,\qquad |x|\leq R,\] where \(a>0\). Since \(U_N\to U\) and \(A<U\), eventually \(r_NA<T_N\). The cap therefore holds on the entire initial physical interval \([0,r_NA]\), as required for the survival estimate below. By Proposition 14, the dilated cold velocity moments are bounded there, and indeed uniformly in position, by \[ \int\langle p\rangle^k C_{r_N}(t,x,p)\,\mathrm dp \leq C_k\left(\frac1{t+l_N}+1\right). \tag{76}\] The escaped tails have the still smaller bound \(C_M r_N^{M+1}t^Me^{-4|p|^2}\) for any fixed \(M>0\). Uniformly decreasing velocity tails hold under these moment bounds.

First restrict to annuli with \(\mu_{N,j}\) in a fixed compact interval \([\varepsilon,R_1]\subset(0,\infty)\). There are only finitely many of them, uniformly in \(N\). Apply Corollary 15 at physical time \(r_Nt\) to a color of scale \(s=r_N\mu_{N,j}\). Its optical depth is bounded, uniformly for \(0\le t\le A\), by \(C_A(r_Nd_s/s+r_N^2+r_Ns_N)+C_{A,M}r_N^{M+1}\). Here \[\frac{r_Nd_s}{s}=d_0r_N^{10}\mu_{N,j}^{9},\qquad r_N^2+r_Ns_N\leq2r_N^2,\] so the corollary gives the same-color comparison \[e^{-\epsilon_N}c_{i,\mathrm{free}}\leq c_i \leq c_{i,\mathrm{free}}+C(d_s/s)m_i, \qquad \epsilon_N\longrightarrow0\] uniformly on these scales. The lower bound covers the full free core because its velocity parameter lies in the compact support of \(J\). After multiplication by \(r_N\), the remainder has velocity mass \[ r_N O(d_s/s^2)=O(d_0r_N^9\mu_{N,j}^8). \tag{77}\] The core moment bound and this estimate show that the dilated actual color and its free core differ by \(o(1)\) in velocity total variation, uniformly on compact regions of these annuli. The same holds with each fixed polynomial velocity weight, using compact support of the free-core parameter and the Gaussian remainder.

We compute the free core exactly. Fix one sign and one such annulus, write \(\mu_N=s/r_N\), and use the velocity parameters \[\bar p=d_s\bar w,\qquad p_3=\sigma+\alpha\tanh\frac{\gamma(z-tp_3)}{\mu_N}+d_sw_3.\] The axial map depends only on \(w_3\). Its derivative is \[\frac{\partial p_3}{\partial w_3} =\frac{d_s} {1+t\alpha\gamma S(\gamma(z-tp_3)/\mu_N)/\mu_N}.\] Thus Equation (50), after dilation, says that the velocity integral of the free core against a test \(\psi\) is \[ \begin{split} \int r_N c_{i,\mathrm{free}}(r_Nt,r_Nx,p)\psi(p)\,\mathrm dp ={}&\int J(w) \frac{K S(\gamma(z-tp_3)/\mu_N)} {\mu_N+t\alpha\gamma S(\gamma(z-tp_3)/\mu_N)} \psi(d_s\bar w,p_3)\\ &\quad\cdot\mathbf 1_E(r_Ny-r_Ntd_s\bar w) \mathbf 1_{\{|r_N(z-tp_3)|<1/8\}}\,\mathrm dw. \end{split} \tag{78}\] On the compact regions under consideration the last axial indicator equals one for all sufficiently large \(N\). Since the implicit velocity map has derivative at least one in \(p_3\), \[|p_3-u_{\mu_N,\sigma}(t,z)|\leq d_s|w_3|.\] After the mesh subselection, \(\mu_N\to\mu>0\), so the velocity test and the smooth amplitude in (78) converge uniformly for \(w\in\mathop{\mathrm{supp}}J\) to \[\psi(u_{\mu,\sigma}e_3) \frac{K S(\gamma(z-tu_{\mu,\sigma})/\mu)} {\mu+t\alpha\gamma S(\gamma(z-tu_{\mu,\sigma})/\mu)}.\]

It remains to average the transverse indicator. In dilated coordinates its period is \[\frac{s^2}{r_N}=r_N\mu_N^2\longrightarrow0.\] The transverse shift of its argument is \(td_s\bar w\), whose ratio to that period is at most \[\frac{Ad_s}{r_N\mu_N^2}=A d_0 r_N^9\mu_N^8\longrightarrow0.\] For the indicator of the fixed periodic stripe pattern, a translation by this vanishing relative amount changes its integral on a bounded set by \(o(1)\), uniformly in its phase and in bounded \(t,w\). One can verify this by summing the lengths of the small intervals around stripe endpoints. The annular margins have dilated width \(O(s^2/r_N)=O(r_N\mu_N^2)\), so their area also tends to zero on the annulus. The outer annular radii converge; their boundaries can likewise be removed in sets of arbitrarily small area.

Finally the periodic stripe indicators converge weak-* to their mean \(\theta\). Explicitly, 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 arbitrary phase \(\zeta\), \[\left|\int f(y_1)(q(y_1/p+\zeta)-\theta)\,\mathrm dy_1\right| \leq p\|Q\|_\infty\|f'\|_1.\] Fubini’s theorem gives the same assertion with the other coordinates and parameters present. Approximation in \(L^1\) extends it to the annular restrictions. Integrating \(J\), whose mass is one, in (78) produces exactly the coefficient \(b_{\mu,\sigma}\) in (72).

These arguments identify the cold limit on all compact positive scale annuli. To remove the lower scale restriction, use (76): on the slab \(t\geq a\), the integral over \(|y|<C\varepsilon\), with bounded \(z\), is at most \(C_{a,A,R,k}\varepsilon^2\). The same estimate holds for the candidate \(B\). Scales larger than a fixed multiple of \(R\) have no main color in the chosen spatial compact set, and their escaped tails vanish. We have proved that for every continuous compactly supported velocity test \(\psi\), \[ \int C_{r_N}(t,x,p)\psi(p)\,\mathrm dp \stackrel{*}{\rightharpoonup} \int\psi(p)B(t,x,\,\mathrm dp) \quad\hbox{in local }L^\infty_{t,x} \tag{79}\] on \((0,U)\times\mathbb R^3\). Indeed the computation first gives weak convergence against smooth compact time-space tests, and the common \(L^\infty\) bound extends it to all \(L^1\) tests. Polynomial velocity tails can also be removed uniformly. If \(l>0\), choose \(\varepsilon<l/2\); all the bounds used above hold down to time zero, so (79) does too.

The hot equation and the size of every error

Under (70), the derivative of \(r_N^{-1}h_N(r_Nt,r_Nx,v)\) is the old transport derivative of \(h_N\). Therefore the rescaled equation is exactly \[ D_th_{r_N} =2P^{b_N}(C_{r_N},h_{r_N}) -h_{r_N}\nu^{b_N}_{C_{r_N}}+E_N, \tag{80}\] where \[\begin{split} E_N={}&r_N^2\bigl[P^{b_N}(h_{r_N},h_{r_N}) -h_{r_N}\nu^{b_N}_{h_{r_N}}\bigr]+S_N,\\ S_N(t,x,v)={}&\sum_{i\ne j} P^{b_N}(c_{i,N},c_{j,N})(r_Nt,r_Nx,v). \end{split}\] There is no additional factor in \(S_N\). Its smallness comes from the physical mixed-cold source estimate: for every sufficiently large fixed \(M\), \[ \begin{split} 0\leq S_N(t,x,v)&\leq C_M r_N^{M-1}t^{M-1}H(v),\\ r_N^2P^{b_N}(h_{r_N},h_{r_N}) &\leq C r_N^2(t+l_N)^2H(v),\\ r_N^2h_{r_N}\nu^{b_N}_{h_{r_N}} &\leq C r_N^2(t+l_N)^2\langle v\rangle H(v). \end{split} \tag{81}\] The second line uses Lemma 9; the last line uses the first velocity moment of \(H\). Thus on every fixed compact time interval, \(E_N\to0\) in local spacetime \(L^1\), with an error bounded by a constant times \(r_N^2+r_N^{M-1}\) after integration over a fixed spatial compact set. Choose \(M>1\).

The main cold sheets and Lemma 6 also give \[P^{b_N}(C_{r_N},H)\leq C\left(\frac1{t+l_N}+1\right)H, \qquad \nu^{b_N}_{C_{r_N}}(v)\leq C\left(\frac1{t+l_N}+1\right)\langle v\rangle.\] The escaped cold component is bounded by a vanishing multiple of \(e^{-4|v|^2}\leq CH\), so its gain is covered by Lemma 9. Equations (71) and (80) now imply, on \(a\leq t\leq A<U\), \[ |D_th_{r_N}(t,x,v)|\leq C_{a,A}\langle v\rangle H(v). \tag{82}\] This is a genuine integrable velocity majorant, uniform in position. If \(l>0\), the same conclusion holds on \(0\leq t\leq A\), since \(l_N\geq l/2\) eventually.

The cap gives weak compactness of \(h_{r_N}\) in local \(L^1\), and (82) permits application of Lemma 19. Subselect on an exhaustion of \((0,U)\times\mathbb R^3\). The resulting limit \(g\) satisfies the bound in (74), and each compactly supported velocity average converges strongly in local \(L^1_{t,x}\).

Passing the cold–hot products

The dilated cold densities concentrate in velocity, so the two-density compactness argument of Section 6 does not apply to \(C_{r_N}\). The needed product passage uses (79) and hot averaging instead.

For compact velocity tests \(\psi,\chi\), put \[a_N(t,x)=\int\psi(p)C_{r_N}(t,x,p)\,\mathrm dp,\qquad \zeta_N(t,x)=\int\chi(v)h_{r_N}(t,x,v)\,\mathrm dv.\] On every compact positive-time region, \(a_N\) is bounded in \(L^\infty\) and converges weak-* to \(a=\int\psi\,\mathrm dB\), while \(\zeta_N\to\zeta=\int\chi g\,\mathrm dv\) strongly in \(L^1\). For a bounded time-space test \(\varphi\), \[\int\varphi(a_N\zeta_N-a\zeta) =\int\varphi a_N(\zeta_N-\zeta) +\int\varphi(a_N-a)\zeta\longrightarrow0.\] Finite sums of separated velocity tests approximate continuous tests uniformly on compact velocity products. The total product mass is uniformly bounded on compact time-space regions by (76) and the cap. We conclude \[ C_{r_N}(t,x,p)h_{r_N}(t,x,v)\,\mathrm dp\,\mathrm dv\,\mathrm dx\,\mathrm dt \rightharpoonup B(t,x,\,\mathrm dp)g(t,x,v)\,\mathrm dv\,\mathrm dx\,\mathrm dt \tag{83}\] against continuous compact velocity tests, with arbitrary smooth compact time-space dependence. Uniform moment bounds in both velocities allow removal of velocity cutoffs in tests with any fixed polynomial growth for which they are used below.

Here is an explicit weak collision test suitable for this convergence. For a continuous output test \(\phi\), let \[\mathcal K_\phi(p,v) =\frac{|p-v|}{2}\int_{S^2} \phi\left(\frac{p+v}{2}+\frac{|p-v|}{2}\omega\right)\,\mathrm d\omega.\] Lemma 3 gives \(\int\phi P(c,h)=\iint c(p)h(v)\mathcal K_\phi(p,v)\,\mathrm dp\,\mathrm dv\). The test \(\mathcal K_\phi\) is continuous in both incoming velocities, including \(p=v\), and is bounded in absolute value by \(2\pi|p-v|\|\phi\|_\infty\). The loss test is \(2\pi|p-v|\phi(v)\), with the same properties. Inserting smooth time-space dependence in \(\phi\) does not change these facts. On bounded velocity products the factor \(\mathbf 1_{\{|p-v|<b_N\}}\) is eventually one; its complement is uniformly negligible by the weighted product tails. Applying (83) proves the required gain and loss limits.

The limiting gain is an integrable function, rather than an additional singular output measure. Indeed each summand of \(B\) is an allowed zero-width axial sheet in Lemma 6; hence on \(a\leq t\leq A\), \[0\leq P(B,g)\leq C_{a,A}H, \qquad 0\leq g\nu_B\leq C_{a,A}\langle v\rangle H.\] Equivalently the weak gain measure is dominated by this integrable function, which also verifies absolute continuity at the velocity axis. Passing (80), using (81), proves the equation in (74).

A positive lower mesh endpoint and the local mild formula

When \(l>0\), (82) holds uniformly down to zero. The rescaled initial hot density, on every fixed spatial compact set for large \(N\), is exactly \[h_{r_N}(0,x,v)=\lambda_N l_N\chi(x/l_N,v).\] The other periodic copies have moved outside that compact set. Subselect \(\lambda_N\to\lambda_0\). Smoothness and compact support of \(\chi\), and \(l_N\to l>0\), give strong local \(L^1\) convergence to (75). Passing the weak transport identity with tests touching time zero is justified by the uniform source majorant and yields that initial trace for \(g\). The cold product argument also extends to zero in this case, as already noted. For \(l=0\) the coefficient can be of order \(1/t\), and the proposition only asserts the equation on the open positive-time interval.

On any \([a,A]\subset(0,U)\), the limiting transport source is locally integrable and has the preceding velocity majorant uniformly in position. Changing to free coordinates as in (69), 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 \(D_tg+\nu_Bg=2P(B,g)\). Its loss coefficient is locally integrable along the path, since \(\nu_B(t,x,v)\leq C_a\langle v\rangle\) for \(t\geq a\). The integrating factor therefore gives, for \(0<a<t<U\), \[ \begin{split} g(t,x,v)={}&g(a,x-(t-a)v,v) \exp\left[-\int_a^t\nu_B(\tau,x-(t-\tau)v,v)\,\mathrm d\tau\right]\\ &+2\int_a^t P(B,g)(s,x-(t-s)v,v) \exp\left[-\int_s^t\nu_B(\tau,x-(t-\tau)v,v)\,\mathrm d\tau\right]\,\mathrm ds. \end{split} \tag{84}\] This is an almost-everywhere identity, or equivalently an identity in local \(L^1\) at the displayed times. It holds with \(a=0\) when \(l>0\). Altering the medium on the null annular interfaces does not change the identity: free transport preserves phase-space measure, so Fubini’s theorem discards the affected set of paths. These statements complete the proof of Proposition 20.

A positive majorizing equation and its kernels

For this section relabel the sequence by \(n\), set \(U_n=T_n/r_n\), and write \[r_n=t_{*,n}+s_{N_n},\quad \ell_n=\frac{s_{N_n}}{r_n},\quad \tau_n=\frac{t_{*,n}}{r_n}=1-\ell_n,\quad h_n=h_{r_n},\quad C_n=C_{r_n},\quad f_n=\frac{h_n}{H}.\] Then \(\ell_n\to l\). In particular \(0\le\ell_n\le1\), \(0\le\tau_n\le1\). The cap gives, on \(0\le t\le U_n\), \[ 0\le f_n(t,x,v)\le t+\ell_n,\qquad \left\|f_n(\tau_n)\right\|_\infty=1. \tag{85}\] Essential supremum norms and pointwise formulas involving \(H\) always ignore the velocity axis, a null set. The mild representatives of the finite approximations will be used at the cap times. Choose Borel representatives when composing densities with the maps below, enforcing their asserted bounds on exceptional null sets as well. Characteristic identities and endpoint choices are still made only on their common full-measure sets.

For a fixed \(a>0\), define the backward Volterra operator \[ (\mathcal V_n F)(t,x,v) = \int_a^t\int F\bigl(s,x-(t-s)v,\xi\bigr) \mathcal K_n\bigl(s,x-(t-s)v,v;\,\mathrm dp\,\,\mathrm d\xi\bigr)\,\mathrm ds . \tag{89}\] Representatives in the finite-temperature iteration. For each finite \(n\), the core measure has a velocity density, \(A_n(s,x,\,\mathrm dp)=a_n(s,x,p)\,\mathrm dp\). Integrating first in \(p\) in (88) shows that, for \(\xi\ne v\), its input marginal is \[\mathcal K_n(s,x,v;\mathbb R^3,\,\mathrm d\xi) =\frac{4H(\xi)}{|\xi-v|H(v)} \left[\int_{(v-p)\cdot(\xi-v)=0} a_n(s,x,p)\,\mathrm dA_p\right]\,\mathrm d\xi.\] There is no atom at \(\xi=v\): each plane-area measure gives that point zero mass, and \(p=v\) is null for the cold density. For a null set in \((s,X,\xi)\), integrate also in the 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 \(\mathcal V_nF(t)\) only on a null set of endpoints, at each fixed terminal time. Finite repetition preserves this property. A countable sequence of lower times \(a\downarrow0\) 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 Lemma 22 give, for a bounded nonnegative \(F\), \[ \|\mathcal V_n^jF(t)\|_\infty \le \|F\|_\infty\frac{\bigl(C\log(t/a)\bigr)^j}{j!}, \qquad a\le t\le1. \tag{90}\] 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

We begin the proof of Proposition 21. The case \(l>0\) is settled directly. For \(l=0\), the cap will force 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 shot parameters converge, \(\lambda_n\to\lambda_\infty\). If \(l>0\) and \(\lambda_\infty>0\), Proposition 20 gives the initial hot trace \[g(0,x,v)=\lambda_\infty l\,\chi(x/l,v).\] This trace is nonzero. More explicitly, the limiting loss frequency obeys \[\nu_B(t,x,v)\le \frac{C(1+|v|)}{t+l}.\] On any bounded velocity set and any finite interval beginning at zero its integral along a flight is finite. The positive mild formula for (74) therefore 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 \(\lambda_\infty=0\), use Lemma 22 from time zero. The initial ratio has norm at most \(\lambda_n\ell_n\|\chi/H\|_\infty\), independently of the spatial scaling of the bump. On \(0\le t\le\tau_n\le1\), eventually \(t+\ell_n\ge l/2\). The positive integral inequality and the elementary scalar integrating-factor estimate give \[\sup_{0\le t\le\tau_n}\|f_n(t)\|_\infty \le \bigl(\lambda_n\ell_n\|\chi/H\|_\infty+\varepsilon_n\bigr) e^{2C/l}\longrightarrow0.\] This contradicts (85). Both alternatives also apply when \(l=1\), even if \(\tau_n\to0\) or some hit occurs at time zero.

It remains to consider \(l=0\), so \(\tau_n\to1\). 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 a compact subset of 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 \(\int_K h_n\to0\). Away from the velocity axis \(H\) has a positive lower bound on compact velocity sets. Consequently \[ f_n\longrightarrow0 \quad\text{in measure on compact subsets of } (0,U)\times\mathbb R^3_x\times\{\bar v\ne0\}. \tag{91}\]

Fix \(0<a<1/4\), so that \(a<\tau_n\) for large \(n\). The positive mild inequality (86) reads \[f_n\le d_n+\mathcal V_nf_n+e_n,\] where \(d_n(t,x,v)= f_n(a,x-(t-a)v,v)\), \(\|d_n\|_\infty\le a+\ell_n\), and \(0\le e_n\le\varepsilon_n\) on this unit-length interval. Iterate the gain term five times. Using positivity and (90) gives \[ f_n(t,x,v) \le (a+\ell_n+\varepsilon_n) \sum_{j=0}^4\frac{(C\log(1/a))^j}{j!} +(\mathcal V_n^5 f_n)(t,x,v), \qquad a\le t\le\tau_n . \tag{92}\] Every formula is used on a common full-measure set of characteristics for the finite approximation.

The term outside \(\mathcal V_n^5f_n\) becomes arbitrarily small when first \(n\to\infty\) and then \(a\downarrow0\). We therefore need to show that the five-fold term tends to zero at endpoints carrying a fixed positive fraction of the cap. We first control those endpoints, then separate axial and nonaxial limiting velocities.

Compactness of the relevant endpoints

Two gains at an axial endpoint

For \(\mu>0\) and \(\sigma\in\{-1,1\}\), denote by \(u_{\mu,\sigma}(t,z)\) the solution of \[u=\sigma+\alpha\tanh\bigl(\gamma(z-tu)/\mu\bigr).\] This notation extends the profile independently of every annular or stripe mask.

Finite-dimensional regularization away from the axis

The next two lemmas make explicit the relation between plane area, redundant angular parameters, and the pullback of sets of small spacetime–velocity volume.

Completion of the nonvanishing argument

Positive histories and their normalization

For \(v,w\in\mathbb R^3\) and \(n\in S^2\), set \[ \Phi(v,w,n)=\frac{v+w}{2}+\frac{|v-w|}{2}n, \qquad \,\mathrm d\varsigma(n)=\frac{\,\mathrm dn}{4\pi}. \tag{101}\] Thus \(\varsigma\) is a probability measure. By Lemma 3, a collision with incident velocities \(v,w\) has rate \(2\pi|v-w|\), and the marginal law of either child is the pushforward of \(\varsigma\) under \(\Phi(v,w,\cdot)\). The two marginal laws agree because \(n\mapsto-n\) preserves \(\varsigma\). Consequently, for a nonnegative velocity measure \(B\) and a density \(f\), \[ \int_{\mathbb R^3}P(B,f)(v)\,\mathrm dv=\int_{\mathbb R^3}\nu_B(v)f(v)\,\mathrm dv. \tag{102}\] The coefficient \(2\) in the gain of (74) therefore counts both children, while the loss removes their parent once.

More generally, fix Borel representatives of the coefficients of a two-beam medium \[\widetilde B(t,x)=\sum_{\sigma=\pm1}b_\sigma(t,x) \delta_{w_\sigma(t,x)},\qquad q_\sigma(t,x,v)=2\pi b_\sigma(t,x)|v-w_\sigma(t,x)|, \qquad \nu=\sum_\sigma q_\sigma.\] All pointwise path conditions below refer to these representatives. On \([a,b]\), a history with ordered collision times \(a<\tau_1<\cdots<\tau_j<b\), signs \(\sigma_i\), and directions \(n_i\) follows free flights between collisions and uses \[V_{\tau_i}=\Phi(V_{\tau_i-},w_{\sigma_i}(\tau_i,X_{\tau_i}),n_i).\] Its weight relative to \(\prod_i\,\mathrm d\tau_i\,\mathrm d\varsigma(n_i)\) is \[ \exp\left(-\int_a^b\nu(s,X_s,V_s)\,\mathrm ds\right) \prod_{i=1}^j2q_{\sigma_i}(\tau_i,X_{\tau_i},V_{\tau_i-}). \tag{103}\] For \(j=0\) the empty product is one.

For the background (72), \(|u_\sigma|\le1+\alpha\) and \[ 0\le b_\sigma(t,x)\le\frac{\theta K}{t\alpha\gamma}, \qquad \nu_B(t,x,v)\le\frac{C(1+|v|)}{t}\quad(t>0). \tag{104}\] At each mesh interface choose the value from the outer adjacent annulus, and on the transverse axis set the medium to zero. These are Borel representatives obeying the same bounds. All bounded finite flights starting at positive time have strictly positive survival. The cap and Lemma 6 give locally integrable collision terms. The local transport formula in Proposition 20 therefore supplies time traces of \(g\) on bounded regions. Testing nonnegative compact functions and taking a time limit shows that \(g(t)\le(t+l)H\) also holds at these traces. We use this representative.

A reference flight with a positive potential

We convert the branching history weights into expectations along a single Markov flight with a positive potential. This is a jump-process Feynman–Kac representation, related to the many-to-one formulas for branching systems; see (Harris and Roberts 2017, sec. 4.1) for general probabilistic context. The finite-history calculation below proves the precise representation for the present collision law.

Fix \[D=5,\qquad E=\{v:|v|\le D\},\qquad B_0=\delta_{e_3}+\delta_{-e_3},\qquad \nu_0(v)=2\pi\bigl(|v-e_3|+|v+e_3|\bigr).\] Let \(J(v,\,\mathrm dv')\) be the single-child law obtained by choosing a sign with probabilities proportional to \(|v-\sigma e_3|\) and then using (101). Put \(p(v)=J(v,E)\).

Three collisions give a volume minorization

Write \(P_t(v,\cdot)\) for the transition law of the preceding velocity process. We establish a Doeblin minorization, the classical common-mass condition for mixing of Markov chains (Doblin 1940). The three-collision geometry proving this condition and the resulting contraction argument are both given below.

A displacement block and measurable perturbations

Fix \(d=1/10\), and choose once and for all \[ 0<\kappa<\min\left\{\frac14,\frac{c_0}{2C_0},\frac{r_0}{4}\right\}. \tag{108}\] In particular a small ball about \(\kappa e_1\) lies in the minorization ball.

To repeat growth in the cold background, we will transport mass through boxes whose transverse distance from the axis grows proportionally to time: \[\mathcal B_t=\{(x,v):|x-\kappa te_1|\le d\kappa t,\ |v|\le D\}, \qquad t>0.\] For a duration \(\Delta>0\), a displacement within \(d\kappa\Delta\) of \(\kappa\Delta e_1\) carries every starting point of \(\mathcal B_t\) into \(\mathcal B_{t+\Delta}\), provided the final velocity remains in \(E\). We first obtain growth with this displacement in the reference medium, then transfer it to the varying cold background.

Uniform choices of the construction parameters

We first reserve one more small geometric tolerance. There is \(\alpha_{\mathrm{acc}}>0\), depending only on \(\kappa\) and Lemma 33, with this property: after a first collision puts a velocity sufficiently close to \(\sigma e_3\), two more collisions, with partners within \(\alpha_{\mathrm{acc}}\) of \(-\sigma e_3,e_3\), have positive angular probability of ending in \[ B(\kappa e_1,d\kappa/4). \tag{112}\] Indeed the inverse-coordinate construction in that lemma, with first output exactly \(\sigma e_3\), maps a positive-measure angular set into \(B(\kappa e_1,d\kappa/8)\), which is inside \(B(0,r_0)\). Keep a compact subset of positive measure in this set. Its outputs have a positive distance from the complement of (112); the intermediate velocities and relative speeds remain in the neighborhoods of that construction. Uniform continuity of the two sphere maps preserves this for sufficiently small perturbations of the first output and partners, including measurable partners at the actual path positions. Take the smaller tolerance for the two possible signs. We also require \(\alpha_{\mathrm{acc}}\) to be less than half the permitted radius of the first-output neighborhood.

The constants \(\kappa,L,J,\varepsilon_*,\alpha_{\mathrm{acc}}\) are now fixed and do not involve any construction parameter. Fix preliminary bounds \(\eta\le1\), \(\alpha\le1\), \(K\ge1\). For \(t>0\), put \[ \Delta=\frac{L\kappa t}{\theta K}, \qquad a=\frac{L\kappa}{\theta K}. \tag{113}\] At a starting point \(x_0=(y_0,z_0)\) in \(\mathcal B_t\), \[(1-d)\kappa t\le|y_0|\le(1+d)\kappa t.\] Every speed-\(D\) excursion over \([t,t+\Delta]\) satisfies \[\big||y|-|y_0|\big|\le D\Delta,\qquad |z|\le d\kappa t+D\Delta.\] Set \(\delta_K=DL/(\theta K(1-d))\). By increasing \(K\), impose \(\delta_K\le1/4\) and \(a\le1/4\). Then \[|y|\ge(1-\delta_K)(1-d)\kappa t.\] For all sufficiently large \(t\), every such excursion is outside the lower mesh hole, since \(l\le1\). On its active annulus \(e^{-\eta}|y|\le\mu\le|y|\), for every mesh phase. Thus, with a fixed constant \(C_1\) depending only on \(d,D\), \[ \begin{split} \left|\frac{|y_0|}{\mu}-1\right|&\le C_1(\eta+\delta_K),\\ \frac{\tau}{\mu}&\le\frac{C_1}{\kappa},\\ \frac{|z-\tau u_\sigma|}{\mu}&\le\frac{C_1}{\kappa} \qquad(t\le\tau\le t+\Delta). \end{split} \tag{114}\] The last estimate uses \(|u_\sigma|\le2\).

Use the reference density factor \(\rho_0=\theta K/|y_0|\), and multiply both time and position units by \(\rho_0\); velocities are unchanged. The reference duration is exactly \[ A=\rho_0\Delta=\frac{L\kappa t}{|y_0|}\in I_L. \tag{115}\] With \(S_\sigma=S(\gamma(z-\tau u_\sigma)/\mu)\), the density ratio in these units is \[ \frac{b_\sigma(\tau,x)}{\rho_0} =\frac{|y_0|}{\mu}\, \frac{S_\sigma}{1+(\tau\alpha\gamma/\mu)S_\sigma}. \tag{116}\] Because \(1-\operatorname{sech}^2 h=\tanh^2h\le h^2\), (114) gives, when the right-hand small quantities are at most one, \[ \left|\frac{b_\sigma}{\rho_0}-1\right| \le C_2\left(\eta+\delta_K+ (\gamma/\kappa)^2+\alpha\gamma/\kappa\right), \qquad |u_\sigma-\sigma|\le\alpha. \tag{117}\] Here \(C_2\) is independent of \(t,l\), and mesh phase.

Choose \(\eta>0\) small, then \[0<\alpha<\min\{1/4,\varepsilon_*,\alpha_{\mathrm{acc}}\},\] then \(\gamma>0\) small, so that the first, third, and fourth terms on the right of (117), including their factor \(C_2\), sum to less than \(\varepsilon_*/2\). Finally increase \(K\) so the \(\delta_K\) term is less than \(\varepsilon_*/2\), the preliminary restrictions hold, and \[ (1+a)^4<3. \tag{118}\] This establishes (110) for all the scaled blocks, with no choice depending on a limit, mesh, or solution.

These choices precede the final small geometric and thermal parameters. Making \(\alpha\gamma\) small may increase the fixed coefficient in (104) and earlier comparison bounds, but introduces no dependence on the approximation index. Propositions 14 and 16 allow \(s_0,d_0\), and then the final short horizon \(T_0\), to be chosen after these fixed constants. We use those choices henceforth.

Access to a late box and the growth contradiction