The general Maxwellian long-time asymptotics conjecture for collision-dominated homoenergetic flows

Let $B$ be a collision cross-section satisfying condition, and let $\gamma$ denote its homogeneity. Let $Q(t)$ and $L_0$ be the matrix-valued quantities appearing in the homoenergetic Boltzmann equation, let $\mu(t)$ be its time-dependent coefficient, and let $\beta(t)$ be the temperature scale. Let $\mathbb{L}$ be the linearized collision operator on the weighted space $L^2(\mathbb{R}^3;e^{-\lvert\xi\rvert^2}d\xi)$. General Maxwellian asymptotics conjecture. Under suitable assumptions on $\gamma$ and $\mu(t)$, there exists a weak solution $g$ in the stated measure-valued solution class and a function $\beta(t)$ such that

$$\beta(t)^{-\frac32}g\left(t,\frac{\xi}{\sqrt{\beta(t)}}\right)\longrightarrow \pi^{-\frac32}e^{-\lvert\xi\rvert^2}$$

in $L^2(\mathbb{R}^3;e^{-\lvert\xi\rvert^2}d\xi)$ as $t\to\infty$. In case 1, if $Q(t)=L_0+O(t^{-1-\delta})$, $\delta>0$, $\operatorname{Tr}(L_0)\ne0$, $a=\frac23\operatorname{Tr}(L_0)$, and $\mu(t)e^{-\gamma at/2}\gg t^{1+\delta}$, then $\beta(t)=Ce^{at}(1+o(1))$. In case 2, if $\gamma>0$, $\operatorname{Tr}(Q(t))=0$, $Q(t)=L_0+o(1)$ with $L_0\ne0$, $\lambda(t)=\int_0^t ds/\mu(s)\to\infty$, $\lambda'(t)/\lambda(t)\to0$, and $b=\langle \xi\cdot L_0\xi,(-\mathbb{L})^{-1}[\xi\cdot L_0\xi]\rangle_w>0$, then $\beta(t)=((4/3)\gamma b\lambda(t))^{-2/\gamma}(1+o(1))$. These asymptotic existence and Maxwellian claims are proposed as the main general collision-dominated result; the paper supplies heuristic support through the Hilbert-expansion strategy, but no proof or resolution is given.