I am currently trying to understand the following article "Thermalization of a rarefied gas with total energy conservation: existence, hypocoercivity, macroscopic limit" (2021) by Favre, Pirner, Schmeiser (https://arxiv.org/pdf/2012.07503). In this article, the authors prove the existence (Theorem 2, page 8) of the following nonlinear PDE system in one dimension:

\begin{align} \partial_t f + v \cdot \partial_x f &= \rho[f] \, M[T] - f, \\ \partial_t T - D \cdot \partial^2_x T &= \int_{\mathbb{R}^d} |v|^2 \big(f- \rho[f] M[T] \big) \, \mathrm{d}v = E[f]-\frac{1}{2} \rho[f] T, \end{align} for a the phase space density $f=f(x,v,t)$ on $\mathbb{T}^1 \times \mathbb{R} \times (0, \infty)$ and temperature $T = T(x,t)$ on $\mathbb{T}^1 \times \mathbb{R}$ with the nondimensionalized heat conductivity $D > 0$, with respect to the initial conditions \begin{aligned} f(x, v, 0) &= f_0(x, v) > 0, \\ T(x, 0) &= T_0(x) > 0. \end{aligned}

$\mathbb{T}^1$ is here the one-dimensional flat torus and $\rho[f](x,t) = \int_{\mathbb{R}} f(x,v,t) \, dv$ the macroscopic densitiy, $E[f](x,t) = \int_{\mathbb{R}} |v|^2f(x,v,t) \, dv$ the macroscopic energy density, \begin{equation} M[T](v) := \left( {\pi T} \right)^{-1/2} \exp \left( - \frac{|v|^2}{T} \right) \end{equation} is the well-known Maxwellian distribution.

It is a kinetic equation of BGK-type with heat-transfer coupling in one dimension.

However, no statement is made about uniqueness. Is this easily observable since the authors do not mention it? If not, how should one approach proving it, given that it is always very difficult with nonlinear problems? Or is that a very hard porblem? Or has this possibly already been shown in other papers? In that case, I would be very grateful for literature references! Thanks in advance.

In my opinion, the main difficulty would be to control the term $$ |M[T_1](v) - M[T_2](v)|$$ e.g. in the form $$ |M[T_1](v) - M[T_2](v)| \leq c(v) |T_1 - T_2|,$$ with $c > 0$ independent of $T_1, T_2.$ This was an issue in the existence result too, so a temperature cut-off was used. I suppose a possibility would be a entropy-dissipation argument, which is already used in the article. For that one would consider the Boltzmann-like-entropy $$ \mathcal{H}[(f_1,T_1),(f_2,T_2)](t) = \int_{\mathbb{T}^d} \left (\int_{\mathbb{R}^d} f_1 \log \left(\frac{f_1}{f_2} \right) \, dv - \log \left(\frac{T_1}{T_2} \right) + \frac{T_1}{T_2} - 1 \right )\, dx.$$ and try to show that the time derivative $\frac{d}{dt}\mathcal{H}(t)$ is monotone decreasing but unfortunately this involves very cumbersome terms.