Exact homogeneous BGK fast relaxation

Adopt Countable Choice, denoted ${\mathrm{A}\mathrm{C}}_{\omega }$, as in the kinetic-density definition. Let $v_{\ast }=1\mathrm{m}{\mathrm{s}}^{-1}$, ${\omega }_3(v):={\sum }_{j=0}^3(\mathrm{\mid }v\mathrm{\mid }\mathrm{/}{v}_{\ast }{)}^j$, let $\tau >0$ be a relaxation time in seconds, and let $ϵ>0$ be dimensionless. Let $f_0:{\mathbb{R}}^3\to [0,\mathrm{\infty })$ be a finite-valued measurable velocity density with units fixed by the kinetic-density definition, with $\int {\omega }_3{f}_{0}dv<\mathrm{\infty }$, number density $n>0$, and strictly positive centered second moment. Let $M=M[f_0]$ be its positive-temperature Gaussian from the single-species BGK definition. Fix a constant reference density $f_{\mathrm{r}\mathrm{e}\mathrm{f}}>0$ with the same units as $f_0$ and assume $\int f_0\mathrm{\mid }\log (f_0\mathrm{/}{f}_{\mathrm{r}\mathrm{e}\mathrm{f}})\mathrm{\mid }dv<\mathrm{\infty }$, with $0\log 0:=0$ and the integrand $f_0\mathrm{\mid }\log (f_0\mathrm{/}{f}_{\mathrm{r}\mathrm{e}\mathrm{f}})\mathrm{\mid }$ assigned value zero where $f_0=0$. Define $f^{ϵ}(t,v):=M(v)+e^{-t\mathrm{/}(ϵ\tau )}(f_0(v)-M(v)),t\ge 0\text{ in seconds}\mathrm{.}$ Then $f^{ϵ}\ge 0$, has the same mass, momentum, and kinetic energy as $f_0$ for every $t$, belongs to $C^1([0,\mathrm{\infty });L^1({\omega }_{3}dv))$, and solves, as an equality in $L^1({\omega }_{3}dv)$, ${\mathrm{\partial }}_t{f}^{ϵ}=\frac{M[f^{ϵ}]-f^{ϵ}}{ϵ\tau },f^{ϵ}(0)=f_0\mathrm{.}$ A solution means a nonnegative $C^1([0,\mathrm{\infty });L^1({\omega }_{3}dv))$ function satisfying this equality at each time. The solution is unique among nonnegative solutions in this function class whose density and temperature remain strictly positive. Moreover, ${\int }_{{\mathbb{R}}^3}{\omega }_3\mathrm{\mid }{f}^{ϵ}(t)-M\mathrm{\mid }dv=e^{-t\mathrm{/}(ϵ\tau )}{\int }_{{\mathbb{R}}^3}{\omega }_3\mathrm{\mid }{f}_0-M\mathrm{\mid }dv,$ and for every $T_{\ast }>0$, ${\int }_0^{T_{\ast }}{\int }_{{\mathbb{R}}^3}{\omega }_3\mathrm{\mid }{f}^{ϵ}-M\mathrm{\mid }dvdt\le ϵ\tau {\int }_{{\mathbb{R}}^3}{\omega }_3\mathrm{\mid }{f}_0-M\mathrm{\mid }dv\mathrm{.}$ Thus the equilibrium limit is exact for each fixed $t>0$ and in spacetime weighted $L^1$, but need not be uniform at $t=0$. This branch has stationary homogeneous Euler moments; it gives no evolving inhomogeneous Euler limit.

Tong, §2.5.1, full lecture notes

Golse, §§3.1–3.2, full chapter