Conserving homogeneous BGK idealization

Adopt Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ as in the BGK definition: for every family $(X_j{)}_{j\in \mathbb{N}}$ of nonempty sets there is a function $a$ with domain $\mathbb{N}$ such that $a(j)\in X_j$ for every $j\in \mathbb{N}$.

Let $v_{\ast }=1\mathrm{m}{\mathrm{s}}^{-1}$ and ${\omega }_3(v)={\sum }_{j=0}^3(\mathrm{\mid }v\mathrm{\mid }\mathrm{/}{v}_{\ast }{)}^j$. A velocity density $g:{\mathbb{R}}^3\to [0,\mathrm{\infty })$ is admissible when it is finite-valued and Lebesgue measurable, $g{d}^{3}v$ is a number density in ${\mathrm{m}}^{-3}$ (so $g$ has units ${\mathrm{s}}^3{\mathrm{m}}^{-6}$), and ${\int }_{{\mathbb{R}}^3}{\omega }_3(v)g(v)d^{3}v<\mathrm{\infty }\mathrm{.}$ The primitive parameters are species mass $m>0$ in kilograms and Boltzmann constant $k_B>0$ in joules per kelvin. For an admissible $g$ with $n[g]:={\int }_{{\mathbb{R}}^3}g{d}^{3}v>0$ and positive centered second moment, define $u[g]:=\frac{1}{n[g]}{\int }_{{\mathbb{R}}^3}vg(v)d^{3}v,T[g]:=\frac{m}{3k_{B}n[g]}{\int }_{{\mathbb{R}}^3}\mathrm{\mid }v-u[g]{\mathrm{\mid }}^{2}g(v)d^{3}v\mathrm{.}$ The domain condition ensures $T[g]>0$. Define its moment-matched Maxwellian by $M[g](v)=n[g]{\left(\frac{m}{2\pi k_{B}T[g]}\right)}^{3\mathrm{/}2}\exp (-\frac{m\mathrm{\mid }v-u[g]{\mathrm{\mid }}^2}{2k_{B}T[g]})\mathrm{.}$ It has the same number, momentum, and kinetic-energy moments as $g$. For a constant relaxation time $\tau >0$ in seconds, adopt the homogeneous collision model $Q_{\mathrm{B}\mathrm{G}\mathrm{K}}(g)=\frac{M[g]-g}{\tau }\mathrm{.}$

Let $f_0$ be an admissible initial density with positive number density and positive centered second moment. For each dimensionless $\epsilon >0$, a trajectory considered in the accelerated model is a continuously differentiable map $f^{\epsilon }\in C^1\text{\hspace{0.17em}⁣}([0,\mathrm{\infty });L^1\text{\hspace{0.17em}⁣}({\mathbb{R}}^3,{\omega }_3(v)d^{3}v)),$ whose states have nonnegative admissible representatives with positive number density and positive centered second moment at every $t\ge 0$, and whose initial trace satisfies $f^{\epsilon }(0)=f_0$ in this weighted $L^1$ space. The derivative is the Banach-space derivative, with the right derivative at $t=0$. Adopt $\frac{d}{dt}{f}^{\epsilon }(t)=\frac{M[f^{\epsilon }(t)]-f^{\epsilon }(t)}{\epsilon \tau },t\ge 0,$ as an equality in $L^1({\mathbb{R}}^3,{\omega }_3(v)d^{3}v)$ at every $t\ge 0$.

The model also permits pointwise solution representatives $\tilde{f}:{\mathbb{R}}^3\times [0,\mathrm{\infty })\to [0,\mathrm{\infty })$ that are jointly measurable, have finite ${\omega }_3$-weighted velocity integral at every time, and satisfy $\tilde{f}(v,0)=f_0(v)$ for almost every $v$. Require their velocity sections to be continuous on $[0,\mathrm{\infty })$ and differentiable on $(0,\mathrm{\infty })$ for almost every $v$, with positive number density and centered second moment at every time. In this solution sense, require one full-measure set of velocities on which ${\mathrm{\partial }}_t\tilde{f}(v,t)=\frac{M[\tilde{f}(\cdot ,t)](v)-\tilde{f}(v,t)}{\epsilon \tau }$ holds for every $t>0$, and require on each compact time interval a velocity-integrable majorant for ${\omega }_3(v)\mathrm{\mid }{\mathrm{\partial }}_t\tilde{f}(v,t)\mathrm{\mid }$ independent of $t$. The Banach-space and pointwise equalities specify two solution senses for the same adopted model; neither asserts solution existence. This is a separate phenomenological relaxation model. It does not replace or follow from the binary Boltzmann postulate, and the relaxation-time form is not asserted to be an exact microscopic collision operator. No empirical premise or measured relaxation time is asserted.

Current draft mathematical definition of the moment-preserving Maxwellian relaxation operator: moment-preserving BGK model.

François Golse and C. David Levermore, Fluid Dynamics from Kinetic Equations, Lecture 2, “The BGK Operator”

David Tong, Lectures on Kinetic Theory, §2.5.1 (relaxation-time approximation and its non-exactness qualification)