Exact unclosed kinetic conservation laws

Let $I$ and the periodic cell $\mathrm{\Omega }$ be as in the kinetic-density definition, and let $f:I\times \mathrm{\Omega }\times {\mathbb{R}}^3\to (0,\mathrm{\infty })$ be continuously differentiable in $(t,x)$ and measurable in $v$. For each compact $I_0\subset I$ and each $\psi \in \{m,m{v}_1,m{v}_2,m{v}_3,m\mathrm{\mid }v{\mathrm{\mid }}^2\mathrm{/}2\}$, assume there are nonnegative functions $G_{\psi ,0,I_0},G_{\psi ,j,I_0}\in L^1({\mathbb{R}}^3,dv)$ for $j=1,2,3$, independent of $(t,x)\in I_0\times \mathrm{\Omega }$, such that for almost every $v$ and every $(t,x)\in I_0\times \mathrm{\Omega }$, $\mathrm{\mid }\psi (v){\mathrm{\partial }}_{t}f(t,x,v)\mathrm{\mid }\le G_{\psi ,0,I_0}(v),\mathrm{\mid }\psi (v)v_j{\mathrm{\partial }}_{x_j}f(t,x,v)\mathrm{\mid }\le G_{\psi ,j,I_0}(v)(j=1,2,3)\mathrm{.}$ At every $(t,x)$ assume $\int {\omega }_3(v)f(t,x,v)dv<\mathrm{\infty }$, so the moments through absolute order three are finite. In addition, assume ${\int }_{I_0}{\int }_{\mathrm{\Omega }}{\int }_{{\mathbb{R}}^3}{\omega }_3(v)f(t,x,v)dvdxdt<\mathrm{\infty }$ for every compact time interval $I_0\subset I$; this makes the moment densities and fluxes locally integrable in space-time. The velocity integral here is over all of ${\mathbb{R}}^3$, without a velocity cutoff. Use the collision set $Q_{\mathrm{e}\mathrm{l}}=\{(v,w,v^{\mathrm{\prime }},w^{\mathrm{\prime }})\in {\mathbb{R}}^{12}:v+w=v^{\mathrm{\prime }}+w^{\mathrm{\prime }},\mathrm{\mid }v{\mathrm{\mid }}^2+\mathrm{\mid }w{\mathrm{\mid }}^2=\mathrm{\mid }{v}^{\mathrm{\prime }}{\mathrm{\mid }}^2+\mathrm{\mid }{w}^{\mathrm{\prime }}{\mathrm{\mid }}^2\}$ with its relative Borel sigma-algebra, and the nonnegative countably additive measure $\nu$ invariant under particle exchange in either pair and incoming/outgoing interchange, together with the operator $Q_{\nu }$ from the elastic-collision definition. At each $(t,x)$ assume $f(t,x,\cdot )\in {\mathcal{D}}_{\nu }$ and $1,v_1,v_2,v_3,\mathrm{\mid }v{\mathrm{\mid }}^2\in {\mathcal{S}}_{f(t,x,\cdot )}$, with the measurability and absolute-convergence conditions of that definition. With $A=f(v)f(w)$, $B=f(v^{\mathrm{\prime }})f(w^{\mathrm{\prime }})$, and $\delta \phi =\phi (v)+\phi (w)-\phi (v^{\mathrm{\prime }})-\phi (w^{\mathrm{\prime }})$, assume for each such test that ${\int }_{{\mathbb{R}}^3}\phi (v)Q_{\nu }(f)(v)dv=\frac{1}{4}{\int }_{Q_{\mathrm{e}\mathrm{l}}}(B-A)\delta \phi d\nu \mathrm{.}$ Suppose pointwise that ${\mathrm{\partial }}_{t}f+v\cdot {\mathrm{\nabla }}_{x}f=Q_{\nu }(f)\mathrm{.}$ Set $n:=\int fdv$, $\rho :=mn$, $u:=n^{-1}\int vfdv$, $c:=v-u$, $T:=m(3k_{B}n{)}^{-1}\int \mathrm{\mid }c{\mathrm{\mid }}^{2}fdv$, $P:=m\int c\otimes cfdv$, $q:=(m\mathrm{/}2)\int \mathrm{\mid }c{\mathrm{\mid }}^{2}cfdv$, and $E:=(m\mathrm{/}2)\int \mathrm{\mid }v{\mathrm{\mid }}^{2}fdv$, using the positive constants $m,k_B$ and Lebesgue measure from the kinetic-density definition. Define $(Pu{)}_i:={\sum }_j{P}_{ij}{u}_j$ and ${\mathrm{\nabla }}_x\cdot A$ rowwise for a matrix field $A$. Then on $I\times \mathrm{\Omega }$, ${\mathrm{\partial }}_t\rho +{\mathrm{\nabla }}_x\cdot (\rho u)=0,$ ${\mathrm{\partial }}_t(\rho u)+{\mathrm{\nabla }}_x\cdot (\rho u\otimes u+P)=0,$ ${\mathrm{\partial }}_{t}E+{\mathrm{\nabla }}_x\cdot (Eu+Pu+q)=0.$ These are exact unclosed moment identities conditional on the displayed equation and integrability assumptions. They do not determine $P$ or $q$ from $\rho ,u,T$ and do not assert existence of a solution.

Tong, §2.4.1, full lecture notes

Golse, §3.1, full chapter