Local kinetic H balance for the elastic Boltzmann model

Let $I\subseteq \mathbb{R}$ be an open time interval in seconds and let $\mathrm{\Omega }={\prod }_{j=1}^3(\mathbb{R}\mathrm{/}{L}_j\mathbb{Z})$ be a periodic spatial cell with $L_j>0$ in metres. Let $v\in {\mathbb{R}}^3$ be measured in metres per second and let $f:I\times \mathrm{\Omega }\times {\mathbb{R}}^3\to (0,\mathrm{\infty })$ be measurable, continuously differentiable in $(t,x)$, and twice continuously differentiable in $v$. Assume $f{d}^{3}x{d}^{3}v$ counts particles, so $f$ has units ${\mathrm{s}}^3{\mathrm{m}}^{-6}$. Put $Q_{\mathrm{e}\mathrm{l}}=\{(v,w,v^{\mathrm{\prime }},w^{\mathrm{\prime }})\in ({\mathbb{R}}^3{)}^4:v+w=v^{\mathrm{\prime }}+w^{\mathrm{\prime }},\mathrm{\mid }v{\mathrm{\mid }}^2+\mathrm{\mid }w{\mathrm{\mid }}^2=\mathrm{\mid }{v}^{\mathrm{\prime }}{\mathrm{\mid }}^2+\mathrm{\mid }{w}^{\mathrm{\prime }}{\mathrm{\mid }}^2\}$, with its relative Borel sigma-algebra, and let $\nu$ be the nonnegative symmetric collision-rate measure on $Q_{\mathrm{e}\mathrm{l}}$ in the adopted model. Fix a constant $f_{\mathrm{r}\mathrm{e}\mathrm{f}}>0$ with the same units as $f$, and put $\mathrm{\ell }=\log (f\mathrm{/}{f}_{\mathrm{r}\mathrm{e}\mathrm{f}})$, which is dimensionless, $A=f(x,v,t)f(x,w,t)$, and $B=f(x,v^{\mathrm{\prime }},t)f(x,w^{\mathrm{\prime }},t)$. Let $v_{\ast }=1\mathrm{m}{\mathrm{s}}^{-1}$ and ${\omega }_3(v)={\sum }_{j=0}^3(\mathrm{\mid }v\mathrm{\mid }\mathrm{/}{v}_{\ast }{)}^j$.

For every compact time interval $J\subset I$, assume there are integrable velocity majorants, independent of $(x,t)\in \mathrm{\Omega }\times J$, for each of $f{\omega }_3(v)$, $f\mathrm{\mid }\mathrm{\ell }\mathrm{\mid }$, $\mathrm{\mid }v\mathrm{\mid }f\mathrm{\mid }\mathrm{\ell }\mathrm{\mid }$, $(1+\mathrm{\mid }\mathrm{\ell }\mathrm{\mid })\mathrm{\mid }{\mathrm{\partial }}_{t}f\mathrm{\mid }$, and $(1+\mathrm{\mid }\mathrm{\ell }\mathrm{\mid })\mathrm{\mid }v\mathrm{\mid }\mathrm{\mid }{\mathrm{\nabla }}_{x}f\mathrm{\mid }$. At every $(x,t)$, assume that $f(x,\cdot ,t)$ belongs to the collision domain and that the weak collision tests $1$ and $\mathrm{\ell }(x,\cdot ,t)$ are measurable and absolutely convergent. For each $t\in I$, assume that the map $x⟼{\int }_{Q_{\mathrm{e}\mathrm{l}}}\mathrm{\mid }(A-B)\log (A\mathrm{/}B)\mathrm{\mid }d\nu$ is measurable and belongs to $L^1(\mathrm{\Omega },dx)$. Assume that $f$ solves the adopted kinetic equation ${\mathrm{\partial }}_{t}f+v\cdot {\mathrm{\nabla }}_{x}f=Q_{\nu }(f)$ for Lebesgue-almost every velocity at every fixed $(x,t)$. This sectionwise solution sense is an additional hypothesis for the pointwise balance below.

Define $h(x,t)={\int }_{{\mathbb{R}}^3}f\log (f\mathrm{/}{f}_{\mathrm{r}\mathrm{e}\mathrm{f}})d^{3}v,J_H(x,t)={\int }_{{\mathbb{R}}^3}vf\log (f\mathrm{/}{f}_{\mathrm{r}\mathrm{e}\mathrm{f}})d^{3}v,$ where $h$ and $J_H$ have units ${\mathrm{m}}^{-3}$ and ${\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$, respectively. Also define $D(x,t)=\frac{1}{4}{\int }_{Q_{\mathrm{e}\mathrm{l}}}(A-B)\log (A\mathrm{/}B)d\nu \mathrm{.}$ Under the collision-rate normalization in the adopted model, $D$ has units ${\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$. Then $D(x,t)\ge 0$ and ${\mathrm{\partial }}_{t}h+{\mathrm{\nabla }}_x\cdot J_H=-D$ throughout $I\times \mathrm{\Omega }$. The identity is conditional on the adopted weak collision model and makes no assertion about reversible many-particle entropy or empirical validity.

MIT OpenCourseWare 8.333, Lecture 9, §III.E (complete H-theorem treatment)

David Tong, §2.3, full lecture notes