Conditional periodic kinetic H theorem

Let $I$ be an open time interval in seconds and let $\mathrm{\Omega }={\prod }_{j=1}^3(\mathbb{R}\mathrm{/}{L}_j\mathbb{Z})$, with fixed periods $L_j>0$ in metres. Give $\mathrm{\Omega }$ the completed, unnormalised pushforward of Lebesgue measure on $C={\prod }_{j=1}^3[0,L_j)$ under the quotient map, as in the kinetic-density definition. Let $f:\mathrm{\Omega }\times {\mathbb{R}}^3\times I\to (0,\mathrm{\infty })$ be $C^1$ in $(x,t)$ and $C^2$ in $v$, with $v$ measured in metres per second and $f$ in the units assigned by the kinetic-density definition. Fix a constant $f_{\mathrm{r}\mathrm{e}\mathrm{f}}>0$ with the same units as $f$. Put $\mathrm{\ell }=\log (f\mathrm{/}{f}_{\mathrm{r}\mathrm{e}\mathrm{f}})$, $A=f(x,v,t)f(x,w,t)$, and $B=f(x,v^{\mathrm{\prime }},t)f(x,w^{\mathrm{\prime }},t)$. Use the elastic 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 weak operator $Q_{\nu }$ from the elastic-collision definition. Assume for each $t$ that $f$, $f\mathrm{\mid }v{\mathrm{\mid }}^2$, and $f\mathrm{\mid }\mathrm{\ell }\mathrm{\mid }$ are integrable over $\mathrm{\Omega }\times {\mathbb{R}}^3$. On each compact time interval assume a common integrable envelope in $(x,v)$ for $\mathrm{\mid }(1+\mathrm{\ell }){\mathrm{\partial }}_{t}f\mathrm{\mid }$; at each $t$ assume $(1+\mathrm{\mid }\mathrm{\ell }\mathrm{\mid })\mathrm{\mid }v\cdot {\mathrm{\nabla }}_{x}f\mathrm{\mid }$ is integrable over $\mathrm{\Omega }\times {\mathbb{R}}^3$. In addition, assume $f(1+\mathrm{\mid }v\mathrm{\mid })\mathrm{\mid }\mathrm{\ell }\mathrm{\mid },(1+\mathrm{\mid }\mathrm{\ell }\mathrm{\mid })\mathrm{\mid }{\mathrm{\partial }}_{t}f\mathrm{\mid },(1+\mathrm{\mid }\mathrm{\ell }\mathrm{\mid })\mathrm{\mid }v\mathrm{\mid }\mathrm{\mid }{\mathrm{\nabla }}_{x}f\mathrm{\mid }\in L^1(I_0\times \mathrm{\Omega }\times {\mathbb{R}}^3,dtdxdv)\text{for every compact time interval }{I}_0\subset I,$ where each velocity integral is over all of ${\mathbb{R}}^3$, without a velocity cutoff. Thus the entropy density, entropy flux, and differentiation terms are integrable on compact time-space regions. Define $D(x,t):=\frac{1}{4}{\int }_{Q_{\mathrm{e}\mathrm{l}}}(A-B)\log (A\mathrm{/}B)d\nu \mathrm{.}$ Assume that $D$ is measurable and locally integrable on $I\times \mathrm{\Omega }$, and that for each $t$ its spatial section belongs to $L^1(\mathrm{\Omega })$. At every $(x,t)$ assume $f(x,\cdot ,t)\in {\mathcal{D}}_{\nu }$ and $1,\mathrm{\ell }(x,\cdot ,t)\in {\mathcal{S}}_{f(x,\cdot ,t)}$, with the measurability and absolute-convergence conditions of the elastic-collision definition. With $\delta \phi =\phi (v)+\phi (w)-\phi (v^{\mathrm{\prime }})-\phi (w^{\mathrm{\prime }})$, assume ${\int }_{{\mathbb{R}}^3}\phi (v)Q_{\nu }(f)(v)dv=\frac{1}{4}{\int }_{Q_{\mathrm{e}\mathrm{l}}}(B-A)\delta \phi d\nu$ for those two tests, and suppose ${\mathrm{\partial }}_{t}f+v\cdot {\mathrm{\nabla }}_{x}f=Q_{\nu }(f)$. Define $h(x,t):={\int }_{{\mathbb{R}}^3}f\log (f\mathrm{/}{f}_{\mathrm{r}\mathrm{e}\mathrm{f}})dv,J_H(x,t):={\int }_{{\mathbb{R}}^3}vf\log (f\mathrm{/}{f}_{\mathrm{r}\mathrm{e}\mathrm{f}})dv,$ $H(t):={\int }_{\mathrm{\Omega }}h(x,t)dx,D_{\mathrm{t}\mathrm{o}\mathrm{t}}(t):={\int }_{\mathrm{\Omega }}D(x,t)dx\mathrm{.}$ Then the local entropy balance holds in distributions: for every $\eta \in C_c^{\mathrm{\infty }}(I\times \mathrm{\Omega })$, ${\int }_I{\int }_{\mathrm{\Omega }}(h{\mathrm{\partial }}_t\eta +J_H\cdot {\mathrm{\nabla }}_x\eta -D\eta )dxdt=0.$ Moreover, $D(x,t)\ge 0$ for almost every $(x,t)$. The periodic global entropy $H$ is differentiable and $\frac{dH}{dt}=-D_{\mathrm{t}\mathrm{o}\mathrm{t}}(t)\le 0.$ The inequality is conditional on the adopted weak collision identity; it is not a derivation from reversible many-particle mechanics or a kinetic-PDE existence theorem.

Tong, §2.3, full lecture notes

Golse, §3.2, full chapter