Periodic Boltzmann model with incoming molecular chaos

Let ${\mathrm{\Omega }}_{\mathrm{p}\mathrm{e}\mathrm{r}}={\prod }_{i=1}^3(0,L_i)$ with opposite faces identified, let $I\subseteq \mathbb{R}$ be an open time interval in seconds, and let $f(x,v,t)\ge 0$ be a one-particle number density for $x\in {\mathrm{\Omega }}_{\mathrm{p}\mathrm{e}\mathrm{r}}$, $v\in {\mathbb{R}}^3$, and $t\in I$. With $x$ in metres and $v$ in $\mathrm{m}{\mathrm{s}}^{-1}$, $f$ has units ${\mathrm{s}}^3{\mathrm{m}}^{-6}$ relative to $d^{3}x{d}^{3}v$; its moments used below are assumed finite. For this postulate, require $f$ to be jointly measurable and periodic in $x$, with $(x,t)\mapsto f(x,v,t)$ continuously differentiable for each fixed $v$ and its derivatives jointly measurable. Require $f$, ${\mathrm{\partial }}_{t}f$, and $v\cdot {\mathrm{\nabla }}_{x}f$ to be locally integrable with respect to product Lebesgue measure. Periodic boundary conditions are imposed in $x$.

Adopt the incoming molecular-chaos closure: at a collision location and time, the incoming two-particle velocity density used in the collision term is $f(x,v,t)f(x,w,t)$. This is a one-sided factorization assumption; no factorization of the outgoing pair and no derivation from reversible many-particle mechanics is asserted.

For the elastic data $Q_{\mathrm{e}\mathrm{l}},\nu$ of Regular single-species elastic collision data, adopt a collision operator $Q_{\nu }(f)$, with units of $f$ per second, as a jointly measurable locally integrable representative whose velocity-wise weak action satisfies, for almost every $(x,t)$ and every measurable test function $\phi$ for which both sides converge absolutely, ${\int }_{{\mathbb{R}}^3}\phi (v)Q_{\nu }(f)(x,v,t)d^{3}v=\frac{1}{4}{\int }_{Q_{\mathrm{e}\mathrm{l}}}(f(x,v^{\mathrm{\prime }},t)f(x,w^{\mathrm{\prime }},t)-f(x,v,t)f(x,w,t))(\phi (v)+\phi (w)-\phi (v^{\mathrm{\prime }})-\phi (w^{\mathrm{\prime }}))d\nu \mathrm{.}$ The adopted kinetic equation is ${\mathrm{\partial }}_{t}f+v\cdot {\mathrm{\nabla }}_{x}f=Q_{\nu }(f),$ as an equality of locally integrable functions almost everywhere on the product domain. These are admissibility conditions for solutions of the adopted model, not an existence assertion. The equation and weak collision identity are model postulates. This item does not assert a hard-sphere Boltzmann–Grad limit, propagation of chaos, or universal equilibration.