Boltzmann equation

Definition

Let $B \colon \R^3 \times S^2 \to [0, \infty)$ be a given function, the collision kernel, where $S^2$ is the unit sphere in $\R^3$. For $v, v_* \in \R^3$ and $\sigma \in S^2$ put $$v' = \frac{v + v_*}{2} + \frac{\abs{v - v_*}}{2} \sigma, \qquad v'_* = \frac{v + v_*}{2} - \frac{\abs{v - v_*}}{2} \sigma.$$ For a function $g \colon \R^3 \to \R$ the collision operator is $$Q(g, g)(v) = \int_{\R^3} \int_{S^2} B(v - v_*, \sigma) \big( g(v') g(v'_*) - g(v) g(v_*) \big) \, d\sigma \, dv_*.$$ The Boltzmann equation is the partial differential equation $$\partial_t f + v \cdot \nabla_x f = Q(f, f)$$ for an unknown function $f \colon [0, \infty) \times \R^3 \times \R^3 \to [0, \infty)$, $(t, x, v) \mapsto f(t, x, v)$. The operator $Q$ acts on the variable $v$, and $\nabla_x$ is the gradient in $x$.