Vlasov equation

Definition

Let $d \ge 1$, let $T > 0$, and let $F$ assign to each function $f \colon [0, T) \times \R^d \times \R^d \to [0, \infty)$ a force field $F[f] \colon [0, T) \times \R^d \to \R^d$. The Vlasov equation with force $F$ is the partial differential equation $$\partial_t f + v \cdot \nabla_x f + F[f](t, x) \cdot \nabla_v f = 0.$$ The unknown is a density $f(t, x, v) \ge 0$, and $\nabla_x$ and $\nabla_v$ are the gradients in $x$ and in $v$. The main case is the Vlasov–Poisson system in $d = 3$, where $\gamma \in \set{1, -1}$ and $$F[f](t, x) = \frac{\gamma}{4\pi} \int_{\R^3} \frac{x - y}{\abs{x - y}^3} \rho(t, y) \, dy, \qquad \rho(t, y) = \int_{\R^3} f(t, y, v) \, dv.$$ The sign $\gamma = 1$ gives a repulsive force, as for a plasma of one species, and $\gamma = -1$ gives an attractive force, as for a self-gravitating system.