Linear transport equations and their characteristic flow

Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n\times \mathbb{R}$ be open and let $a:\mathrm{\Omega }\to {\mathbb{R}}^n$, $c:\mathrm{\Omega }\to \mathbb{R}$, and $f:\mathrm{\Omega }\to \mathbb{R}$. The scalar first-order equation

$u_t(x,t)+a(x,t)\cdot {\mathrm{\nabla }}_{x}u(x,t)+c(x,t)u(x,t)=f(x,t)$

is a linear transport equation.

Fix $(x_0,t_0)\in \mathrm{\Omega }$. A characteristic through $(x_0,t_0)$ is a solution $X(\cdot )$ of the ODE

$X^{\mathrm{\prime }}(s)=a(X(s),s),X(t_0)=x_0,$

on an interval containing $t_0$. The corresponding space-time curve is $s\mapsto (X(s),s)$. If these initial-value problems have unique solutions for varying initial data, write $X(s;t_0,x_0)$ for the solution through $(x_0,t_0)$. The resulting maps $x_0\mapsto X(s;t_0,x_0)$, on the domains where they are defined, form the characteristic flow of the transport field. Without uniqueness there are characteristics, but no single-valued characteristic flow.