This paper continues work in the authors’ paper [Mediterr. J. Math. 6, No. 4, 367–402 (2009; Zbl 1198.47055)]. The subject is well-posedness (in the sense of semigroup theory) of the general transport equation
\[ \partial_t f({\mathbf x}, t) + {\mathcal F}({\mathbf x}) \cdot \nabla _{\mathbf x}f({\mathbf x}, t) = 0 \quad ({\mathbf x} \in \Omega, \;t > 0), \]
where \({\mathcal F}({\mathbf x})\) is a regular field in a domain \(\Omega\) of \(n\)-dimensional Euclidean space. The initial and boundary conditions are
\[ f({\mathbf x}, 0) = f_0({\mathbf x}) \quad ({\mathbf x} \in \Omega) \, , \qquad f|_{\Gamma_-}({\mathbf y}, t) = H(f|_{\Gamma_+})({\mathbf y}, t) \quad ({\mathbf y} \in \Gamma_- \, , \;t > 0), \]
where the \(\Gamma_\pm\) are boundaries of the phase space and the boundary operator \(H\) is a linear, possibly unbounded operator between the trace spaces \(L^1(\Gamma_\pm, \mu_\pm(dx))\); here, \(\mu_-\) (resp., \(\mu_+)\) is a suitably defined positive Borel measure in \(\Gamma_-\) (resp., \(\Gamma_+)\). The authors study fine properties of the traces and extend previous results on dissipative and multiplicative boundary conditions. Finally, they extend theorems on boundary operators of norm 1 to more general fields and measures.