An $L^p$--approach to the well-posedness of transport equations associated to a regular field

Abstract

We investigate transport equations associated to a Lipschitz field on some subspace of ${\mathbb{R}}^N$ endowedwith a general measure $\mu$ in $L^p$-spaces $1<p<\infty$, extending the results obtained in two previous contributions of the author in the $L^1$-context. We notably prove the well-posedness of boundary-value transport problems with a large variety of boundary conditions. New explicit formula for the transport semigroup are in particular given.