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

TL;DR

This paper extends the well-posedness results for transport equations associated with Lipschitz fields in $L^{p}$ spaces, providing new explicit formulas for the transport semigroup and addressing boundary-value problems.

Contribution

It introduces an $L^{p}$-approach to transport equations, generalizing previous $L^{1}$ results and offering explicit formulas for the associated semigroup.

Findings

Proved well-posedness of boundary-value transport problems with various boundary conditions

Extended previous $L^{1}$ results to $L^{p}$ spaces for $1<p< finite$

Derived explicit formulas for the transport semigroup

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.