Trace theorems and spatial continuity properties for the solutions of the transport equation

Abstract

This paper is first concerned with the trace problem for the transport equation. We prove the existence and the uniqueness of the traces as well as the well posedness of the initial- and boundary-value problem for the transport equation for any $L^p$ data ($p\in ]1,+\infty]$). In a second part, we use our study of the trace problem to prove that any solution to the transport equation is, roughly speaking, continuous with respect to the spatial variable along the direction of the transport vector field with values in a suitable $L^q$ space in the other variables. We want to emphasize the fact that we do not need to suppose any time regularity on the vector field defining the transport. This point is crucial in view of applications to fluid mechanics for instance. One of the main tools in our study is the theory of renormalized solutions of Di Perna and Lions.