An $$L^{p}$$-Approach to the Well-Posedness of Transport Equations Associated with a Regular Field: Part I

Abstract

Transport equations associated with a Lipschitz field $$\mathscr {F}$$ on some subspace of $${\mathbb {R}}^N$$ endowed with some general measure $$\mu $$ are considered. Our aim is to extend the results obtained in two previous contributions (Arlotti et al. in Mediterr J Math 6:367–402, 2009, Mediterr J Math 8:1–35, 2011) in the $$L^{1}$$-context to $$L^{p}$$-spaces $$1< p <\infty $$. This is the first part of a two-part contribution (see in Arlotti and Lods An $$L^{p}$$-approach to the well-posedness of transport equations associated with a regular field—part II, Mediterr. J. Math. 16:145, 2019, for the second part) and we here establish the general mathematical framework we are dealing with and notably prove trace formula and uniqueness of boundary value transport problems with abstract boundary conditions. The abstract results of this first part will be used in the Part II of this work (Arlotti and Lods in Meditter J Math 16:145, 2019) to deal with general initial and boundary value problems and semigroup generation properties.