  In this work, we demonstrate well-posedness and regularisation by noise
results for a class of geometric transport equations that contains, among
others, the linear transport and continuity equations. This class is known as
linear advection of $k$-forms. In particular, we prove global existence and
uniqueness of $L^p$-solutions to the stochastic equation, driven by a spatially
$\alpha$-H\"older drift $b$, uniformly bounded in time, with an integrability
condition on the distributional derivative of $b$, and sufficiently regular
diffusion vector fields. Furthermore, we prove that all our solutions are
continuous if the initial datum is continuous. Finally, we show that our class
of equations without noise admits infinitely many $L^p$-solutions and is hence
ill-posed. Moreover, the deterministic solutions can be discontinuous in both
time and space independently of the regularity of the initial datum. We also
demonstrate that for certain initial data of class $C^\infty_{0},$ the
deterministic $L^p$-solutions blow up instantaneously in the space
$L^{\infty}_{loc}$. In order to establish our results, we employ
characteristics-based techniques that exploit the geometric structure of our
equations.
