  A uniform bounded variation estimate for finite volume approximations of the
nonlinear scalar conservation law $\partial_t \alpha +
\mathrm{div}(\boldsymbol{u}f(\alpha)) = 0$ in two and three spatial dimensions
with an initial data of bounded variation is established. We assume that the
divergence of the velocity $\mathrm{div}(\boldsymbol{u})$ is of bounded
variation instead of the classical assumption that
$\mathrm{div}(\boldsymbol{u})$ is zero. The finite volume schemes analysed in
this article are set on nonuniform Cartesian grids. A uniform bounded variation
estimate for finite volume solutions of the conservation law $\partial_t \alpha
+ \mathrm{div}(\boldsymbol{F}(t,\boldsymbol{x},\alpha)) = 0$, where
$\mathrm{div}_{\boldsymbol{x}}\boldsymbol{F} \not=0$ on nonuniform Cartesian
grids is also proved. Such an estimate provides compactness for finite volume
approximations in $L^p$ spaces, which is essential to prove the existence of a
solution for a partial differential equation with nonlinear terms in $\alpha$,
when the uniqueness of the solution is not available. This application is
demonstrated by establishing the existence of a weak solution for a model that
describes the evolution of initial stages of breast cancer proposed by S. J.
Franks et al. [14]. The model consists of four coupled variables: tumour cell
concentration, tumour cell velocity--pressure, and nutrient concentration,
which are governed by a hyperbolic conservation law, viscous Stokes system, and
Poisson equation, respectively. Results from numerical tests are provided and
they complement theoretical findings.
