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

Strong bounded variation estimates for the multi-dimensional finite  volume approximation of scalar conservation laws and application to a tumour  growth model