Global higher integrability of weak solutions of porous medium systems

Abstract: We establish higher integrability up to the boundary for the gradient of solutions to porous medium type systems, whose model case is given by \begin{equation*} \partial_t u-\Delta(|u|^{m-1}u)=\mathrm{div}\,F\,, \end{equation*} where $m>1$. More precisely, we prove that under suitable assumptions the spatial gradient $D(|u|^{m-1}u)$ of any weak solution is integrable to a larger power than the natural power $2$. Our analysis includes both the case of the lateral boundary and the initial boundary.