Higher integrability for the porous medium system

The gradient of weak solutions to porous medium-type equations or systems possesses a higher integrability than the one assumed in the pure notion of a solution. This holds true both in the degenerate range $m>1$ and in the singular range $0<m<1$. The critical and sub-critical singular case, i.e. when $0<m\le \frac{(N-2)_+}{N+2}$, presents further difficulties, which have been recently settled. I will discuss the problem in its generality, focusing in particular on the latest results. The critical and subcritical case is a joint work with V. B\"ogelein, F. Duzaar and. N. Liao (University of Salzburg, Austria), but previous contributions are also due to S. Schwarzacher, R. Korte, C. Scheven.